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\newfontfamily\ReaderSymbols{Segoe UI Symbol}
\makeatletter
\let\ReaderOriginalAnchorStart\hyper@anchorstart
\let\ReaderOriginalAnchorEnd\hyper@anchorend
\expandafter\def\csname ReaderAliases@section*.1\endcsname{\ReaderOriginalAnchorStart{detecting-regularity-without-choosing-coordinates}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.2\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-001}\label{AN03-GEO-001}\endgroup\ReaderOriginalAnchorStart{an03-geo-001-local-estimates-and-amplitude-reduction}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-001-local-estimates-and-amplitude-reduction}\label{an03-geo-001-local-estimates-and-amplitude-reduction}\endgroup\ReaderOriginalAnchorStart{1-local-estimates-and-amplitude-reduction}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{1-local-estimates-and-amplitude-reduction}\label{1-local-estimates-and-amplitude-reduction}\endgroup\ReaderOriginalAnchorStart{local-estimates-and-amplitude-reduction}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.7\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-002}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-002}\label{AN03-GEO-002}\endgroup\ReaderOriginalAnchorStart{an03-geo-002-kernels-exponential-tests-and-coordinate-transport}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-002-kernels-exponential-tests-and-coordinate-transport}\label{an03-geo-002-kernels-exponential-tests-and-coordinate-transport}\endgroup\ReaderOriginalAnchorStart{2-kernels-exponential-tests-and-coordinate-transport}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{2-kernels-exponential-tests-and-coordinate-transport}\label{2-kernels-exponential-tests-and-coordinate-transport}\endgroup\ReaderOriginalAnchorStart{kernels-exponential-tests-and-coordinate-transport}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.15\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-003}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-003}\label{AN03-GEO-003}\endgroup\ReaderOriginalAnchorStart{an03-geo-003-the-second-symbol-term-and-the-density-it-measures}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-003-the-second-symbol-term-and-the-density-it-measures}\label{an03-geo-003-the-second-symbol-term-and-the-density-it-measures}\endgroup\ReaderOriginalAnchorStart{3-the-second-symbol-term-and-the-density-it-measures}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{3-the-second-symbol-term-and-the-density-it-measures}\label{3-the-second-symbol-term-and-the-density-it-measures}\endgroup\ReaderOriginalAnchorStart{the-second-symbol-term-and-the-density-it-measures}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.20\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-004}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-004}\label{AN03-GEO-004}\endgroup\ReaderOriginalAnchorStart{an03-geo-004-assembling-an-operator-and-its-principal-symbol}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-004-assembling-an-operator-and-its-principal-symbol}\label{an03-geo-004-assembling-an-operator-and-its-principal-symbol}\endgroup\ReaderOriginalAnchorStart{4-assembling-an-operator-and-its-principal-symbol}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{4-assembling-an-operator-and-its-principal-symbol}\label{4-assembling-an-operator-and-its-principal-symbol}\endgroup\ReaderOriginalAnchorStart{assembling-an-operator-and-its-principal-symbol}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.23\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-005}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-005}\label{AN03-GEO-005}\endgroup\ReaderOriginalAnchorStart{an03-geo-005-properness-products-and-global-summation}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-005-properness-products-and-global-summation}\label{an03-geo-005-properness-products-and-global-summation}\endgroup\ReaderOriginalAnchorStart{5-properness-products-and-global-summation}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{5-properness-products-and-global-summation}\label{5-properness-products-and-global-summation}\endgroup\ReaderOriginalAnchorStart{properness-products-and-global-summation}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.26\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-006}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-006}\label{AN03-GEO-006}\endgroup\ReaderOriginalAnchorStart{an03-geo-006-inverses-with-global-and-conic-error-control}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-006-inverses-with-global-and-conic-error-control}\label{an03-geo-006-inverses-with-global-and-conic-error-control}\endgroup\ReaderOriginalAnchorStart{6-inverses-with-global-and-conic-error-control}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{6-inverses-with-global-and-conic-error-control}\label{6-inverses-with-global-and-conic-error-control}\endgroup\ReaderOriginalAnchorStart{inverses-with-global-and-conic-error-control}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.29\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-007}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-007}\label{AN03-GEO-007}\endgroup\ReaderOriginalAnchorStart{an03-geo-007-fourier-directions-and-the-precise-kernel-relation}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-007-fourier-directions-and-the-precise-kernel-relation}\label{an03-geo-007-fourier-directions-and-the-precise-kernel-relation}\endgroup\ReaderOriginalAnchorStart{7-fourier-directions-and-the-precise-kernel-relation}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{7-fourier-directions-and-the-precise-kernel-relation}\label{7-fourier-directions-and-the-precise-kernel-relation}\endgroup\ReaderOriginalAnchorStart{fourier-directions-and-the-precise-kernel-relation}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.33\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-008}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-008}\label{AN03-GEO-008}\endgroup\ReaderOriginalAnchorStart{an03-geo-008-what-every-input-can-detect}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-008-what-every-input-can-detect}\label{an03-geo-008-what-every-input-can-detect}\endgroup\ReaderOriginalAnchorStart{8-what-every-input-can-detect}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{8-what-every-input-can-detect}\label{8-what-every-input-can-detect}\endgroup\ReaderOriginalAnchorStart{what-every-input-can-detect}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.36\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-009}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-009}\label{AN03-GEO-009}\endgroup\ReaderOriginalAnchorStart{an03-geo-009-elliptic-tests-and-intrinsic-wavefront}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-009-elliptic-tests-and-intrinsic-wavefront}\label{an03-geo-009-elliptic-tests-and-intrinsic-wavefront}\endgroup\ReaderOriginalAnchorStart{9-elliptic-tests-and-intrinsic-wavefront}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{9-elliptic-tests-and-intrinsic-wavefront}\label{9-elliptic-tests-and-intrinsic-wavefront}\endgroup\ReaderOriginalAnchorStart{elliptic-tests-and-intrinsic-wavefront}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.39\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-010}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-010}\label{AN03-GEO-010}\endgroup\ReaderOriginalAnchorStart{an03-geo-010-convergence-when-singular-directions-are-fixed}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-010-convergence-when-singular-directions-are-fixed}\label{an03-geo-010-convergence-when-singular-directions-are-fixed}\endgroup\ReaderOriginalAnchorStart{10-convergence-when-singular-directions-are-fixed}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{10-convergence-when-singular-directions-are-fixed}\label{10-convergence-when-singular-directions-are-fixed}\endgroup\ReaderOriginalAnchorStart{convergence-when-singular-directions-are-fixed}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.42\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-011}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-011}\label{AN03-GEO-011}\endgroup\ReaderOriginalAnchorStart{an03-geo-011-sobolev-and-besov-spaces-in-a-moving-chart}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-011-sobolev-and-besov-spaces-in-a-moving-chart}\label{an03-geo-011-sobolev-and-besov-spaces-in-a-moving-chart}\endgroup\ReaderOriginalAnchorStart{11-sobolev-and-besov-spaces-in-a-moving-chart}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{11-sobolev-and-besov-spaces-in-a-moving-chart}\label{11-sobolev-and-besov-spaces-in-a-moving-chart}\endgroup\ReaderOriginalAnchorStart{sobolev-and-besov-spaces-in-a-moving-chart}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.46\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-012}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-012}\label{AN03-GEO-012}\endgroup\ReaderOriginalAnchorStart{an03-geo-012-microlocal-scales-and-their-unattained-thresholds}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-012-microlocal-scales-and-their-unattained-thresholds}\label{an03-geo-012-microlocal-scales-and-their-unattained-thresholds}\endgroup\ReaderOriginalAnchorStart{12-microlocal-scales-and-their-unattained-thresholds}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{12-microlocal-scales-and-their-unattained-thresholds}\label{12-microlocal-scales-and-their-unattained-thresholds}\endgroup\ReaderOriginalAnchorStart{microlocal-scales-and-their-unattained-thresholds}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.69\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-013}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-013}\label{AN03-GEO-013}\endgroup\ReaderOriginalAnchorStart{an03-geo-013-bundles-anti-duals-and-kernels-between-different-manifolds}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-013-bundles-anti-duals-and-kernels-between-different-manifolds}\label{an03-geo-013-bundles-anti-duals-and-kernels-between-different-manifolds}\endgroup\ReaderOriginalAnchorStart{13-bundles-anti-duals-and-kernels-between-different-manifolds}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{13-bundles-anti-duals-and-kernels-between-different-manifolds}\label{13-bundles-anti-duals-and-kernels-between-different-manifolds}\endgroup\ReaderOriginalAnchorStart{bundles-anti-duals-and-kernels-between-different-manifolds}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.76\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-KERNEL-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-KERNEL-001}\label{AN03-GEO-KERNEL-001}\endgroup\ReaderOriginalAnchorStart{test-spaces-bounded-supports-and-the-strong-dual}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.81\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-KERNEL-002}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-KERNEL-002}\label{AN03-GEO-KERNEL-002}\endgroup\ReaderOriginalAnchorStart{product-tests-and-a-convergent-fourier-expansion}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.91\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-KERNEL-003}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-KERNEL-003}\label{AN03-GEO-KERNEL-003}\endgroup\ReaderOriginalAnchorStart{separate-continuity-gives-a-fixed-support-estimate}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.94\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-KERNEL-004}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-KERNEL-004}\label{AN03-GEO-KERNEL-004}\endgroup\ReaderOriginalAnchorStart{construction-of-the-local-scalar-kernel}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.99\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-KERNEL-005}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-KERNEL-005}\label{AN03-GEO-KERNEL-005}\endgroup\ReaderOriginalAnchorStart{the-global-kernel-and-continuity-into-the-strong-dual}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.103\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-KERNEL-006}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-KERNEL-006}\label{AN03-GEO-KERNEL-006}\endgroup\ReaderOriginalAnchorStart{the-exact-elementary-distribution-operations}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.108\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-CONV-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-CONV-001}\label{AN03-GEO-CONV-001}\endgroup\ReaderOriginalAnchorStart{convolution-with-its-actual-support-and-derivative-bounds}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.114\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-CONV-002}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-CONV-002}\label{AN03-GEO-CONV-002}\endgroup\ReaderOriginalAnchorStart{the-full-schwartz-estimate-for-a-compact-convolution-factor}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.118\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-CONV-003}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-CONV-003}\label{AN03-GEO-CONV-003}\endgroup\ReaderOriginalAnchorStart{smooth-parameter-pairings-and-distributional-approximation}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.124\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-CONV-004}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-CONV-004}\label{AN03-GEO-CONV-004}\endgroup\ReaderOriginalAnchorStart{compact-fourier-pairings-and-the-receiving-inverse-identities}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.136\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-EX-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-EX-001}\label{AN03-GEO-EX-001}\endgroup\ReaderOriginalAnchorStart{an03-geo-ex-001-four-calculations-that-separate-the-assertions}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-ex-001-four-calculations-that-separate-the-assertions}\label{an03-geo-ex-001-four-calculations-that-separate-the-assertions}\endgroup\ReaderOriginalAnchorStart{14-four-calculations-that-separate-the-assertions}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{14-four-calculations-that-separate-the-assertions}\label{14-four-calculations-that-separate-the-assertions}\endgroup\ReaderOriginalAnchorStart{four-calculations-that-separate-the-assertions}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.137\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-PS-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-PS-001}\label{AN03-GEO-PS-001}\endgroup\ReaderOriginalAnchorStart{an03-geo-ps-001-exercises-with-complete-solutions}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{an03-geo-ps-001-exercises-with-complete-solutions}\label{an03-geo-ps-001-exercises-with-complete-solutions}\endgroup\ReaderOriginalAnchorStart{15-exercises-with-complete-solutions}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{15-exercises-with-complete-solutions}\label{15-exercises-with-complete-solutions}\endgroup\ReaderOriginalAnchorStart{exercises-with-complete-solutions}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.138\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-000}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-000}\label{AN03-GEO-FOUNDATION-000}\endgroup\ReaderOriginalAnchorStart{original-chart-bundle-and-inverse-maps}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.139\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-001}\label{AN03-GEO-FOUNDATION-001}\endgroup\ReaderOriginalAnchorStart{the-actual-smooth-partition-and-every-denominator-derivative}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.142\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-002}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-002}\label{AN03-GEO-FOUNDATION-002}\endgroup\ReaderOriginalAnchorStart{gluing-the-actual-finite-rank-bundle}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.147\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-003}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-003}\label{AN03-GEO-FOUNDATION-003}\endgroup\ReaderOriginalAnchorStart{every-tensor-dual-metric-and-density-factor}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.152\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-004}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-004}\label{AN03-GEO-FOUNDATION-004}\endgroup\ReaderOriginalAnchorStart{constructing-the-inverse-without-changing-the-map-or-norms}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.159\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-005}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-005}\label{AN03-GEO-FOUNDATION-005}\endgroup\ReaderOriginalAnchorStart{every-higher-inverse-derivative-and-the-implicit-map}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.164\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FOUNDATION-006}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FOUNDATION-006}\label{AN03-GEO-FOUNDATION-006}\endgroup\ReaderOriginalAnchorStart{exact-receiving-scope-and-remaining-work}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.165\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-000}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-000}\label{AN03-GEO-FLOW-000}\endgroup\ReaderOriginalAnchorStart{original-finite-coordinate-flows-and-their-geodesic-receiving-map}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.166\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-001}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-001}\label{AN03-GEO-FLOW-001}\endgroup\ReaderOriginalAnchorStart{the-original-integral-equation-and-its-complete-local-solution}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.172\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-002}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-002}\label{AN03-GEO-FLOW-002}\endgroup\ReaderOriginalAnchorStart{ordered-linear-transport-on-the-original-two-sided-interval}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.177\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-003}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-003}\label{AN03-GEO-FLOW-003}\endgroup\ReaderOriginalAnchorStart{all-parameter-derivatives-of-the-linear-inverse}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.180\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-004}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-004}\label{AN03-GEO-FLOW-004}\endgroup\ReaderOriginalAnchorStart{the-first-actual-nonlinear-parameter-derivative}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.183\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-005}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-005}\label{AN03-GEO-FLOW-005}\endgroup\ReaderOriginalAnchorStart{every-nonlinear-parameter-derivative-including-its-original-blocks}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.186\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-006}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-006}\label{AN03-GEO-FLOW-006}\endgroup\ReaderOriginalAnchorStart{variable-initial-times-without-replacing-the-original-time-coordinate}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.190\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-007}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-007}\label{AN03-GEO-FLOW-007}\endgroup\ReaderOriginalAnchorStart{continuation-the-exact-composition-law-and-coordinate-transport}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.194\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-008}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-008}\label{AN03-GEO-FLOW-008}\endgroup\ReaderOriginalAnchorStart{the-actual-geodesic-vector-field-and-smooth-endpoint-map}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.200\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-009}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-009}\label{AN03-GEO-FLOW-009}\endgroup\ReaderOriginalAnchorStart{the-unaltered-metric-identity-all-first-jets-and-minimizing-radius}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.204\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-010}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-010}\label{AN03-GEO-FLOW-010}\endgroup\ReaderOriginalAnchorStart{every-transformed-divergence-and-density-contribution}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.208\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-FLOW-011}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-FLOW-011}\label{AN03-GEO-FLOW-011}\endgroup\ReaderOriginalAnchorStart{exact-course-use-and-its-remaining-interfaces}\ReaderOriginalAnchorEnd}
\expandafter\def\csname ReaderAliases@section*.211\endcsname{\ReaderOriginalAnchorStart{AN03-GEO-WAVEFRONT-000}\ReaderOriginalAnchorEnd\begingroup\def\@currentHref{AN03-GEO-WAVEFRONT-000}\label{AN03-GEO-WAVEFRONT-000}\endgroup\ReaderOriginalAnchorStart{original-wavefront-and-pullback-foundations}\ReaderOriginalAnchorEnd}
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\begin{document}

\section{Detecting regularity without choosing
coordinates}\label{detecting-regularity-without-choosing-coordinates}

An operator has two kinds of locality. Its kernel determines which input
points can influence an output point. Its symbol determines which
cotangent directions can carry an irregularity. We construct these two
structures together, then use them to measure regularity of sections of
arbitrary finite-rank complex bundles. All manifolds below are
Hausdorff, second countable, smooth, and without boundary; compactness
is imposed only where stated.

The analytic inputs are \href{euclidean-symbol-calculus.pdf}{From symbol
estimates to operators on every Sobolev scale}, Sections2--5 and8--9,
and \href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at
a moving scale}, Sections7--8. The convention is \(D=-i\partial\), with
inverse Fourier factor \((2\pi)^{-n}\). Local symbols have estimates on
each compact base set, with no uniformity across a noncompact manifold.
Unless parameters are displayed, \(S^m=S^m_{1,0}\) and
\(\Psi^m=\Psi^m_{1,0}\).

The geometric construction uses the following analytic facts.
\textbf{Distributions and kernels} supplies distributions as continuous
functionals on compact smooth densities, their restriction and
multiplication by smooth functions, tensor products, Fourier
transformation, the finite-order estimate on compact supports, and the
scalar Schwartz kernel theorem for continuous
\(C_c^\infty(U)\to\mathcal D'(V)\) maps on Euclidean open sets. The
kernel theorem includes its identification with separately continuous
bilinear test pairings, jointly continuous on every pair of fixed
compact-support test spaces; Sections 13.1--13.6 prove this scalar
theorem, its fixed-support joint estimates, and continuity into the
strong distribution dual, together with the exact elementary
distribution operations. The test spaces carry their usual smooth
Fréchet topologies at fixed support and their inductive-limit topology
over supports; distributions carry the strong dual topology.
\textbf{Manifolds and bundles} supplies smooth atlases, the chain rule,
inverse function theorem, locally finite smooth partitions of unity with
compact supports subordinate to open covers, and finite-rank bundle
gluing. \textbf{Uniform boundedness} and the closed graph theorem are
proved in Sections 14.1--14.6 of
\href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments}. Those sections prove completeness of
the exact fixed-support smooth test spaces and smooth output spaces, a
common finite-order estimate for arbitrary pointwise bounded functional
families, and the closed graph theorem between Fréchet spaces, including
a Banach domain. Lebesgue change of variables, Fubini, dominated
convergence, and Fourier inversion/Plancherel have the same exact entry
contracts as \href{euclidean-symbol-calculus.pdf}{From symbol estimates
to operators on every Sobolev scale}. We prove the geometric kernel
adapter and the wavefront results used below; no general theorem about
multiplying arbitrary distributions is assumed.

\subsection{1. Local estimates and amplitude
reduction}\label{local-estimates-and-amplitude-reduction}

For an open set \(U\subset\mathbb R^n\),
\(a\in S^m_{\rho,\delta}(U\times\mathbb R^n)\) means that \[
\sup_{x\in K,\xi}\langle\xi\rangle^{-m+\rho|\alpha|-\delta|\beta|}
|\partial_\xi^\alpha\partial_x^\beta a(x,\xi)|<\infty
\quad(K\Subset U),
\qquad 0<\rho\leq1,\quad0\leq\delta<1.
\tag{G1}
\] These seminorms define the local Fréchet topology, using a compact
exhaustion. On an open conic set
\(\mathcal C\subset U\times(\mathbb R^n\setminus0)\), the same estimates
are required along \((x,t\xi)\), \(t\geq1\), for every compact subset of
\(\mathcal C\). They make no assertion about approaching the zero
section outside such compact sets. A symbol defined at all frequencies
must also be smooth there.

Local asymptotic summation follows with support control. Given \(a_j\)
of orders \(m_j\to-\infty\), take a locally finite partition
\(\theta_l\) of the base, or of a relatively compact angular/base cover
in the conic case. Apply Section 2 of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale} to each compactly supported piece
\(\theta_l a_j\), extended into its coordinate product by zero,
obtaining \(b_l\) with support contained in
\(\bigcup_j\operatorname{supp}(\theta_l a_j)\). Then \(b=\sum_l b_l\) is
locally finite. On each fixed compact set only finitely many \(l\)
occur, so every remainder satisfies (G1) of order
\(M_k=\max_{j\geq k}m_j\). The support assertion survives the locally
finite sum. On a conic cover the cutoffs are homogeneous of degree zero
above a fixed frequency; the bounded-frequency discrepancy is smooth and
irrelevant to asymptotic orders. Differences of two sums are of every
negative order. This also proves continuity of restrictions and the
local form of the symbol recognition statement in Section 2 of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}.

We need an amplitude version of the Euclidean calculus. Suppose
\(c(z,w,\eta)\), compactly supported in \((z,w)\), has estimates \[
|\partial_\eta^\alpha\partial_z^\beta\partial_w^\gamma c|
\leq C_{\alpha\beta\gamma}
\langle\eta\rangle^{m-\rho|\alpha|+\delta|\beta|+d|\gamma|},
\qquad 0\leq d\leq\rho,\quad d<1.
\tag{G2}
\] The kernel \((2\pi)^{-n}\int e^{i(z-w)\cdot\eta}c(z,w,\eta)\,d\eta\)
has a left symbol \[
b(z,\xi)=\left[e^{iD_w\cdot D_\eta}c(z,w,\eta)\right]_{w=z,\eta=\xi}.
\tag{G3}
\] When \(d\leq\delta\) and \(\delta\leq\rho\), it belongs to
\(S^m_{\rho,\delta}\). More precisely its remainder after \(|\alpha|<N\)
is \[
b-\sum_{|\alpha|<N}\frac{\partial_\eta^\alpha D_w^\alpha c(z,z,\xi)}{\alpha!}
\in S^{m-N(\rho-d)}_{\rho,\delta}.
\tag{G4}
\] Every seminorm uses finitely many constants in (G2), and the bound
remains valid at \(d=\rho\), where it gives no order gain.

Here are the estimates behind this adapter. For fixed \(z\), apply
Section 8 of \href{gauss-transform-estimates.pdf}{Quadratic Fourier
multipliers at a moving scale} with the metric
\(g_{(w,\eta)}(v,\nu)=\langle\eta\rangle^{2d}|v|^2+\langle\eta\rangle^{-2\rho}|\nu|^2\)
and weight \(\langle\eta\rangle^m\). Its slow variation and
temperateness are exactly the parameter verification in Section 4 of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}, with \(\delta\) there replaced by \(d\); its
quadratic parameter is \(\frac12\langle\eta\rangle^{d-\rho}\). The
theorem gives (G4) before restricting \(w=z\). Differentiate in \(z\)
first, using the modified weight
\(\langle\eta\rangle^{m+\delta|\beta|}\), and in \(w,\eta\) using their
exact modified weights. Differentiation after the diagonal restriction
is a sum of these derivatives. Since \(d\leq\delta\), each total base
derivative costs at most \(\delta\). This proves all of (G1), including
the remainder, rather than just a bound for values. Regularize \(c\) in
all variables by cutoffs tending to one. Formula (G3) for the
regularized kernels follows by Fourier inversion: the multiplier
\(e^{i\omega\cdot\theta}\) becomes the kernel
\((2\pi)^{-n}e^{-i(w-z)\cdot(\eta-\xi)}\). The Gauss bounds and
distributional continuity identify the limits. They also justify all
parameter derivatives. Thus no stationary phase theorem is hidden in
(G3).

\subsection{2. Kernels, exponential tests, and coordinate
transport}\label{kernels-exponential-tests-and-coordinate-transport}

For a global symbol \(a\in S^m\), its kernel is \[
K_a(x,y)=(2\pi)^{-n}\int e^{i(x-y)\cdot\xi}a(x,\xi)\,d\xi.
\tag{G5}
\] It is \(C^j\) if \(m+j+n<0\). Indeed, differentiating a total of at
most \(j\) times gives finitely many integrands bounded by
\(C\langle\xi\rangle^{m+j}\); these are integrable under the strict
inequality, uniformly on compact sets. Dominated convergence gives the
claimed derivatives. Off the diagonal, apply
\(e^{i(x-y)\xi}=|x-y|^{-2L}(-\Delta_\xi)^L e^{i(x-y)\xi}\), integrate by
parts in frequency, and take \(2L>m+j+n\). The coefficients and their
derivatives are bounded on every compact set separated from the
diagonal, giving smoothness there. For the parameters in (G1), frequency
differentiation instead gains \(\rho\) per derivative; sufficiently many
integrations still prove off-diagonal smoothness.

An order-minus-infinity symbol maps \(\mathcal S'\) into \(C^\infty\),
although its output need not decrease at spatial infinity. At fixed
\(x\), the differentiated expression
\(\partial_x^\beta(e^{ix\cdot\xi}a(x,\xi))\) is a Schwartz function of
\(\xi\), smoothly depending on \(x\) in that topology. Pair it with
\(\widehat u\). The compact-uniform Schwartz seminorms justify all
output derivatives and continuity. The same reasoning applies to local
smoothing symbols on every compact output set.

Exponential functions recover a symbol without being test functions: \[
\operatorname{Op}(a)(e^{ix\cdot\xi}u)
=e^{ix\cdot\xi}a(x,D+\xi)u,
\qquad
\operatorname{Op}(a)e^{ix\cdot\xi}=e^{ix\cdot\xi}a(x,\xi).
\tag{G6}
\] The first identity follows by the Fourier translation rule on
\(\mathcal S\), then on \(\mathcal S'\) by the continuous extensions in
Section 5 of \href{euclidean-symbol-calculus.pdf}{From symbol estimates
to operators on every Sobolev scale}. For the second, choose
\(v\in\mathcal S\) with \(\widehat v\in C_c^\infty\) and \(v(0)=1\). The
functions \(v(\varepsilon x)\) tend to one in \(\mathcal S'\): multiply
a Schwartz test function by their bounded pointwise differences and use
dominated convergence. The integrand
\(a(x,\xi+\varepsilon\theta)e^{i\varepsilon x\theta}\widehat v(\theta)\)
must retain the inverse Fourier factor. The exact identity is {\small
\[
 e^{-ix\cdot\xi}\operatorname{Op}(a)
       (e^{i(\cdot)\cdot\xi}v(\varepsilon\cdot))(x)
 =(2\pi)^{-n}\int a(x,\xi+\varepsilon\theta)
                e^{i\varepsilon x\cdot\theta}\widehat v(\theta)\,d\theta
 \longrightarrow a(x,\xi),\qquad
 (2\pi)^{-n}\int\widehat v(\theta)\,d\theta=v(0)=1.
 \tag{G6a}
\]
} Compact support of \(\widehat v\) permits every derivative limit on
compact base sets. Derivatives falling on the exponential retain their
powers of \(\varepsilon\theta\); the limit with no such derivative is
the corresponding derivative of \(a(x,\xi)\) times the displayed mass
one. Without the factor, even \(a=1\) would give the limit \((2\pi)^n\),
not one. This proves (G6) with its distributional meaning.

Let \(\kappa:U\to V\) be a smooth diffeomorphism and suppose that
\(\operatorname{Op}(a)\) has compact kernel support inside
\(U\times U\). Define its transport on functions by \[
A_\kappa f=(A(f\circ\kappa))\circ\kappa^{-1}.
\tag{G7}
\] It has compact kernel support in \(V\times V\). For
\(1-\rho\leq\delta\leq\rho\), \(a\in S^m_{\rho,\delta}\) implies
\(A_\kappa=\operatorname{Op}(a_\kappa)\) with a global
\(S^m_{\rho,\delta}\) symbol, zero outside a compact subset of \(V\) in
its base variable. In particular the allowed range has
\(\rho\geq\frac12\), and \(\rho=\frac12\) forces \(\delta=\frac12\).

\textbf{Proof with the Jacobians retained.} Cut the kernel into a part
sufficiently close to the diagonal and a smooth compact remainder. The
latter has a smoothing symbol by Fourier transformation in \(x-y\). For
the first part use small convex coordinate neighborhoods and write \[
\kappa(x)-\kappa(y)=L(x,y)(x-y),\qquad
L(x,y)=\int_0^1\kappa'(y+t(x-y))\,dt.
\tag{G8}
\] After shrinking the neighborhoods, \(L\) is invertible with all
derivatives bounded, as are those of its inverse. Put
\(z=\kappa(x),w=\kappa(y)\), and change frequency by
\(\xi=L(x,y)^T\eta\). The transported amplitude is \[
c(z,w,\eta)=\chi(x,y)a(x,L(x,y)^T\eta)
\frac{|\det L(x,y)|}{|\det\kappa'(y)|},
\tag{G9}
\] where \(\chi=1\) near the relevant diagonal. On that diagonal the
determinant ratio is one. Differentiating a frequency-linear argument
once in \(x\) or \(y\) introduces a factor \(O(|\eta|)\) and one
frequency derivative of \(a\), hence costs \(1-\rho\). A direct base
derivative of \(a\) costs \(\delta\). Repeated chain and product rules
therefore give (G2) with \(d=1-\rho\) for the \(w\) derivatives and with
\(\delta\) for the \(z\) derivatives, because \(1-\rho\leq\delta\).
Ordinary derivatives of the cutoff and determinants cost zero.
Compact-set norm comparability
\(\langle L^T\eta\rangle\asymp\langle\eta\rangle\) controls every real
order. Apply (G3). A finite cover of the compact diagonal part completes
the proof, including \(\rho=\delta=\frac12\). {\ReaderSymbols\char"220E}

This chain-rule argument also proves a useful symbol pullback statement.
If \(F(x,\xi)=(f(x),M(x)\xi)\), with smooth \(f\), smooth invertible
\(M\), and \(F(\mathcal C_1)\subset\mathcal C_2\), then
\(F^*:S^m_{\rho,\delta}(\mathcal C_2)\to S^m_{\rho,\delta}(\mathcal C_1)\)
is continuous when \(1-\rho\leq\delta\). For a compact ray generator its
image is compact; the smallest and largest singular values of \(M\) have
positive finite bounds. Every chain-rule term has the required
derivative budget, exactly as in (G9). No assumption that \(f\) is a
diffeomorphism is needed for this symbol statement.

For the classical class the full coordinate expansion is \[
a_\kappa(\kappa(x),\eta)
\sim\sum_\alpha\frac{\partial_\xi^\alpha a(x,\kappa'(x)^T\eta)}{\alpha!}
\left.D_y^\alpha e^{i r_x(y)\cdot\eta}\right|_{y=x},
\quad
r_x(y)=\kappa(y)-\kappa(x)-\kappa'(x)(y-x).
\tag{G10}
\] The exact value before expansion is
\(e^{-i\kappa(x)\cdot\eta}A(e^{i\kappa(\cdot)\cdot\eta})(x)\); a cutoff
equal to one near the compact kernel support makes this input
unambiguous. In (G10) the second factor is a polynomial in \(\eta\) of
degree at most \(\lfloor|\alpha|/2\rfloor\): every factor \(\eta\) comes
with a derivative of \(r_x\), and its value survives at \(y=x\) only
after at least two derivatives. The factors for \(|\alpha|=0,1\) are
respectively one and zero. Consequently the term has order at most
\(m-|\alpha|/2\).

For completeness, the coefficients and the remainder can be checked
without assuming that the exponential has small derivatives. Expand (G3)
to arbitrary order using (G4) applied to (G9). Each coefficient is a
finite linear combination of frequency derivatives of \(a\), evaluated
at \(\kappa'(x)^T\eta\), with polynomial coefficients in \(\eta\) and
jets of \(\kappa\). To identify these finite-jet expressions, take \(a\)
to be a polynomial in frequency with arbitrary coefficients at the fixed
base point. Then \(A=a(x,D)\) is a differential operator and the product
rule applied to
\(e^{i\kappa(y)\cdot\eta}=e^{i\kappa(x)\cdot\eta}e^{i(y-x)\cdot\kappa'(x)^T\eta}e^{ir_x(y)\cdot\eta}\)
gives precisely (G10). Polynomial jets span every finite jet space, so
these identities identify the coefficients for an arbitrary smooth
symbol as well. Sorting the finite expressions by the number of
frequency derivatives minus their polynomial degree is legitimate: only
finitely many terms have this number below any prescribed bound. The
discarded amplitude remainder after sufficiently many terms of (G4) and
the discarded coefficient terms both have arbitrarily low order. Thus
truncating (G10) at \(|\alpha|<2N\) has remainder in \(S^{m-N}\); the
same proof applies after any prescribed derivatives.

When \(1-\rho\leq\delta<\rho\), the same formula is an asymptotic series
in \(S_{\rho,\delta}\). The term with index \(\alpha\) has order at most
\(m-\rho|\alpha|+\lfloor|\alpha|/2\rfloor\). Using (G4) with
\(d=1-\rho\), truncation at \(|\alpha|<2N\) has remainder in
\(S^{m-N(2\rho-1)}_{\rho,\delta}\), hence also in
\(S^{m-N(\rho-\delta)}_{\rho,\delta}\). The parameter \(2\rho-1\) is
positive in this strict range. The equality-case invariance already
proved uses (G3), and does not rely on a decreasing-order series or
imply an elliptic inverse theorem at \(\delta=\rho\).

\subsection{3. The second symbol term and the density it
measures}\label{the-second-symbol-term-and-the-density-it-measures}

Equation (G10) first gives
\(a_\kappa(\kappa(x),\eta)-a(x,\kappa'(x)^T\eta)\in S^{m-1}\). Suppose
that \(a\sim a_m+a_{m-1}+\cdots\) is classical with step one. Define \[
s(a)=a_{m-1}+\frac{i}{2}\sum_j\partial_{x_j}\partial_{\xi_j}a_m.
\tag{G11}
\] For operators on functions, this is generally not a scalar on the
cotangent bundle. If \(J=|\det\kappa'|\), then \[
s(a_\kappa)(\kappa(x),\eta)
=s(a)(x,\kappa'(x)^T\eta)
-\frac12\sum_j(\partial_{\xi_j}a_m)(x,\kappa'(x)^T\eta)
\frac{D_{x_j}J}{J}.
\tag{G12}
\] Here is an index verification of the correction. Write
\(M_{kj}=\partial_j\kappa_k\), \(N=M^{-1}\), and \(\xi=M^T\eta\). The
order-\(m-1\) term added by (G10) is
\(-\frac i2\sum_{j,l,k}(\partial_{\xi_j}\partial_{\xi_l}a_m)\,\eta_k\partial_j\partial_l\kappa_k\).
In \(\sum_k\partial_{z_k}\partial_{\eta_k}[a_m(x,M^T\eta)]\), the chain
rule produces \(\sum_j\partial_{x_j}\partial_{\xi_j}a_m\), the same
Hessian term with coefficient \(+1\), and
\(\sum_{j,k,l}N_{lk}(\partial_lM_{kj})\partial_{\xi_j}a_m\). The Hessian
terms cancel after multiplication by \(i/2\). Since mixed derivatives
commute,
\(\sum_{k,l}N_{lk}\partial_lM_{kj}=\sum_{k,l}N_{lk}\partial_jM_{kl}=\partial_j\log J\),
by the determinant derivative formula. The remainder is
\((i/2)\sum_j\partial_{\xi_j}a_m\,\partial_j\log J\), which is (G12).
Orientation reversal changes the sign of \(\det M\) only by a locally
constant factor, so its logarithmic derivative is that of \(J\).

The density line bundle \(\Omega\) has local generator \(|dx|\). For any
\(z\in\mathbb C\), define \(\Omega^z\) using positive Jacobians raised
to \(z\): \(t^z=e^{z\log t}\) for \(t>0\). The real logarithm makes
\((t_1t_2)^z=t_1^zt_2^z\), proving the cocycle relation, including
complex exponents. Components transform by
\(u_\kappa(\kappa(x))=J(x)^{-z}u(x)\). Thus an operator on
\(z\)-densities is transported as the function operator conjugated in
the old coordinates by \(J^{-z}AJ^z\). Its symbol differs from \(a\),
modulo \(S^{m-2}\), by \[
z\sum_j\partial_{\xi_j}a\,\frac{D_jJ}{J}.
\tag{G13}
\] Indeed Section 5 of \href{euclidean-symbol-calculus.pdf}{From symbol
estimates to operators on every Sobolev scale} applied to multiplication
by \(J^z\) gives \(J^{-z}(aJ^z+\sum_j\partial_{\xi_j}a D_jJ^z)\) through
the first correction; all higher frequency derivatives have order at
most \(m-2\). The identity \(D_jJ^z=zJ^zD_jJ/J\) proves (G13). At
\(z=\frac12\) it cancels (G12).

There is a version without homogeneity. For scalar half-density
operators, the local quantity \[
\sigma_{[2]}(A)=a+\frac i2\sum_j\partial_{x_j}\partial_{\xi_j}a
\pmod {S^{m-2}}
\tag{G14}
\] agrees on overlaps as a scalar symbol class on \(T^*X\). Repeat the
preceding calculation with \(a\) in place of \(a_m\): omitted terms
contain at least two net frequency losses, so they are of order \(m-2\).
A partition of unity patches the local quantities to a global symbol,
and any two patches differ by \(S^{m-2}\) because they agree in each
chart modulo that space. Its homogeneous term of degree \(m-1\), when
present, is (G11). Formula (G12) also shows invariance for function
operators at double zeros of \(a_m\), and under transformations with
locally constant Jacobian. The refined assertion (G14) is specifically
for scalar half-densities; a varying vector-bundle frame introduces its
own first-order correction.

\subsection{4. Assembling an operator and its principal
symbol}\label{assembling-an-operator-and-its-principal-symbol}

Let \(A:C_c^\infty(U)\to C^\infty(U)\) be continuous. Suppose
\(\phi A\psi\) is the global quantization of an \(S^m\) symbol for every
\(\phi,\psi\in C_c^\infty(U)\). Then \[
A=\operatorname{Op}(a)+R,\qquad a\in S^m(U\times\mathbb R^n),
\qquad K_R\in C^\infty(U\times U).
\tag{G15}
\] The symbol is unique modulo local \(S^{-\infty}\).

To prove this, choose a locally finite smooth partition \(\theta_j\)
with compact supports. Compactness and local finiteness imply that each
support meets only finitely many others. Let \(a_{jk}\) be the symbol of
\(\theta_jA\theta_k\). By (G6) it vanishes in the base variable outside
\(\operatorname{supp}\theta_j\). Sum \(a_{jk}\) only for intersecting
pairs of supports. The sum is locally finite in the base, and on a fixed
compact set the finitely many symbols satisfy the required bounds. The
kernel difference is the locally finite sum
\(\sum'\theta_j(x)K_A(x,y)\theta_k(y)\) over disjoint pairs. Each term
is smooth by (G5); this proves (G15). If a compactly supported smooth
kernel is given, its left symbol is its Fourier transform in \(y\) after
multiplication by \(e^{-ix\cdot\xi}\). Integrating by parts in \(y\),
and differentiating in \(x,\xi\), proves every smoothing seminorm. Apply
this to \(\phi(A-\operatorname{Op}(a))\psi\), with \(\psi=1\) near
\(\operatorname{supp}\phi\). The symbol of
\(\phi\operatorname{Op}(a)(1-\psi)\) is smoothing by off-diagonal
integration by parts; hence local uniqueness follows. Conversely the
symbol and smooth kernel in (G15) clearly satisfy the localized
hypothesis.

Define \(\Psi^m(X)\) by this localized condition in coordinate charts,
together with smoothness of the kernel off the diagonal. Equivalently
one can impose the localized condition in every chart and for every pair
of compact cutoffs: that condition already implies off-diagonal
smoothness. One atlas suffices if off-diagonal smoothness is included.
To verify this, localize a kernel near any diagonal point and transport
its compact localization by (G7)--(G10); the remainder in (G15) stays
smooth under a diffeomorphism. Away from the diagonal the assumed
smoothness handles pairs of different charts. The kernel is a
distributional density in its input variable, so it is intrinsically a
section with coefficients in \(1\boxtimes\Omega\), not a scalar function
on \(X\times X\).

The fiber-linear pullback proved in Section 2 defines \(S^m(T^*X)\)
independently of charts. Local principal symbols patch and give \[
\Psi^m(X)/\Psi^{m-1}(X)
\simeq S^m(T^*X)/S^{m-1}(T^*X).
\tag{G16}
\] For existence, take real compact smooth \(\theta_j\) subordinate to
charts with \(\sum_j\theta_j^2=1\), and a cutoff \(\psi_j=1\) near
\(\operatorname{supp}\theta_j\). Quantize a given cotangent symbol in
each chart as \(A_j=\theta_j\operatorname{Op}(\psi_j a_j)\theta_j\),
transporting back and extending by zero. The sum is locally finite and
properly supported: over a fixed compact set only finitely many
\(\operatorname{supp}\theta_j\) occur in either projection. Its
principal symbol is \(\sum_j\theta_j^2a=a\). A square partition exists
by dividing any nonnegative partition \(\chi_j\) by
\((\sum_l\chi_l^2)^{1/2}\). On overlaps (G10) identifies principal
classes. Their vanishing is equivalent in each local representation to
order \(m-1\), proving the kernel of (G16) and injectivity. This also
proves that \(\Psi^{-\infty}(X)\) is exactly the smooth-kernel class, by
applying (G15) at every order and the integrability threshold of (G5).

\subsection{5. Properness, products, and global
summation}\label{properness-products-and-global-summation}

Write \(S=\operatorname{supp}K_A\subset X\times X\). The operator is
\textbf{properly supported} if both projections \(S\to X\) are proper.
For each compact \(K\subset X\), this says that there is a compact
\(K'\) such that \[
\operatorname{supp}u\subset K\ \Longrightarrow\ \operatorname{supp}Au\subset K',
\qquad
u=0\text{ near }K'\ \Longrightarrow\ Au=0\text{ near }K.
\tag{G17}
\] The implications initially apply to test functions and then to the
distributional extensions on their domains. Properness gives (G17) by
the two kernel projections, enlarging the resulting compact sets to
compact neighborhoods. Conversely, apply the support bounds to all test
functions supported in a compact neighborhood of \(K\). If the kernel
were nonzero outside the corresponding product set, its restriction to a
small product of coordinate neighborhoods would pair nontrivially with
some product of test functions, contradicting a support bound. Product
test functions detect kernels by the scalar kernel theorem. Thus each
projection has compact inverse images. This proves the equivalence,
including the distinction between vanishing on a set and on a
neighborhood used for distributions.

An arbitrary \(A\in\Psi^m\) extends continuously
\(\mathcal E'(X)\to\mathcal D'(X)\): on each compact output set the
local expression (G15) acts on compactly supported distributions by
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale} and by pairing the smooth remainder. For a
properly supported operator and arbitrary \(u\in\mathcal D'(X)\), define
its output near a compact set \(K\) using \(A(\chi u)\), where
\(\chi=1\) near the corresponding input compact set \(K'\). Formula
(G17) proves independence of \(\chi\), and compatibility on overlapping
output sets patches the result. Each output seminorm is controlled by
finitely many input seminorms after this localization, proving
continuity. Proper operators likewise map \(C^\infty\to C^\infty\),
\(C_c^\infty\to C_c^\infty\), and \(\mathcal E'\to\mathcal E'\).

Every \(A\in\Psi^m\) is a properly supported operator plus a
smooth-kernel operator. In the proof of (G15), retain only the
intersecting partition pairs as an operator sum, now working
intrinsically. The union of their compact kernel support rectangles has
both projections proper, since each compact base set meets only finitely
many partition supports and each such support has finitely many
neighbors. The omitted terms are a locally finite smooth kernel.
Equivalently this construction gives a smooth properly supported
function on \(X\times X\), equal to one near the diagonal, by summing
the retained products \(\theta_j(x)\theta_k(y)\). Multiplying any kernel
by that function gives a common properness convention.

If \(A\in\Psi^m\) and \(B\in\Psi^{m'}\) are proper, then
\(AB\in\Psi^{m+m'}\) is proper, and its principal symbol is
\(\sigma_m(A)\sigma_{m'}(B)\), with the displayed order of factors. The
relation \(S_A\circ S_B\) containing the product kernel support is
closed and proper in both projections: above a compact output set the
possible intermediate points lie in a compact set by properness of
\(A\), and the possible input points then lie in a compact set by
properness of \(B\). The same reasoning with the projections reversed
proves the other direction, and these compactness statements prove
closure by convergent subsequences.

For the local analytic assertion choose compact cutoffs \(\phi,\psi\) in
a chart and \(\chi=1\) near both supports. Then
\(\phi AB\psi=(\phi A\chi)(\chi B\psi)+\phi A(1-\chi^2)B\psi\). The
first term is given by Section 5 of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}. The second has a smooth kernel: properness
confines its intermediate variable to a compact set, and the separated
cutoffs make both factors smooth in the relevant external-variable
neighborhoods. Differentiation under the compact integral, or action of
a localized operator on a smooth kernel in one variable, proves every
derivative. The same argument shows that smooth proper kernels form a
two-sided ideal among proper pseudodifferential operators. It also
covers products where the same support conditions make composition
meaningful; we do not compose arbitrary nonproper operators on all
distributions.

Finally let \(A_j\in\Psi^{m_j}(X)\), with \(m_j\) decreasing to
\(-\infty\). There exists a proper \(A\in\Psi^{m_0}\) such that \[
A-\sum_{j<k}A_j\in\Psi^{m_k}(X)\quad(k\geq0).
\tag{G18}
\] First replace every \(A_j\) by a representative using the same proper
kernel cutoff just constructed; this changes only smooth remainders.
Choose a locally finite partition of a neighborhood of the diagonal by
compact chart-product pieces. For each such piece, its localized kernel
has a symbol of order \(m_j\) by (G3). Sum these symbols over \(j\)
using Section 1, then quantize and multiply by a cutoff equal to one
near the corresponding diagonal piece. The difference introduced by that
last cutoff is smooth. Summing the pieces gives (G18), since any compact
product meets finitely many pieces and hence every remainder is a finite
sum of symbols of order \(m_k\) and smooth kernels. Apply the common
proper cutoff once more. It does not change any quotient by smoothing,
proving properness and all remainder claims. This argument uses local
finiteness, not a finite atlas or a uniform geometry assumption.

\subsection{6. Inverses with global and conic error
control}\label{inverses-with-global-and-conic-error-control}

A scalar \(A\in\Psi^m(X)\) is elliptic when its principal class in (G16)
has an inverse of order \(-m\). In coordinates this is equivalent to a
lower bound \[
|a(x,\xi)|\geq c_K\langle\xi\rangle^m
\quad(x\in K,\ |\xi|\geq R_K)
\tag{G19}
\] on each compact set. For a square matrix symbol, use its smallest
singular value, or equivalently
\(\|a^{-1}\|\leq C_K\langle\xi\rangle^{-m}\). The reciprocal estimates
of Section 9 of \href{euclidean-symbol-calculus.pdf}{From symbol
estimates to operators on every Sobolev scale} prove the implication
from (G19) to an inverse class by taking local frequency cutoffs and
patching. Conversely \(ab=1+r\), \(r\in S^{-1}\), gives \(|ab|\geq1/2\)
above a compact-dependent radius and hence (G19). The same argument for
square matrices uses the convergent finite-dimensional matrix Neumann
series for \(I+r\).

For proper elliptic \(A\), choose a proper \(B_0\) quantizing the
inverse class. Then \(R=I-AB_0\), \(L=I-B_0A\) have order \(-1\). Sum
\(B_0R^j\) and \(L^jB_0\) by (G18), obtaining right and left
parametrices \(B_R,B_L\). The finite identities \[
A\sum_{j<N}B_0R^j=I-R^N,
\qquad
\left(\sum_{j<N}L^jB_0\right)A=I-L^N
\tag{G20}
\] and the orders of their tails prove smoothing errors. Since smooth
proper kernels form an ideal, \(B_L-B_R=B_L(I-AB_R)+(B_LA-I)B_R\) is
smoothing. Thus either is a proper two-sided parametrix, of order
\(-m\). This proof requires no convergence of an operator Neumann series
and no uniform ellipticity constant on a noncompact manifold.

A nonzero covector \(\gamma=(x_0,\xi_0)\) is noncharacteristic if (G19)
holds in some open conic neighborhood of it, at sufficiently large
frequency. Equivalently \(ab-1\in S^{-1}\) there for some conic
\(b\in S^{-m}\). The reciprocal argument is local on a smaller conic
neighborhood, so it proves this equivalence. Adding an order-\(m-1\)
symbol cannot destroy the lower bound after shrinking and increasing the
radius; coordinate changes preserve it by (G10). The complement
\(\operatorname{Char}_m(A)\) is closed and conic. For a homogeneous
leading symbol it is its zero set off the zero section. More generally
the full local symbol has \textbf{microlocal order \(k\)} at a covector
if it is in \(S^k\) on a conic neighborhood. Formula (G10), applied
after a conic cutoff and with its separated remainder, and uniqueness in
(G15), show that this notion is independent of chart and full-symbol
representative. Order minus infinity means one neighborhood on which
every negative-order estimate holds.

We record the conic form of the calculus used here. If symbols \(a,b\)
are prescribed on a cone and the usual composition is formed after
extending them with cutoffs, its differential expansion is the usual one
on every smaller cone; changes of extension have smoothing effects
there. To prove it, choose a classical degree-zero cutoff \(q\) equal to
one on the smaller cone, supported in the larger one, and a second
cutoff equal to one near its support. In the oscillatory formula for
composition, terms away from those supports have either separated base
points, handled by frequency integration by parts, or separated
frequency directions. In the latter case
\(|\eta-\xi|\geq c(|\eta|+|\xi|)\); integration by parts in the
intermediate base variable supplies arbitrarily many factors of this
denominator, while its derivatives increase a symbol order by strictly
less than one. Split into dyadic frequency shells to sum the resulting
estimates. Derivatives of any fixed order are handled by performing more
integrations. In the remaining region the global remainder theorem of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale} applies. This proves all differentiated conic
estimates, including changes of cutoffs. It also shows that rapidly
decreasing conic symbols form a two-sided microlocal ideal.

At a noncharacteristic point construct a local right inverse as follows.
Choose \(b_0\), supported in the elliptic cone, equal there on a smaller
cone to the reciprocal of \(a\), with a high-frequency cutoff. Extend
\(r=1-a\circ b_0\), restricted to that smaller cone, to a global symbol
of order \(-1\), using a further cone cutoff. The formal sum
\(\sum b_0\circ r^{\circ j}\), made by local summation, gives an inverse
modulo smoothing on a still smaller cone by (G20). The same construction
on the left gives \(b_L\); the identity comparing the two inverses shows
they agree modulo smoothing on their common cone. Proper quantization
yields a single \(B\in\Psi^{-m}\) with both errors microlocally
smoothing at the original point. Either such one-sided condition for a
given \(B\) implies noncharacteristicity by the first remainder of the
composition formula, hence implies the other condition by comparison
with the inverse just constructed.

For \(1-\rho\leq\delta<\rho\), all constructions in this item hold with
gain \(\varepsilon=\rho-\delta>0\), with \(S^{m-\varepsilon}\) in the
principal quotient and \(S^{-\varepsilon}\) in the inverse-class
condition. The coordinate error from Section 2 has order at most
\(m-(2\rho-1)\leq m-\varepsilon\), so these quotient symbols are
intrinsic. Replace powers of order \(-j\) by order \(-j\varepsilon\) in
(G20). At \(\delta=\rho\), closure and coordinate invariance survive;
the zero gain does not furnish this principal quotient, conic inverse
argument, or elliptic regularity assertion.

\subsection{7. Fourier directions and the precise kernel
relation}\label{fourier-directions-and-the-precise-kernel-relation}

On a coordinate open set, a nonzero covector \((x_0,\xi_0)\) is absent
from \(\operatorname{WF}(u)\) if there is \(\chi\in C_c^\infty\), equal
to one near \(x_0\), such that \(\widehat{\chi u}\) decreases faster
than every inverse power in a cone about \(\xi_0\). A cutoff merely
nonzero at \(x_0\) gives the same definition by multiplying with its
smooth local reciprocal. The basic cutoff estimate is worth spelling
out. If a compact distribution has polynomially bounded Fourier
transform and that transform decreases rapidly in a cone \(V\), then
multiplying by a compact smooth function preserves rapid decrease in
every cone whose angular closure lies in \(V\). In the convolution
formula split into \(\eta\in V\), where the input decreases rapidly, and
its complement. For an output frequency in the smaller cone and a
complementary \(\eta\), \(|\xi-\eta|\geq c(|\xi|+|\eta|)\); rapid decay
of the cutoff transform beats both the polynomial input bound and any
desired output weight. Integrating these bounds proves the assertion.
Spatial localization then proves that this definition is local, has a
closed conic wavefront set, and gives smoothness exactly when every
direction is regular: a finite angular cover of the unit sphere provides
rapid Fourier decrease in all directions, followed by Fourier inversion.

For a compactly localized pseudodifferential kernel, use coordinates
\((x,v)=(x,x-y)\). Before localizing in \(v\), its Fourier transform in
\((x,v)\) is \[
\mathcal F_{x,v}\big[\phi(x)K_a(x,x-v)\big](\zeta,\eta)
=\mathcal F_x(\phi a)(\zeta,\eta).
\tag{G21}
\] For classical symbols the right side is bounded by
\(C_L\langle\zeta\rangle^{-L}\langle\eta\rangle^m\), after any fixed
frequency derivatives. With general parameters the power in the last
factor becomes \(m+\delta L\). Where \(|\zeta|\geq c|\eta|\), choose
\(L\) large; the gain \(1-\delta>0\) gives rapid decay. Multiplication
by a compact \(v\)-cutoff preserves that decay on smaller cones by the
preceding convolution estimate. Together with off-diagonal smoothness
this proves \[
\operatorname{WF}(K_A)\subset
\{(x,x;\xi,-\xi):\xi\ne0\}.
\tag{G22}
\] If \(a\) is smoothing on a conic neighborhood of \((x_0,\xi_0)\), the
same calculation, splitting \(a\) with a conic cutoff, shows that the
corresponding point is absent from (G22).

The converse also follows from (G21); here are the estimates needed for
it. If the kernel is regular near \((x_0,v=0;0,\xi_0)\), choose compact
cutoffs in \(x,v\), equal to one on smaller neighborhoods. Its full
Fourier transform is rapidly decreasing when \(|\zeta|<c|\eta|\) and
\(\eta\) is in a fixed smaller cone about \(\xi_0\). Recover the symbol
of the cutoff kernel by inverse Fourier transformation in \(\zeta\). On
that region rapid decrease gives every desired \(\eta\) bound, even
after multiplying by a fixed power of \(\zeta\). On
\(|\zeta|\geq c|\eta|\), the estimate following (G21) gives the same
bound by increasing \(L\). Frequency derivatives can be taken before the
estimates: they correspond to multiplying the compact kernel by powers
of \(v\), which preserves its conic regularity. The recovered symbol
therefore satisfies every negative-order estimate near \((x_0,\xi_0)\).
Removing the \(v\)-cutoff changes the symbol by a smoothing symbol, by
off-diagonal integration by parts and compact localization in both
original variables. Thus (G22) is precisely the diagonal relation of the
nonsmoothing directions of the full symbol.

Denote that closed conic subset of \(T^*X\setminus0\) by
\(\operatorname{WF}(A)\). For the moment the assertion is local;
coordinate invariance will follow below. The prime on a kernel relation
reverses its input covector: \[
\operatorname{WF}'(K_A)
=\{(x,\xi;y,-\eta):(x,y;\xi,\eta)\in\operatorname{WF}(K_A)\}
=\{(\gamma,\gamma):\gamma\in\operatorname{WF}(A)\}.
\tag{G23}
\] This sign is essential: the identity operator has kernel
\(\delta(x-y)\), whose untwisted covectors are \((\xi,-\xi)\).

\subsection{8. What every input can
detect}\label{what-every-input-can-detect}

For proper \(A\), and also for arbitrary \(A\in\Psi^m\) on compactly
supported inputs, \[
\operatorname{WF}(Au)\subset\operatorname{WF}(A)\cap\operatorname{WF}(u).
\tag{G24}
\] We give a proof using cutoffs, so no unstated general
kernel-composition theorem is needed. If a direction is absent from
\(\operatorname{WF}(A)\), choose a classical conic cutoff \(Q\)
supported in that regular symbol cone, elliptic at the point. Then
\(QA\) is smoothing there, and by restricting the support of \(Q\) it is
smoothing everywhere on the output neighborhood. The symbol estimates of
Section 6 prove this. If instead the direction is absent from
\(\operatorname{WF}(u)\), choose compact spatial cutoffs and a smooth
frequency cutoff \(q\), homogeneous above a ball and equal to one in a
smaller cone, such that \(q(D)(\chi u)\) is smooth. Choose a proper
\(Q\) equal to \(q(D)\chi\) near the point and a smaller conic cutoff
\(P\). The full symbol of \(PA(I-Q)\) is smoothing by the
separated-support estimates. Thus \(PAu=PAQu+PA(I-Q)u\) is smooth. Since
\(P\) is a product of a spatial cutoff and an angular Fourier cutoff,
this says directly that \(Au\) is Fourier-regular at the point. For an
arbitrary distribution under a proper operator, (G17) reduces this
argument to a compact input. For a global symbol and
\(u\in\mathcal S'\), the separated spatial remainder \(\phi A(1-\chi)u\)
is smooth: its kernel, after each output derivative, is Schwartz in the
input variable, as repeated frequency integrations in (G5) show. This
also proves global tempered pseudolocality and shrinkage of singular
support.

The cutoff proof works for \(1-\rho\leq\delta\leq\rho\), including
equality. One does not need an asymptotic expansion of two general
equality-class symbols to separate a classical cutoff from a symbol. In
\(a\circ q\) the differentiated terms gain \(\rho\) per paired
derivative, since base derivatives of \(q\) cost zero; in \(q\circ a\)
they gain \(1-\delta\). Both are positive. The same mixed-metric Gauss
argument as Section 4 of \href{euclidean-symbol-calculus.pdf}{From symbol
estimates to operators on every Sobolev scale} proves the remainders
with these respective gains. It follows, by inserting nested classical
cutoffs, that the microsupport of a defined product satisfies \[
\operatorname{WF}(AB)\subset\operatorname{WF}(A)\cap\operatorname{WF}(B).
\tag{G25}
\] For example if \(A\) is smoothing near a direction, insert \(Q=1\) on
a smaller cone: \(QAB\) is smoothing because \(QA\) is smoothing. If
\(B\) is smoothing there, insert a second cutoff \(Q_1=1\) near \(Q\);
write \(QAB=QAQ_1B+QA(1-Q_1)B\), where the first term has a smoothing
right factor in that cone and the second is separated. This proves the
general equality-case assertion as well as the classical one.

There is an exact converse to (G24). For a closed conic
\(\Gamma\subset T^*X\setminus0\), the following conditions are
equivalent:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  The full symbol of \(A\) is smoothing off \(\Gamma\).
\item
  \(\operatorname{WF}'(K_A)\subset\{(\gamma,\gamma):\gamma\in\Gamma\}\).
\item
  Every \(u\in\mathcal E'(X)\) satisfies
  \(\operatorname{WF}(Au)\subset\Gamma\cap\operatorname{WF}(u)\).
\end{enumerate}

We proved the first equivalence in (G21)--(G23), and its implication to
the third in (G24). Suppose the third holds and choose a compactly based
classical cutoff \(Q\) whose microsupport lies in a small closed cone
disjoint from \(\Gamma\), with symbol one near a chosen covector. For
every compact distribution, \(AQ u\) has wavefront contained both in
\(\Gamma\) and in \(\operatorname{WF}(Q)\), so it is smooth. We justify
that its kernel is smooth, a fact stronger than smoothing individual
test functions.

Localize both variables in compact coordinate sets. For a fixed input
compact set \(K\), let \(E'_{K,N}\) be the Banach space of distributions
supported in \(K\) with norm the dual norm of \(C^N\) test functions on
a fixed compact neighborhood. The restricted map
\(AQ:E'_{K,N}\to C^\infty\) has closed graph: convergence in that Banach
norm implies distributional convergence, the original operator is
continuous into distributions, and a smooth limit has the same
distributional limit. Sections 14.5--14.6 of the Banach foundation
lesson prove this closed graph step with the exact dual norm and every
output seminorm, and therefore make this map continuous. For input
points in the interior of \(K\), the map \(y\mapsto\delta_y\) is \(C^r\)
into \(E'_{K,N}\) when \(N\geq r+1\). Taylor's formula, with one extra
uniformly bounded derivative on the unit ball of \(C^N\), proves this
assertion and its difference-quotient derivatives. Consequently
\((x,y)\mapsto (AQ\delta_y)(x)\) has every prescribed mixed derivative:
choose \(r,N\) sufficiently large and use continuity into every output
\(C^k\) seminorm. It is the kernel, since integration of
\(\delta_y f(y)\) reproduces a test function and the operator is
continuous. Hence that kernel is smooth.

Its symbol is therefore smoothing after compact localization. The conic
cutoff calculus gives \(a\circ q-a\in S^{-\infty}\) near the selected
covector, since \(q=1\) there to all orders. Thus \(a\) is smoothing
there. This proves the converse and the minimality of
\(\operatorname{WF}(A)\) among all sets allowed in (G24).

\subsection{9. Elliptic tests and intrinsic
wavefront}\label{elliptic-tests-and-intrinsic-wavefront}

For every fixed real \(m\) and \(u\in\mathcal D'(X)\), \[
\operatorname{WF}(u)=
\bigcap_{\substack{A\in\Psi^m(X)\ \mathrm{proper}\\Au\in C^\infty(X)}}
\operatorname{Char}_m(A).
\tag{G26}
\] If a direction is Fourier-regular, choose an order-\(m\) conic symbol
supported in a regular neighborhood and elliptic at the direction, for
instance a degree-zero cutoff times \(\langle\xi\rangle^m\), with
compact base support. Proper quantization and (G24) make \(Au\) smooth
globally. Conversely, if \(Au\) is smooth and \(A\) is noncharacteristic
at the direction, its conic inverse from Section 6 gives
\(u=BAu+(I-BA)u\). The first term is smooth because \(B\) is proper, and
the second is regular at the direction by (G24). This proves (G26) for
the specified fixed order, not just for order zero.

Conjugating the test operators in (G26) by a diffeomorphism preserves
properness, their order, their action on smooth functions, and their
noncharacteristic set under the cotangent map, by Section 2 and Section
6. Distributions themselves pull back by the change-of-variables formula
\((\kappa^*u)(f|dx|)=u((f\circ\kappa^{-1})|\det(\kappa^{-1})'|\,|dz|)\);
it is continuous because composition and multiplication are continuous
on compact test-function spaces. Thus (G26) proves that the Fourier
definition transforms by the cotangent map. This establishes its
intrinsic meaning on \(X\), and also that of (G23), without assuming a
general pullback theorem for distributions with wavefront constraints.
Smooth changes of bundle frame preserve it componentwise, by (G24) for
multiplication and by the inverse frame transformation.

For a proper \(A\in\Psi^m\), \[
\operatorname{WF}(u)\subset\operatorname{WF}(Au)\cup\operatorname{Char}_m(A).
\tag{G27}
\] At a point outside the right side, choose a proper order-zero cutoff
\(C\) elliptic there with \(CAu\) smooth, using (G26). The product
\(CA\) is noncharacteristic there, and (G26) applied to it proves
regularity of \(u\). Equivalently use its conic inverse. Together (G24)
and (G27) say that a noncharacteristic operator neither creates nor
removes a smoothness defect at that direction. The same proof applies in
the strict coordinate-invariant \((\rho,\delta)\) range using the
positive-gain conic inverse from Section 6.

\subsection{10. Convergence when singular directions are
fixed}\label{convergence-when-singular-directions-are-fixed}

Let \(\Gamma\) be closed and conic off zero. Set
\(\mathcal D'_\Gamma(X)=\{u:\operatorname{WF}(u)\subset\Gamma\}\). We
use the following sequential convergence convention, defined here as
follows. In each chart, \(u_j\to u\) means weak distributional
convergence and \[
\sup_{\xi\in V}\langle\xi\rangle^N
|\widehat{\phi(u_j-u)}(\xi)|\longrightarrow0
\tag{G28}
\] for every \(N\geq0\), compact smooth \(\phi\), and closed frequency
cone \(V\) for which
\((\operatorname{supp}\phi\times V)\cap\Gamma=\varnothing\) off zero.
Replacing \(\langle\xi\rangle^N\) by \(|\xi|^N\), \(N\geq1\), gives the
same convention: weak convergence and uniform boundedness imply uniform
Fourier convergence on bounded frequency sets. We make a statement about
sequences, not an identification of all locally convex topologies that
share the same convergent sequences.

The equivalent operator test is \[
u_j\longrightarrow u\text{ weakly in }\mathcal D',
\qquad
Au_j\longrightarrow Au\text{ in }C^\infty(X)
\tag{G29}
\] for every proper pseudodifferential \(A\) with
\(\operatorname{WF}(A)\cap\Gamma=\varnothing\).

We first isolate a uniformity fact. A weakly convergent sequence of
distributions has, on each fixed compact input set, a common
finite-order bound by (BF9) in Section 14.3 of the Banach foundation
lesson, after the test-space completeness proof in Section 14.2.
Moreover it converges uniformly on compact families of test functions in
that space: cover the family by finitely many small neighborhoods in the
controlling seminorm, and combine the common bound with convergence at
the finitely many centers. In particular a fixed smooth proper kernel
sends weakly convergent sequences to \(C^\infty\)-convergent sequences,
since its differentiated input test functions over a compact output set
form compact families.

Assume (G28) and fix an output compact set for \(A\). Properness
restricts the input to a fixed compact set. Up to a smooth kernel, split
the symbol into finitely many pieces whose angular/base supports have
closures disjoint from \(\Gamma\); this is possible by compactness of
the cosphere over the output compact set. Use a spatial cutoff \(\phi\)
and a slightly larger good frequency cone for each piece. In the Fourier
formula for its action on \(\phi(u_j-u)\), the part in that cone is
bounded, after \(k\) output derivatives, by a constant times the
seminorm (G28) with \(N>m+k+n+1\). It therefore tends to zero. The
complementary frequency part has separated angular supports and is a
smooth kernel after the conic integration-by-parts argument in Section
6. The uniformity fact handles that part, as well as the initial smooth
remainder. Finitely many such bounds prove every output \(C^k\) seminorm
in (G29).

Conversely fix \(\phi,V\) in (G28). Enlarge its spatial support slightly
and its angular cone slightly while remaining disjoint from \(\Gamma\);
if necessary a finite partition in the spatial variable achieves such an
enlargement. Choose \(\chi=1\) near the enlarged spatial support and a
classical angular cutoff \(q=1\) near \(V\), supported in the enlarged
cone above frequency one. Make \(A=\chi q(D)\phi\) proper by the common
near-diagonal kernel cutoff. Its microsupport avoids \(\Gamma\), so
(G29) gives convergence of its compactly supported outputs in every
smooth norm. The properness correction is a smooth kernel on the fixed
input support and tends to zero in the same norms. Thus
\(\chi q(D)\phi(u_j-u)\to0\) in \(C_c^\infty\). The omitted output tail
\((1-\chi)q(D)\phi(u_j-u)\) has a Schwartz kernel in the output variable
and compact support in the input variable: repeated frequency
integration by parts gives arbitrary spatial decay, uniformly with
derivatives and input derivatives, because the two spatial supports are
separated. The uniformity fact therefore makes this tail tend to zero in
\(\mathcal S\). We conclude \(q(D)\phi(u_j-u)\to0\) in \(\mathcal S\).
Taking its Fourier transform and restricting to \(V\) above frequency
one gives (G28). Bounded frequencies were already controlled by weak
convergence. This proves the equivalence and, by its intrinsic operator
formulation, its coordinate independence.

One may replace convergence in (G28) by uniform boundedness of those
seminorms for all \(N\), retaining weak distributional convergence. To
see this, control a desired order \(N\) tail by the uniform
order-\(N+1\) bound divided by the frequency radius; on a bounded
frequency set use uniform convergence. This is a statement about the
entire sequence, including its limit, and does not weaken the
requirement that every \(u_j\) and \(u\) lie in \(\mathcal D'_\Gamma\).

\subsection{11. Sobolev and Besov spaces in a moving
chart}\label{sobolev-and-besov-spaces-in-a-moving-chart}

Let \(\mathcal B^s_p=B^s_{2,p}\), \(1\leq p\leq\infty\), be the dyadic
Hilbert-based Besov space of Section 8 of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}. Explicitly, with sharp annular Fourier
projections \(\Pi_j\), \[
\|u\|_{\mathcal B^s_p}
=\left\|(2^{js}\|\Pi_j u\|_{L^2})_{j\geq0}\right\|_{\ell^p},
\qquad \mathcal B^s_2=H^s
\tag{G30}
\] up to equivalent norms. At \(p=\infty\) the norm is a supremum. This
is the scale also denoted \({}^pH_{(s)}\), and the exponent \(p\) is not
a spatial \(L^p\) exponent.

We prove invariance under compactly localized coordinate changes for
every real \(s\) and every displayed \(p\). Let
\(Tu(z)=\phi(z)\,u(\kappa^{-1}(z))\), with compact smooth \(\phi\)
inside a coordinate image, inserting an input cutoff equal to one on its
inverse image. For an integer \(k\geq0\), the chain rule and change of
variables bound its \(H^k\) norm by the input \(H^k\) norm: by
Plancherel, the sum of the \(L^2\) norms of derivatives of orders at
most \(k\) is an equivalent norm. All coefficient functions and
Jacobians are bounded on the fixed compact supports. The \(L^2\)
transpose of \(T\) is the reverse coordinate pullback times the smooth
density Jacobian and cutoffs, so it has the same positive-integer
estimates. Duality gives the estimates on \(H^{-k}\). This duality
follows directly from the weighted Fourier pairing:
\(\|u\|_{H^{-k}}=\sup_{\|v\|_{H^k}\leq1}|\langle u,v\rangle|\), first
for Schwartz functions and then by completion.

Choose integers \(k_-<s<k_+\). The annular two-endpoint argument (E39)
of Section 8 of \href{euclidean-symbol-calculus.pdf}{From symbol
estimates to operators on every Sobolev scale}, with order shift zero,
now yields \[
T:\mathcal B^s_p\longrightarrow\mathcal B^s_p
\quad\text{continuously},\qquad1\leq p\leq\infty.
\tag{G31}
\] Its proof uses a summable convolution sequence in the dyadic indices
and consistency of \(T\) on the two Sobolev endpoints. Distributional
pullback supplies that consistency. At \(p=\infty\) the input annular
sums are taken in the lower Sobolev space, not incorrectly asserted to
converge in the Besov norm. Applying (G31) to the inverse diffeomorphism
proves equivalence of chart norms. Thus this argument also proves
real-order Sobolev coordinate invariance, without importing it as an
extra theorem.

Define \(\mathcal B^{s,\mathrm{loc}}_p(X)\) by requiring every compact
coordinate cutoff of a distribution, expressed in that chart and
extended by zero, to have finite norm (G30). Its topology consists of
those seminorms. Define \(\mathcal B^{s,\mathrm{comp}}_p(X)\) by adding
compact support, with the inductive-limit topology over compact
supports. Formula (G31), multiplication boundedness from
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}, and finite partitions on each compact set show
independence of coordinates and cutoff families. The same definitions
apply to finitely many components of a vector bundle; changing frame
multiplies by a smooth invertible matrix, bounded on every compact set
together with its derivatives, and hence preserves these local spaces.
No globally chosen bundle norm or volume form is required.

If \(A\in\Psi^m\) is proper, then \[
A:\mathcal B^{s,\mathrm{loc}}_p(X)\to\mathcal B^{s-m,\mathrm{loc}}_p(X),
\qquad
A:\mathcal B^{s,\mathrm{comp}}_p(X)\to\mathcal B^{s-m,\mathrm{comp}}_p(X)
\tag{G32}
\] continuously. Fix an output compact set and use properness to choose
an input cutoff. A finite chart partition and (G15) express the
localized map as finitely many global symbols of order \(m\), plus a
compactly localized smooth kernel.
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale} proves the desired bound for each symbol. A
compact smooth kernel maps any fixed lower Sobolev space into any higher
one: its Fourier transform decreases rapidly in both variables, and
Cauchy--Schwarz with weights proves this bound. Applying the same dyadic
endpoint argument proves its Besov bound. This handles every summand and
every local output seminorm. Properness also bounds output support on
each fixed input support, which proves the compact-space continuity.
Without properness, the same proof gives the meaningful map
\(\mathcal B^{s,\mathrm{comp}}_p\to\mathcal B^{s-m,\mathrm{loc}}_p\),
and a local symbol on \(U\) maps
\(\mathcal S'(\mathbb R^n)\to\mathcal D'(U)\), as follows by multiplying
each output by a compact cutoff and using
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}.

For proper elliptic \(A\), its parametrix gives the converse regularity
implication in (G32). For example if \(u\in\mathcal D'(X)\) and
\(Au\in\mathcal B^{s-m,\mathrm{loc}}_p\), write \(u=BAu+Ru\) with \(R\)
a smooth proper kernel. The first term belongs to
\(\mathcal B^{s,\mathrm{loc}}_p\) by (G32), and the second is smooth. If
the input \(u\) is already compactly supported, this gives the
corresponding compact-space regularity assertion. It does not infer
compact support of \(u\) from compact support of \(Au\). For Sobolev
spaces, one can equivalently test \(H^{s,\mathrm{loc}}\) by a proper
elliptic operator of order \(s\) mapping to \(L^2_{\mathrm{loc}}\).
Existence follows by quantizing a positive cotangent weight of order
\(s\), constructed in charts and patched; any such operator gives the
equivalence by (G32) and its parametrix. With general
coordinate-invariant parameters (G32) is valid through equality, while
its elliptic converse uses \(\delta<\rho\).

\subsection{12. Microlocal scales and their unattained
thresholds}\label{microlocal-scales-and-their-unattained-thresholds}

Fix \(s\in\mathbb R\) and \(p\in[1,\infty]\). We say that \(u\) is
\(\mathcal B^s_p\) at a base point \(x\) if \(u=v+w\), with
\(v\in\mathcal B^{s,\mathrm{loc}}_p\) and \(w\) smooth near \(x\). We
say that it is \(\mathcal B^s_p\) at a nonzero covector \(\gamma\) if
such a decomposition has \(\gamma\notin\operatorname{WF}(w)\). These are
properties of \(u\): two possible decompositions need not have the same
individual terms. At a base point the definition is equivalent to
\(\phi u\in\mathcal B^{s,\mathrm{loc}}_p\) for some compact smooth
\(\phi\) nonzero at that point. One implication follows by multiplying
the decomposition with a cutoff supported where \(w\) is smooth; the
other by setting \(v=\chi u\) with \(\chi=1\) near the point, using a
local reciprocal of \(\phi\) and multiplication boundedness.

The equivalent covector test, for any fixed real \(m\), is that there
exists a proper \(A\in\Psi^m\), noncharacteristic at \(\gamma\), with \[
Au\in\mathcal B^{s-m,\mathrm{loc}}_p(X).
\tag{G33}
\] Given the decomposition, take a compactly based conic operator of
order \(m\), elliptic at \(\gamma\), whose microsupport avoids
\(\operatorname{WF}(w)\). Then \(Av\) has the required regularity by
(G32), and \(Aw\) is smooth by (G24). Conversely a conic parametrix
gives \(u=BAu+(I-BA)u\), which is exactly a decomposition of the
stipulated kind. Thus this test is independent of the chosen order and
coordinates, and works at the Besov endpoints as well as in the Hilbert
case.

For any proper \(A\in\Psi^m\), \[
u\text{ is }\mathcal B^s_p\text{ at }\gamma
\ \Longrightarrow\ Au\text{ is }\mathcal B^{s-m}_p\text{ at }\gamma.
\tag{G34}
\] Apply \(A\) to a defining decomposition. Equation (G32) handles its
regular part, and (G24) handles its wavefront-regular remainder. If
\(A\) is noncharacteristic at \(\gamma\), the converse follows by
applying its conic parametrix and (G34), since the remaining error is
regular at \(\gamma\). In the strict \((\rho,\delta)\) range the same
argument works with its conic inverse; the forward implication uses only
mapping and pseudolocality and therefore also holds at equality.

If \(u\) has \(\mathcal B^s_p\) regularity at every nonzero cotangent
vector over \(x_0\), then it has that regularity at \(x_0\). To prove
the nontrivial direction, use (G33) with \(m=0\). The elliptic sets of
finitely many tests \(A_1,\ldots,A_N\) cover the cosphere over \(x_0\),
by compactness. Their order-zero principal representatives \(a_j\)
satisfy \(\sum_j|a_j(x,\xi)|^2\geq c>0\) in a neighborhood of that
cosphere, above a common radius: shrink finitely many neighborhoods and
use compactness once more. Choose proper order-zero \(C_j\) with
principal symbol \(\overline{a_j}\), and set \(P=\sum_j C_jA_j\). Then
\(Pu\in\mathcal B^{s,\mathrm{loc}}_p\) and \(P\) is elliptic over a
neighborhood of \(x_0\). Its conic inverse can be chosen uniformly over
that compact cosphere, or constructed from a local reciprocal there by
Section 6. Thus \(BP-I\) is smoothing over that whole spatial
neighborhood. Multiplying by a smaller spatial cutoff yields local
regularity of \(u\). The reverse implication follows at once from the
definition. No summation over infinitely many directions, nor
reflexivity of the Besov space, occurs.

Define the extended real Sobolev thresholds \[
s_u(x)=\sup\{s:u\text{ is }H^s\text{ at }x\},\qquad
s_u^*(x,\xi)=\sup\{s:u\text{ is }H^s\text{ at }(x,\xi)\}.
\tag{G35}
\] Each is lower semicontinuous: if its value exceeds \(t\), choose a
regularity order \(s>t\) below that value and keep the corresponding
cutoff or decomposition on its open regularity neighborhood. Thus the
set where the threshold exceeds \(t\) is open. Both definitions allow
\(+\infty\); finite-order compact distribution estimates imply some
local negative Sobolev regularity, so the value is never \(-\infty\) at
an ordinary base point or covector. Indeed a compactly localized
distribution has \(|\widehat u(\xi)|\leq C\langle\xi\rangle^M\) for some
\(M\); the defining Fourier integral for its squared \(H^s\) norm is
finite whenever \(2s+2M+n<0\).

There is an exact relation \[
s_u(x)=\inf_{\xi\ne0}s_u^*(x,\xi).
\tag{G36}
\] Local membership implies membership in every direction, giving
\(\leq\). If \(t\) is strictly below the infimum, for each direction
choose an order strictly between \(t\) and its threshold. Sobolev
monotonicity implies membership at order \(t\) in that direction. The
finite-cosphere argument just proved gives local membership at order
\(t\), hence \(s_u(x)\geq t\). Let \(t\) approach the infimum; this also
handles an infinite infimum. This reasoning does not claim membership at
the threshold itself.

For \(A\in\Psi^m\) proper, (G34) gives \[
s_{Au}^*(x,\xi)\geq s_u^*(x,\xi)-m,
\qquad
s_{Au}^*(x,\xi)=s_u^*(x,\xi)-m
\quad\text{off }\operatorname{Char}_m(A).
\tag{G37}
\] The equality follows by applying the inverse regularity implication
at every real order; subtraction of a finite \(m\) from \(+\infty\) is
interpreted as \(+\infty\). Identical definitions and proofs give
thresholds for each fixed Besov exponent \(p\), but equality of
thresholds never identifies endpoint membership for different \(p\).
Equations (G30)--(G37) supply the microlocal continuation of the global
and local-cutoff Besov assertions established in
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale}.

\subsubsection{12.1. Editorial extension: every positive sequence
exponent and the same
thresholds}\label{editorial-extension-every-positive-sequence-exponent-and-the-same-thresholds}

The restrictions on the sequence exponent in (G30)--(G34) can be
removed. The original statements remain above; this extension proves the
actual coordinate, bundle and microlocal maps for \(0<p\le\infty\). It
also proves that the supremum thresholds are independent of this
sequence exponent, while keeping endpoint membership distinct. This is a
standard consequence of the exact estimates, with no novelty claim.

Keep every original sharp annulus \(A_0=\{|\xi|<1\}\),
\(A_j=\{2^{j-1}\le|\xi|<2^j\}\) for \(j\ge1\), and projection \(\Pi_j\).
For a finite coordinate fiber \(H=\mathbb C^r\), with its original
Euclidean Hilbert norm, put \[
 N_{s,p,H}(u)=\left(\sum_{j\ge0}
       (2^{js}\|\Pi_j u\|_{L^2(H)})^p\right)^{1/p}
       \quad(0<p<\infty),\qquad
 N_{s,\infty,H}(u)=\sup_{j\ge0}2^{js}\|\Pi_j u\|_{L^2(H)}.
 \tag{GB1}
\] The actual domain is the \(H\)-valued tempered distributions whose
Fourier transforms are locally \(L^2(H)\) and have this finite quantity.
For \(0<p<1\), its distance is \(N_{s,p,H}(u-v)^p\). The inequality
\((a+b)^p\le a^p+b^p\), proved in BQ1--BQ2 of
\href{euclidean-symbol-calculus.pdf}{the Euclidean chapter}, gives the
metric triangle inequality.

The scalar reconstruction BQ3 applies to the finitely many original
components, keeping their full vector: \[
 u(\varphi)=(2\pi)^{-n}\sum_{j\ge0}
       \int_{A_j}F_j(\xi)\widehat\varphi(-\xi)\,d\xi,\qquad
 v_j=2^{js}(2\pi)^{-n/2}\|F_j\|_{L^2(H)}.
 \tag{GB2}
\] For \(j\ge1\), its vector norm is bounded by
\((2\pi)^{-n/2}\omega_n^{1/2}2^L C_L
2^{j(n/2-s-L)}v_j\), where
\(C_L=\sup_\xi\langle\xi\rangle^L|\widehat\varphi(-\xi)|\) and an
integer \(L>n/2-s\) is chosen. The low term is bounded by
\((2\pi)^{-n/2}v_0\|\widehat\varphi\|_{L^2(A_0)}\). These are the same
Cauchy--Schwarz bounds on the original vector-valued integrals. They
prove absolute convergence, continuity into tempered distributions, and
identification of the Fourier transform with the locally \(L^2\)
function \(\sum F_j\). Plancherel retains \((2\pi)^{-n/2}\) in each
piece. Component Hilbert limits and increasing finite sums give BQ4 with
this vector norm; for infinity take the supremum of the component
limits. Thus every domain in (GB1) is complete. BQ6's finite-overlap
proof uses the Hilbert triangle inequality and the scalar \(p\)-power
inequality, so its two displayed constants apply unchanged to these
original vectors.

Here are the coordinate bounds, with the entire derivative and measure
factors. Let \(\kappa:U\to V\) be the original smooth coordinate
diffeomorphism, \(\psi=\kappa^{-1}\), \(\phi\in C_c^\infty(V)\), and
\(\chi\in C_c^\infty(U)\) real and equal to one near
\(\psi(\operatorname{supp}\phi)\). Let \(M(z)\) be an arbitrary smooth
\(f\times e\) matrix on \(V\). Its localized map is \[
 Tu(z)=A(z)u(\psi(z)),\qquad
 A(z)=\phi(z)M(z)\chi(\psi(z)),\qquad
 J_\kappa(x)=|\det D\kappa(x)|.
 \tag{GB3}
\] It is extended by zero outside \(V\). This is the original pullback
applied to \(\chi u\), followed by the displayed multiplication; its
scalar distribution pairing includes the Jacobian in (GK26). Because
\(\chi=1\) on a neighborhood of the relevant compact set,
\(\chi(\psi(z))=1\) wherever \(\phi\) or any derivative of \(\phi\) is
nonzero. Its derivative contributions there are exactly zero. This
proves agreement with the original map (G31), rather than dropping an
input cutoff without explanation.

For a multi-index \(\gamma\), retain a labeled list \(h_1,\ldots,h_k\),
\(k=|\gamma|\), containing \(\gamma_i\) copies of the original
coordinate direction \(e_i\). Let \(D_B\) mean differentiation in the
directions with labels in \(B\), and \(\mathfrak P(S)\) all partitions
of a label set \(S\) into nonempty blocks; the empty set has its one
empty partition. For \(|\alpha|\le k\), the entire coefficient of
\((\partial^\alpha u)(\psi(z))\) is \[
 Q_{\gamma,\alpha}(z)=
 \sum_{J\subset\{1,\ldots,k\}}
 \sum_{\Pi\in\mathfrak P(J^c)}
 \sum_{\substack{\nu:\Pi\to\{1,\ldots,n\}\\
       \#\{B:\nu(B)=i\}=\alpha_i\ (\text{all }i)}}
       (D_J A)(z)\prod_{B\in\Pi}D_B\psi^{\nu(B)}(z).
 \tag{GB4}
\] The sum over \(\nu\) is zero unless \(|\Pi|=|\alpha|\). In \(D_J A\)
the full product rule is \[
 D_J A=\sum_{J_1\sqcup J_2\sqcup J_3=J}
        (D_{J_1}\phi)(D_{J_2}M)D_{J_3}(\chi\circ\psi).
 \tag{GB5}
\] For a nonempty \(J_3\), its last derivative is the full chain sum
\(\sum_{\Theta\in\mathfrak P(J_3)}
D^{|\Theta|}\chi(\psi)[D_B\psi:B\in\Theta]\); for an empty \(J_3\) it is
\(\chi(\psi)\). The finite-coordinate chain and product rules, proved
with these same labeled partitions in Sections16.1 and16.5, give exactly
\(\partial^\gamma Tu=\sum_{|\alpha|\le|\gamma|}
Q_{\gamma,\alpha}(\partial^\alpha u)\circ\psi\). Repeated coordinate
directions keep every multiplicity. The matrix coefficient acts on the
left and is never commuted with another matrix.

For an integer \(N\ge0\), set \[
 c_{N,\alpha}=\frac{N!}{(N-|\alpha|)!\alpha!},\qquad
 \|u\|_{H^N(H)}^2=(2\pi)^{-n}
       \int\langle\xi\rangle^{2N}\|\widehat u(\xi)\|_H^2\,d\xi
       =\sum_{|\alpha|\le N}c_{N,\alpha}
          \|\partial^\alpha u\|_{L^2(H)}^2.
 \tag{GB6}
\] The last equality is the full multinomial expansion of
\((1+\sum_i\xi_i^2)^N\) and Plancherel: the derivative transform is
\((i\xi)^\alpha\widehat u\), whose squared norm keeps precisely
\(\xi^{2\alpha}\). If the coordinate opens are empty, or the localized
coefficient is identically zero, the actual map is zero and every
endpoint bound is zero. Otherwise choose a nonempty compact
\(K\Subset U\) containing \(\psi(\operatorname{supp}\phi)\). Let
\(J_*=\sup_K J_\kappa\) and
\(t_\gamma=\#\{\alpha:|\alpha|\le|\gamma|\}=\binom{n+|\gamma|}{n}\). All
norms of \(Q\) below are supremums on \(V\), finite by their compact
support. Change of variables retains \(dz=J_\kappa(x)\,dx\);
Cauchy--Schwarz in the finite \(\alpha\)-sum therefore gives \[
 \begin{aligned}
 \|Tu\|_{H^N(\mathbb C^f)}&\le E_N(T)
                       \|u\|_{H^N(\mathbb C^e)},\\
 E_N(T)^2&=
 J_*\sum_{|\gamma|\le N}c_{N,\gamma}t_\gamma
      \sum_{|\alpha|\le|\gamma|}
        \frac{\|Q_{\gamma,\alpha}\|_\infty^2}{c_{N,\alpha}}.
 \end{aligned}
 \tag{GB7}
\] To verify the bound, square the displayed derivative sum, bound it by
\(t_\gamma\sum_\alpha\|Q_{\gamma,\alpha}\|_\infty^2
\|(\partial^\alpha u)\circ\psi\|^2\), integrate with the original
Jacobian, and use
\(\|\partial^\alpha u\|_2^2\le c_{N,\alpha}^{-1}\|u\|_{H^N}^2\) for
every retained \(\alpha\). This proves the full finite constant,
including \(N=0\). A zero coefficient contributes zero to that sum.

The actual Hilbert adjoint on the original coordinate measures is \[
 T^*v(x)=
 \chi(x)J_\kappa(x)\overline{\phi(\kappa(x))}
          M(\kappa(x))^*v(\kappa(x)).
 \tag{GB8}
\] Substitution in the compact integral proves this identity, including
its order and its whole positive Jacobian. Applying (GB4)--(GB7) to this
reverse map, with \(\psi\) replaced by \(\kappa\) and with the entire
coefficient in (GB8), gives its positive-integer constants \(E_N(T^*)\).
Weighted Fourier duality retains
\((2\pi)^{-n}\int\widehat u\cdot\overline{\widehat v}\,d\xi\);
Cauchy--Schwarz and the inverse weights show
\(\|u\|_{H^{-N}}=\sup_{\|v\|_{H^N}\le1}|(u,v)|\). The reverse inequality
is obtained by the actual weighted Fourier vector, or its truncated
\(L^2\) approximations. Thus
\(\|Tu\|_{H^{-N}}\le E_N(T^*)\|u\|_{H^{-N}}\). The completions agree
with the original distribution pullback, since all formulas agree on
Schwartz functions and converge into distributions. These give
consistent bounds at every integer endpoint, with no assumption of
real-order coordinate invariance.

For any real \(s\), choose integers \(k_-<s<k_+\), and write \(M_-,M_+\)
for these proved endpoint norms. Put \[
 \begin{aligned}
 a_t&=\min(2^{-t},2^{t/2}),&
 b_t&=\max(2^{-t},2^{t/2}),\\
 C&=\max(M_-b_{k_-}/a_{k_-},
                  M_+b_{k_+}/a_{k_+}),&
 c_l&=\min(2^{l(s-k_-)},2^{l(s-k_+)}).
 \end{aligned}
 \tag{GB9}
\] On every original annulus \(2^{-j}\langle\xi\rangle\in[1/2,\sqrt2]\),
including \(j=0\). Applying both endpoint estimates to \(\Pi_j u\) gives
\[
 2^{ks}\|\Pi_kT\Pi_j u\|_2
       \le Cc_{k-j}\,2^{js}\|\Pi_j u\|_2.
 \tag{GB10}
\] For finite annular sums apply the target Hilbert triangle inequality.
For \(0<p<1\), take the \(p\)-th power, use its subadditivity, and sum
over both nonnegative indices. For \(1\le p<\infty\), use the sequence
triangle inequality on the translated sequences; for infinity use the
supremum directly. The complete resulting constants are \[
 \begin{aligned}
 N_{s,p,\mathbb C^f}(Tu)&\le CD_pN_{s,p,\mathbb C^e}(u),\\
 D_p^p&=1+\frac{2^{-p(s-k_-)}}{1-2^{-p(s-k_-)}}
             +\frac{2^{-p(k_+-s)}}{1-2^{-p(k_+-s)}}
                         &&(0<p<1),\\
 D_p&=1+\frac{2^{-(s-k_-)}}{1-2^{-(s-k_-)}}
             +\frac{2^{-(k_+-s)}}{1-2^{-(k_+-s)}}
                         &&(1\le p\le\infty).
 \end{aligned}
 \tag{GB11}
\] Indeed the two tails are respectively \(l<0\) and \(l>0\), and the
retained \(l=0\) term is one. For an arbitrary input in (GB1), \[
 \|u\|_{H^{k_-}}^2\le
       \frac{b_{k_-}^2}{1-2^{-2(s-k_-)}}
          N_{s,p,H}(u)^2.
 \tag{GB12}
\] Sum the disjoint original Fourier squares and bound each weighted
annular norm by its sequence norm to obtain this inequality; the same
geometric tail proves convergence of the input partial sums in
\(H^{k_-}\). Hence \(Tu\) is the already defined endpoint distribution.
Each fixed output projection converges in \(L^2\). Increasing finite
sums, or a supremum, pass (GB11) to that actual limit. This proves all
positive sequence exponents, including infinity, without claiming
Besov-norm density at infinity. Applying the same proof to a localized
inverse coordinate map and inverse frame matrix gives the actual inverse
local maps and the equivalence of chart norms.

Consequently the local and compact spaces in Section11 extend to every
\(0<p\le\infty\), on the same distributions and bundles. For \(p<1\),
use the finite local quasi-norm conditions \(N_{s,p}(\phi u)\), rather
than calling them locally convex seminorms. With \(q=\min(1,p)\) and
\(q=1\) at infinity, finite sums satisfy \[
 N_{s,p}\left(\sum_{\ell=1}^L u_\ell\right)^q
       \le\sum_{\ell=1}^L N_{s,p}(u_\ell)^q.
 \tag{GB13}
\] For \(p<1\) this follows from the Hilbert triangle inequality on each
annulus and scalar subadditivity; otherwise it is the norm triangle
inequality. Compactness makes each local partition sum finite. Thus
(GB11), its inverse and (GB13) prove independence of every chart, cutoff
and smooth invertible finite-rank frame, preserving all transition
matrices.

The mapping theorem (G32) and its elliptic converse hold for every such
\(p\). For each localized symbol, apply the consistent Sobolev endpoints
already proved in the Euclidean chapter, with shift \(m\); (GB9)--(GB12)
then has \(a_{k_\pm-m}\) in its denominator and \(2^{k(s-m)}\) on its
output. The geometric sums are unchanged. A compactly localized smooth
kernel is bounded from every integer \(H^{k_\pm}\) to \(H^{k_\pm-m}\):
its full two-variable Fourier transform decreases to every order, and
weighted Cauchy--Schwarz retains the two original inverse factors. The
same endpoint argument receives its actual distributional map. A finite
chart partition and (GB13) combine these maps, with bound
\((\sum_\ell C_\ell^q)^{1/q}\) when the individual bounds are
\(C_\ell\). Properness gives the same input and output compact sets as
(G17), and hence both local and compact maps. Without properness it
gives exactly compact input to local output. The positive-gain
parametrix gives \(u=BAu+Ru\) as before; multiplication by a local
cutoff makes the smooth \(Ru\) belong to every space (GB1). This proves
the converse, with compact support of the input still required for its
compact-space version. In the coordinate-invariant range
\(1-\rho\le\delta\le\rho\), \(\delta<1\), the forward mapping holds
through equality; the inverse assertion still requires \(\delta<\rho\).

The decomposition, fixed-order test and both microlocal implications
(G33)--(G34) also extend to every \(p\). Given \(u=v+w\), choose the
same compact conic test of order \(m\), elliptic at the selected
covector and supported away from \(\operatorname{WF}(w)\). The extended
(G32) sends \(v\) into the stated order \(s-m\) space, and (G24) makes
the image of \(w\) smooth. Conversely the same proper conic parametrix
gives \(v=BAu\) and \(w=(I-BA)u\); the first has order \(s\) regularity
and the second has no wavefront at that covector. Applying any proper
operator to this decomposition proves the forward implication, and its
conic inverse proves the noncharacteristic converse. No convexity or
sequence reflexivity is used.

The finite-cosphere argument is valid for all these exponents too. In
one fixed local frame, choose finitely many order-zero tests \(A_j\)
whose elliptic sets cover the cosphere over the original point. Their
principal matrices have \(\sum_j a_j^*a_j\ge cI\) on a smaller
base/cosphere neighborhood: at each direction at least one original
smallest singular value is positive; the finite cover and compactness
give the common \(c>0\). Quantize \(a_j^*\) in that same frame with the
original compact cutoffs, obtaining \(C_j\), and use the actual ordered
sum \(P=\sum_j C_jA_j\). By (GB13), \(Pu\) has the required local Besov
regularity. Its displayed principal matrix is invertible there, so the
proper local inverse yields base-point regularity. Thus regularity at
all nonzero directions over a point is exactly regularity at that point,
for arbitrary finite rank as well as for scalars.

Finally there is a stronger exact threshold statement. For any
\(0<p,r\le\infty\) and \(\epsilon>0\), the same original annuli give \[
 \begin{aligned}
 N_{s-\epsilon,r,H}(u)&\le
       (1-2^{-\epsilon r})^{-1/r}N_{s,p,H}(u)
                                  &&(0<r<\infty),\\
 N_{s-\epsilon,\infty,H}(u)&\le N_{s,p,H}(u).
 \end{aligned}
 \tag{GB14}
\] Each \(v_j=2^{js}\|\Pi_j u\|_2\) is at most its sequence norm for
every \(p\), and the entire \(r\)-power sum is bounded by
\(\sum_{j\ge0}2^{-j\epsilon r}\sup_jv_j^r\); this is precisely the
constant in (GB14), with its low term retained. Apply it to each compact
cutoff for local membership and to the regular part of a microlocal
decomposition for directional membership. If one exponent admits an
order greater than \(t\), choose a loss \(\epsilon\) smaller than the
difference from that order to \(t\). Every other exponent then admits
order \(t\). Reversing the exponents proves that all supremum thresholds
coincide, including \(+\infty\). In particular they are exactly the
Sobolev thresholds (G35), since the \(p=2\) norm is equivalent to the
original \(H^s\) norm. Their lower semicontinuity, base/cosphere infimum
identity (G36), and the proper-operator inequality and elliptic equality
(G37) therefore hold for every positive sequence exponent with these
same thresholds. There is still no assertion of membership at the
supremum.

The retained point-mass example makes the endpoint distinction exact.
For \(n\ge1\), \(\widehat{\delta_0}=1\), so \[
 \|\Pi_0\delta_0\|_2=(2\pi)^{-n/2}\omega_n^{1/2},\qquad
 \|\Pi_j\delta_0\|_2=(2\pi)^{-n/2}
       \bigl(\omega_n(1-2^{-n})\bigr)^{1/2}2^{jn/2}
       \quad(j\ge1).
 \tag{GB15}
\] These are the original annular volumes and Plancherel factor. At
\(s=-n/2\) the tail sequence is the same strictly positive constant,
hence belongs to infinity and to no finite positive sequence exponent.
At \(s<-n/2\), every finite \(p>0\) has the full norm \[
 N_{s,p}(\delta_0)^p=(2\pi)^{-np/2}\omega_n^{p/2}
 \left(1+(1-2^{-n})^{p/2}
       \frac{2^{p(s+n/2)}}{1-2^{p(s+n/2)}}\right).
 \tag{GB16}
\] At \(s>-n/2\) the annular sequence is unbounded. Thus every global
supremum is \(-n/2\), although its endpoint depends on the exponent.
Dimension zero has one Fourier piece with empty determinant and inverse
factor one, and all finite-coordinate maps reduce to the actual matrix
map at a point. The local discrete manifold statements follow on finite
compact sets; wavefront sets are empty and the regularity thresholds are
\(+\infty\). No sphere of negative dimension, zero denominator,
infinite-rank bundle extension or assertion about spatial \(L^p\)
exponents is introduced.

\subsection{13. Bundles, anti-duals, and kernels between different
manifolds}\label{bundles-anti-duals-and-kernels-between-different-manifolds}

Let \(E,F\to X\) be smooth complex vector bundles of finite ranks
\(e,f\), which need not be equal. An operator \(A\in\Psi^m(X;E,F)\) is
represented in local frames by an \(f\times e\) matrix of operators in
\(\Psi^m\), with smooth kernel off the diagonal. A new pair of frames
changes that matrix by \(G_F A G_E^{-1}\); multiplication by these
smooth matrix functions and Section 5 of
\href{euclidean-symbol-calculus.pdf}{From symbol estimates to operators
on every Sobolev scale} preserve its class. Coordinate changes are
handled by Section 2 componentwise. Its principal symbol therefore is a
section class \[
\sigma_m(A)\in
S^m(T^*X;\operatorname{Hom}(\pi^*E,\pi^*F))
\big/S^{m-1}(T^*X;\operatorname{Hom}(\pi^*E,\pi^*F)).
\tag{G38}
\] The leading transformation is \(a\mapsto G_FaG_E^{-1}\), since
differentiated transition matrices occur only in lower orders. The
construction proving (G16), performed in local frames, proves
surjectivity onto these symbol classes and identifies the lower-order
kernel. Matrix products keep their order, so
\(\sigma(BA)=\sigma(B)\sigma(A)\) for compatible bundles. Properness,
summation, (G32), and (G34) all apply componentwise and patch by local
finiteness. Two-sided ellipticity means that (G38) is fiberwise
invertible; that condition forces equal ranks, but it was not imposed on
the definition, mapping, composition, or adjoint theorems. The same
constructions define \(\Psi^m_{\rho,\delta}(X;E,F)\) in the entire
coordinate-invariant range, with the positive-gain principal quotient as
described in Section 6 when \(\delta<\rho\).

Half-densities provide an intrinsic pairing. For scalar half-densities
\(u,v\) with compact overlap of supports, set
\((u,v)=\int_Xu\overline v\); the product is a density, so change of
variables makes the integral independent of coordinates and of
orientation. Every scalar \(A\in\Psi^m(X;\Omega^{1/2},\Omega^{1/2})\)
has an adjoint \(A^*\) in the same class satisfying \[
(Au,v)=(u,A^*v),\qquad u,v\in C_c^\infty(X;\Omega^{1/2}).
\tag{G39}
\] Its kernel is \(\overline{K_A(y,x)}\), with the half-density factors
interchanged. Locally \href{euclidean-symbol-calculus.pdf}{From symbol
estimates to operators on every Sobolev scale} gives the adjoint symbol
\(a^\dagger=\overline a+\sum_j\partial_{\xi_j}D_{x_j}\overline a\pmod {S^{m-2}}\).
This is of order \(m\), proves the identity for compact tests by Fourier
inversion, and defines the distributional adjoint by continuity.
Transposing kernel support preserves properness if present; without
properness the compact-test identity remains well-defined. The principal
symbol is conjugated. For the refined class, \[
a^\dagger+\frac i2\sum_j\partial_{x_j}\partial_{\xi_j}a^\dagger
=\overline a-\frac i2\sum_j\partial_{x_j}\partial_{\xi_j}\overline a
\pmod {S^{m-2}},
\tag{G40}
\] which is exactly the complex conjugate of (G14).

For bundles, let \(E^*\) denote the \textbf{anti-dual}: its elements are
conjugate-linear functionals on \(E\). Use
\(\langle u,v\rangle=\overline{v(u)}\), linear in \(u\) and
conjugate-linear in \(v\). In components, if vectors change by
\(u'=G_Eu\), the anti-dual components change by
\(v'=\overline{G_E}^{-T}v\), since
\((v')^T\overline{u'}=v^T\overline u\). Thus \[
A^*\in\Psi^m(X;F^*\otimes\Omega^{1/2},E^*\otimes\Omega^{1/2})
\quad\text{when}\quad
A\in\Psi^m(X;E\otimes\Omega^{1/2},F\otimes\Omega^{1/2}).
\tag{G41}
\] The local kernel is the conjugate transpose with variables reversed.
The anti-dual transformation law just computed makes these kernels agree
on overlaps. The principal symbol is the fiberwise adjoint,
characterized by the pairing, and (G39) holds for compact smooth \(E\)-
and \(F^*\)-valued half-densities. No Hermitian metric identifying \(E\)
with \(E^*\) has been silently chosen.

There is also a kernel statement with different source and target
manifolds. For smooth finite-rank bundles \(E\to X\), \(F\to Y\),
continuous linear maps \[
C_c^\infty(X;E\otimes\Omega_X^{1/2})
\longrightarrow\mathcal D'(Y;F\otimes\Omega_Y^{1/2})
\tag{G42}
\] correspond bijectively to distributions on \(Y\times X\) with
coefficients in \[
\operatorname{Hom}(E_x,F_y)\otimes\Omega_{Y\times X}^{1/2}.
\tag{G43}
\] To prove the adapter, trivialize both bundles and density lines on
relatively compact charts. Each component of (G42) is a continuous
scalar test-to-distribution map and has a unique scalar kernel by the
scalar kernel theorem proved in Sections 13.1--13.5. Under new frames
that kernel is multiplied by \(G_F(y)\) on the left and \(G_E(x)^{-1}\)
on the right. Under new coordinates the scalar kernel density factor in
the input variable combines with the two half-density component laws to
give one half-density in each variable. These are exactly the transition
functions of (G43), since
\(\Omega_{Y\times X}^{1/2}\simeq\Omega_Y^{1/2}\boxtimes\Omega_X^{1/2}\).
Uniqueness of the scalar kernel makes the local distributions agree, so
they glue to one global kernel.

Conversely take a kernel as in (G43). Pair its input half-density factor
with the input half-density and its \(\operatorname{Hom}\) factor with
the input section; this leaves a distributional target half-density.
More concretely, to pair the result with a compact target dual test
section, localize the product of the two compact supports into finitely
many chart products and use the scalar kernel pairing there. The
transformation rules prove independence of the localization. The scalar
kernel theorem supplies continuity on each test-function support space,
and the inductive-limit definition of \(C_c^\infty\) supplies global
continuity. The two constructions are inverse because they agree on all
pairs of scalar test functions in every chart. For topology, this is the
same kernel identification with continuous bilinear test pairings as the
declared scalar theorem, transported by finite local matrices and
locally finite gluing; we do not replace it by an unspecified
operator-norm topology. This proves the geometric and bundle content of
(G42)--(G43); Sections 13.1--13.6 supply its scalar foundation with the
actual support and strong-dual estimates.

\subsubsection{13.1. Test spaces, bounded supports, and the strong
dual}\label{test-spaces-bounded-supports-and-the-strong-dual}

Let \(U\subset\mathbb R^p\) and \(V\subset\mathbb R^q\) be open. Scalar
distributions are complex-linear functionals on scalar tests; the
pairing in this proof is bilinear, without a complex conjugate. The
Lebesgue coordinate density is included in each test pairing; we write
only its coefficient. A scalar kernel pairs with \(\Phi(y,x)\,dy\,dx\).
For a compact \(K\subset U\), retain the original spaces and seminorms
\[
 \mathcal D_K(U)=\{f\in C^\infty(U):\operatorname{supp}f\subset K\},
 \qquad p_N(f)=\max_{|\alpha|\leq N}\sup_U|\partial^\alpha f|.
 \tag{GK1}
\] They are complete by Section 14.2 of the Banach foundation lesson.
Choose a compact exhaustion \(K_j\subset\operatorname{int}K_{j+1}\) of
\(U\), with interiors covering \(U\), as constructed there. Every
compact subset lies in some \(K_j\): finitely many of these nested
interiors cover it. Give \(\mathcal D(U)=\bigcup_j\mathcal D_{K_j}(U)\)
the finest locally convex topology making every inclusion continuous.
This is its test-function inductive-limit topology. A seminorm on this
union is continuous exactly when its restriction to each
\(\mathcal D_{K_j}\) is continuous. Indeed adding such a seminorm to the
topology still leaves every inclusion continuous, so the defining finest
topology already contains it. Likewise a linear map from this union to a
locally convex space is continuous exactly when all its restrictions are
continuous: apply the seminorm statement to the target's continuous
seminorms. This specifies the topology used below, rather than using
only the global derivative seminorms.

A subset \(H\) is bounded if every continuous seminorm has finite
supremum on \(H\). It is bounded in \(\mathcal D(U)\) exactly when all
its supports lie in one compact \(K\subset U\) and \[
 \sup_{f\in H}p_N(f)<\infty \quad\text{for every }N.
 \tag{GK2}
\] To prove the support assertion, suppose no \(K_j\) contains all the
supports. Choose \(f_j\in H\) and \(x_j\in U\setminus K_j\) with
\(f_j(x_j)\ne0\). Such a point exists because the nonzero set is dense
in the support and the complement of \(K_j\) is open. The points
eventually leave every compact subset. Set \(a_j=j/|f_j(x_j)|\) and \[
 P(f)=\sup_j a_j|f(x_j)|.
 \tag{GK3}
\] For each compactly supported \(f\) only finitely many terms can be
nonzero. On each \(\mathcal D_K\) this is a finite maximum of continuous
evaluations, bounded by a finite constant times \(p_0\); hence \(P\) is
a continuous test-space seminorm. But \(P(f_j)\geq j\), contradicting
boundedness. Each global \(p_N\) is also continuous by its restrictions,
proving (GK2). Conversely (GK2) makes \(H\) bounded in the fixed-support
Fréchet space: every neighborhood contains finitely many seminorm
constraints, and a single sufficiently large dilation contains \(H\).
Its continuous inclusion into \(\mathcal D(U)\) preserves boundedness.
This proves both directions.

Define \(\mathcal D'(U)\) as the continuous dual and its strong topology
by the seminorms \[
 Q_H(u)=\sup_{f\in H}|\langle u,f\rangle|,
 \qquad H\subset\mathcal D(U)\text{ bounded}.
 \tag{GK4}
\] A scalar functional is a distribution exactly when its restriction to
every \(\mathcal D_K\) obeys \(|u(f)|\leq C_Kp_{N_K}(f)\) for some
finite \(N_K,C_K\). Continuity gives finitely many seminorm constraints;
their increasing order reduces them to one \(p_N\), and rescaling gives
this inequality, including the zero-seminorm case. The inequality
implies continuity in the reverse direction. Formula (GK2) makes every
(GK4) finite. Singletons of tests are bounded, so their evaluations are
continuous in the strong dual. The same definitions apply on \(V\) and
\(V\times U\).

\subsubsection{13.2. Product tests and a convergent Fourier
expansion}\label{product-tests-and-a-convergent-fourier-expansion}

We need a density statement with its support and all derivative orders
retained. Let \[
 I=\prod_{i=1}^p(a_i,b_i),\quad \ell_i=b_i-a_i>0,\qquad
 J=\prod_{j=1}^q(c_j,d_j),\quad s_j=d_j-c_j>0,
 \quad \overline I\subset U,\quad\overline J\subset V.
 \tag{GK5}
\] Write \(x\in I\), \(y\in J\), and \[
 e_k(x)=\exp\!\left(2\pi i\sum_i k_i(x_i-a_i)/\ell_i\right),\qquad
 e_l(y)=\exp\!\left(2\pi i\sum_j l_j(y_j-c_j)/s_j\right).
 \tag{GK6}
\] For \(\Phi\in\mathcal D(J\times I)\), extend it periodically; it is
smooth because it is zero near every face. Its coefficients, in
output-then-input order, are \[
 c_{l,k}(\Phi)=\frac1{|J||I|}
       \int_J\int_I\Phi(y,x)e_{-l}(y)e_{-k}(x)\,dx\,dy,
 \qquad |I|=\prod_i\ell_i,\quad |J|=\prod_j s_j.
 \tag{GK7}
\] Let \[
 L=1-\sum_i\left(\frac{\ell_i}{2\pi}\right)^2\partial_{x_i}^2
       -\sum_j\left(\frac{s_j}{2\pi}\right)^2\partial_{y_j}^2 .
 \tag{GK8}
\] Integration by parts has no face terms. For every integer \(r\geq0\),
\[
 c_{l,k}(\Phi)=
  \frac{\int_J\int_I(L^r\Phi)(y,x)e_{-l}(y)e_{-k}(x)\,dx\,dy}
       {|J||I|(1+|k|^2+|l|^2)^r},
\] \[
 L^r=\sum_{h+|\alpha|+|\beta|=r}
 \frac{r!(-1)^{|\alpha|+|\beta|}}{h!\alpha!\beta!}
 \prod_i\left(\frac{\ell_i}{2\pi}\right)^{2\alpha_i}
 \prod_j\left(\frac{s_j}{2\pi}\right)^{2\beta_j}
 \partial_x^{2\alpha}\partial_y^{2\beta}.
 \tag{GK9}
\] Here the sum includes \(h=r,\alpha=\beta=0\); no constant term is
suppressed. Taking the absolute value of the numerator yields the
coefficient estimate with this entire finite differential operator. In
particular the coefficients decay faster than every power of
\((1+|k|^2+|l|^2)^{1/2}\).

We justify the periodic inversion needed here. On a circle of its actual
length \(\ell\), the Fejér kernel is \[
 F_N(t)=\frac1{N+1}\left|\sum_{\nu=0}^N e^{2\pi i\nu t/\ell}\right|^2
 =\sum_{|k|\leq N}\left(1-\frac{|k|}{N+1}\right)e^{2\pi ikt/\ell}.
 \tag{GK10}
\] It is nonnegative and \(\ell^{-1}\int_0^\ell F_N=1\), by integrating
the finite sum. Away from \(t=0\) modulo \(\ell\), the finite geometric
sum gives \(F_N(t)\leq[(N+1)\sin^2(\pi t/\ell)]^{-1}\). Thus the mass
outside circular distance \(\delta\), \(0<\delta<\ell/2\), is at most
\([(N+1)\sin^2(\pi\delta/\ell)]^{-1}\). The product kernels over all
\(p+q\) circles, with their respective factors \(1/\ell_i\) and
\(1/s_j\) in convolution, have total mass one. Their mass outside a
product of small circular intervals is bounded by the sum of these
one-coordinate tail bounds. Uniform continuity on the compact period box
then proves uniform convergence of their convolutions to any continuous
periodic function: inside the small intervals bound the difference by
its modulus of continuity, and outside bound it by twice the supremum
times that tail mass.

For a smooth function the same argument applies to each derivative,
since differentiating the periodic convolution differentiates the smooth
function under a finite integral. These convolutions are the finite
Fourier sums with coefficient multiplier \[
 \prod_i\left(1-\frac{|k_i|}{N+1}\right)_+
 \prod_j\left(1-\frac{|l_j|}{N+1}\right)_+ .
 \tag{GK11}
\] On the other hand (GK9), with \(r\) as large as needed, makes the
unweighted Fourier series and each of its differentiated series
absolutely uniformly convergent. The convergence of the lattice sums
follows directly by grouping vectors in shells
\(2^h\leq|(k,l)|<2^{h+1}\): there are at most \((2^{h+2}+1)^{p+q}\)
vectors, whereas a power of exponent exceeding \(p+q\) gives a summable
geometric bound. Dominated convergence for these absolutely summable
series makes the sums with (GK11) converge to the unweighted sum. The
preceding convolution limit identifies that sum with \(\Phi\). Uniform
convergence of all derivative series also verifies termwise
differentiation, by the fundamental theorem of calculus on coordinate
segments.

Choose \(\sigma\in\mathcal D(I)\), \(\tau\in\mathcal D(J)\), equal to
one on neighborhoods of the respective projections of
\(\operatorname{supp}\Phi\). Multiplication gives \[
 \Phi(y,x)=\sum_{l\in\mathbb Z^q,\ k\in\mathbb Z^p}
       c_{l,k}(\Phi)\,\tau(y)e_l(y)\,\sigma(x)e_k(x).
 \tag{GK12}
\] Its finite partial sums converge in every smooth seminorm with
support in the fixed compact
\(\operatorname{supp}\tau\times\operatorname{supp}\sigma\). This follows
from the derivative convergence just proved and the full finite product
rule for the two cutoffs. Therefore this is convergence in a
fixed-support test space.

For an arbitrary compactly supported test on \(V\times U\), finitely
many boxes \(I_i,J_j\) of the form (GK5) cover its compact projections.
Choose bumps \(0\leq\beta_i\leq1\) supported in \(I_i\), each equal to
one on a smaller set, with those smaller sets covering the input
projection. Define \[
 \chi_i=\beta_i\prod_{h<i}(1-\beta_h),\qquad
 \sum_i\chi_i=1-\prod_i(1-\beta_i).
 \tag{GK13}
\] The last function is one near that projection; the identity follows
by telescoping successive products. Construct \(\eta_j\) on the output
projection in the same way. Then
\(\Phi=\sum_{j,i}\eta_j(y)\chi_i(x)\Phi(y,x)\), with each piece
compactly supported in \(J_j\times I_i\). Apply (GK12) to every piece.
Finite sums of product tests are consequently dense in
\(\mathcal D(V\times U)\), with convergence in one fixed compact-support
space for each given test. This proof uses only elementary smooth bumps
on boxes, finite products, scalar integration and the displayed Fourier
series; it does not assume a kernel theorem.

\subsubsection{13.3. Separate continuity gives a fixed-support
estimate}\label{separate-continuity-gives-a-fixed-support-estimate}

Let \(B:\mathcal D(U)\times\mathcal D(V)\to\mathbb C\) be bilinear and
separately continuous. Fix compacts \(K\subset U\), \(H\subset V\).
Write \(p_N,q_M\) for their increasing derivative seminorms. There are
finite orders \(N,M\) and a constant \(C\) such that \[
 |B(f,g)|\leq C p_N(f)q_M(g),
 \qquad f\in\mathcal D_K(U),\quad g\in\mathcal D_H(V).
 \tag{GK14}
\] Here is the complete argument. For positive integers \(m\) and
integers \(M\geq0\), put \[
 E_{m,M}=\{f\in\mathcal D_K:|B(f,g)|\leq m q_M(g)
                                      \text{ for every }g\in\mathcal D_H\}.
 \tag{GK15}
\] Every set is closed, by continuity in \(f\) for each \(g\). They
cover \(\mathcal D_K\), because each continuous functional
\(B(f,\cdot)\) has a finite-order bound. Completeness and the
complete-metric Baire theorem proved in Section 6 of the Banach
foundation lesson show that one \(E_{m,M}\) has interior. Choose \(f_0\)
in that interior and \(r>0,N\) such that \(f_0+h\in E_{m,M}\) when
\(p_N(h)<r\). Subtract the two inequalities for \(f_0+h\) and \(f_0\):
\(|B(h,g)|\leq2m q_M(g)\). Rescaling \(h\ne0\) to \(r h/(2p_N(h))\)
gives (GK14) with \(C=4m/r\). If the seminorm is zero, use arbitrary
positive rescaling to obtain zero; on these test spaces \(p_N\) already
separates points. This proves joint continuity on each pair of fixed
supports, without claiming a single order works for every support.

In fact continuity on every pair of fixed supports is sufficient here in
place of global separate continuity. For fixed \(g\), its support is
contained in one \(H\), and (GK14) on each \(K\) makes
\(f\mapsto B(f,g)\) continuous on the inductive limit. The reverse
variable follows in the same way.

\subsubsection{13.4. Construction of the local scalar
kernel}\label{construction-of-the-local-scalar-kernel}

Retain \(I,J\) and choose smaller boxes \(I_0,J_0\) with
\(\overline I_0\subset I\), \(\overline J_0\subset J\). Fix cutoffs
\(\sigma,\tau\) equal to one near these smaller closures and supported
in \(I,J\). For \(\Phi\in\mathcal D(J_0\times I_0)\), define \[
 T_{J_0,I_0}(\Phi)=\sum_{l,k}c_{l,k}(\Phi)
                          B(\sigma e_k,\tau e_l).
 \tag{GK16}
\] The series is absolutely convergent. To verify this with all its
constants, apply (GK14) to
\(\operatorname{supp}\sigma,\operatorname{supp}\tau\). The product rule
gives the bounds \[
 p_N(\sigma e_k)\leq A_N(k):=
 \max_{|\gamma|\leq N}
 \sum_{\alpha\leq\gamma}\binom{\gamma}{\alpha}
 \|\partial^\alpha\sigma\|_\infty
 \prod_i\left(\frac{2\pi|k_i|}{\ell_i}\right)^{\gamma_i-\alpha_i},
\] \[
 q_M(\tau e_l)\leq B_M(l):=
 \max_{|\delta|\leq M}
 \sum_{\beta\leq\delta}\binom{\delta}{\beta}
 \|\partial^\beta\tau\|_\infty
 \prod_j\left(\frac{2\pi|l_j|}{s_j}\right)^{\delta_j-\beta_j}.
 \tag{GK17}
\] Zero powers are one, including a zero frequency. These are bounded by
constants times \((1+|k|)^N,(1+|l|)^M\), respectively, as is seen by
expanding the displayed finite sums. Combining (GK9), (GK14) and (GK17)
yields \[
 |T_{J_0,I_0}(\Phi)|
 \leq \frac{C}{|J||I|}
       \int_J\int_I|L^r\Phi(y,x)|\,dx\,dy
       \sum_{l,k}\frac{A_N(k)B_M(l)}
                         {(1+|k|^2+|l|^2)^r},
 \qquad 2r>N+M+p+q.
 \tag{GK18}
\] The shell estimate in Section 13.2 proves the sum finite with
precisely this strict inequality. The full expression for \(L^r\) in
(GK9) bounds its integral by a finite sum of the original derivative
supremums of order at most \(2r\), times \(|J||I|\). Thus (GK18) proves
distributional continuity on every compact support in the smaller box.

For a product \(\Phi(y,x)=g(y)f(x)\), \(f\in\mathcal D(I_0)\),
\(g\in\mathcal D(J_0)\), Fubini gives \(c_{l,k}(\Phi)=c_l(g)c_k(f)\),
with their respective volume factors from (GK7). The cutoff Fourier sums
for \(f,g\) converge in their fixed-support test spaces. Joint
continuity (GK14) and the absolute convergence of (GK16) therefore give
\[
 T_{J_0,I_0}(g\otimes f)=B(f,g).
 \tag{GK19}
\] If different boxes or cutoffs give two local kernels, both give this
value on every product test in the intersection. Intersections of the
smaller coordinate boxes are again products of boxes, possibly empty.
The product-test density in Section 13.2 and distributional continuity
prove equality on the entire intersection. This establishes independence
of the construction.

\subsubsection{13.5. The global kernel and continuity into the strong
dual}\label{the-global-kernel-and-continuity-into-the-strong-dual}

The smaller boxes of Section 13.4 cover \(V\times U\). Their local
kernels glue as follows. For a compactly supported \(\Phi\), choose
finitely many input and output boxes and the functions \(\chi_i,\eta_j\)
of (GK13), and put \[
 T(\Phi)=\sum_{j,i}T_{J_j,I_i}(\eta_j(y)\chi_i(x)\Phi(y,x)).
 \tag{GK20}
\] Here \(I_i,J_j\) denote smaller boxes, each equipped with its larger
box and local kernel. For a second choice, insert both product
partitions. Their sums are one near \(\operatorname{supp}\Phi\), so
expanding gives the same finite double refinement. On each refined
support the local distributions agree by Section 13.4. This proves
independence, hence linearity by choosing one partition for the union of
finitely many test supports. For any fixed compact support, choose one
such finite partition once. The finitely many estimates (GK18), followed
by the full product rule for \(\eta_j\chi_i\Phi\), give a finite-order
bound there. Thus \(T\in\mathcal D'(V\times U)\). Partitioning \(f,g\)
in (GK19) proves \(T(g\otimes f)=B(f,g)\) for arbitrary tests.
Product-test density proves uniqueness.

Conversely, if \(T\in\mathcal D'(V\times U)\), define
\(B(f,g)=T(g\otimes f)\). On a fixed product of supports, the
distribution estimate of order \(R\) gives \[
 |B(f,g)|\leq C
 \max_{|\alpha|+|\beta|\leq R}
       \|\partial_x^\alpha f\|_\infty
       \|\partial_y^\beta g\|_\infty
 \leq C p_R(f)q_R(g).
 \tag{GK21}
\] This proves separate continuity on the inductive limits and joint
continuity at fixed supports. It also defines a distribution \(Af\) by
\(\langle Af,g\rangle=B(f,g)\). For a bounded set
\(H\subset\mathcal D(V)\), Section 13.1 supplies a common compact output
support and bounds for every \(q_R\). For a fixed input compact \(K\),
(GK21) consequently gives \[
 Q_H(Af)\leq C p_R(f)\sup_{g\in H}q_R(g),\qquad f\in\mathcal D_K(U).
 \tag{GK22}
\] Thus \(A:\mathcal D_K(U)\to\mathcal D'(V)\) is continuous for every
strong-dual seminorm. The inductive-limit criterion proves continuity of
\(A:\mathcal D(U)\to\mathcal D'(V)\) in the strong topology.

If we start instead with such a continuous linear \(A\), then
\(B(f,g)=\langle Af,g\rangle\) is separately continuous: for fixed
\(g\), its evaluation is a continuous strong-dual seminorm; for fixed
\(f\), \(Af\) is a distribution. Sections 13.3--13.5 give its unique
kernel. These constructions are inverse by their values on every pair of
tests. We have proved the exact bijection among scalar distributions on
\(V\times U\), separately continuous bilinear test pairings, and
continuous linear test-to-strong-distribution maps. We assert the
displayed continuity statements, with their actual support-dependent
orders; no extra topology on the space of all operators is being
substituted.

If an open set is empty, its test and distribution spaces are zero and
the assertion reduces to the unique zero map. If a coordinate dimension
is zero, its nonempty Euclidean open set is the one-point space, its
test space is \(\mathbb C\), and the sums, volume factors and products
over its empty coordinate list are respectively one term, one and one.
The same formulas give the asserted bijection.

\subsubsection{13.6. The exact elementary distribution
operations}\label{the-exact-elementary-distribution-operations}

Restriction from \(U\) to an open \(W\subset U\) is
\(\langle u|_W,f\rangle=\langle u,\widetilde f\rangle\), with the
compact test extended by zero. That extension is smooth and continuous
on each fixed-support space, so restriction is well defined. Extension
sends bounded test sets to bounded test sets by (GK2); hence restriction
is also continuous between the strong distribution duals.

For \(a\in C^\infty(U)\), define
\(\langle au,f\rangle=\langle u,af\rangle\), and define
\(\langle\partial^\alpha u,f\rangle=(-1)^{|\alpha|}\langle u,\partial^\alpha f\rangle\).
Both test operations preserve compact supports and bounded sets, by the
complete finite product rule and the derivative seminorms. They
therefore give continuous strong-dual operations. Differentiating once
and using the test product rule gives
\(\partial_j(au)=(\partial_j a)u+a\partial_j u\), including both signs
from the pairing definition. Induction using Pascal's identity proves
the complete multi-index formula \[
 \partial^\alpha(au)=
   \sum_{\beta\leq\alpha}\binom{\alpha}{\beta}
                (\partial^\beta a)\partial^{\alpha-\beta}u.
 \tag{GK23}
\]

If \(\operatorname{supp}u\subset K\subset U\) is compact, choose
\(\zeta\in\mathcal D(U)\) equal to one on a neighborhood of \(K\). Then
\(u\) acts on every \(g\in C^\infty(U)\) by \(u(\zeta g)\). The value is
independent of \(\zeta\). To see the locality being used, a compact test
supported outside \(\operatorname{supp}u\) has a finite cover by boxes
on which \(u\) vanishes. The finite bump construction (GK13) splits the
test into tests on those boxes, proving its value zero. The difference
of two cutoff products is supported outside a neighborhood of \(K\), so
this locality proves independence. The distribution estimate on
\(\operatorname{supp}\zeta\), with its finite order \(N\), gives the
entire bound \[
 |u(g)|\leq C
 \max_{|\gamma|\leq N}
 \sum_{\alpha\leq\gamma}\binom{\gamma}{\alpha}
       \|\partial^\alpha\zeta\|_\infty
       \sup_{\operatorname{supp}\zeta}
                    |\partial^{\gamma-\alpha}g|.
 \tag{GK24}
\] For \(U=\mathbb R^p\) this is a bound by a finite collection of the
original Schwartz seminorms, so the compactly supported distribution is
tempered. For a general \(U\), \(f\mapsto u(\zeta f|_U)\) similarly
gives its canonical compactly supported extension to \(\mathbb R^p\),
independent of the chosen cutoff. Its support is exactly the original
support: tests near that compact set recover \(u\), and tests away from
it give zero. This proves the finite-order statement without replacing
the original compact set or discarding the cutoff derivatives.

For \(u\in\mathcal D'(U)\), \(v\in\mathcal D'(V)\), the bilinear form
\(B(f,g)=u(f)v(g)\) is separately continuous. Its kernel, just proved,
is their product distribution, denoted \(v\otimes u\) in
output-then-input order. There is also the exact iterated formula \[
 \langle v\otimes u,\Phi\rangle
    =u\!\left(x\mapsto v(\Phi(\,\cdot\,,x))\right)
    =v\!\left(y\mapsto u(\Phi(y,\,\cdot\,))\right).
 \tag{GK25}
\] Indeed the first inner pairing is a smooth compactly supported
function of \(x\): on a fixed product support the finite-order bound for
\(v\), applied to the Taylor remainder in each \(x\)-direction with all
\(y\)-derivatives through that order, proves differentiation under the
pairing of every order. Its support is in the input projection. The
finite-order bounds of \(u,v\) make the resulting functional continuous
on each product support. It agrees with \(u(f)v(g)\) on product tests,
so density proves the first equality; reversing the roles proves the
second. No interchange of undefined integrals is used.

Its support is exactly
\(\operatorname{supp}v\times\operatorname{supp}u\). Outside this
product, each point has a product neighborhood with one distribution
zero, and (GK24), or product-test density, makes the product zero there.
At a point in the product, every product neighborhood contains tests
\(f,g\) with \(u(f)\ne0,v(g)\ne0\), by the definition of support. Their
product test gives a nonzero value. Every neighborhood contains such a
product neighborhood, proving the reverse inclusion, including zero
distributions.

For a smooth diffeomorphism \(H:W\to U\), scalar pullback is \[
 \langle H^*u,f\rangle
  =\left\langle u,(f\circ H^{-1})|\det DH^{-1}|\right\rangle .
 \tag{GK26}
\] The transformed support is \(H(\operatorname{supp}f)\), a compact
subset of \(U\). Every derivative of the transformed test is a finite
sum of chain and product terms containing the derivatives of
\(f,H^{-1}\) and the entire positive absolute Jacobian. On a fixed
compact, all these latter coefficients are bounded. Thus this test map
is continuous and carries bounded sets to bounded sets, proving
strong-dual continuity of pullback. The full change-of-variables formula
verifies (GK26) for a smooth scalar function; applying the determinant
chain rule to two diffeomorphisms proves the composition law and that
the inverse pullback is its inverse. Orientation reversal changes no
absolute Jacobian. This is a diffeomorphic pullback; no pullback for
arbitrary smooth maps or multiplication of two arbitrary distributions
is asserted.

These arguments prove the scalar kernel, restriction, product and
coordinate operations used in Section 13. Fourier operations still use
their precise Schwartz or compact-support domains and the Fourier proofs
named at the beginning of the lesson. The finite matrix and half-density
adapter (G42)--(G43) retains those original factors and transition laws.

\subsubsection{13.7. Convolution with its actual support and derivative
bounds}\label{convolution-with-its-actual-support-and-derivative-bounds}

We use scalar distributions paired with the original coordinate density
\(dx\) on \(\mathbb R^n\), and \(D_j=-i\partial_j\). Let
\(u\in\mathcal D'(\mathbb R^n)\), and let \(s\) have compact support
\(K\). Fix \(\zeta\in\mathcal D\), equal to one near \(K\), and set
\(L=\operatorname{supp}\zeta\). The constant \(C_s\) and order \(N_s\)
below are those in the distribution estimate for \(s\) on \(L\). For
\(\phi\in\mathcal D\) define \[
 F_s\phi(x)=s_y\bigl(\phi(x+y)\bigr),\qquad
 (u*s)(\phi)=u_x(F_s\phi(x)).
 \tag{GC1}
\] The inner pairing means \(s_y(\zeta(y)\phi(x+y))\). Its value is
independent of the chosen cutoff by the locality proved in Section 13.6.
Taylor's formula in an \(x_j\)-direction, with all \(y\)-derivatives
through order \(N_s\), shows that it is smooth and that every
\(x\)-derivative passes under the pairing. The remainder divided by the
increment tends uniformly to zero on each compact parameter set and on
\(L\), including those finitely many derivatives; the finite-order
estimate therefore makes its paired remainder tend to zero.

If \(\operatorname{supp}\phi\subset P\) is compact, then
\(\operatorname{supp}F_s\phi\subset P-K\subset P-L\). Indeed when
\(x\notin P-K\), the function \(y\mapsto\phi(x+y)\) vanishes near \(K\).
The same holds for all nearby \(x\), because the two compact sets are
disjoint. Thus locality proves the support assertion. For every
\(\alpha\), the entire finite-order estimate is \[
 \sup_x|\partial_x^\alpha F_s\phi(x)|
 \leq C_s\max_{|\gamma|\leq N_s}
 \sum_{\eta\leq\gamma}\binom{\gamma}{\eta}
 \|\partial^\eta\zeta\|_\infty
 \sup_{z\in\mathbb R^n}|\partial^{\alpha+\gamma-\eta}\phi(z)|.
 \tag{GC2}
\] No derivative of the cutoff is omitted. This proves that
\(F_s:\mathcal D_P\to\mathcal D_{P-L}\) is continuous. Applying the
finite-order bound for \(u\) on the fixed compact \(P-L\) proves that
(GC1) is a distribution, with a bound using orders through \(N_u+N_s\).
It also proves that \(u\mapsto u*s\) is continuous for the strong
distribution topology: (GC2) sends every bounded test set to a bounded
test set with common compact support, so the strong seminorm
\(Q_H(u*s)\) is exactly \(Q_{F_sH}(u)\).

The set \(\operatorname{supp}u+K\) is closed. If \(x_j+y_j\) converges,
with \(x_j\in\operatorname{supp}u\) and \(y_j\in K\), a subsequence of
\(y_j\) converges in \(K\), and then \(x_j\) converges in the closed
support of \(u\). If a test support avoids this sum, the inner function
in (GC1) vanishes near \(\operatorname{supp}u\). Consequently \[
 \operatorname{supp}(u*s)\subset\operatorname{supp}u+K.
 \tag{GC3}
\] This is an inclusion; cancellation may make it strict.

Choose also \(\rho\in\mathcal D\) equal to one near \(P-L\). The smooth
compact test \(\rho(x)\zeta(y)\phi(x+y)\) permits both iterated pairings
in (GK25). Its value in the order \(u_xs_y\) is (GC1). In the reverse
order, for every \(y\in L\) the support of \(x\mapsto\phi(x+y)\) lies in
\(P-L\), so \(\rho\) can be removed. This proves commutativity and
independence of both auxiliary cutoffs. Differentiating the output
distribution and differentiating either factor give exactly \[
 \partial^\alpha(u*s)
    =(\partial^\alpha u)*s=u*(\partial^\alpha s),\qquad
 D^\alpha(u*s)=(D^\alpha u)*s=u*(D^\alpha s).
 \tag{GC4}
\] For the first equality, the factor \((-1)^{|\alpha|}\) in the test
definition of \(\partial^\alpha u\) differentiates \(\phi(x+y)\) in
\(x\). For the second, the same factor differentiates it in \(y\); those
derivatives equal the output derivatives. The extension of
\(\partial^\alpha s\) to smooth functions still satisfies that formula,
since derivatives of its cutoff are supported away from \(K\) and pair
to zero. Multiplying by the full constant \((-i)^{|\alpha|}\) gives the
\(D\) identities. Taking \(s=\delta_0\) in (GC1) proves
\(u*\delta_0=u\).

Let now \(t\) also have compact support, with cutoff support \(L_t\).
For a test supported in \(P\), choose an \(x\)-cutoff equal to one near
\(P-L-L_t\), and insert the two compact-factor cutoffs. Their product
times \(\phi(x+y+z)\) is a compact smooth test on the full product. All
its iterated pairings exist; the finite-order bounds justify parameter
derivatives at every step. Repeated (GK25) identifies the different
orders. More explicitly, the triple functionals are continuous on every
fixed product support by the three finite-order estimates and agree on
all products of three tests. Applying the product-test density of
Section 13.2 first to two variables and then to the third proves their
equality on every such test. Both definitions of iterated convolution
therefore have the same value \[
 ((u*s)*t)(\phi)=(u*(s*t))(\phi)
       =u_xs_yt_z\bigl(\phi(x+y+z)\bigr).
 \tag{GC5}
\] Here \(s*t\) has compact support in \(K+\operatorname{supp}t\) by
(GC3), so its pairing with a smooth function is already defined. Removal
of each cutoff is valid on precisely the compact support just described.
Commutativity then proves associativity for any ordering in which all
but at most one factor have compact support. This argument does not
assign a convolution to an arbitrary pair of noncompact distributions.

\subsubsection{13.8. The full Schwartz estimate for a compact
convolution
factor}\label{the-full-schwartz-estimate-for-a-compact-convolution-factor}

Use the original seminorms
\(p_{\alpha,\beta}(\phi)=\sup_x|x^\alpha\partial^\beta\phi(x)|\), with
every monomial and derivative retained. For the same compact factor
\(s\), the function in (GC1) satisfies \[
\begin{aligned}
 p_{\alpha,\beta}(F_s\phi)
 \leq{}&C_s\max_{|\gamma|\leq N_s}
 \sum_{\eta\leq\gamma}\binom{\gamma}{\eta}
          \|\partial^\eta\zeta\|_\infty\\
 &\times\sum_{\kappa\leq\alpha}\binom{\alpha}{\kappa}
    \sup_{y\in L}|y^{\alpha-\kappa}|
             p_{\kappa,\beta+\gamma-\eta}(\phi).
\end{aligned}
 \tag{GC6}
\] To prove it, apply (GK24) in the \(y\)-variable to
\(x^\alpha\partial_x^\beta\phi(x+y)\), treating \(x^\alpha\) as constant
during those derivatives. Then use the full expansion \[
 x^\alpha=\sum_{\kappa\leq\alpha}\binom{\alpha}{\kappa}
       (x+y)^\kappa(-y)^{\alpha-\kappa}.
 \tag{GC7}
\] Taking absolute values and suprema gives (GC6). The parameter
argument from Section 13.7 proves smoothness. The bound proves
membership in \(\mathcal S\) and continuity of
\(F_s:\mathcal S\to\mathcal S\), using a finite list of the original
seminorms for each requested one. No weighted factor has been absorbed
into a replacement seminorm.

For \(u\in\mathcal S'\), formula (GC1) now defines a continuous
functional on \(\mathcal S\). On compact tests it is the convolution
already constructed; hence \[
 u\in\mathcal S',\quad s\text{ compactly supported}
       \quad\Longrightarrow\quad u*s\in\mathcal S'.
 \tag{GC8}
\] It is continuous in \(u\) for the strong Schwartz dual: a bounded
Schwartz set has a finite supremum for every original seminorm, and
(GC6) carries it to another bounded set. The exact dual seminorm is
again \(T_H(u*s)=T_{F_sH}(u)\). Completeness and this description of
Schwartz boundedness were proved in Sections 4.1 and 4.5 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.

\subsubsection{13.9. Smooth parameter pairings and distributional
approximation}\label{smooth-parameter-pairings-and-distributional-approximation}

Let \(k(x,y)\) be smooth on an open set containing \(Y\times K\), where
\(Y\subset\mathbb R^a\) is open and \(s\) has compact support \(K\).
Near any fixed compact parameter neighborhood \(Y_0\Subset Y\),
compactness gives a \(y\)-neighborhood of \(K\) and a slightly larger
parameter neighborhood on whose product \(k\) is smooth. Choose
\(\zeta\) supported in that \(y\)-neighborhood and equal to one near
\(K\). The function \[
 h(x)=s_y(k(x,y)),\qquad
 \partial_x^\alpha h(x)=s_y(\partial_x^\alpha k(x,y))
 \tag{GC9}
\] is well defined and smooth on \(Y\). Independence of the cutoff
follows from locality. For every \(\alpha\) its actual bound on \(Y_0\)
is \[
 \sup_{x\in Y_0}|\partial_x^\alpha h(x)|
 \leq C_s\max_{|\gamma|\leq N_s}
 \sum_{\eta\leq\gamma}\binom{\gamma}{\eta}
 \|\partial^\eta\zeta\|_\infty
 \sup_{\substack{x\in Y_0\\y\in L}}
        |\partial_x^\alpha\partial_y^{\gamma-\eta}k(x,y)|.
 \tag{GC10}
\] For the derivative assertion, apply the fundamental theorem of
calculus to the difference quotient in a parameter direction. The
difference from \(\partial_{x_j}k\), and each \(y\)-derivative through
order \(N_s\), tends uniformly to zero on the fixed product compact.
Estimate (GC10) therefore proves differentiation under the pairing.
Iterating proves every order. If the kernel has a singular set, the
argument applies locally whenever this product neighborhood avoids that
set; this is the precise hypothesis needed for off-singularity
smoothness.

In particular, when \(v\) is compactly supported and a distribution
\(f\) is represented by a smooth function near every difference \(x-y\),
with \(x\in Y\) and \(y\in\operatorname{supp}v\), the restriction of
\(f*v\) to \(Y\) is the smooth function \(v_y(f(x-y))\). To verify
equality as distributions, take a test supported in a compact subset of
\(Y\). All relevant differences form a compact set in the smooth region.
Insert a cutoff in that region, use (GK25) to exchange the two
finite-order pairings, and use the original Lebesgue substitution
\(z=x+y\), with absolute Jacobian one. This gives exactly the integral
of the stated smooth function against the test. The parts of \(f\)
outside that region pair to zero because the inner test vanishes there.
This proves the claimed representation, rather than merely the
smoothness of a separately defined function.

For an arbitrary \(u\in\mathcal D'\) and \(r\in\mathcal D\), the
convolution \(u*r\) is smooth. On a compact output neighborhood, all
\(y\)-arguments in \(r(x-y)\) lie in one compact set. Insert a cutoff
there into \(u\), making it a compact distribution, and apply (GC9). The
same change of variables and iterated pairing identify the result with
(GC1). Thus \[
 (u*r)(x)=u_y(r(x-y)),\qquad
 \partial^\alpha(u*r)(x)=u_y(\partial^\alpha r(x-y)).
 \tag{GC11}
\] For the useful approximation statement, let \(r\in\mathcal D\) have
integral one, and retain the full scaled function
\(r_\varepsilon(y)=\varepsilon^{-n}r(y/\varepsilon)\),
\(0<\varepsilon\leq1\). For a bounded test set \(H\), Section 13.1 gives
a common compact support \(P\) and finite bounds for every derivative.
The tests \(F_{r_\varepsilon}\phi\) are supported in the common compact
\(P-\{tz:0\leq t\leq1,\ z\in\operatorname{supp}r\}\). The substitution
\(y=\varepsilon z\) retains both the scaled factor and the Jacobian,
yielding \[
 F_{r_\varepsilon}\phi(x)-\phi(x)
       =\int r(z)\bigl(\phi(x+\varepsilon z)-\phi(x)\bigr)\,dz.
 \tag{GC12}
\] The fundamental theorem of calculus along the actual segment gives,
for every \(\alpha\), \[
 \sup_x|\partial^\alpha(F_{r_\varepsilon}\phi-\phi)(x)|
 \leq\varepsilon\sum_{j=1}^n
       \left(\int|r(z)|\,|z_j|\,dz\right)
       \sup_x|\partial^{\alpha+e_j}\phi(x)|.
 \tag{GC13}
\] Apply the fixed finite-order estimate for \(u\) on that common
compact and then take the supremum over \(\phi\in H\). Its right side
tends to zero. Therefore the smooth distributions \(u*r_\varepsilon\)
converge to \(u\) in the strong distribution topology. This proves a
stronger statement than convergence on each individual test, with the
same original smoothing functions and no positivity requirement on
\(r\).

\subsubsection{13.10. Compact Fourier pairings and the receiving inverse
identities}\label{compact-fourier-pairings-and-the-receiving-inverse-identities}

Retain the original transform
\(\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx\) and inverse factor
\((2\pi)^{-n}\). For a compact distribution \(s\) in the frequency
variable, (GC9) proves that \[
 H(x)=(2\pi)^{-n}s_\xi(e^{ix\cdot\xi}),\qquad
 \partial_x^\alpha H(x)
        =(2\pi)^{-n}s_\xi((i\xi)^\alpha e^{ix\cdot\xi})
 \tag{GC14}
\] is smooth. All pairings include the cutoff \(\zeta(\xi)\) used in
(GK24). For clarity, every derivative entering that finite-order bound
is the complete expression \[
 \partial_\xi^\nu\bigl((i\xi)^\alpha e^{ix\cdot\xi}\bigr)
 =i^{|\alpha|}
   \sum_{\substack{\lambda\leq\nu\\\lambda\leq\alpha}}
   \binom{\nu}{\lambda}\frac{\alpha!}{(\alpha-\lambda)!}
   \xi^{\alpha-\lambda}(ix)^{\nu-\lambda}e^{ix\cdot\xi}.
 \tag{GC15}
\] Combining (GC15) with (GK24), with \(\nu=\gamma-\eta\), gives \[
\begin{aligned}
 |\partial_x^\alpha H(x)|\leq{}&(2\pi)^{-n}C_s
 \max_{|\gamma|\leq N_s}
 \sum_{\eta\leq\gamma}\binom{\gamma}{\eta}
       \|\partial^\eta\zeta\|_\infty\\
 &\times\sum_{\substack{\lambda\leq\gamma-\eta\\\lambda\leq\alpha}}
   \binom{\gamma-\eta}{\lambda}\frac{\alpha!}{(\alpha-\lambda)!}
   \sup_{\xi\in L}|\xi^{\alpha-\lambda}|
   |x^{\gamma-\eta-\lambda}|.
\end{aligned}
 \tag{GC16}
\] Thus every derivative has polynomial growth of degree at most the
original order \(N_s\), with its entire constant displayed. The powers
of \(i\) in (GC15) have modulus one for real \(x,\xi\); no sign or power
is lost from the identity.

The smooth function in (GC14) represents the inverse Fourier
distribution. Indeed, for \(\phi\in\mathcal S\), \[
 \int H(x)\phi(x)\,dx
  =(2\pi)^{-n}s_\xi\left(\int e^{ix\cdot\xi}\phi(x)\,dx\right)
  =(2\pi)^{-n}s_\xi\bigl(\widehat\phi(-\xi)\bigr)
  =\langle\mathcal F^{-1}s,\phi\rangle.
 \tag{GC17}
\] To justify the first equality, truncate the \(x\)-integral to compact
boxes and approximate there by Riemann sums in the smooth
\(\xi\)-topology on \(L\). Uniform convergence of every derivative
through order \(N_s\) permits the finite-order pairing. The tail of each
such derivative is bounded by a constant times
\(\int_{|x|>R}(1+|x|)^{N_s}|\phi(x)|\,dx\), which tends to zero by the
original Schwartz decay. Bound (GC16) also gives convergence of the
outer integral. The last expression is precisely the transpose
definition of the inverse transform at the declared Fourier entry base.
Hence (GC17) proves the asserted identification.

The same argument with \(e^{-iy\cdot\xi}\), without an inverse
prefactor, proves that \(\widehat s(\xi)=s_y(e^{-iy\cdot\xi})\) is
smooth. Every frequency derivative has polynomial growth of degree at
most \(N_s\), by the full version of (GC15) with the negative phase.
Multiplication by \(\widehat s\) consequently preserves \(\mathcal S\):
the complete product rule for \(x^\alpha\partial^\beta(\widehat s\phi)\)
is a sum over every \(\eta\leq\beta\), with coefficient
\(\binom{\beta}{\eta}\); its polynomial growth is bounded by the finite
multinomial expansion of \((1+\sum_j|x_j|)^{N_s}\) and the corresponding
original monomial seminorms of \(\phi\).

For \(u\in\mathcal S'\) and compact \(s\), the exact transform identity
is \[
 \widehat{u*s}=\widehat s\,\widehat u.
 \tag{GC18}
\] On a Schwartz test \(\psi\), its proof is the test identity \[
 F_s(\widehat\psi)(x)
   =s_y\left(\int e^{-i(x+y)\cdot\xi}\psi(\xi)\,d\xi\right)
   =\int e^{-ix\cdot\xi}\widehat s(\xi)\psi(\xi)\,d\xi
   =\widehat{\widehat s\psi}(x).
 \tag{GC19}
\] The compact-factor finite-order bound justifies passing its pairing
through the integral, exactly as in (GC17); the additional
\(x\)-derivatives require only further powers of \(\xi\), all integrable
against the Schwartz test. Both sides are Schwartz functions by (GC6)
and the multiplier bound just proved. Pairing (GC19) with \(u\), using
the transpose definition
\(\langle\widehat u,\psi\rangle=\langle u,\widehat\psi\rangle\), proves
(GC18). No pairing of an arbitrary distribution against a non-Schwartz
function is used.

The noncompact Schwartz input in (L14) also requires its actual
convolution, which we now construct. If \(u\in\mathcal S'\) and
\(r\in\mathcal S\), define \[
 h(x)=u_y(r(x-y)),\qquad
 \partial_x^\eta h(x)=u_y(\partial^\eta r(x-y)).
 \tag{GC20}
\] For each fixed \(x\) the argument is Schwartz. Its full bound in the
original seminorms is \[
 p_{\alpha,\beta}^{(y)}(\partial_x^\eta r(x-y))
 \leq\sum_{\kappa\leq\alpha}\binom{\alpha}{\kappa}
       |x^{\alpha-\kappa}|p_{\kappa,\beta+\eta}(r).
 \tag{GC21}
\] This follows from \(y=x-(x-y)\) by the complete multinomial rule;
\(\partial_y^\beta\) supplies the sign \((-1)^{|\beta|}\). Taylor's
formula for a parameter increment, including each \(y\)-derivative and
monomial weight in (GC21), bounds the difference-quotient remainder by
the increment times a finite sum of the next derivative seminorms of
\(r\), uniformly on each compact parameter set. Thus the map into
\(\mathcal S\) is smooth in every original seminorm. Applying the
continuous functional \(u\) proves (GC20), and its finite-seminorm
estimate combined with (GC21) proves polynomial growth of every
derivative. In particular \(h\) is a tempered distribution.

Its distributional formula is \((u*r)(\phi)=u(F_r\phi)\), where
\(F_r\phi(x)=\int r(y)\phi(x+y)\,dy\) and \[
 p_{\alpha,\beta}(F_r\phi)
 \leq\sum_{\kappa\leq\alpha}\binom{\alpha}{\kappa}
       \left(\int |r(y)|\,|y^{\alpha-\kappa}|\,dy\right)
                  p_{\kappa,\beta}(\phi).
 \tag{GC22}
\] The same complete expansion as (GC7) proves the bound, and
differentiation under this absolutely convergent integral proves
smoothness. To identify this functional with the function \(h\),
integrate \(\phi(x)r(x-y)\) first over a compact \(x\)-box. Riemann sums
converge in every Schwartz seminorm in \(y\), by (GC21) and parameter
continuity. The seminorm of the tail is bounded by a finite sum of
\(p_{\kappa,\beta}(r)\int_{|x|>R}|\phi(x)|\,|x^{\alpha-\kappa}|\,dx\),
tending to zero. Completeness of the Schwartz space, proved in Section
4.1 of the cited Weyl-product lesson, supplies the integral in that
topology. The ordinary substitution in the scalar integral gives
\(F_r\phi(y)=\int r(x-y)\phi(x)\,dx\). Applying \(u\) verifies the
identification. Formula (GC22) proves continuity into the strong
Schwartz dual for fixed \(r\).

For this Schwartz factor the same Fourier identity holds: \[
 \widehat{u*r}=\widehat r\,\widehat u.
 \tag{GC23}
\] Here \(\widehat r\) is Schwartz by the declared Fourier theorem, so
its product with a Schwartz test is Schwartz by the entire finite
product rule. Fubini in the two absolutely integrable functions \(r(y)\)
and \(\psi(\xi)\) proves \(F_r\widehat\psi=\widehat{\widehat r\psi}\);
both sides lie in \(\mathcal S\) by (GC22). Pairing with \(u\) proves
(GC23), with no inverse factor introduced into a forward transform. If
\(r\) is compact smooth, this definition agrees with (GC1). Thus its
domains extend the earlier construction exactly.

Here are the exact receiving maps in
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates}. Its compact frequency
distribution \(T-b_0\) gives the smooth correction \(G-F_0\) by
(GC14)--(GC17), with the original inverse factor. Its zero extension of
\(g\in L^p(V)\), for \(V\subset B_1\), is a compact distribution
supported in \(\overline B_1\): Hölder on that bounded set bounds its
test pairing, including when \(\overline V\) has irregular boundary.
Thus \(G*g\) is defined by (GC1). Since \(q(D)G=\delta_0\), (GC4) and
the delta identity give \(q(D)(G*g)=g\). For \(v\in C_c^\infty(V)\),
they give \(G*q(D)v=(q(D)G)*v=v\). Restriction to \(V\) proves both
original identities (L12), with their stated domains.

In its homogeneous-equation argument, \(q(D)F_0=\delta_0+\omega\) and
compactness of \(\eta w\) give exactly \[
 F_0*q(D)(\eta w)
       =\eta w+\omega*(\eta w).
 \tag{GC24}
\] The commutator \([q(D),\eta]w\) has compact support outside the
region where \(\eta=1\). The first kernel is smooth at every resulting
nonzero difference, so (GC9)--(GC11) prove local smoothness there; the
second kernel \(\omega\) is smooth everywhere. Rearranging (GC24) proves
the original assertion that \(w\) is smooth, without imposing a
homogeneity formula on the fundamental solution. For every original
Schwartz input \(v\), (GC23) and \(q b_0=1+\widehat\omega\) prove the
full formula \((D^\alpha F_0)*q(D)v=D^\alpha v+(D^\alpha\omega)*v\),
hence (L14). Its later extension still uses the separately stated
\(L^p\) multiplier and approximation bounds. These distribution proofs
supply the declared distribution-calculus interface; they do not replace
those other analytic estimates.

\subsection{14. Four calculations that separate the
assertions}\label{four-calculations-that-separate-the-assertions}

\textbf{A density correction in one dimension.} Work on \(x>0\) with
\(z=\log x\) and \(A=D_x\). On functions, (G7) gives
\(A_\kappa=e^{-z}D_z\). Its leading symbol is \(e^{-z}\eta\), so its
subprincipal symbol is \(-\frac i2e^{-z}\). Here \(J=x^{-1}\), and (G12)
gives the same value: \(-\frac12D_x\log J=-\frac i{2x}\). On
half-densities, (G13) adds \(+\frac i{2x}\), giving
\(A^{(1/2)}_\kappa=e^{-z}D_z+\frac i2e^{-z}\). Its subprincipal symbol
is zero, agreeing with \(D_x\). These calculations can be localized by
compact cutoffs equal to one on the interval under discussion.

\textbf{One smoothing kernel, two support failures.} On \(\mathbb R\),
the smooth kernel \(K(x,y)=1\) sends a compact distribution \(u\) to the
constant \(u(1)\). It is smoothing and belongs locally to every
\(\Psi^m\), but neither kernel projection is proper. It does not send
general distributions to distributions by this integral formula: the
proposed value on the constant input one would require integrating over
the whole line. Multiplying the kernel by a compact smooth function of
\(x-y\) makes it proper and still smooth; its integral then acts on
every distribution. This distinguishes smoothness from support control
without confusing either with symbol order.

\textbf{The same threshold, different endpoint membership.} For the
point mass \(\delta_0\) in \(\mathbb R^n\), \(\widehat{\delta_0}=1\) and
\(\|\Pi_j\delta_0\|_2\asymp2^{jn/2}\) for \(j\geq1\). Consequently
\(\delta_0\in H^s\) exactly when \(s<-n/2\), while
\(\delta_0\in B^{-n/2}_{2,\infty}\) and
\(\delta_0\notin B^{-n/2}_{2,p}\) for finite \(p\). Each nonzero
direction at the origin has Sobolev threshold \(-n/2\), because
restricting the constant Fourier transform to any cone of positive angle
gives the same divergence. The thresholds in (G36) coincide, and the
endpoint still has to be checked separately. This is a standard
diagnostic example, not a novelty claim.

\textbf{A characteristic operator can improve the threshold.} Let
\(u=\delta_0(x_1)\otimes f(x')\) in a coordinate product, with smooth
nonzero \(f\). Multiplication by \(x_1\) is an order-zero proper
operator and sends \(u\) to zero. It is characteristic over \(x_1=0\),
so the strict improvement from finite input regularity to smooth output
is consistent with (G37). In contrast an elliptic order-zero matrix
acting between equal-rank bundles preserves the threshold at every point
of its elliptic set, regardless of the local frame.

\subsection{15. Exercises with complete
solutions}\label{exercises-with-complete-solutions}

\textbf{1. A complex density cocycle.} Let three charts have positive
transition Jacobians \(J_{12},J_{23}\). For \(z=\frac12+i\tau\), verify
the density transition law on a triple overlap. Determine the modulus of
a local transition factor and explain why a complex logarithm branch is
unnecessary.

\textbf{Solution.} The chain rule gives \(J_{13}=J_{23}J_{12}\), with
the factors evaluated at the appropriate common point. All numbers are
positive, so \(\log J_{13}=\log J_{23}+\log J_{12}\) in the real
logarithm. Exponentiating after multiplication by \(-z\) gives
\(J_{13}^{-z}=J_{23}^{-z}J_{12}^{-z}\), exactly the component cocycle.
Its modulus is \(J_{13}^{-1/2}\), because \(|e^{-i\tau\log J_{13}}|=1\).
The positivity supplied by the absolute Jacobian makes the real
logarithm single valued even on orientation-reversing overlaps.

\textbf{2. Proper representatives need not preserve an exact operator.}
For the kernel \(K(x,y)=1\) on \(\mathbb R\), choose
\(h\in C_c^\infty(\mathbb R)\), equal to one near zero, and replace
\(K\) by \(h(x-y)K\). Prove properness and identify the exact type of
error. Does this authorize replacing the original operator on a fixed
compact input without recording an error?

\textbf{Solution.} If \(\operatorname{supp}h\subset[-R,R]\), the new
kernel support satisfies \(|x-y|\leq R\). Above a compact set in either
variable, the other variable lies in its closed \(R\)-neighborhood,
which is compact. Thus both projections are proper. The difference
kernel \(1-h(x-y)\) is smooth, so the error is smoothing, but is
generally nonzero. For a smooth input of nonzero integral supported far
from a chosen output point, the proper replacement vanishes there while
the original output is its nonzero integral. Hence the replacement is an
identity modulo smoothing, not an exact operator identity.

\textbf{3. Recover the endpoint obstruction with a logarithmic
sequence.} Let \(v_j\) be \(L^2\) Fourier functions on disjoint dyadic
annuli with \(\|v_j\|_2=2^{-js_0}(1+j)^{-\beta}\), and let \(u\) be the
tempered distribution with Fourier transform \(\sum_{j\geq1}v_j\).
Determine its global \(B^{s_0}_{2,p}\) membership and its supremum of
global Sobolev orders.

\textbf{Solution.} Polynomial growth of the annular \(L^2\) norms makes
the sum a tempered distribution: pairing each annulus with a Schwartz
function and applying Cauchy--Schwarz yields a summable geometric
majorant. With the sharp annular projections in (G30), its order-\(s_0\)
Besov sequence is \((2\pi)^{-n/2}(1+j)^{-\beta}\), by Plancherel in the
stated Fourier convention. For finite \(p\), membership holds exactly
when \(\beta p>1\); for \(p=\infty\), exactly when \(\beta\geq0\). The
squared \(H^s\) norm is comparable to the sum of
\(2^{2j(s-s_0)}(1+j)^{-2\beta}\), by disjoint Fourier supports and
comparability of \(\langle\xi\rangle\) with \(2^j\) on each annulus.
This series is summable for every \(s<s_0\), and not summable for any
\(s>s_0\), since its terms then fail to tend to zero. Thus the supremum
is \(s_0\) for every real \(\beta\), while endpoint Hilbert membership
holds precisely when \(\beta>1/2\). This is a global calculation and
makes no unsupported claim about the spatial location of its wavefront.

\textbf{4. Why a compact output does not imply a compact input.} Give a
proper elliptic operator on a noncompact manifold and a noncompactly
supported smooth input whose output is compactly supported. Locate the
hypothesis that prevents this from contradicting the compact-space
converse in (G32).

\textbf{Solution.} On \(\mathbb R\), take \(A=D_x\), a proper elliptic
differential operator of order one, and \(u=1\). Its output is zero,
whose support is empty and hence compact, while \(u\) has support all of
\(\mathbb R\). The converse in (G32) asserts input regularity from
output regularity; its compact-space version assumes at the outset that
the input distribution has compact support. A parametrix leaves a smooth
remainder, which can carry a noncompact smooth solution such as this
one.

\textbf{5. An unequal-rank symbol.} Let \(E\) and \(F\) have local ranks
two and three, and let a principal symbol be
\(a(x,\xi)=\langle\xi\rangle^m\begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}\).
Compute the symbol of the half-density adjoint and explain which
statements of Section 13 apply without modification, and which two-sided
assertion cannot hold.

\textbf{Solution.} In dual frames the adjoint symbol is
\(\langle\xi\rangle^m\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}\), since
the order \(m\) is real and the displayed matrix has real entries. The
local class, its patching law, Sobolev/Besov mapping, proper-support
extension, compatible composition, and the adjoint theorem all allow
these ranks. Multiplying the adjoint symbol by the original gives
\(\langle\xi\rangle^{2m}I_2\), whereas the reverse product is
\(\langle\xi\rangle^{2m}\operatorname{diag}(1,1,0)\). Thus there cannot
be a two-sided inverse from a three-dimensional fiber to a
two-dimensional fiber. The failure is finite-dimensional algebra, not a
regularity defect.

\textbf{6. The boundary of coordinate invariance.} For a symbol
\(a(\xi)\in S^0_{\rho,0}\), consider \(b(x,\xi)=a(M(x)\xi)\) with smooth
invertible nonconstant \(M\). Derive the order cost of one base
derivative. Then classify
\((\rho,\delta)=(2/5,1/5),(3/4,1/4),(1/2,1/2)\) with respect to the
coordinate theorem and the inverse construction in this lesson.

\textbf{Solution.} The chain rule gives
\(\partial_{x_j}b=\sum_k(\partial_{\xi_k}a)(M(x)\xi)(\partial_{x_j}M(x)\xi)_k\).
On a compact base set, the first factor costs \(-\rho\) and the second
costs one, so the bound is of order \(1-\rho\). This estimate is the
reason for the requirement \(\delta\geq1-\rho\); the calculation is not
by itself an assertion that every particular symbol fails below that
bound. For \((2/5,1/5)\), this condition fails because \(1/5<3/5\), so
the general coordinate theorem does not apply, even though the Euclidean
symbol class is defined. For \((3/4,1/4)\), both
\(1-\rho\leq\delta\leq\rho\) and \(\delta<\rho\) hold, giving coordinate
invariance and an inverse gain \(\rho-\delta=1/2\). For \((1/2,1/2)\),
coordinate invariance holds through the amplitude estimate, but the
inverse gain is zero. No decreasing-order parametrix construction
follows there.

\subsection{16. Original chart, bundle and inverse
maps}\label{original-chart-bundle-and-inverse-maps}

These are complete standard foundation proofs for the course's original
smooth manifolds and finite-coordinate maps. No novelty is claimed. The
original spaces, norms, coordinates, matrices and all transition factors
remain. The independent inputs are the full finite linear algebra in
Polynomial and contour tools Section 10, the original scalar and higher
finite-coordinate calculus in Metric and topological foundations Section
13, the norm completeness proved there and in Banach estimates, and the
full original chart construction MG1--MG9 in the divergence chapter. A
manifold means its given Hausdorff, second-countable smooth atlas; a
bundle means the finite-rank locally trivial complex vector space
structure specified below. These definitions do not assume a missing
inverse, gluing or partition theorem.

\subsubsection{16.1. The actual smooth partition and every denominator
derivative}\label{the-actual-smooth-partition-and-every-denominator-derivative}

The construction MG1--MG5 works in the original smooth atlas. Its
coordinate bumps, including the full exponential and positive
denominator in MG3 and the actual three radii and quadratic denominator
in MG4, are smooth. Composition with the given smooth charts and
extension by zero preserve every derivative order because each support
is compact inside its chart. The original shell construction proves
local finiteness independently of any bundle or inverse-function
theorem. Consequently its actual functions are

\[
 S(x)=\sum_{i\in I}\beta_i(x),\qquad S(x)\geq1,\qquad
 \phi_i(x)=\frac{\beta_i(x)}{S(x)},\qquad
 \sum_{i\in I}\phi_i(x)=1 .
 \tag{BG1}
\]

Every sum is finite on a neighborhood of each point. The compact
supports stay in their assigned original cover members. There is no
global compactness or finite-atlas assumption.

Here is the full higher derivative formula, including every denominator
contribution. For labeled coordinate directions \(h_1,\ldots,h_N\), let
\(\mathfrak P(J)\) be the set of all partitions of the finite label set
\(J\) into nonempty blocks. The empty set has its one empty partition.
In a block \(B\), retain the directions with those actual labels,
writing \(D_B S=D^{|B|}S[h_b:b\in B]\). Then

\[
 \begin{aligned}
 D^N\phi_i[h_1,\ldots,h_N]
 =\sum_{J\subset\{1,\ldots,N\}}
 &D^{|J|}\beta_i[h_j:j\in J]\\
 {}\times\sum_{\Pi\in\mathfrak P(J^c)}
 &(-1)^{|\Pi|}|\Pi|!\,S^{-|\Pi|-1}
       \prod_{B\in\Pi}D_BS .
 \end{aligned}
 \tag{BG2}
\]

For \(N=0\), this is exactly the original quotient in BG1. To prove the
formula, differentiate the reciprocal once:
\(D(S^{-1})[h]=-S^{-2}DS[h]\). At any following differentiation a new
label either joins exactly one existing block by differentiating that
derivative of \(S\), or forms a singleton by differentiating the
reciprocal coefficient. The latter coefficient changes from
\((-1)^k k!S^{-k-1}\) to \((-1)^{k+1}(k+1)!S^{-k-2}\). Every partition
of the enlarged label set has exactly one predecessor of one of these
two kinds, determined by the block containing the new label. This proves
the reciprocal partition formula by induction. The complete product rule
chooses the labels \(J\) assigned to \(\beta_i\), giving BG2. Repeated
equal coordinate directions retain all their multiplicities because the
labels remain distinct. On any compact subchart, finitely many original
supports occur and all derivatives on the right have finite bounds;
\(S\geq1\) keeps every original denominator away from zero.

The original compact cutoff MG5--MG7 is smooth by the same reasoning:
its sum has finitely many selected original compact supports and equals
one on a neighborhood of the compact set by the locally finite
closed-support argument. The original locally positive radius minorant,
original metric and absolute density MG7--MG9 are smooth at every order
when the given atlas is smooth. Their original sums, metric matrices,
square roots and absolute Jacobians remain those of MG7--MG9. On an
empty manifold these assertions are vacuous with the empty functions; in
dimension zero the given Hausdorff second-countable manifold is discrete
and its countable singleton charts give the same construction with empty
derivative lists.

\subsubsection{16.2. Gluing the actual finite-rank
bundle}\label{gluing-the-actual-finite-rank-bundle}

Let \(\{U_i\}_{i\in I}\) be an open cover of the original smooth
manifold \(X\). The original rank \(r\geq0\) is fixed on the component
under discussion; a locally constant rank is treated component by
component. Suppose given smooth matrices on every overlap, with exactly
the convention

\[
 G_{ji}:U_i\cap U_j\longrightarrow {\rm GL}(r,\mathbb C),
 \qquad
 G_{ii}=I_r,\qquad
 G_{kj}(x)G_{ji}(x)=G_{ki}(x).
 \tag{BG3}
\]

These are the data to be glued, not an assumed gluing theorem. In
particular \(G_{ij}G_{ji}=I_r=G_{ji}G_{ij}\); the actual inverses are
given by the reversed transition matrices. Form the disjoint union of
the original \(U_i\times\mathbb C^r\), and impose

\[
 (i,x,v_i)\sim(j,x,v_j)
 \quad\Longleftrightarrow\quad
 v_j=G_{ji}(x)v_i .
 \tag{BG4}
\]

BG3 proves reflexivity, symmetry and transitivity, with the displayed
multiplication order. Let \(E\) be this quotient with its quotient
topology, \(q\) the quotient map, and \(\pi[(i,x,v_i)]=x\). The last map
is well defined and continuous: on each disjoint-union component its
composition with \(q\) is the continuous projection, and the defining
quotient criterion proves continuity.

For each original \(U_i\), define

\[
 \begin{aligned}
 \Phi_i:\pi^{-1}(U_i)&\longrightarrow U_i\times\mathbb C^r,\\
 \Phi_i[(j,x,v_j)]&=(x,G_{ij}(x)v_j),\\
 \Phi_i^{-1}(x,v_i)&=q(i,x,v_i).
 \end{aligned}
 \tag{BG5}
\]

The cocycle proves independence of the representative, and the two
displayed maps are inverse. The set \(\pi^{-1}(U_i)\) is open. The
restriction of a quotient to an open saturated preimage is a quotient
map: the inverse image of an open subset of that restriction is open in
the open preimage and therefore open in the whole disjoint union. On
each component of this preimage, \(\Phi_iq\) is
\((x,v_j)\mapsto(x,G_{ij}(x)v_j)\), continuous. Hence \(\Phi_i\) is
continuous by this restricted quotient criterion. Its displayed inverse
is the continuous inclusion followed by \(q\). Thus BG5 is a
homeomorphism onto the entire original product, without a missing
bijection-to-homeomorphism step.

If two points in \(E\) have different base points, disjoint original
base neighborhoods separate them by \(\pi\). If their base points agree,
one chart BG5 contains both; its Hausdorff product separates them by
open subsets of that open chart. Thus \(E\) is Hausdorff. The original
countable-base argument in MG1 selects a countable subcover of the
\(U_i\). Each \(\pi^{-1}(U_i)\) in that subcover is second countable by
BG5, since its base is an open subset of a second-countable manifold and
its fiber is finite-dimensional. The union of those countably many
open-chart bases is a countable base for \(E\).

Use the original base coordinates in BG5 and the original real and
imaginary fiber coordinates. On an overlap its full change of
coordinates is

\[
 (x_i,v_i)\longmapsto
       \bigl(\kappa_{ji}(x_i),G_{ji}(x)v_i\bigr).
 \tag{BG6}
\]

It and its inverse are smooth by the full coordinate product and chain
rules; the inverse retains \(\kappa_{ij}\) and \(G_{ij}\). This supplies
a smooth atlas on the quotient. Fiber addition and complex scalar
multiplication are the original operations in each \(\mathbb C^r\). BG3
makes them independent of the chart, and BG6 proves their smoothness. We
have constructed the complete smooth finite-rank vector bundle.

A section is equivalent to its actual smooth component functions \(v_i\)
with \(v_j=G_{ji}v_i\). In one direction this follows from BG5--BG6.
Conversely these functions define the same quotient point on every
overlap by BG4; their local smoothness in the open charts gives a global
smooth section. The same argument glues any family of compatible local
sections over arbitrary open sets, uniquely. Compatible local
fiber-linear maps glue too: with input transitions \(G^E\), output
transitions \(G^F\) and local matrix \(A_i\), the exact condition is
\(A_jG^E_{ji}=G^F_{ji}A_i\). It gives one well-defined smooth quotient
map, and its local matrices recover the original family. If all local
matrices are invertible, their actual inverses satisfy the reverse
transition law and glue to the inverse map. Any other bundle with the
same trivializations and transition data receives this map and its
inverse, so the construction is unique up to the uniquely specified
bundle isomorphism. This proves existence, gluing and the full inverse
statement rather than assuming a bundle object from its cocycle.

For rank zero, every fiber is the singleton \(\mathbb C^0\), every
matrix is \(I_0\), its determinant is one and every empty product has
value one. BG4--BG6 give \(E=X\), with its zero-dimensional fibers and
the unique zero sections. Empty base sets give the empty bundle.

\subsubsection{16.3. Every tensor, dual, metric and density
factor}\label{every-tensor-dual-metric-and-density-factor}

The same construction applies to the actual derived transitions. For two
bundles, the tensor transition has all of its original entries:

\[
 (v^E\otimes v^F)^{ab}_j
   =\sum_{c=1}^{r_E}\sum_{d=1}^{r_F}
       (G^E_{ji})_{ac}(G^F_{ji})_{bd}
       (v^E\otimes v^F)^{cd}_i .
 \tag{BG7}
\]

The tensor of the two cocycle identities, with both finite sums
expanded, proves its cocycle in the same order. Direct sums use the full
block-diagonal matrices. For a fiber-linear map from \(E\) to \(F\) and
for the linear and conjugate-linear duals, respectively, the transitions
are

\[
 A_j=G^F_{ji}A_iG^E_{ij},\qquad
 \ell_j=(G^E_{ij})^T\ell_i,\qquad
 m_j=\overline{G^E_{ij}}^{\,T}m_i .
 \tag{BG8}
\]

Indeed \(v_j=G^E_{ji}v_i\), so the first equation is exactly equality of
the original map in both frames. A linear functional has value
\(\ell_i^Tv_i\); a conjugate-linear functional has value
\(m_i^T\overline{v_i}\). Substitution gives the last two equations
including the full conjugation and transpose. Each pairwise composition
keeps every matrix factor and proves the corresponding cocycle. These
are exact maps to the actual Hom, dual and anti-dual fibers, not merely
formally similar transition lists.

Take the smooth nonnegative partition BG1 subordinate to
trivializations, retaining its actual partition index set I\_phi and its
assignment a(i) to the original trivialization U\_a(i). This assignment
may repeat a trivialization; no grouping changes the original compact
supports. In each original frame choose its actual positive Hermitian
matrix \(H_i(x)\), for example the identity in that particular original
frame. In frame \(j\), the resulting global metric has the entire matrix

\[
 H_j(x)=\sum_{i\in I_\phi}\phi_i(x)\,
                  G_{a(i),j}(x)^*H_{a(i)}(x)G_{a(i),j}(x).
 \tag{BG9}
\]

Terms are defined on their original trivialization overlap and extended
by zero where their compact base support permits it. Each such extension
is smooth, and the sum is locally finite. Every term is nonnegative on
the actual vector \(v_j\). At each point one \(\phi_i\) is positive; if
\(v_j\ne0\), the inverse matrix \(G_{a(i),j}\) makes that term strictly
positive. Thus the metric is positive. Its change of frame is
\(H_k=G_{jk}^*H_jG_{jk}\): insert BG3 into every term in BG9, retaining
both ordered outer factors. On a compact subchart the continuous value
\(v^*H_j(x)v\) on the original unit sphere has a positive minimum and a
finite maximum, so the original fiber norm has exact positive local
comparison bounds. Rank-zero assertions are vacuous; no positive
eigenvalue is asserted for an empty matrix.

For original smooth coordinates \(x_i\) and \(x_j=\kappa_{ji}(x_i)\),
tangent components change by \(D\kappa_{ji}\); cotangent components
change by \((D\kappa_{ij})^T\), from evaluation on the original tangent
vector. The chain rule proves both inverse products. The actual scalar
density and half-density coefficients have transitions

\[
 \rho_j(x_j)=\rho_i(x_i)|\det D\kappa_{ij}(x_j)|,\qquad
 h_j(x_j)=h_i(x_i)|\det D\kappa_{ij}(x_j)|^{1/2}.
 \tag{BG10}
\]

The full determinant chain rule proves both cocycles, including
orientation reversal. The determinants never vanish because both actual
inverse coordinate maps are given; their sign is locally constant, so
the absolute determinant is smooth. Its positive square root is smooth
by the already proved scalar calculus. Tensoring BG10 with BG3 gives the
full section and kernel factors of the original bundle-valued density
and half-density spaces. In particular a Hom-valued half-density kernel
from \(X\) to \(Y\) keeps \(G_F(y)\) on the left, \(G_E(x)^{-1}\) on the
right, and one positive half-density Jacobian in each original variable.
Pairing the input half-density with a test half-density gives the
complete input density. Their multiplication is exactly the product of
the two original square roots, so no input Jacobian is dropped. This is
the bundle gluing and pairing map required by the existing scalar kernel
construction; it does not assume a new scalar kernel theorem.

\subsubsection{16.4. Constructing the inverse without changing the map
or
norms}\label{constructing-the-inverse-without-changing-the-map-or-norms}

Let \(E,F\) be the original real finite-dimensional normed coordinate
spaces of the same dimension \(n\geq1\), with their original norms
\(\|\cdot\|_E,\|\cdot\|_F\). Let \(f:O\subset E\to F\) be \(C^r\),
\(1\leq r\leq\infty\), on the original open set, and let \(a\in O\) have
the actual invertible derivative \(A=Df(a):E\to F\). We keep
\(f,A,a,f(a)\) and both norms, and never replace \(A\) by an identity or
\(f\) by a changed coordinate expression.

Choose \(0<\theta<1\). Continuity of the actual derivative gives \(R>0\)
such that the original closed ball \(\overline B_E(a,R)\) is inside
\(O\) and

\[
 \sup_{x\in\overline B_E(a,R)}
       \|I_E-A^{-1}Df(x)\|_{E\to E}\leq\theta .
 \tag{IV1}
\]

The closed ball is complete in the original norm by the existing
finite-coordinate completeness proof. For \(y\in F\), use the actual
correction map

\[
 T_y(x)=x+A^{-1}(y-f(x)),\qquad
 \varepsilon=\frac{(1-\theta)R}{\|A^{-1}\|_{F\to E}},\qquad
 V=B_F(f(a),\varepsilon).
 \tag{IV2}
\]

The denominator is positive for \(n\geq1\). The full segment fundamental
theorem gives \(\|T_y(x)-T_y(z)\|_E\leq\theta\|x-z\|_E\) for \(x,z\) in
the original closed ball. Also
\(\|T_y(x)-a\|_E\leq\theta R+\|A^{-1}\|_{F\to E}\|y-f(a)\|_F<R\) for
\(y\in V\). Thus the whole original closed ball maps into itself.

Start \(x_0=a\), \(x_{m+1}=T_y(x_m)\). All iterates remain in that ball.
The entire contraction estimate is

\[
 \begin{aligned}
 \|x_{m+1}-x_m\|_E
   &\leq\theta^m\|A^{-1}(y-f(a))\|_E,\\
 \|x_{m+k}-x_m\|_E
   &\leq\|A^{-1}(y-f(a))\|_E
                         \sum_{\nu=m}^{m+k-1}\theta^\nu .
 \end{aligned}
 \tag{IV3}
\]

The finite sum and its convergent geometric tail prove that the sequence
is Cauchy. Completeness gives its limit \(h(y)\) in the original closed
ball. Continuity gives \(T_y(h(y))=h(y)\), hence \(f(h(y))=y\). Letting
\(k\) tend to infinity in IV3 retains the exact tail bound
\(\|h(y)-x_m\|_E\leq\theta^m(1-\theta)^{-1}\|A^{-1}(y-f(a))\|_E\). At a
fixed point, \(\|h(y)-a\|_E\leq(1-\theta)^{-1}\|A^{-1}(y-f(a))\|_E<R\).
If two fixed points existed, their distance would be at most \(\theta\)
times itself; since \(1-\theta>0\), they agree. These arguments prove
every existence and uniqueness step of the contraction construction.

For any \(x,z\) in the original closed ball, the same full segment
formula gives

\[
 \begin{aligned}
 A^{-1}(f(x)-f(z))
 &=x-z+\int_0^1
      A^{-1}\bigl(Df(z+t(x-z))-A\bigr)(x-z)\,dt,\\
 \|f(x)-f(z)\|_F
 &\geq\frac{1-\theta}{\|A^{-1}\|_{F\to E}}\|x-z\|_E .
 \end{aligned}
 \tag{IV4}
\]

Every ordered matrix factor and the entire integral remain. The reverse
triangle inequality proves the second line. Put
\(U=B_E(a,R)\cap f^{-1}(V)\), an original open neighborhood of \(a\).
The map \(f:U\to V\) is injective by IV4 and surjective by the
constructed interior fixed points. Thus \(h\) is its actual inverse, and
IV4 gives its precise Lipschitz bound
\(\|A^{-1}\|_{F\to E}/(1-\theta)\). In particular both maps are
continuous.

For \(x\) in the ball set \(B_x=I_E-A^{-1}Df(x)\). Retain both finite
product identities \((I_E-B_x)\sum_{m=0}^N B_x^m=I_E-B_x^{N+1}
=\bigl(\sum_{m=0}^N B_x^m\bigr)(I_E-B_x)\). The full operator-norm tail
is at most \(\theta^{N+1}/(1-\theta)\). Operator completeness therefore
gives both actual inverse products and

\[
 Df(x)^{-1}=\left(\sum_{m=0}^{\infty}B_x^m\right)A^{-1},
 \qquad
 \|Df(x)^{-1}\|_{F\to E}
       \leq\frac{\|A^{-1}\|_{F\to E}}{1-\theta}.
 \tag{IV5}
\]

The rightmost original inverse factor has not been absorbed into a
changed derivative. The uniformly convergent series also proves
continuous dependence on \(x\), since every finite matrix power is
continuous.

For \(y,y+k\in V\), write \(x=h(y)\), \(v=h(y+k)-h(y)\). The original
differentiability remainder gives

\[
 \begin{aligned}
 k&=Df(x)v+r_x(v),\qquad
          \frac{\|r_x(v)\|_F}{\|v\|_E}\longrightarrow0,\\
 v&=Df(x)^{-1}k-Df(x)^{-1}r_x(v).
 \end{aligned}
 \tag{IV6}
\]

The remainder is zero at \(v=0\). IV4 bounds \(\|v\|_E\) by
\(\|A^{-1}\|_{F\to E}(1-\theta)^{-1}\|k\|_F\); IV5 then makes the last
remainder \(o(\|k\|_F)\). Consequently \(Dh(y)=Df(h(y))^{-1}\), and its
continuity follows from IV5 and continuity of \(h\). This proves \(C^1\)
regularity of the actual inverse before any higher smoothness is used.

\subsubsection{16.5. Every higher inverse derivative and the implicit
map}\label{every-higher-inverse-derivative-and-the-implicit-map}

Matrix inversion is smooth on the original invertible matrices: the full
cofactor inverse proved in Polynomial and contour tools Section10 is a
matrix of polynomial entries divided by its actual nonzero determinant.
The proved finite product, reciprocal and chain rules therefore apply at
every available order. Differentiating both original inverse products
gives \(D(M^{-1})[v]=-M^{-1}DM[v]M^{-1}\). For every integer \(N\geq1\),
iteration retains every ordered factor:

\[
 D^N(M^{-1})[v_1,\ldots,v_N]
 =\sum_{k=1}^N(-1)^k
   \sum_{\substack{(I_1,\ldots,I_k)\ {\rm ordered}\\
                   I_j\ne\varnothing,\ 
                   I_1\sqcup\cdots\sqcup I_k=\{1,\ldots,N\}}}
 M^{-1}D_{I_1}M\,M^{-1}\cdots
             D_{I_k}M\,M^{-1}.
 \tag{IV7}
\]

Each differentiation either adds its label to one existing derivative
block or inserts a singleton block at one of the inverse factors. The
sign changes precisely in the latter case. Every ordered partition of
the new label set has one such predecessor, proving IV7 by induction
with no commutation of matrix factors. Repeated equal directions keep
all labeled multiplicities.

Starting from the proved \(C^1\) inverse, the identity
\(Dh=(Df\circ h)^{-1}\) and this smooth inversion map prove \(h\in C^r\)
by induction. Precisely, if \(h\in C^s\) with \(s<r\), then \(Df\) is
\(C^{r-1}\), so the right side is \(C^s\); hence \(Dh\) is \(C^s\) and
\(h\) is \(C^{s+1}\). This covers every finite order and all orders when
\(r=\infty\).

For \(2\leq N\leq r\), set \(D_Bh(y)=D^{|B|}h(y)[v_b:b\in B]\) with
every original direction label retained, and differentiate the exact
composition \(f(h(y))=y\) by the full higher chain rule. The partition
with one block is \(Df(h(y))D^Nh\); all other partitions keep their
original derivatives. Thus the complete inverse derivative is

\[
 D^Nh(y)[v_1,\ldots,v_N]
 =-Df(h(y))^{-1}
   \sum_{\substack{\Pi\in\mathfrak P(\{1,\ldots,N\})\\|\Pi|\geq2}}
 D^{|\Pi|}f(h(y))
       [D_Bh(y):B\in\Pi].
 \tag{IV8}
\]

The full higher chain rule itself follows by the same labeled-partition
induction: differentiating a block derivative adds the new label to that
block, while differentiating the outer derivative creates its singleton
block. This accounts for every partition once. The derivatives of the
outer map are symmetric multilinear maps, so the bracket order across
blocks can be read in increasing least-label order without changing a
matrix product. All lower inverse derivatives, repeated-direction
multiplicities and the original left inverse matrix remain in IV8.

For the actual implicit problem \(f(s,x)=y\), where \(s\) lies in its
original finite-coordinate parameter space and \(D_xf(s_0,x_0)=A\) is
invertible, apply the just proved inverse construction to the unchanged
augmented map \(L(s,x)=(s,f(s,x))\). At the original point its two block
matrices are

\[
 DL=\begin{pmatrix}I&0\\ B&A\end{pmatrix},
 \qquad
 (DL)^{-1}=\begin{pmatrix}I&0\\-A^{-1}B&A^{-1}\end{pmatrix},
 \qquad B=D_sf(s_0,x_0).
 \tag{IV9}
\]

Multiplying in both orders proves the inverse, including the lower-left
term. Use the original product norm and an open product contained in the
constructed image neighborhood. The first component of \(L^{-1}(s,y)\)
must be exactly \(s\), because \(L\)'s first component is \(s\). Its
second component \(H(s,y)\) is therefore the unique local solution of
the actual original implicit equation. It has the proved \(C^r\)
regularity and

\[
 D_yH=(D_xf)^{-1},\qquad
 D_sH=-(D_xf)^{-1}D_sf,
 \tag{IV10}
\]

with every derivative of \(f\) evaluated at the original \((s,H(s,y))\).
All higher derivatives are IV8 applied to the entire augmented map and
then its second projection; this retains its full parameter blocks and
ordered inverse factors. No implicit-function theorem was assumed in
constructing this map.

If the solved coordinate dimension is zero, its original domain and
range are the singleton spaces and the local inverse is the identity
there; the implicit augmented map is exactly the identity on its
parameter component. Its empty matrix determinant is one, and no
division by the zero operator norm is made. Empty open domains have no
base point and make no local assertion.

\subsubsection{16.6. Exact receiving scope and remaining
work}\label{exact-receiving-scope-and-remaining-work}

The original smooth atlas and finite-rank bundle clauses receive
BG1--BG10 with their actual transition matrices, component maps, duals
and density factors. The smooth inverse-function entry receives
IV1--IV10 in the original finite-coordinate norms. The existing scalar
distribution and kernel constructions then have the precise bundle
gluing and half-density pairing map they require; their scalar proofs
remain separate independent providers. Smooth coordinate charts already
given by an atlas need no new inverse assumption, since both original
coordinate maps are part of that atlas.

These proofs do not by themselves establish the nonlinear local flow
construction, complete exponential-chart/minimizing-radius calculations,
wavefront-qualified arbitrary-map pullback or submersion converse, or
complex contour and Laplace/Euler interfaces. Those require their actual
remaining calculations.

\subsection{17. Original finite-coordinate flows and their geodesic
receiving
map}\label{original-finite-coordinate-flows-and-their-geodesic-receiving-map}

These are standard foundation proofs. Their inputs are the original
finite norm completeness, compactness, scalar integration and full
finite-coordinate calculus already proved in the course, and the actual
inverse and implicit maps BG1--BG10/IV1--IV10 in geometry Section16.
Every original state space, norm, time endpoint, parameter, vector
field, matrix order and derivative contribution remains.

\subsubsection{17.1. The original integral equation and its complete
local
solution}\label{the-original-integral-equation-and-its-complete-local-solution}

Let \(E,P\) be the original real finite-dimensional normed spaces, with
their given norms. Let \(W\subset\mathbb R\times E\times P\) be open and
let \(F:W\to E\) be \(C^r\), \(1\leq r\leq\infty\). Fix the original
point \((t_0,a,p_0)\in W\). The equation and datum are

\[
 \partial_tu(t)=F(t,u(t),p),\qquad u(t_0)=x.
 \tag{NF1}
\]

Choose positive \(d,\rho,\delta\) such that the compact rectangle
\(K=[t_0-d,t_0+d]\times\overline B_E(a,\rho)\times\overline B_P(p_0,\delta)\)
lies in \(W\). An open neighborhood contains a sufficiently small
product of balls; their closures are compact in the original norms and
can still be chosen inside that neighborhood. Let the actual bounds be

\[
 M=\sup_K\|F\|_E,\qquad
 L=\sup_K\|D_xF\|_{E\to E},\qquad
 Q=\sup_K\|D_pF\|_{P\to E}.
 \tag{NF2}
\]

They are finite by continuity and compactness. Choose \(0<\theta<1\) and
\(0<h<d\) with \(hM\leq\rho/4\) and \(hL\leq\theta\). Such an \(h\)
exists also when either bound is zero; no division by a zero bound is
used. Keep the original interval \(I=[t_0-h,t_0+h]\), the state ball,
and \(x\in B_E(a,\rho/4)\), \(p\in B_P(p_0,\delta)\). On continuous
paths in the original closed state ball set

\[
 (\Phi_{x,p}w)(t)=x+\int_{t_0}^{t}F(s,w(s),p)\,ds .
 \tag{NF3}
\]

All integrals retain their original oriented endpoints. A
finite-coordinate continuous integral exists coordinatewise by the
proved Riemann integral. Its norm is at most the integral of the norm:
the finite tagged sums have this bound by the triangle inequality; their
vector limits and the scalar integral limit preserve it. The
continuous-path space is complete in
\(\|w\|_\infty=\sup_{t\in I}\|w(t)\|_E\). Indeed a uniform Cauchy
sequence converges pointwise by the original completeness of \(E\), its
Cauchy estimates pass uniformly to the limit, and the triangle
inequality with one continuous approximant proves continuity. The paths
with values in the closed ball form a closed subset of that complete
space.

NF2--NF3 show that \(\Phi_{x,p}w\) stays within distance \(\rho/2\) of
\(a\). The full state-segment fundamental theorem gives
\(\|F(s,v,p)-F(s,w,p)\|_E\leq L\|v-w\|_E\) inside the original convex
ball. Thus
\(\|\Phi_{x,p}v-\Phi_{x,p}w\|_\infty\leq hL\|v-w\|_\infty\leq\theta\|v-w\|_\infty\).
Starting with the actual constant path \(u_0(t)=x\), define
\(u_{m+1}=\Phi_{x,p}u_m\). For every \(m\geq0\) and \(k\geq1\), retain

\[
 \begin{aligned}
 \|u_{m+1}-u_m\|_\infty
   &\leq\theta^m\|u_1-u_0\|_\infty,\qquad
       \|u_1-u_0\|_\infty\leq hM,\\
 \|u_{m+k}-u_m\|_\infty
   &\leq\|u_1-u_0\|_\infty
                         \sum_{\nu=m}^{m+k-1}\theta^\nu,\\
 \|u-u_m\|_\infty
   &\leq\frac{\theta^m}{1-\theta}\|u_1-u_0\|_\infty .
 \end{aligned}
 \tag{NF4}
\]

The finite sum proves the Cauchy property, completeness supplies \(u\),
and continuity of \(\Phi\) gives NF3 with \(w=u\). The last line follows
by taking the limit in the full finite-tail estimate. The fundamental
theorem then proves NF1, including at \(t_0\). Two fixed points have
distance at most \(\theta\) times that distance and therefore coincide.

For another \(C^1\) solution with the same datum, restrict to an
interval on which both solutions lie in a common compact rectangle. On a
sufficiently short subinterval beginning at any time at which they
agree, the same contraction estimate forces agreement. Their equality
set is closed by continuity and open by this two-sided local argument.
On their connected common time interval containing the datum it is
therefore the whole interval. This proves local uniqueness even for a
solution not initially confined to the particular closed ball used to
construct \(u\).

The original parameter differences satisfy

\[
 \|u(\,\cdot\,;x',p')-u(\,\cdot\,;x,p)\|_\infty
 \leq\frac{\|x'-x\|_E+hQ\|p'-p\|_P}{1-\theta}.
 \tag{NF5}
\]

To prove it, subtract the two full equations NF3, use the state and
parameter segment formulas with their original derivative bounds, and
bring the \(hL\) term to the left. Convexity of the original parameter
ball keeps every intermediate point in \(K\). Continuity in time and
this uniform estimate give joint continuity in \((t,x,p)\). The retained
buffer \(\rho/2\) to the boundary of the larger state ball permits the
full parameter Taylor comparisons below.

\subsubsection{17.2. Ordered linear transport on the original two-sided
interval}\label{ordered-linear-transport-on-the-original-two-sided-interval}

Let \(A:I\to\mathcal L(E,E)\) be continuous and \(B:I\to E\) be
continuous. For the same original \(t_0\), define

\[
 (\mathcal V_Az)(t)=\int_{t_0}^t A(s)z(s)\,ds,\qquad
 z=B+\mathcal V_Az.
 \tag{NF6}
\]

For \(L_A=\sup_I\|A(s)\|\), nested integration proves
\(\|\mathcal V_A^m\|\leq(hL_A)^m/m!\). In detail the \(m\)-fold
application has the full integral
\(\int_{t_0}^t ds_1\int_{t_0}^{s_1}ds_2\cdots\int_{t_0}^{s_{m-1}}ds_m\)
of \(A(s_1)\cdots A(s_m)z(s_m)\). Taking absolute values reverses the
bounds when \(t<t_0\); the resulting scalar nested integral of one is
\(|t-t_0|^m/m!\), by induction from the scalar power integral. Every
oriented sign remains in the original operator integral. Norm
completeness therefore supplies

\[
 \begin{aligned}
 \mathcal R_A&=\sum_{m=0}^{\infty}\mathcal V_A^m,&
 (I-\mathcal V_A)\mathcal R_A
   &=I=\mathcal R_A(I-\mathcal V_A),\\
 z&=\mathcal R_AB,&
 \|\mathcal R_A\|&\leq
        \sum_{m=0}^{\infty}\frac{(hL_A)^m}{m!}
        =\exp(hL_A).
 \end{aligned}
 \tag{NF7}
\]

The inverse products follow from both finite identities
\((I-\mathcal V_A)\sum_{m=0}^N\mathcal V_A^m
=I-\mathcal V_A^{N+1}
=(\sum_{m=0}^N\mathcal V_A^m)(I-\mathcal V_A)\) and the factorial bound.
This proves uniqueness as well as existence. When \(hL_A\leq\theta\),
the additional geometric bounds \(\|\mathcal R_A\|\leq(1-\theta)^{-1}\)
and \(\|\sum_{m>N}\mathcal V_A^m\|\leq\theta^{N+1}/(1-\theta)\) hold
with their complete factors. Neither estimate replaces the original
operator or the factorial estimate.

For a matrix coefficient acting on a finite-dimensional original fiber,
the fundamental matrix is

\[
 Y(t)=I_E+\sum_{m=1}^{\infty}
   \int_{t_0}^{t}ds_1\int_{t_0}^{s_1}ds_2
       \cdots\int_{t_0}^{s_{m-1}}ds_m\,
        A(s_1)A(s_2)\cdots A(s_m).
 \tag{NF8}
\]

The order is exactly the displayed one. Its factorial bound proves
uniform convergence. Its integral equation and the fundamental theorem
give \(Y'=AY\), \(Y(t_0)=I_E\). Construct \(Z'=-ZA\), \(Z(t_0)=I_E\), by
the same integral argument, now on the original matrix space with right
multiplication. Differentiating \(ZY\) gives zero with the two full
terms \(-ZAY+ZAY\), so \(ZY=I_E\). Finite-dimensional injectivity and
surjectivity imply \(YZ=I_E\) as well. Thus this constructed \(Z\) is
the actual inverse at every original time.

For \(z'=Az+b\), \(z(t_0)=c\), both full maps are

\[
 z(t)=Y(t)\left(c+\int_{t_0}^{t}Y(s)^{-1}b(s)\,ds\right),
 \qquad
 Y(t)^{-1}z(t)=c+\int_{t_0}^{t}Y(s)^{-1}b(s)\,ds .
 \tag{NF9}
\]

The product rule and the proved inverse equation verify both identities,
including their initial values and multiplication order. Uniqueness
follows from NF7. Arbitrary complex matrices are handled on their
original real and imaginary coordinates, with the same complex matrix
products; no diagonalization, self-adjointness or commutation is used.

\subsubsection{17.3. All parameter derivatives of the linear
inverse}\label{all-parameter-derivatives-of-the-linear-inverse}

Let \(\eta\) range in an open finite-dimensional original normed
parameter space. Suppose \(\eta\mapsto A_\eta\) is \(C^s\) into
continuous matrix or operator paths in the original supremum norm.
Integration in NF6 is a bounded linear map of \(A\), with norm at most
\(h\), so \(\eta\mapsto\mathcal V_\eta\) is \(C^s\) and every derivative
retains that integral. Locally bounded \(L_A\) gives locally bounded
inverses by NF7. Their exact difference identity is

\[
 \mathcal R_{\eta'}-\mathcal R_\eta
   =\mathcal R_{\eta'}(\mathcal V_{\eta'}-\mathcal V_\eta)
                         \mathcal R_\eta.
 \tag{NF10}
\]

Multiplication on the left by \(I-\mathcal V_{\eta'}\) and on the right
by \(I-\mathcal V_\eta\) verifies the identity directly. It proves norm
continuity. Insert the full differentiability remainder of
\(\mathcal V\) into NF10; continuity and the local inverse bound make
its remaining error \(o(\|\eta'-\eta\|)\). Consequently
\(D\mathcal R[v]=\mathcal R(D\mathcal V[v])\mathcal R\). This proof uses
no Banach inverse-function theorem. Bounded operator multiplication has
the required product rule: expand the actual two-factor increment,
retaining the bilinear increment product whose norm is bounded by the
product of its two increment norms. The remainder is therefore of second
order. Induction gives its full higher product rule.

For every integer \(N\geq1\) allowed by the parameter differentiability,
the entire inverse derivative is

\[
 D^N\mathcal R[v_1,\ldots,v_N]
 =\sum_{k=1}^N
   \sum_{\substack{(I_1,\ldots,I_k)\ {\rm ordered}\\
                   I_j\ne\varnothing,\ 
                   I_1\sqcup\cdots\sqcup I_k=\{1,\ldots,N\}}}
 \mathcal R D_{I_1}\mathcal V\,\mathcal R\cdots
                       D_{I_k}\mathcal V\,\mathcal R .
 \tag{NF11}
\]

Here \(D_I\) retains precisely the labeled directions in \(I\). A new
differentiation either joins an existing derivative block or
differentiates one inverse factor and inserts its new singleton block
there. Every ordered partition of the enlarged label set has exactly one
predecessor, determined by the new label's block. This proves NF11 and
\(C^s\) regularity by induction, with all repeated-direction
multiplicities and every noncommuting factor retained. For \(N=0\) the
value is the full inverse \(\mathcal R\).

If \(B_\eta\) is \(C^s\) in the continuous-path norm, NF7 and the full
product rule therefore make \(z_\eta=\mathcal R_\eta B_\eta\) \(C^s\) in
that same norm. For coefficient paths of the form
\(A_\eta(t)=A(t,\eta)\), whose finite-coordinate derivatives are jointly
continuous, their path-valued derivatives exist uniformly on compact
parameter neighborhoods. The segment Taylor remainder is bounded by the
uniform oscillation of the next continuous derivative on the compact
time/parameter product, which tends to zero. This proves the asserted
path-valued regularity, rather than merely pointwise differentiability.

\subsubsection{17.4. The first actual nonlinear parameter
derivative}\label{the-first-actual-nonlinear-parameter-derivative}

Write \(\eta=(x,p)\), retaining its given product-space norm and both
coordinate projections. For a direction \(v=(v_x,v_p)\), the candidate
derivative \(z_v\) solves

\[
 \begin{aligned}
 A_\eta(t)&=D_xF(t,u_\eta(t),p),\\
 B_{\eta,v}(t)&=v_x+
           \int_{t_0}^tD_pF(s,u_\eta(s),p)v_p\,ds,\\
 z_v&=B_{\eta,v}+\mathcal V_{A_\eta}z_v
       =\mathcal R_{A_\eta}B_{\eta,v}.
 \end{aligned}
 \tag{NF12}
\]

NF7 supplies this entire linear solution and uniqueness. It is linear in
the original direction, and continuous as an operator from the original
parameter norm to the path norm: keep the original projection norms
\(c_x,c_p\), so \(\|z_v\|_\infty\leq(1-\theta)^{-1}(c_x+hQc_p)\|v\|\).
No original norm is replaced by a coordinate norm in this estimate.

To prove that it is the derivative, let
\(\Delta\eta=(\Delta x,\Delta p)\) and
\(\Delta u=u_{\eta+\Delta\eta}-u_\eta\). The exact segment expansion of
the original field is

\[
 \begin{aligned}
 &F(t,u_\eta+\Delta u,p+\Delta p)-F(t,u_\eta,p)\\
 &\quad=A_\eta(t)\Delta u+
               D_pF(t,u_\eta,p)\Delta p+\epsilon_\eta(t),\\
 &\|\epsilon_\eta\|_\infty
   \leq\omega_\eta\bigl(\|\Delta u\|_\infty+\|\Delta p\|_P\bigr)
             \bigl(\|\Delta u\|_\infty+\|\Delta p\|_P\bigr),
       \qquad \omega_\eta(q)\longrightarrow0\quad(q\downarrow0).
 \end{aligned}
 \tag{NF13}
\]

For the bound, subtract the derivatives at the start of each full
state/parameter segment from their values along it and integrate.
Uniform continuity on a compact rectangle containing these segments
gives the displayed modulus. NF5, with both original projection factors,
bounds the argument of the modulus by
\(((c_x+hQc_p)/(1-\theta)+c_p)\|\Delta\eta\|\). Subtract NF12 for
\(v=\Delta\eta\) from the exact difference of NF3. Its residual is
\(\Delta u-z_{\Delta\eta}=\mathcal V_{A_\eta}(\Delta u-z_{\Delta\eta})
+\int_{t_0}^{\,\cdot\,}\epsilon_\eta(s)\,ds\). NF7 bounds it by
\(h(1-\theta)^{-1}\|\epsilon_\eta\|_\infty=o(\|\Delta\eta\|)\). This
proves the full path-valued Fréchet derivative. Joint continuity of the
field derivatives, NF5 and NF10 give continuity of the derivative
operator. Thus \(u_\eta\) is \(C^1\) into continuous paths before any
higher dependence is used.

\subsubsection{17.5. Every nonlinear parameter derivative, including its
original
blocks}\label{every-nonlinear-parameter-derivative-including-its-original-blocks}

Assume \(u_\eta\) is \(C^s\) in the path norm, with \(s<r\). The
path-valued maps \(A_\eta=D_xF(\,\cdot\,,u_\eta,p)\) and \(B_{\eta,v}\)
in NF12 are \(C^s\), because \(F\) is \(C^r\) and \(s\leq r-1\). To
justify the composition statement in the path norm, apply the full
finite-coordinate chain and product rules at each time. The derivatives
are finite sums of continuous products of the original derivatives of
\(F\) and those of \(u\). All their Taylor remainder bounds are uniform
on the compact rectangle, by the same segment and uniform-continuity
argument as NF13. The bound remains uniform for \(v\) in the unit ball
of its original finite-dimensional direction space. NF10--NF11 then show
that \(Du_\eta=\mathcal R_{A_\eta}B_{\eta,\cdot}\) is \(C^s\) as an
operator-valued map. Hence \(u_\eta\) is \(C^{s+1}\). Induction proves
the full \(C^r\) parameter theorem, and every order for \(r=\infty\).

Here is its exact higher equation. Retain labeled directions
\(v_1,\ldots,v_N\) in the original parameter space, and put
\(\zeta_\eta(t)=(u_\eta(t),p)\). For a nonempty block \(B\) of labels
use

\[
 D_B\zeta_\eta(t)=
 \begin{cases}
 (D u_\eta(t)[v_b],(v_b)_p),&B=\{b\},\\
 (D^{|B|}u_\eta(t)[v_b:b\in B],0),&|B|\geq2.
 \end{cases}
 \tag{NF14}
\]

For \(2\leq N\leq r\), the whole \(N\)-th derivative equation is

\[
 \begin{aligned}
 D^Nu_\eta(t)[v_1,\ldots,v_N]
 &=\int_{t_0}^t A_\eta(s)
                  D^Nu_\eta(s)[v_1,\ldots,v_N]\,ds\\
 &\quad+\int_{t_0}^t
   \sum_{\substack{\Pi\in\mathfrak P(\{1,\ldots,N\})\\|\Pi|\geq2}}
   D_{(x,p)}^{|\Pi|}F(s,u_\eta(s),p)
                 [D_B\zeta_\eta(s):B\in\Pi]\,ds .
 \end{aligned}
 \tag{NF15}
\]

The initial \(N\)-th derivative is zero because the original datum \(x\)
is linear in \(\eta\). The one-block term of the full chain rule is
\(D_xF\,D^Nu\), since the parameter component of NF14 is zero for that
block. Every other partition remains explicitly in NF15. Differentiating
a block appends the new direction to that block; differentiating the
outer derivative creates its singleton block. Every partition is
obtained once, proving the full chain formula with no omitted
multiplicity. The outer derivatives are symmetric multilinear maps on
the original state/parameter product; the order of their arguments may
be fixed by the least label in each block. Their values do not authorize
commuting any linear transport factors. NF7 applied to the entire
displayed inhomogeneous integral gives the actual unique derivative,
including all pure-state, pure-parameter and mixed derivatives.

All derivatives just constructed are jointly continuous in time and
parameters. NF1 gives the time derivative. More explicitly, the first
time derivative of each parameter derivative of total order at most
\(r-1\) is its full chain-rule derivative of \(F(t,u_\eta(t),p)\). Its
right side is a finite continuous sum of the just proved parameter
derivatives and the original field derivatives. Differentiating these
identities in time gives all further mixed time/parameter derivatives of
total order at most \(r\), inductively using the complete product and
chain rules. At a given total order only field derivatives and solution
derivatives of lower total order occur on the right of the time
equation. Their continuity proves the next derivatives and permits
equality of the mixed derivatives by the proved finite-coordinate
calculus. The case of no time derivative and \(r\) parameter derivatives
was already constructed in the path norm. Thus \(u(t;x,p)\) is jointly
\(C^r\), not only separately differentiable.

\subsubsection{17.6. Variable initial times without replacing the
original time
coordinate}\label{variable-initial-times-without-replacing-the-original-time-coordinate}

Use the proved solution with the fixed original \(t_0\). Write
\(U(t;t_0,y,p)\) for it. Near \((t_0,a,p_0)\), the actual finite map

\[
 \mathcal L(\tau,y,p)
   =(\tau,U(\tau;t_0,y,p),p)
 \tag{NF16}
\]

is \(C^r\). At \(\tau=t_0\) its state derivative is the original
identity because \(U(t_0;t_0,y,p)=y\), its state derivative in \(p\) is
zero, and its time derivative is \(F(t_0,y,p)\). Its full derivative is
block triangular with diagonal identities; its inverse keeps the lower
state/time block \(-F(t_0,y,p)\). The proved finite inverse/implicit
theorem therefore constructs its actual local inverse, with second
component \(Y(\tau,x,p)\). The unchanged original-time solution is

\[
 U(t;\tau,x,p)=U(t;t_0,Y(\tau,x,p),p),\qquad
 U(\tau;t_0,Y(\tau,x,p),p)=x .
 \tag{NF17}
\]

It is jointly \(C^r\), solves the original differential equation in the
original \(t\), and has the original datum at the original time
\(\tau\). Uniqueness was proved in Section17.1. Both maps in NF17 are
explicit; no translated time equation or changed vector field is
substituted for NF1.

Let \(M(t)=D_yU(t;t_0,y,p)\). Differentiating the already proved
equation gives \(M'=D_xF(t,U,p)M\), \(M(t_0)=I_E\). NF8 proves that it
is invertible with the constructed opposite-order inverse. Differentiate
the second identity of NF17 and then its first identity to obtain the
full initial-time derivative

\[
 \begin{aligned}
 D_\tau Y&=
  -D_yU(\tau;t_0,Y,p)^{-1}F(\tau,x,p),\\
 D_\tau U(t;\tau,x,p)&=
  -D_yU(t;t_0,Y,p)
   D_yU(\tau;t_0,Y,p)^{-1}F(\tau,x,p)\\
  &=-D_xU(t;\tau,x,p)F(\tau,x,p).
 \end{aligned}
 \tag{NF18}
\]

The last equality follows by differentiating NF17 in \(x\):
\(D_xY=D_yU(\tau;t_0,Y,p)^{-1}\). All original inverse factors and
endpoint evaluations appear before the comparison. Higher initial-time
derivatives are the full finite composition and inverse partition
formulas applied to NF16--NF17; they keep the entire
time/state/parameter blocks.

\subsubsection{17.7. Continuation, the exact composition law and
coordinate
transport}\label{continuation-the-exact-composition-law-and-coordinate-transport}

At every point of a solution, the local construction above applies with
that original point and its original time as datum. Its local
continuations agree by uniqueness. The union of all continuation
intervals is an interval: each contains the original datum time, so two
such intervals overlap; their agreeing solutions glue. This gives the
unique maximal solution on that interval. If two overlapping
continuations are first described through different chains of local
rectangles, the same uniqueness argument on the connected time overlap
makes them equal. No choice of a chart or of a time subdivision changes
the solution.

The full admissible domain of \((t,\tau,x,p)\) is open and the solution
is jointly \(C^r\) there. For an actual trajectory segment from \(\tau\)
to \(t\), its compact graph is covered by finitely many construction
rectangles with smaller time intervals. Subdivide that original time
segment finely enough that each successive piece lies in one such
rectangle: a Lebesgue number follows from compactness by taking finitely
many smaller neighborhoods whose closures remain in the covering
neighborhoods, or by the elementary compact interval subdivision
argument. Each local solution map is defined on an open neighborhood of
its intermediate datum. Continuity of the finite preceding composition
lets one shrink the original data neighborhood so that all intermediate
data stay in these open neighborhoods. The finite composition then
exists near the given endpoint and is \(C^r\), and uniqueness identifies
it with the maximal solution. Slightly moving both endpoint times uses
the first and last rectangles. This proves openness and full local
regularity throughout the actual domain.

Whenever all displayed solutions are defined along their connected
common intervals,

\[
 U(t;s,U(s;\tau,x,p),p)=U(t;\tau,x,p),\qquad
 U(\tau;t,U(t;\tau,x,p),p)=x.
 \tag{NF19}
\]

Both sides of the first identity solve NF1 with the same datum at \(s\);
uniqueness proves it, and the second is its endpoint case. For an
autonomous field write \(\varphi_t(x,p)=U(t;0,x,p)\). NF19 gives the
local flow law \(\varphi_t(\varphi_s(x,p),p)=\varphi_{t+s}(x,p)\), with
their actual domains, and inverse \(\varphi_{-t}\). This is a local
diffeomorphism, since both maps are the constructed smooth inverse maps
on open sets. It is not an assertion that every field has a globally
defined flow.

Under an original smooth coordinate diffeomorphism \(\kappa\), retain
the transformed field
\(\widetilde F(t,\kappa(x),p)=D\kappa(x)F(t,x,p)\). The chain rule
proves that \(\kappa(U(t;\tau,x,p))\) solves its equation with datum
\(\kappa(x)\). Applying uniqueness gives

\[
 \widetilde U(t;\tau,\kappa(x),p)
       =\kappa(U(t;\tau,x,p)).
 \tag{NF20}
\]

This is the exact map between coordinate presentations. For a smooth
vector field on the given manifold, these transformations glue local
chart solutions and all their parameter derivatives. Their domains are
open by the finite-chain proof. For a manifold with boundary, the same
assertion in boundary charts needs an actual smooth extension of the
coefficients to an open coordinate neighborhood; no extension theorem is
hidden in NF20. The collar lesson's stated coefficient extension or an
explicitly given extension supplies that separate condition.

The determinant of the original state variation has all of its factors:

\[
 \det D_xU(t;\tau,x,p)
   =\exp\left(\int_\tau^t
             \operatorname{tr}D_xF(s,U(s;\tau,x,p),p)\,ds\right).
 \tag{NF21}
\]

Its derivative follows from the full cofactor determinant derivative and
the two actual inverse products of NF8: for \(M'=AM\),
\((\det M)'=(\det M)\operatorname{tr}(M^{-1}AM)
=(\det M)\operatorname{tr}A\). The trace equality is the finite sum
\(\sum_{i,j,k}(M^{-1})_{ij}A_{jk}M_{ki}
=\sum_{j,k}A_{jk}\sum_i M_{ki}(M^{-1})_{ij}
=\sum_jA_{jj}\). The scalar integrating-factor proof and the initial
determinant one give NF21. The original determinant, oriented time
integral and original trace are all retained. Positivity of this
determinant is the local orientation statement; no state norm or
coordinate volume has been silently changed.

If the original state dimension is zero, the unique state is its zero
vector, \(F\) is zero and the solution is constant. Every state inverse
is the unique \(I_0\), every empty matrix determinant is one, and NF21
is \(1=\exp(0)\). No positive state comparison constant or division by a
zero operator norm is needed. A zero-dimensional parameter space has no
parameter directions; its derivative terms in the original formulas are
zero. If \(W\) is empty there is no datum and no local assertion.

\subsubsection{17.8. The actual geodesic vector field and smooth
endpoint
map}\label{the-actual-geodesic-vector-field-and-smooth-endpoint-map}

Return to the original positive metric \(H=(g_{ij})=G^{-1}\) and its
original Christoffel coefficients HG2. On the original state space of
coordinate positions and velocities its full vector field is

\[
 \mathcal F(x,v)=
 \left(v,\left(-\sum_{j,k=1}^n
                       \Gamma^i_{jk}(x)v^jv^k\right)_{i=1}^n\right).
 \tag{NG1}
\]

The given original \(G=(g^{ij})\) is real symmetric, smooth and positive
definite, \(H=G^{-1}=(g_{ij})\), and the complete connection formula is

\[
 \Gamma^i_{jk}=\frac12\sum_l g^{il}
             (\partial_jg_{kl}+\partial_kg_{jl}-\partial_lg_{jk}).
                                                                    \tag{NG0}
\]

Use the given position and velocity norms with their complete product
norm and projection constants. This is a smooth finite-coordinate field
by the full inverse-matrix and calculus proofs; no initial metric is
made into an identity. NF1--NF20 now construct and uniquely continue the
actual geodesic \((x(t;v,y),\dot x(t;v,y))\), smoothly in all initial
coordinates and times.

At \((y,0)\) the field vanishes, so the constant position and zero
velocity solve the equation for every real time while remaining in the
original chart. Openness of the full flow domain, over the compact time
interval \([0,1]\) of this constant trajectory, gives one neighborhood
of \((y,0)\) on which solutions exist throughout \([0,1]\). In detail
use finitely many open product neighborhoods of its points; intersect
their initial-data neighborhoods and use uniqueness to identify all
overlapping local solutions. The endpoint \(\gamma(v,y)=x(1;v,y)\) is
therefore jointly smooth near each original zero velocity.

The exact initial-velocity variation at zero solves

\[
 \dot V=W,\qquad \dot W=0,\qquad
 V(0)=0,\quad W(0)=w,\qquad
 V(t)=tw,\quad W(t)=w .
 \tag{NG2}
\]

Every derivative in the quadratic velocity expression in NG1 vanishes at
zero velocity; its position derivative also vanishes there. Thus NG2 is
the actual NF12 variation equation. The initial-position variation is
\((V,W)=(z,0)\). Consequently \(\gamma(0,y)=y\) and the actual original
derivative \(D_v\gamma(0,y)=I\) is proved. Scaling the original curve's
time by \(t\) gives a curve with initial velocity \(tv\) and the same
complete quadratic equation; the chain rule contributes both factors of
\(t\). Uniqueness gives \(x(t;v,y)=\gamma(tv,y)\) whenever those
original curve segments are defined.

Under \(z=\kappa(x)\), the velocity changes by
\(\dot z^a=\sum_i\partial_i\kappa^a\,v^i\). The second derivative
retains both contributions. Hence the transformed Christoffel
coefficients are precisely

\[
 \widetilde\Gamma^a_{bc}(\kappa(x))
 =\sum_{i,j}
  \left(\sum_k\partial_k\kappa^a(x)\Gamma^k_{ij}(x)
                  -\partial_{ij}\kappa^a(x)\right)
   \partial_b(\kappa^{-1})^i(\kappa(x))
   \partial_c(\kappa^{-1})^j(\kappa(x)).
 \tag{NG3}
\]

Both Hessian and original connection terms remain. They agree with HG2
for the actual transported metric
\(\widetilde g_{bc}=\sum_{i,j}g_{ij}\partial_b(\kappa^{-1})^i\partial_c(\kappa^{-1})^j\).
Here is a direct justification. HG2 is symmetric in its two lower
indices and satisfies
\(\partial_k g_{ij}=\sum_\ell g_{\ell j}\Gamma^\ell_{ki}+\sum_\ell g_{i\ell}\Gamma^\ell_{kj}\);
substitution gives the full six-term right side
\(\frac12(\partial_kg_{ij}+\partial_ig_{kj}-\partial_jg_{ki}
+\partial_kg_{ji}+\partial_jg_{ki}-\partial_ig_{kj})\), which equals the
original derivative. Apply the product and chain rules to the full
displayed transported metric: its two differentiated coordinate factors
yield the two Hessian contributions, and its differentiated \(g_{ij}\)
yields the two original connection sums. The identity for the second
derivative of \(\kappa\kappa^{-1}\) changes these Hessian contributions
into the negative Hessian term in NG3. Thus NG3 is symmetric and has the
same metric-compatibility identity for \(\widetilde g\). Conversely any
symmetric compatible coefficients \(C^\ell_{jk}\) satisfy
\(2\sum_\ell g_{i\ell}C^\ell_{jk}
=\partial_jg_{ik}+\partial_kg_{ij}-\partial_ig_{jk}\), by adding the
first two compatibility identities and subtracting the third.
Multiplication by the actual inverse \(G\) gives exactly HG2 and proves
uniqueness. NG3 therefore has the required actual metric formula. NF20
now proves the coordinate-independent geodesic and endpoint maps on the
actual tangent bundle.

For completeness the full differentiated metric and the
inverse-coordinate Hessian comparison used above are, with
\(K^i_b=\partial_b(\kappa^{-1})^i\),

\[
 \begin{aligned}
 \partial_d\widetilde g_{bc}
 &=\sum_{i,j,k}(\partial_kg_{ij})(\kappa^{-1}(z))
                           K^k_dK^i_bK^j_c\\
 &\quad+\sum_{i,j}g_{ij}(\kappa^{-1}(z))
       \left(\partial_{db}(\kappa^{-1})^iK^j_c
                        +K^i_b\partial_{dc}(\kappa^{-1})^j\right),\\
 \sum_k\partial_k\kappa^a(\kappa^{-1}(z))
                           \partial_{bc}(\kappa^{-1})^k
 &=-\sum_{i,j}\partial_{ij}\kappa^a(\kappa^{-1}(z))
                                             K^i_bK^j_c .
 \end{aligned}
 \tag{NG3a}
\]

The second line is the full second derivative of
\(\kappa(\kappa^{-1}(z))=z\). Substitution of the six original
compatibility terms into the first line retains the two connection
products and both Hessian terms; substitution of the second line
supplies precisely NG3. This explicitly gives every coordinate
derivative contribution in the comparison.

\subsubsection{17.9. The unaltered metric identity, all first jets and
minimizing
radius}\label{the-unaltered-metric-identity-all-first-jets-and-minimizing-radius}

The finite inverse theorem applies to \((v,y)\mapsto(\gamma(v,y),y)\)
with its actual derivative. Near every \((0,y)\) it gives the full
smooth inverse. In a local base neighborhood a positive velocity radius
works uniformly after shrinking; all needed existence, injectivity and
derivative bounds are open conditions on a compact smaller base
neighborhood. Use the actual continuous metric norm
\(r_y(v)=(v^tH(y)v)^{1/2}\) to choose the velocity balls. On the smaller
base neighborhood its original coordinate norm comparison constants
bound both inclusions, so a sufficiently small metric ball fits the
original inverse neighborhood.

These local radii have a locally positive lower bound after taking a
locally finite cover by smaller base neighborhoods and assigning each
its positive radius. The original smooth positive-minorant proof MG7
gives a smaller smooth positive function \(R(y)\) below these allowed
radii. Its support assignment and all original metric factors are
retained. The map on \(r_y(v)<R(y)\) is injective in each fiber by its
local inverse, and points over different centers have different second
coordinates. It is a local diffeomorphism and injective, hence has an
open image and a smooth inverse: its locally defined inverses agree on
every overlap by injectivity. This supplies a neighborhood of the zero
section and a neighborhood of the diagonal without assuming a uniform
global injectivity radius. For compact centers the positive smooth
radius has a positive minimum, and finitely many original chart bounds
give the stated uniform constants.

Write \(\widetilde H(v,y)=(D_v\gamma)^tH(\gamma(v,y))D_v\gamma\). Its
actual center value is \(H(y)\). Metric compatibility, proved above,
gives constant geodesic energy; torsion freeness gives
\(\nabla_t\partial_s q=\nabla_s\partial_tq\) for a velocity variation
\(q(t,s)=x(t;v+sw,y)\). The two covariant derivatives have equal
mixed-coordinate derivatives, and their original Christoffel sums agree
because the lower indices are symmetric. Keeping both product terms
therefore gives

\[
 \begin{aligned}
 \frac d{dt}\langle\dot q,\partial_s q\rangle_H
 &=\langle\nabla_t\dot q,\partial_s q\rangle_H
       +\langle\dot q,\nabla_t\partial_s q\rangle_H\\
 &=\tfrac12\partial_s\langle\dot q,\dot q\rangle_H
       =\langle v,w\rangle_{H(y)},\\
 \langle D_v\gamma(v,y)v,D_v\gamma(v,y)w\rangle_H
 &=\langle v,w\rangle_{H(y)},\\
 \sum_k\widetilde g_{jk}(v,y)v^k&=\sum_k g_{jk}(y)v^k.
 \end{aligned}
 \tag{NG4}
\]

The first term vanishes by the actual geodesic equation, the energy is
the original initial energy \((v+sw)^tH(y)(v+sw)\), and the initial
varied position is zero. Integration on the original time interval
\([0,1]\) gives the second line. No orthonormal frame or rescaled
velocity enters this identity.

Let \(A_{ijk}=\partial_k\widetilde g_{ij}(0,y)\). Symmetry and the
quadratic terms of the last line in NG4 give \(A_{ijk}=A_{jik}\) and
\(A_{ijk}=-A_{ikj}\). The full sequence
\(A_{ijk}=-A_{ikj}=-A_{kij}=A_{kji}=A_{jki}=-A_{jik}=-A_{ijk}\) makes
every coefficient zero. The complete integral Taylor formula is
therefore

\[
 \begin{aligned}
 \widetilde g_{ij}(v,y)-g_{ij}(y)
 &=\sum_{k,l}v^kv^l\int_0^1
            (1-t)\partial_{kl}\widetilde g_{ij}(tv,y)\,dt,\\
 \widetilde G-G(y)
 &=-G(y)(\widetilde H-H(y))\widetilde G,\\
 r_y(v)^2&=v^tH(y)v,\qquad
 \operatorname{grad}_{\widetilde H}r_y(v)=v/r_y(v)
                         \quad(v\ne0).
 \end{aligned}
 \tag{NG5}
\]

The inverse comparison follows by multiplying both original inverse
identities; its full left and right factors remain. Every finite
collection of center derivatives of the first line has the same entire
integral with its differentiated coefficients; compact-center bounds
follow from smoothness on the actual compact charts. Its first velocity
derivative is \(O(|v|)\), and the entire matrix differences are
\(O(|v|^2)\), with the original norm comparison constants. The gradient
identity follows by multiplying \(dr_y=H(y)v/r_y\) by the actual
\(\widetilde G\) and using NG4. Its squared length is
\(v^tH(y)v/r_y^2=1\).

Choose \(0<R_1<R_0<R_2\) whose closed original metric ball of radius
\(R_2\) lies in the inverse neighborhood at the given center. Its closed
image is compact and the image of \(r_y<R_0\) has boundary exactly the
image of \(r_y=R_0\): the diffeomorphism on the larger ball preserves
relative interiors and boundaries, and compactness excludes an extra
boundary point outside that larger image. The radial curve from the
center to \(v\) has length \(r_y(v)\). Any piecewise \(C^1\) curve
remaining in this normal image and ending at \(v\) has length at least
\(r_y(v)\), since away from zero \(|(r_y\circ c)'|\leq\|\dot c\|_H\). To
handle the center without assuming differentiability of the radius
there, fix \(0<\epsilon<r_y(v)\), take the last crossing of radius
\(\epsilon\) before the endpoint, integrate the inequality on the
following segment, and let \(\epsilon\) decrease to zero. If there are
further zero-radius points, choose that last crossing; compactness of
the time interval guarantees it. The inequality then holds on the whole
remaining positive-radius segment.

If a curve to a point with \(r_y(v)<R_1\) leaves the outer image
\(r_y<R_0\), its segment to its first boundary crossing has length at
least \(R_0\), by approaching that crossing from inside and using the
same radius argument. It is strictly longer than the retained radial
curve. Thus curves which leave and return cannot lower the infimum. The
actual Riemannian distance and square are

\[
 s(\gamma(v,y),y)=r_y(v),\qquad
 s(\gamma(v,y),y)^2=v^tH(y)v
                     \quad(r_y(v)<R_1).
 \tag{NG6}
\]

The radii can again be chosen locally uniformly in the center and
reduced by MG7's positive minorant; compact-center subsets have uniform
positive lower radii. The inverse endpoint map is jointly smooth, so the
square is jointly smooth near the diagonal and the positive square root
is smooth off the diagonal. No smoothness of the distance itself at the
diagonal is claimed.

\subsubsection{17.10. Every transformed divergence and density
contribution}\label{every-transformed-divergence-and-density-contribution}

Keep the whole original coordinate operator and its distinct measure:

\[
 P=-\sum_{j,k=1}^n\partial_j\bigl(g^{jk}(x)\partial_k\bigr)I_r
       +\sum_{j=1}^n b^j(x)\partial_j+c(x).                    \tag{NG7a}
\]

\[
                   d\mu(x)=\rho(x)\,dx,
 \qquad \rho(x)=(\det G(x))^{-1/2}.                            \tag{NG7b}
\]

Here \(b^j,c\) are smooth complex matrices acting on the left. Their
original rank and every original coordinate sum remain. These data
define the receiving operator independently of an elliptic or geodesic
conclusion.

For the actual original operator H1 keep \(x=\gamma(v,y)\),
\(C=D_v\gamma\), \(J=|\det C|>0\). The original coordinate measure is
\(dx=J\,dv\); the full Riemannian measure is \(\rho(\gamma(v,y))J\,dv\).
Its density is not identified with coordinate measure or suppressed. The
principal and lower-order coefficient transports are

\[
 \begin{aligned}
 \widetilde G&=C^{-1}G(\gamma(v,y))C^{-t},\\
 \widetilde b^{\,i}
 &=\sum_j(C^{-1})_{ij}b^j(\gamma(v,y))
      -\sum_k\widetilde g^{ik}J^{-1}\partial_kJ\,I_r,\\
 \widetilde c&=c(\gamma(v,y)).
 \end{aligned}
 \tag{NG7}
\]

To prove the divergence law, pair the original divergence against a
smooth compact test and integrate by parts. The complete two gradient
transforms are \(C^{-t}\partial_v\) and the full original substitution
factor is \(J\). Their product gives
\(\int J\,(\partial_v\psi)^t\widetilde G\,\partial_v f\,dv\).
Integration by parts in \(v\) identifies the transported divergence as
\(J^{-1}\sum_{i,k}\partial_i(J\widetilde g^{ik}\partial_k f)\). This
equality of test integrals is equality of smooth coefficients, because a
nonzero continuous difference has a compact test of the same sign or a
complex component with nonzero pairing. Expanding the complete
derivative retains both terms
\(\sum_{i,k}\partial_i(\widetilde g^{ik}\partial_kf)
+\sum_{i,k}J^{-1}\partial_iJ\,\widetilde g^{ik}\partial_kf\). The
original minus sign in H1 and symmetry of \(\widetilde G\) yield
precisely the second line of NG7, with
\(J^{-1}\partial_kJ=\partial_k\log J\). The first-order original
matrices transform by the full state chain rule and continue to act on
the left; the zeroth-order matrices are the stated compositions. Since
the actual \(C(0,y)=I\), \(J(0,y)=1\). Smoothness and every compact
parameter bound follow from the proved endpoint map, full
inverse/cofactor formulas and the positive determinant. This proves HG10
with every Jacobian contribution, rather than identifying the original
coordinate operator with a differently measured Laplacian.

\subsubsection{17.11. Exact course use and its remaining
interfaces}\label{exact-course-use-and-its-remaining-interfaces}

NF1--NF21 prove local existence, uniqueness, maximal continuation on its
actual domain, jointly \(C^r\) parameter and initial-time dependence,
every labeled higher variation, actual flow inverses, the ordered linear
equation and full variation-of-constants maps. NG1--NG7 supply the
original geodesic equation's dependence, endpoint derivative, unaltered
normal-coordinate metric, first jets, true local minimizing radius and
complete transformed drift and density. Their independently proved
inverse and partition inputs are the actual earlier geometry/calculus
providers.

The elementary scalar ray integral HS2--HS8 and matrix series HM2--HM7
remain their original proofs. Their center and parameter derivatives
have the uniform compact bounds proved there; NF6--NF15 now give an
additional general original-time linear provider. Bundle frame
covariance of an amplitude requires the exact transformed operator and
its normalized transport equation, not only existence of a smooth bundle
or of a curve. Any remaining such receiving calculation must be checked
in its actual matrix convention. The nonlinear flow entry also supplies
the stated local base used by the collar construction; the collar's
global injectivity, coefficient extension and boundary support arguments
remain their own full assertions.

Complex Cauchy/residue/maximum principles, Laplace/Euler continuations,
wavefront-qualified arbitrary-map pullbacks and source-formula
comparisons are separate interfaces. This standard foundation proof
makes no claim to establish those assertions.

\subsection{References}\label{references}

The coordinate amplitude construction is the Kuranishi change of phase
variables discussed in
\href{https://web.math.ku.dk/~grubb/dist8n.pdf}{Grubb's author-hosted
Chapter 8}; the separate roles of operator microsupport and manifold
assembly can also be compared in
\href{https://math.mit.edu/~rbm/iml/Chapter5.pdf}{Melrose's
microlocalization chapter} and
\href{https://math.mit.edu/~rbm/iml/Chapter6.pdf}{manifold chapter}.

\subsection{Further questions}\label{further-questions}

The proofs suggest two further research directions. Quantizing
bundle-valued principal symbols with extra geometric structure requires
tracking the frame correction omitted by the scalar half-density
refinement. Studying propagation beyond elliptic directions requires
information about the characteristic symbol that is absent from the
inequality (G27). Those are separate problems; neither is inferred from
elliptic inversion alone.

\subsection{18. Original wavefront and pullback
foundations}\label{original-wavefront-and-pullback-foundations}

This section proves the actual coordinate, product, qualified smooth-map
and submersion pullbacks and their complete wavefront relations. It
supplies strong smooth parameter dependence and differential elliptic
regularity with the full original symbol, then compares the whole
odd-wave kernel at nonzero cone points and its vertex. The test topology
and scalar operations are proved in Sections13.1--13.6,
\href{prerequisite-bridges.pdf}{the original Fourier inverses} retain
their factors, and \href{conormal-transmission.pdf}{the nonlinear
completed measure theorem} is proved in CX35a--c.~The finite-symbol and
conic operator proofs in Sections1/5/6/8 and
\href{euclidean-symbol-calculus.pdf}{the Euclidean calculus} are used on
their stated domains. The exact original wave formulas are retained from
\href{wave-hadamard-kernels.pdf}{the wave kernel chapter}.

\subsubsection{18.1. The original Fourier estimate and every
cutoff}\label{the-original-fourier-estimate-and-every-cutoff}

Use the original convention

\[
\widehat f(\xi)=\int_{\mathbb R^d}e^{-ix\cdot\xi}f(x)\,dx,\qquad
f(x)=(2\pi)^{-d}\int_{\mathbb R^d}e^{ix\cdot\xi}\widehat f(\xi)\,d\xi.
\tag{WF1}
\]

For a compactly supported distribution \(v\), its Fourier transform is
\(v(e^{-ix\cdot\xi})\), with a compact smooth cutoff equal to one on its
support inserted in that pairing. The full finite-order test estimate
and product rule give \(|\widehat v(\xi)|\leq C(1+|\xi|)^m\) for some
actual integer \(m\). Every frequency derivative has the same type of
estimate, with its original multiplier \((-ix)^\alpha\) in the test. The
estimate keeps the derivatives of the inserted support cutoff. A compact
smooth test has rapidly decreasing transform by repeated integration by
parts with \(1-\Delta_x\), whose full multiplier is \(1+|\xi|^2\). These
facts follow on the original coordinates.

For \(b\in C_c^\infty\), inverse Fourier transformation of the test and
the finite-order bound justify

\[
\widehat{bv}(\xi)=(2\pi)^{-d}
       \int_{\mathbb R^d}\widehat b(\xi-\eta)\widehat v(\eta)\,d\eta.
\tag{WF2}
\]

The integral is absolutely convergent: take the Schwartz decrease of
\(\widehat b\) beyond the original polynomial order plus \(d\). It also
converges in every test seminorm needed in the pairing, so the
distribution can be passed through the integral. No Fourier multiplier
constant is suppressed.

Suppose \(\widehat v\) decreases rapidly in an open cone \(V\), and the
angular closure of a smaller cone \(W\) lies in \(V\). For
\(\xi\in W,\eta\notin V\), angular separation gives
\(|\xi-\eta|\geq c(|\xi|+|\eta|)\) for some \(c>0\). To prove this,
restrict to \(|\xi|+|\eta|=1\); the two closed angular sets are
disjoint, so the continuous difference has a positive minimum. Scaling
restores the original lengths. Split WF2 at \(\eta\in V\). On that
region both original transforms decrease rapidly, and
\(1+|\xi|\leq(1+|\xi-\eta|)(1+|\eta|)\). Taking their exponents larger
than \(N+d+1\) gives the requested output power \((1+|\xi|)^{-N}\) and
an integrable residual power in \(\eta\). On the complement, take the
cutoff exponent larger than \(N+m+d+1\); angular separation beats the
full polynomial input bound. Thus multiplication preserves rapid
decrease on \(W\).

Consequently the definition of \(\operatorname{WF}(u)\) using a cutoff
equal to one near the original base point is unchanged if the cutoff is
only nonzero there: multiply by its actual smooth reciprocal on a
smaller neighborhood and use WF2. It is local. Its complement is open in
the nonzero cotangent variables by the same smaller-neighborhood and
smaller-cone construction, so the wavefront set is closed and conic. If
all directions are regular over a compactly localized neighborhood, a
finite cover of the original unit sphere and the cutoff estimate give
rapid decrease everywhere. WF1 then proves smoothness with every
derivative. Conversely compactly localized smooth functions have that
decrease, so absence of all wavefront directions is exactly smoothness.

\subsubsection{18.2. A full nonstationary integral
bound}\label{a-full-nonstationary-integral-bound}

Let \(K\) be a fixed compact coordinate set and
\(I(\xi,\eta)=\int a(x)e^{i\Phi(x,\xi,\eta)}dx\), with \(a\) smooth and
supported in \(K\). In each application below the original phase is
linear in the frequency parameters, and its coordinate derivatives on
\(K\) satisfy \(|\partial_x^\alpha\Phi|\leq C_\alpha(|\xi|+|\eta|)\).
Suppose its actual gradient satisfies
\(|\nabla_x\Phi|\geq c(|\xi|+|\eta|)\) on \(K\), for
\(|\xi|+|\eta|\geq1\). Retain the entire operators

\[
\begin{aligned}
L&=\sum_j\frac{\partial_{x_j}\Phi}{i|\nabla_x\Phi|^2}\partial_{x_j},
&Le^{i\Phi}&=e^{i\Phi},\\
L^ta&=-\sum_j\partial_{x_j}
       \left(\frac{\partial_{x_j}\Phi}{i|\nabla_x\Phi|^2}a\right),
&I(\xi,\eta)&=\int (L^t)^Na(x)e^{i\Phi(x,\xi,\eta)}dx .
\end{aligned}
\tag{WF3}
\]

The superscript \(t\) denotes the bilinear integration transpose, not a
conjugate adjoint. Boundary terms vanish because the original amplitude
is compactly supported. Every derivative of each coefficient of \(L\) is
bounded by \(C_\alpha(|\xi|+|\eta|)^{-1}\): the numerator has one
frequency power, the denominator has two, and each differentiated
quotient has the same net power, with the positive lower gradient bound
retained. The full iterated product and chain rules in the calculus
chapter give a finite sum at every \(N\), keeping every differentiated
coefficient and amplitude. They therefore prove

\[
|I(\xi,\eta)|\leq C_N(1+|\xi|+|\eta|)^{-N}
                \max_{|\alpha|\leq N}\|\partial^\alpha a\|_\infty
\tag{WF4}
\]

on the separated region, enlarging \(C_N\) for bounded frequencies.
Every fixed additional derivative of the amplitude or phase parameters
has the corresponding estimate after increasing the number of
integrations; its original polynomial frequency factors must be included
before choosing \(N\). The constants depend on the actual compact set,
gradient margin and finitely many original derivatives, rather than a
replaced phase.

\subsubsection{18.3. The exact diffeomorphism covector
map}\label{the-exact-diffeomorphism-covector-map}

Let \(\kappa:X\to Y\) be a smooth diffeomorphism between coordinate
opens of dimension \(d\), and put \(\phi=\kappa^{-1}\). The already
constructed scalar pullback retains its full test Jacobian:

\[
\langle\kappa^*u,f\rangle
 =\langle u,(f\circ\phi)|\det D\phi|\rangle.
\tag{WF5}
\]

Fix \(x_0\), \(y_0=\kappa(x_0)\), and \(\xi_0\ne0\), and put
\(\eta_0=D\phi(y_0)^T\xi_0\). Suppose \((y_0,\eta_0)\) is regular for
\(u\). Choose an original compact cutoff \(\chi\) equal to one on a
neighborhood of \(y_0\) so that \(v=\chi u\) has rapidly decreasing
transform on a cone \(V\) about \(\eta_0\). Shrink an \(x\)-cutoff \(b\)
about \(x_0\) until its image lies in that neighborhood. Its transform
after pullback is exactly

\[
\widehat{b\kappa^*u}(\xi)
=(2\pi)^{-d}\int\widehat v(\eta)I(\xi,\eta)\,d\eta,\qquad
I(\xi,\eta)=\int b(\phi(y))|\det D\phi(y)|
        e^{i(y\cdot\eta-\phi(y)\cdot\xi)}dy.
\tag{WF6}
\]

For each fixed \(\xi\), this is the Fourier inverse formula applied to a
compact smooth test, and is absolutely convergent at large \(\eta\) by
WF3. Its phase gradient is the original \(\eta-D\phi(y)^T\xi\). On the
compact support, both \(D\phi^T\) and its inverse have uniform finite
norm bounds. Thus when \(|\eta|\) is below a sufficiently small multiple
of \(|\xi|\), or above a sufficiently large multiple, the gradient is
bounded below by \(c(|\xi|+|\eta|)\). For the remaining comparable
lengths, continuity of \(D\phi\) and sufficiently small base/direction
neighborhoods place \(D\phi(y)^T\xi\) in an angular cone whose closure
lies in \(V\). If \(\eta\notin V\), compact angular separation gives the
same lower bound. This compactness argument retains the full derivative
matrix and its original covectors.

On those separated regions WF4 beats the polynomial bound for
\(\widehat v\), with \(N\) larger than that order, the requested decay
exponent and \(d\). On the remaining region \(\eta\in V\) and \(|\eta|\)
comparable with \(|\xi|\), use the rapid input decrease and the original
constant bound \(|I|\leq\int|b(\phi(y))||\det D\phi(y)|dy\). The
integration volume is at most \(C(1+|\xi|)^d\); choosing the input decay
exponent larger than that volume power and the requested output exponent
proves rapid output decrease. Hence regularity at \((y_0,\eta_0)\)
implies regularity at \((x_0,\xi_0)\). Applying this proved implication
to the actual inverse diffeomorphism and using WF5's composition law
proves both inclusions, giving the exact equality

\[
\operatorname{WF}(\kappa^*u)
 =\{(x,D\kappa(x)^T\eta):(\kappa(x),\eta)\in\operatorname{WF}(u)\}.
\tag{WF7}
\]

Invertibility keeps every displayed output covector nonzero. Both
original Jacobians remain in WF5--WF6; they do not change the covector
relation.

\subsubsection{18.4. Products and the exact projection
converse}\label{products-and-the-exact-projection-converse}

For compact localizations \(v\) on \(\mathbb R^a\) and \(w\) on
\(\mathbb R^b\), their already constructed tensor product has

\[
\widehat{v\otimes w}(\xi,\eta)=\widehat v(\xi)\widehat w(\eta).
\tag{WF8}
\]

Here the order of the two factors is input-coordinate order, as
indicated by the variables; converting to the geometric chapter's
output-then-input notation requires that actual permutation. The
equality follows by its two original iterated test pairings and the
factored exponential.

Write \(Z_u=\{(x,0):x\in\operatorname{supp}u\}\). Then

\[
\operatorname{WF}(u\otimes v)
\subset
\bigl((\operatorname{WF}(u)\cup Z_u)
      \times(\operatorname{WF}(v)\cup Z_v)\bigr)
          \setminus\{(\xi,\eta)=(0,0)\}.
\tag{WF9}
\]

Outside the product support the tensor distribution vanishes, by the
proved exact support formula. At a point inside it but outside the
displayed covector set, at least one nonzero component is a regular
direction for its factor. Near that pair direction its length is at
least a fixed positive fraction of the total frequency length. Its
factor in WF8 decreases to every order, while the other has its original
polynomial bound. Their product therefore decreases to every order in
that product cone. Product spatial cutoffs are equal to one near the
base point, so this proves the claimed inclusion with the correct
original Fourier definition. WF2 then handles all smaller nonproduct
cutoffs. No general equality for two arbitrary singular factors is
asserted.

For the original projection \(\pi(x,y)=x\), define \(\pi^*u=u\otimes1\)
by \(\langle\pi^*u,f\rangle=\langle u,x\mapsto\int f(x,y)dy\rangle\).
The integral test is smooth with compact support and every
\(x\)-derivative is its actual integrated derivative, so this is a
continuous distribution. The constant factor is smooth, and WF9 gives
one inclusion in

\[
\operatorname{WF}(\pi^*u)
 =\{(x,y;\xi,0):(x,\xi)\in\operatorname{WF}(u)\}.
\tag{WF10}
\]

For the converse suppose \(\pi^*u\) is regular at \((x_0,y_0;\xi_0,0)\).
By WF2 choose product cutoffs \(\alpha(x),\beta(y)\) supported in that
regular base neighborhood, both equal to one near their respective
points, with \(\beta\geq0\) and \(\int\beta>0\). Their full transform is
\(\widehat{\alpha u}(\xi)\widehat\beta(\eta)\). The regular cone
contains \((\xi,0)\) for all \(\xi\) in a smaller cone about \(\xi_0\).
Its restriction at \(\eta=0\) is the original nonzero scalar
\((\int\beta)\widehat{\alpha u}(\xi)\). Division by that scalar proves
rapid decrease of the original factor, contradicting singularity at
\((x_0,\xi_0)\). This proves the exact converse, not only the forward
inclusion.

\subsubsection{18.5. Constructing the pullback for the original smooth
map}\label{constructing-the-pullback-for-the-original-smooth-map}

Let \(F:X\subset\mathbb R^b\to Y\subset\mathbb R^a\) be smooth. Its
precise normal set and the required condition are

\[
N_F=\{(F(x),\eta):\eta\ne0,\ DF(x)^T\eta=0\},
\qquad N_F\cap\operatorname{WF}(u)=\varnothing.
\tag{WF11}
\]

We construct the map under this original transversality condition; no
pullback at a forbidden covector is assumed. Near any fixed \(x_0\), the
unit directions in the kernel of \(DF(x_0)^T\) form a compact set,
possibly empty. WF11 makes every such direction regular at \(F(x_0)\). A
finite regular cone cover, a common sufficiently small target cutoff
\(\chi\), and WF2 give an open cone \(V_{\rm reg}\) on which
\(\widehat{\chi u}\) decreases to every order, containing an angular
neighborhood of those kernel directions. On the compact complementary
unit directions \(DF(x_0)^T\eta\) has a positive lower norm. Continuity
allows a fixed original neighborhood \(U_0\) of \(x_0\) and a constant
\(c>0\) with \(|DF(x)^T\eta|\geq c|\eta|\) there for every complementary
direction. Shrink \(U_0\) so its image lies in the region where
\(\chi=1\).

For \(f\in C_c^\infty(U_0)\), put \(v=\chi u\) and define

\[
\langle F^*u,f\rangle
=(2\pi)^{-a}\int_{\mathbb R^a}\widehat v(\eta)
                   \left(\int_{U_0}f(x)e^{iF(x)\cdot\eta}dx\right)d\eta.
\tag{WF12}
\]

On \(V_{\rm reg}\), the rapid input decrease and \(\int|f|\) give an
integrable bound. On its complement, the actual phase gradient is
\(DF(x)^T\eta\), so WF4 gives a bound
\(C_N(1+|\eta|)^{-N}\max_{|\alpha|\leq N}\|\partial^\alpha f\|_\infty\).
Multiplying by the original polynomial bound for \(\widehat v\) and
choosing \(N>m+a\) proves absolute convergence. It also proves a
continuous finite-order distribution estimate on each original compact
test support, with its complete coordinate and map derivative constants.
For smooth \(u\), WF1 and absolute testing show that this is exactly the
smooth scalar function \(u(F(x))\).

Here is the needed independence and locality, rather than an assumed
gluing rule. Use the original Cartesian Gaussian regularization \[
g_\epsilon(y)=(4\pi\epsilon)^{-a/2}
      e^{-|y|^2/(4\epsilon)},\qquad
v_\epsilon=g_\epsilon*v,\qquad
\widehat v_\epsilon(\eta)=e^{-\epsilon|\eta|^2}\widehat v(\eta).
\tag{WF13}
\] Its full mass and Fourier identity are proved in the
integration/Fourier chapters. The two absolute bounds for WF12 dominate
uniformly for \(0<\epsilon\leq1\), since
\(0<e^{-\epsilon|\eta|^2}\leq1\). Thus WF12 is the limit of the actual
smooth pullbacks \(v_\epsilon\circ F\).

If two target cutoffs equal one near \(F(\operatorname{supp}f)\), their
difference times \(u\) is a compact distribution \(w\) vanishing near
that compact image. The distance between that image and
\(\operatorname{supp}w\) is positive. Its original compact finite-order
estimate, applied to every derivative of \(g_\epsilon(F(x)-y)\), shows
uniform convergence to zero on that image, including every fixed
\(x\)-derivative: the full differentiated Gaussian has finitely many
polynomial factors in the original difference coordinates and inverse
powers of \(\epsilon\), times the same exponential. For distance at
least \(d>0\), each such factor is bounded by
\(C\epsilon^{-M}e^{-d^2/(8\epsilon)}\), tending to zero. The inserted
support cutoff derivatives also remain in this estimate. Consequently
the two limits agree on \(f\).

The same argument handles restriction to smaller original neighborhoods.
For a compact test crossing several \(U_0\)'s, choose a finite smooth
partition subordinate to them and sum WF12. On an overlap the preceding
common Gaussian comparison gives equality; refinements therefore have
the same sum. This constructs a unique local distribution \(F^*u\) on
all of \(X\) with these formulas, preserving the original map. It proves
its locality and compatibility with restriction. At a diffeomorphism,
the smooth Gaussian approximations and the full measure substitution
show agreement with WF5, including its actual inverse Jacobian. At the
projection the same approximations and the integrated test show
agreement with WF10's construction.

Its precise wavefront bound is

\[
\operatorname{WF}(F^*u)
\subset\{(x,DF(x)^T\eta):(F(x),\eta)\in\operatorname{WF}(u)\}.
\tag{WF14}
\]

For completeness fix a nonzero \(\xi_0\) outside the displayed set over
\(x_0\). The singular unit target directions over \(F(x_0)\) are
compact. WF11 says their images under \(DF(x_0)^T\) never vanish, and
the chosen output direction is different from every resulting direction.
Compactness yields positive margins for both assertions. Choose a closed
angular neighborhood \(\Gamma\) of that singular set retaining these
margins. Its complementary directions have a finite regular cone cover;
WF2 again chooses a common target cutoff so that \(\widehat v\)
decreases to every order outside \(\Gamma\). Shrink both original base
and output direction neighborhoods so the margins hold for all \(x\) in
the support of a cutoff \(b\) and every allowed output \(\xi\). If the
singular set is empty, local smoothness already proved in Section18.1
supplies the conclusion directly.

The localized output transform is WF12 with
\(f(x)=b(x)e^{-ix\cdot\xi}\). Its inner phase is exactly
\(F(x)\cdot\eta-x\cdot\xi\), with gradient \(DF(x)^T\eta-\xi\). For
\(\eta\in\Gamma\), the two compact margins give
\(|DF(x)^T\eta-\xi|\geq c(|\eta|+|\xi|)\): normalize their combined
lengths to one; if the continuous difference had zero minimum, the
covectors would have the excluded common direction, or a nonzero
singular target direction would map to zero. WF4 therefore beats the
full polynomial input bound and gives every output inverse power after
integration in \(\eta\).

For \(\eta\notin\Gamma\), use the rapidly decreasing input. Split at
\(|\eta|\leq c_0|\xi|\), with \(c_0>0\) smaller than the reciprocal of
twice the actual compact bound for \(\|DF^T\|\). On that region the
gradient has norm at least \(|\xi|/2\), so WF4 and the rapid input bound
give every output power. On the other region \(|\eta|>c_0|\xi|\), the
original bound for the inner integral is \(\int|b|\); taking the input
decay exponent larger than the requested output exponent plus \(a+1\)
gives every output power after integrating the full tail. This proves
WF14 with the original Fourier constant still present in WF12. It proves
a bound, and does not claim a converse for an arbitrary smooth map.

For a submersion \(F\), the normal set is empty, so the construction
applies to every \(u\). Retain a nonzero \(a\)-row derivative minor at
\(x_0\) and complete \(F\)'s coordinates by the remaining original
\(x\)-coordinates. The resulting map \(\kappa(x)=(F(x),x_J)\) has its
actual nonzero determinant. The proved inverse/implicit theorem gives a
local diffeomorphism; in these coordinates \(F=\pi\circ\kappa\).
Gaussian comparison in WF12 proves \(F^*u=\kappa^*(\pi^*u)\). WF7 and
the exact WF10 converse therefore give

\[
\operatorname{WF}(F^*u)
 =\{(x,DF(x)^T\eta):(F(x),\eta)\in\operatorname{WF}(u)\}
 \quad\text{for a submersion}.
\tag{WF15}
\]

The derivative identity is the original chain rule
\(DF^T=D\kappa^T D\pi^T\). Its injectivity on target covectors is
precisely full row rank, so every right-hand covector is nonzero. Every
dimension-zero case has its point mass and empty Fourier factor; when
the target has dimension zero, its distributions are smooth constants
and both wavefront sets are empty.

\subsubsection{18.6. Absence of parameter-normal covectors gives the
actual smooth
family}\label{absence-of-parameter-normal-covectors-gives-the-actual-smooth-family}

Let \(u\) be a distribution on an open product in the original variables
\((t,x)\in\mathbb R^p\times\mathbb R^q\), with no wavefront covector
\((t,x;\tau,0)\), \(\tau\ne0\). Every slice embedding \(j_t(x)=(t,x)\)
satisfies WF11, because its transpose sends \((\tau,\xi)\) to \(\xi\).
Thus the already constructed restriction \(u_t=j_t^*u\) exists.

Fix compact spatial test support and a small compact parameter
neighborhood. The compact base set and the compact unit sphere in the
original \(\tau\)-coordinates have a finite regular cone/base cover. Use
its finite spatial partition and common smaller parameter neighborhood.
After each compact localization \(v\), WF2 gives a constant \(\delta>0\)
such that \(\widehat v(\tau,\xi)\) decreases to every order when
\(|\xi|\leq\delta|\tau|\); on the whole space it has the original
polynomial bound \(C(1+|\tau|+|\xi|)^m\). Summing the finitely many
localized formulas retains all their cutoff derivatives.

For a spatial test \(\psi\), the actual pairing on the neighborhood
where the parameter cutoff is one is

\[
\langle u_t,\psi\rangle
=(2\pi)^{-(p+q)}
 \iint e^{it\cdot\tau}\widehat v(\tau,\xi)\widehat\psi(-\xi)
                           \,d\xi\,d\tau,\qquad
\partial_t^\alpha e^{it\cdot\tau}=(i\tau)^\alpha e^{it\cdot\tau}.
\tag{WF16}
\]

In \(|\xi|\leq\delta|\tau|\), the rapid decrease of \(\widehat v\)
dominates every original factor \(\tau^\alpha\) and both integration
dimensions, while \(|\widehat\psi|\leq\int|\psi|\). In the complementary
region, \(|\tau|\leq|\xi|/\delta\). The polynomial bound for \(v\),
multiplied by the Schwartz bound \[
|\widehat\psi(\xi)|\leq
(1+|\xi|^2)^{-N}
\int |(1-\Delta_x)^N\psi(x)|dx,
\tag{WF17}
\] is integrable after both integrations when \(2N>m+|\alpha|+p+q\). The
integral in WF17 is bounded by finitely many original test derivative
seminorms on the fixed support, with every binomial and derivative from
\((1-\Delta)^N\) retained in that full operator. These bounds are
uniform on bounded sets of tests. For each derivative order, take a
corresponding \(N\); no one fixed finite test order for all parameter
derivatives is inferred.

Dominated differentiation therefore gives every derivative in WF16.
Taylor's remainder for \(e^{ih\cdot\tau}\), bounded by a constant times
\(|h|^2|\tau|^2\), and the same bounds with two additional frequency
powers prove convergence of difference quotients uniformly on every
bounded set of spatial tests. Continuity uses the same dominated bounds.
This is precisely smoothness into the strong distribution topology,
whose seminorms are suprema of pairings over bounded test sets. It is
stronger than only the smoothness of each single pairing.

To verify that WF16 is the original pullback, apply WF13 to \(v\), then
restrict the smooth Gaussian approximations. Their transforms acquire
the full factor \(e^{-\epsilon(|\tau|^2+|\xi|^2)}\), bounded by one. The
just proved absolute envelopes allow its limit and every fixed parameter
derivative in WF16, agreeing with WF12 for the slice map. Finally
testing against any original compact smooth \(\varphi(t,x)\) and
applying these absolute bounds permits integration in \(t\), and
recovers the original distribution pairing by WF1. Thus the family
reconstructs \(u\), rather than an unrelated family with the same
notation. Finite localization and restriction compatibility paste these
actual families on the whole parameter domain. For \(p=0\) there is no
parameter derivative; for \(q=0\) the no-normal condition is absence of
all nonzero covectors and Section18.1 proves the required scalar
smoothness.

\subsubsection{18.7. Differential elliptic regularity with the entire
original
symbol}\label{differential-elliptic-regularity-with-the-entire-original-symbol}

The actual finite-symbol composition and its complete remainder are
proved in the Euclidean chapter, E19--E22. Local support-controlled
summation and conic separation are proved in the geometric chapter,
Sections1/5/6, and its G24 proves that a proper operator has
\(\operatorname{WF}(Av)\subset\operatorname{WF}(A)\cap\operatorname{WF}(v)\).
We use these proved maps on their actual domains. In particular the
composition product below is its full oscillatory product, rather than
multiplication of a leading term. The positive remainder gain used here
is one, for the original class \(S^m_{1,0}\).

In fixed original coordinates and frames let the actual differential
operator between bundles of the same finite rank be

\[
\begin{aligned}
P(x,D)&=\sum_{|\alpha|\leq m}A_\alpha(x)D^\alpha,
&D_j&=-i\partial_{x_j},\\
a(x,\xi)&=\sum_{|\alpha|\leq m}A_\alpha(x)\xi^\alpha,
&p_m(x,\xi)&=\sum_{|\alpha|=m}A_\alpha(x)\xi^\alpha .
\end{aligned}
\tag{WF18}
\]

Here \(m\) is a nonnegative integer. Every coefficient and every
lower-order term remains in \(a\), which is the working symbol
throughout. In a scalar case the matrices are one by one. At a
noncharacteristic original covector \((x_0,\xi_0)\), \(\xi_0\ne0\), the
square matrix \(p_m(x_0,\xi_0)\) is invertible. Homogeneity, continuity
of all its entries and finite-dimensional inverse continuity give a
compact base neighborhood and a closed angular neighborhood of
\(\xi_0/|\xi_0|\) on which \(\|p_m(x,\xi)^{-1}\|\leq C|\xi|^{-m}\).
These are estimates of the original principal polynomial, with the
original Euclidean matrix norms. For each lower coefficient the compact
coefficient bounds retain \(\|A_\alpha(x)\||\xi^\alpha|\), and their
complete finite sum is at most
\(\sum_{|\alpha|<m}C_\alpha|\xi|^{|\alpha|}\). For \(m\geq1\), choose
the frequency radius so

\[
\left\|p_m(x,\xi)^{-1}
       \sum_{|\alpha|<m}A_\alpha(x)\xi^\alpha\right\|
\leq C\sum_{|\alpha|<m}C_\alpha|\xi|^{|\alpha|-m}
\leq\frac12 .
\tag{WF19}
\]

The ordered identity \[
a^{-1}
=\left(I+p_m^{-1}\sum_{|\alpha|<m}A_\alpha\xi^\alpha\right)^{-1}
        p_m^{-1}
\tag{WF20}
\] and the actual convergent finite-dimensional geometric series for the
first inverse prove invertibility of the full \(a\) and
\(\|a^{-1}\|\leq2C|\xi|^{-m}\) there. The whole lower-term sum remains
in WF19--WF20. For \(m=0\) the sum is empty and \(a=A_0\); its original
inverse is bounded on that base neighborhood at all frequencies.
Increasing the radius above one permits the usual symbol estimates with
\(\langle\xi\rangle=(1+|\xi|^2)^{1/2}\), retaining the exact
inequalities \(|\xi|\leq\langle\xi\rangle\leq\sqrt2|\xi|\). No rescaled
replacement for \(a\) is introduced.

Every derivative of this original inverse is obtained from
differentiating \(aa^{-1}=I\). Write a joint multi-index
\(\gamma=(\beta,\alpha)\) for original base and frequency derivatives,
and \(\nu=(\nu_x,\nu_\xi)\). The complete recurrence, with its actual
matrix order, is

\[
\partial_x^\beta\partial_\xi^\alpha a^{-1}
=-a^{-1}
 \sum_{\substack{0<\nu\leq(\beta,\alpha)}}
  \binom{(\beta,\alpha)}{\nu}
  (\partial_x^{\nu_x}\partial_\xi^{\nu_\xi}a)
  (\partial_x^{\beta-\nu_x}
        \partial_\xi^{\alpha-\nu_\xi}a^{-1}) .
\tag{WF21}
\]

For derivative order zero use the just proved bound. In each summand at
positive total order the final inverse derivative has smaller total
order. Induction, the original polynomial derivative estimate
\(\|\partial_x^{\nu_x}\partial_\xi^{\nu_\xi}a\|
\leq C_\nu\langle\xi\rangle^{m-|\nu_\xi|}\), and the exact three
exponents in that ordered summand give \[
\langle\xi\rangle^{-m}
\langle\xi\rangle^{m-|\nu_\xi|}
\langle\xi\rangle^{-m-|\alpha-\nu_\xi|}
=\langle\xi\rangle^{-m-|\alpha|}.
\tag{WF22}
\] Every binomial, differentiated original coefficient and inverse
factor is retained in WF21; the finite sum supplies its actual constant.
Thus the full inverse is in \(S^{-m}_{1,0}\) on this cone.

Choose original base/angular/high-frequency cutoffs whose product
\(\theta\) is supported strictly in that invertibility region and equals
one on a smaller conic neighborhood for sufficiently large frequency.
Extend \(b_0=\theta a^{-1}\) by zero outside it. This is smooth across
its support boundary because \(\theta\) vanishes on a neighborhood of
that boundary. All its derivative estimates follow from the full sum \[
\partial^{(\beta,\alpha)}b_0
 =\sum_{\nu\leq(\beta,\alpha)}
   \binom{(\beta,\alpha)}{\nu}
   (\partial^\nu\theta)
   (\partial^{(\beta,\alpha)-\nu}a^{-1}).
\tag{WF23}
\] Above the cutoff radius \(ab_0=b_0a=I\) on the smaller cone. On that
cone the actual full composition products have \(\ell=I-b_0\circ a\) and
\(r=I-a\circ b_0\) of order \(-1\). This follows from E21 with \(N=1\),
and includes the whole composition remainder. The entire higher
expansion is \[
b_0\circ a-
 \sum_{|\gamma|<N}
  \frac{(\partial_\xi^\gamma b_0)(D_x^\gamma a)}{\gamma!}
 \in S^{-N}_{1,0}
\tag{WF24}
\] on that smaller cone; every original \(A_\alpha\) contributes through
\(D_x^\gamma a\). Extend the restricted errors with further cutoffs
equal to one on a still smaller cone, and form their powers there. The
proved conic separated-support calculus makes every change of extension
smoothing on the final cone. It does not set any nonzero off-cone
remainder equal to zero.

The ordered finite inverse identities on that conic calculus are \[
\left(\sum_{j=0}^{N-1}\ell^{\circ j}\circ b_0\right)\circ a
   =I-\ell^{\circ N},\qquad
a\circ\left(\sum_{j=0}^{N-1}b_0\circ r^{\circ j}\right)
   =I-r^{\circ N}.
\tag{WF25}
\] The notation in this display denotes the original conic germs; after
the cutoff extensions each equality has the corresponding conic
smoothing discrepancy, already proved by the separation estimates. The
finite identities themselves follow by associativity and telescoping,
with the left powers on the left and the right powers on the right. Each
\(j\)-term has order \(-m-j\). Support-controlled local summation gives
\(b_L,b_R\) such that their differences from the first \(N\) terms have
order \(-m-N\). Composing their actual tails with \(a\), and using WF25,
proves both errors have order \(-N\) for every \(N\) on one fixed final
cone. Hence their errors are smoothing there. The exact ordered
comparison \[
b_L-b_R=b_L\circ(I-a\circ b_R)
                 +(b_L\circ a-I)\circ b_R
\tag{WF26}
\] and the conic smoothing ideal show that either sum is a two-sided
inverse there. Proper quantization yields a proper \(B\) of order \(-m\)
with \(I-BP\) and \(I-PB\) smoothing in that original covector
neighborhood. A compact coordinate localization makes the original
differential operator proper for this construction; the separated
exterior-input kernel is smooth on the smaller output neighborhood, as
proved in G5/G17. Thus this local construction applies to every
distribution on the original open domain, without a support assumption
on the whole input.

If \(Pu\) is regular at the original covector, the exact distribution
identity \[
u=B(Pu)+(I-BP)u
\tag{WF27}
\] and G24 prove regularity of both terms there: the first because
proper \(B\) does not enlarge the wavefront of its input, and the second
because its operator symbol is smoothing on that cone. No smoothness of
\(Pu\) away from the selected cone is assumed. Conversely G24 for the
original differential operator gives
\(\operatorname{WF}(Pu)\subset\operatorname{WF}(u)\). Consequently \[
\operatorname{WF}(u)\subset
 \operatorname{WF}(Pu)\cup\operatorname{Char}_m(P),\qquad
\operatorname{WF}(Pu)\setminus\operatorname{Char}_m(P)
=\operatorname{WF}(u)\setminus\operatorname{Char}_m(P),
\tag{WF28}
\] where \(\operatorname{Char}_m(P)=\{(x,\xi):\xi\ne0,\ p_m(x,\xi)
\text{ is not invertible}\}\). The latter set is the actual
characteristic set for an order-\(m\) differential operator: an inverse
principal matrix implies the full lower bound by WF19--WF20, while a
singular principal matrix has a nonzero kernel vector whose full
\(a(x,t\xi)\) norm is bounded by the retained lower sum \(O(t^{m-1})\),
ruling out an order-\(m\) lower bound on that ray. For \(m=0\)
singularity of \(A_0\) rules it out directly.

The construction uses the original coordinate and frame maps. The
derivative transpose law in WF7 and the full bundle coefficient
transport in the geometric chapter preserve the displayed regular
directions; multiplication by an invertible smooth frame matrix and its
inverse preserves componentwise wavefront by the cutoff lemma. Thus WF28
applies intrinsically to the differential bundle operator, with all its
original coefficients. The argument supplies no zero-gain parametrix at
the equality endpoint \(\delta=\rho\); the gain used in every tail
estimate was exactly one.

For the original wave operator in the wave chapter,
\(P=\partial_t^2-\sum_{j=1}^n\partial_{x_j}^2\), the convention
\(D=-i\partial\) gives its entire symbol \[
a(t,x;\tau,\xi)=-\tau^2+\sum_{j=1}^n\xi_j^2,\qquad
\operatorname{Char}_2(P)=
\{(t,x;\tau,\xi):(\tau,\xi)\ne(0,0),\quad
                        \tau^2=\sum_{j=1}^n\xi_j^2\}.
\tag{WF29}
\] For \(n\geq1\), every characteristic covector has a nonzero spatial
component when its time component is nonzero. A parameter-normal
covector for the spatial slice parameter \(x\), namely
\((\tau,\xi)=(0,\xi)\) with \(\xi\ne0\), is not characteristic: its
original symbol value is \(\sum_j\xi_j^2>0\). The next section verifies
the exact receiving recurrence before using this fact.

\subsubsection{18.8. The entire original odd-wave receiving
comparison}\label{the-entire-original-odd-wave-receiving-comparison}

Retain \(n\geq1\), \(\nu\in\mathbb N_0\) and every original constant
from W11/W14/W15/W17. Put \(L=\partial_t^2-\sum_j\partial_{x_j}^2\), and
let \(\mathcal R(t,x)=(-t,x)\). The odd family is the actual
distribution difference

\[
\begin{aligned}
E_\nu&=\nu!R_{\nu+1},&
A_\nu&=2^{-2\nu-1}\pi^{(1-n)/2},&
a_\nu&=\nu+(1-n)/2,\\
W_\nu&=E_\nu-\mathcal R^*E_\nu,&
q(t,x)&=t^2-\sum_{j=1}^n x_j^2,\\
LE_0&=\delta_{(0,0)},&
LE_\nu&=\nu E_{\nu-1}\quad(\nu\geq1),&
L^{\nu+1}E_\nu&=\nu!\delta_{(0,0)}.
\end{aligned}
\tag{WF30}
\]

These full distribution identities, including the point source, are
proved in W5/W11/W17 by the original Fourier--Laplace transforms. The
last equality follows by induction and retains each integer factor
\(\nu(\nu-1)\cdots1=\nu!\); for \(\nu=0\) the empty product is one. The
reflection has derivative \(\operatorname{diag}(-1,I_n)\) and absolute
determinant one, so its pullback preserves the point mass and commutes
with every term of \(L\). Thus \[
L^{\nu+1}W_\nu
=\nu!\delta_{(0,0)}-\nu!\mathcal R^*\delta_{(0,0)}
=0 .
\tag{WF31}
\] The two source terms are retained explicitly before their equality is
used. Elliptic regularity WF28 for the actual operator \(L^{\nu+1}\),
whose full constant symbol is \((-\tau^2+\sum_j\xi_j^2)^{\nu+1}\),
excludes every nonnull covector at the vertex. It does not assert that
either retarded source term alone has that exclusion.

At a nonzero cone point \(q=0\), \(t\ne0\) and \(dq=2(t,-x)\ne0\). The
source gives the entire local expression
\(\operatorname{sgn}(t)A_\nu\chi_+^{a_\nu}(q)\), where the sign is
constant on a sufficiently small neighborhood. The map \(q\) is a
submersion there. The one-variable distribution \(\chi_+^{a_\nu}\) is
nonzero, real, supported in the nonnegative half-line, and homogeneous
of the original degree \(a_\nu\), as proved by its Euler integral and
derivative continuation in the wave chapter. It cannot be smooth at
zero: smoothness and vanishing on the negative half-line would make
every derivative at zero vanish; the resulting Taylor estimate of any
order, compared along positive dilations with its original homogeneity,
would force it to vanish on the positive half-line. A distribution
supported only at zero is a finite sum of delta derivatives by the
compact finite-order/Taylor argument and is smooth only if zero; those
nonzero continuation values are therefore singular as well. This
contradicts nonzero \(\chi_+^{a_\nu}\). It has no other singular base
point. For a real cutoff, its transform has equal magnitude in opposite
directions, so failure of rapid decay in one of the two one-dimensional
directions implies failure in both. The equivalence between smoothness
and absence of every direction in Section18.1 proves \[
\operatorname{WF}(\chi_+^{a_\nu})
=\{(0,\lambda):\lambda\ne0\}.
\tag{WF32}
\]

Here is the point-support assertion used in that argument. Let a
one-variable distribution \(T\) be supported at zero and have finite
order \(M\) on a fixed compact neighborhood. If every derivative of a
test \(f\) through order \(M\) vanishes at zero, its Taylor remainders
satisfy \(f^{(j)}(s)=o(|s|^{M-j})\), \(0\leq j\leq M\). Choose a fixed
compact cutoff \(\zeta=1\) near zero and set
\(\zeta_\epsilon(s)=\zeta(s/\epsilon)\). The full derivative product is
\((\zeta_\epsilon f)^{(k)}
=\sum_{j=0}^k\binom{k}{j}\epsilon^{-j}
\zeta^{(j)}(s/\epsilon)f^{(k-j)}(s)\). On its support each summand is
\(o(\epsilon^{M-k})\), hence tends to zero for every \(k\leq M\).
Support locality gives \(T(f)=T(\zeta_\epsilon f)\), and the actual
order-\(M\) bound proves this value is zero. Subtracting the entire
Taylor polynomial of an arbitrary test, with the fixed cutoff, therefore
gives \[
T(f)=\sum_{j=0}^M\frac{T(\zeta(s)s^j)}{j!}f^{(j)}(0),
\qquad
T=\sum_{j=0}^M
\frac{(-1)^jT(\zeta(s)s^j)}{j!}\delta_0^{(j)} .
\tag{WF32a}
\] All signs and factorials follow from
\(\delta_0^{(j)}(f)=(-1)^jf^{(j)}(0)\). A smooth function supported at a
point is zero by continuity, so any nonzero point-supported continuation
value is singular.

The exact submersion converse WF15 and the smooth nonzero factor and its
inverse now give every and only nonzero multiple of \(dq\) over each
nonzero cone point. In original coordinates these satisfy \[
t^2=\sum_jx_j^2,\qquad
\tau^2=\sum_j\xi_j^2,\qquad
\tau x+t\xi=0,\qquad(\tau,\xi)\ne(0,0).
\tag{WF33}
\] Indeed \((\tau,\xi)=2\lambda(t,-x)\) proves all three equations.
Conversely \(t\ne0\) and the last vector equation give
\(\xi=-(\tau/t)x\); \(\tau=0\) would then force the excluded zero
covector, and the multiple is the original \(\lambda=\tau/(2t)\). Away
from the cone the source's function or continued distribution expression
is smooth, so these directions exhaust its nonvertex wavefront.

For any nonzero null covector \((\tau,\xi)\), the original base points
\((t_s,x_s)=s(\tau,-\xi)\), \(s>0\), are nonzero cone points with
\(dq(t_s,x_s)=2s(\tau,\xi)\). At each of them the same retained covector
lies in the wavefront by the exact converse just proved. Closedness from
Section18.1 as \(s\downarrow0\) includes it over the vertex. Together
with WF31 and WF28 this proves the full original OE22 equality \[
 \operatorname{WF}(W_\nu)=
 \{(t,x;\tau,\xi):
   (\tau,\xi)\ne0,\ t^2=|x|^2,
   \tau^2=|\xi|^2,\ \tau x+t\xi=0\}.
 \tag{WF34}
\]

In this exact set \(\tau=0\) forces \(\xi=0\), which is excluded. Thus
the embedding \(j_x(t)=(t,x)\) has its entire normal set disjoint from
the wavefront, including at \(x=0\). WF12 constructs every original
time-distribution slice, and Section18.6 with parameter \(x\) proves a
strong smooth map \(x\mapsto W_\nu(\,\cdot\,,x)\). Differentiation in
time commutes with that map: its transposed test derivative preserves
bounded test sets, as proved in GK23, so
\(x\mapsto\partial_tW_\nu(\,\cdot\,,x)\) is strong smooth as well. These
slices agree with the explicit OE10--OE17 construction: for \(x\ne0\)
both are the same source expression, and both are continuous into
distributions at zero, so their difference at zero is the limit of their
zero difference away from zero. Every factorial and finite-test-order
bound in OE11/OE18 remains intact; the present argument never replaces
those bounds by a negative derivative order or by one fixed finite order
for all spatial derivatives.

All clauses of the original distribution foundation now have written
proofs here or in the exact elementary distribution, finite-symbol and
conic-calculus sections named above. Its original wave-kernel
application has been compared in full, including both cancelling point
sources and every nonvertex/vertex covector.

\end{document}