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\begin{document}

\ReaderArmAliases{\ReaderOriginalAnchorStart{boundary-energy-local-inverses-and-harmonic-data}\ReaderOriginalAnchorEnd}
\section{Boundary energy, local inverses, and harmonic
data}\label{boundary-energy-local-inverses-and-harmonic-data}

A boundary condition plays two distinct roles. It selects an energy
space before a solution has pointwise values, and it removes one of the
normal modes in a local solver. We develop both roles and connect them
through a regularity argument that starts with only one derivative. The
harmonic case then turns interior solutions into operators acting on
boundary data.

The order of construction is intentional: weak existence does not depend
on boundary smoothness; regularity does. A separate normal-mode
calculation records the precise extension from real to complex quadratic
principal symbols. Higher-order systems and arbitrary boundary
conditions require their own complementing-condition theory.

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\subsection{1. Spaces, conventions, and exact entry
contracts}\label{spaces-conventions-and-exact-entry-contracts}

Put \(D_j=-i\partial_j\), and use
\(\widehat u(\xi)=\int e^{-ix\cdot\xi}u(x)\,dx\) with inverse factor
\((2\pi)^{-n}\). Thus \(\sum D_j^2=-\Delta\). Inner products are linear
in the first variable. On \(\mathbb R^d\), \[
 \|u\|_{H^s}^2=(2\pi)^{-d}\int (1+|\xi|^2)^s|\widehat u(\xi)|^2\,d\xi. \tag{B1}
\] For an integer \(k\geq0\), \(H^k(V)\) denotes the space of \(L^2\)
functions with all weak derivatives through order \(k\) in \(L^2(V)\).
On a half-space it is the restriction of the whole-space space, with
equivalent norm; a proof appears below. For nonintegral \(s\), the
restriction space carries the quotient norm. We write \(\gamma u\) for
the boundary trace and \(\nu\) for the \textbf{exterior} unit normal. An
equation with Lipschitz leading coefficients and an initially \(H^1\)
solution uses the weak multiplication rule from Section 7 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates}: \(aD_jv=D_j(av)-(D_ja)v\).

The next four groups of facts identify exactly what the arguments use.

\begin{itemize}
\tightlist
\item
  \textbf{Fourier and Sobolev facts.} Fourier inversion and Plancherel
  extend to all of \(L^2\), with the normalization (B1); Fourier
  transformation commutes with weak derivatives. The full-space
  extension is the initial Plancherel paragraph in the proof in Section
  5 of \href{weyl-metric-products.pdf}{Two measuring scales, one Weyl
  product} (no selfadjoint-generator result from that item is needed).
  Its argument, from the Schwartz identity in Section 2 of
  \href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
  convex separation}, is: an isometry on a dense subspace extends to its
  completion, and its dense image is closed and therefore all of
  \(L^2\). The same argument gives partial Fourier transformation with
  an additional \(L^2\) variable, first on finite sums of product
  functions and then by density. Weighted completions define (B1). No
  pointwise Fourier transform is required for a general \(L^2\)
  function.
\item
  \textbf{Weak derivatives and localization.} The integration,
  weak-derivative, smooth approximation, and Lipschitz product results
  in \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates} are used with \(p=2\). Smooth
  changes of variables on compact coordinate patches, finite partitions
  of unity, and finite-dimensional linear algebra are entry facts. We
  supply below the precise Sobolev consequences of boundary flattening
  that are needed here.
\item
  \textbf{Local elliptic inverses.} Section 2 of
  \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates} supplies a fundamental solution
  for any homogeneous elliptic scalar polynomial, local bounds for
  convolution through its order, and the two convolution inverse
  identities on their stated domains. Section 7 of
  \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates} supplies the Lipschitz
  commutator, and Section 8 of
  \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates} supplies interior \(H^2\)
  regularity from an \(H^1\) solution with \(L^2\) principal part. Our
  boundary argument proves the additional trace and reflection
  assertions instead of treating the interior theorem as a boundary
  theorem.
\item
  \textbf{Boundary operator calculus.} Only the layer-potential portion
  uses the order \(-1\) operators, local quantization, Sobolev mapping,
  and smooth compact-manifold assembly proved in
  \href{euclidean-symbol-calculus.pdf}{Symbols, operators and Sobolev
  scales} and \href{geometric-microlocal-calculus.pdf}{Detecting
  regularity without choosing coordinates}, and compact-perturbation
  Fredholm theory in \href{fredholm-stability.pdf}{Finite defects under
  perturbation}. The particular kernels, jumps, and boundary data spaces
  must still be checked below.
\end{itemize}

All Sobolev estimates are complex-linear unless an energy argument
explicitly assumes a real symmetric positive coefficient matrix. Bounds
may depend on a fixed coordinate patch and coefficient bounds. A local
inverse makes no assertion of uniqueness without conditions on the rest
of the boundary.

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\subsection{2. Traces, lifts, and a boundary-free energy
domain}\label{traces-lifts-and-a-boundary-free-energy-domain}

Write \(x=(z,t)\in\mathbb R^{n-1}\times(0,\infty)\),
\(\lambda(\zeta)=(1+|\zeta|^2)^{1/2}\). The following elementary
estimate is the source of the half derivative at the boundary. For a
smooth scalar function \(h\) decreasing at infinity, \[
 \lambda |h(t)|^2
 \leq\int_t^\infty\bigl(\lambda^2|h(r)|^2+|h'(r)|^2\bigr)\,dr. \tag{B2}
\] Indeed integrate \(-(|h|^2)'\) and use
\(2\lambda |h||h'|\leq\lambda^2|h|^2+|h'|^2\). The same inequality holds
for an \(H^1\) function on the ray by approximation, or by choosing a
sequence tending to infinity on which its value tends to zero.
Integrating (B2) in \(\zeta\) proves \[
 \sup_{t\geq0}\|u(\cdot,t)\|_{H^{1/2}}\leq C\|u\|_{H^1(\mathbb R^n_+)}. \tag{B3}
\] Here and below the assertion about slices refers to their continuous
representative, not to an arbitrary representative on sets of measure
zero.

We justify the approximation and the weak-space interpretation. If
\(u,\partial_tu\in L^2\), testing against products of a test function in
\(z\) and one in \(t\) gives the one-dimensional weak derivative
identity for almost every tangential slice. A one-dimensional function
with integrable weak derivative equals an absolutely continuous
primitive plus a constant: subtract its integral primitive; the
remaining distribution has derivative zero, as follows by testing
against the derivative of every compactly supported test function of
integral zero. Reflecting this representative evenly across zero
therefore gives an \(H^1\) function. Its tangential derivatives are even
reflections and its normal derivative is the signed reflection of
\(\partial_tu\); integration by parts on the two rays cancels the
boundary terms. Fubini yields a bounded even extension on \(H^1\).
Whole-space convolution and cutoffs now give approximation by
restrictions of \(C_c^\infty(\mathbb R^n)\). This proves both the
weak-derivative characterization of the restriction space and the
density used in (B3). The case \(n=1\) has no tangential variable and
uses exactly the same one-dimensional argument.

For smooth approximants the map \(t\mapsto u(\cdot,t)\) is continuous
into \(H^{1/2}\). Bound (B3) makes these maps uniformly Cauchy on the
closed ray, proving bounded continuity for every \(H^1\) function. In
particular \(\gamma:H^1(\mathbb R^n_+)\to H^{1/2}(\mathbb R^{n-1})\) is
bounded. More generally, for every integer \(k\geq1\), \[
 \gamma:H^k(\mathbb R^n_+)\longrightarrow H^{k-1/2}(\mathbb R^{n-1}),
 \qquad \widehat{L\varphi}(\zeta,t)=e^{-t\lambda(\zeta)}\widehat\varphi(\zeta) \tag{B4}
\] are bounded maps with \(\gamma L=I\). To prove the trace bound, apply
(B2) after multiplying by \(\lambda^{2k-2}\); the two resulting
integrals are controlled by the tangential derivatives of orders at most
\(k\) and their first normal derivatives. For the lift, each derivative
of total order \(j\leq k\) has squared integral at most
\(C\int\lambda^{2j-1}|\widehat\varphi|^2\), because
\(\int_0^\infty e^{-2t\lambda}dt=(2\lambda)^{-1}\). This proves (B4),
including its inhomogeneous low frequencies.

For completeness, the restriction and weak norms agree at every integer
\(k\). On the negative ray set \(Eu(z,t)=\sum_{j=1}^{k}c_j u(z,-jt)\),
choosing the unique solution of \(\sum_{j=1}^{k}c_j(-j)^\ell=1\),
\(0\leq\ell<k\). The matrix is Vandermonde. Derivatives through order
\(k-1\) match at zero; repeated integration by parts introduces no
interface delta through order \(k\). Change of variable on each
reflected term bounds the extension in \(H^k\). Smooth approximation
follows by this extension and convolution. In particular tangential
differentiation commutes with the trace whenever the indicated
derivatives lie in \(H^1\).

Let \(u^0\) be extension by zero. Integration by parts, first on smooth
approximants and then using (B3), gives the distribution identities \[
 \partial_t u^0=(\partial_tu)^0+\gamma u\otimes\delta_0,
 \qquad \partial_{z_j}u^0=(\partial_{z_j}u)^0. \tag{B5}
\] Consequently \(u^0\in H^1(\mathbb R^n)\) if and only if
\(\gamma u=0\). For necessity, an \(L^2\) function cannot be a nonzero
distribution supported on a hyperplane: that hyperplane has Lebesgue
measure zero. Conversely (B5) lists all its first derivatives as \(L^2\)
functions. These formulas also show why a nonzero boundary value cannot
simply be extended by zero in an energy argument.

For \textbf{any} bounded open \(\Omega\subset\mathbb R^n\), define \[
 V_\Omega=\overline{C_c^\infty(\Omega)}^{\ H^1(\mathbb R^n)}, \tag{B6}
\] identifying its elements with their restrictions to \(\Omega\). Every
such element vanishes almost everywhere outside \(\Omega\). This is the
homogeneous Dirichlet domain, even when the boundary is irregular.
Multiplication by a smooth function with bounded value and first
derivatives preserves it, by the product estimate and approximation.

If \(\partial\Omega\) is \(C^1\), three descriptions agree: (B6);
functions in \(H^1(\Omega)\) with zero local flattened trace; and
\(H^1(\mathbb R^n)\) functions supported in \(\overline\Omega\). Here is
the local-to-global proof. A \(C^1\) flattening and its inverse have
bounded first derivatives and Jacobians on a smaller compact patch. The
chain rule and change of variables bound composition in \(H^1\),
initially for smooth functions and then for weak functions by
approximation. A zero-trace function extends by zero across the
flattened plane by (B5). Translate that extension a distance
\(\varepsilon\) into the upper half-space and convolve at radius less
than \(\varepsilon/2\); after a fixed smaller cutoff this gives smooth
functions supported strictly on the interior side, converging in
\(H^1\). Pulling them back gives compactly supported \(C^1\) functions
in \(\Omega\); ordinary convolution at a distance smaller than their
distance from the boundary approximates each by \(C_c^\infty(\Omega)\).
A finite partition of unity and interior convolution prove inclusion in
(B6). Conversely (B3) and the composition estimate give zero trace for
limits of tests. Finally a whole-space \(H^1\) function supported in
\(\overline\Omega\) has identical traces from both sides of each
flattened plane, by the slice continuity proof; its exterior trace is
zero and so is its interior trace. A \(C^1\) boundary has measure zero,
since finitely many graph patches have measure zero by Fubini. Thus
there is no additional \(H^1\) function carried only by the boundary. No
support characterization for an arbitrary irregular domain has been
used.

Inhomogeneous Dirichlet data is removed by a \textbf{lift}, not by a
pointwise substitution without a function-space bound. If
\(\varphi\in H^{3/2}\) on a flat boundary and \(\Phi=L\varphi\in H^2\),
then \[
 Pu=f,\quad\gamma u=\varphi
 \quad\Longleftrightarrow\quad
 P(u-\Phi)=f-P\Phi,\quad\gamma(u-\Phi)=0. \tag{B7}
\] Bounded coefficients suffice for \(P\Phi\in L^2\). For a
divergence-form energy equation the corresponding lift is
\(H^{1/2}\to H^1\), and the adjusted forcing is a continuous functional
on \(V_\Omega\), generally in \(H^{-1}\), not necessarily \(L^2\).
Smooth compact-boundary lifts are obtained from (B4) in charts and a
partition of unity. At regularity \(C^{1,1}\), meaning that the first
chart derivatives are Lipschitz, the same \(H^{3/2}\to H^2\) assertion
holds: integer \(H^2\) composition uses the bounded weak second
derivatives of the charts, while the boundary \(H^{3/2}\) norm is
\(H^1\) plus the half-order norm of first derivatives. That half-order
norm is equivalent to \(\iint |v(x)-v(y)|^2/|x-y|^{d+1}\,dx\,dy\), as
Fourier transformation and the substitution \(h=x-y\) show. Bi-Lipschitz
changes of variables preserve this integral up to constants, and
multiplication by a bounded Lipschitz function is bounded in it after
separating differences and integrating near and away from the diagonal.
These facts justify the stated chart norm without assuming a smooth
boundary at this step.

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\subsection{3. The energy solution before boundary
regularity}\label{the-energy-solution-before-boundary-regularity}

Let \(\Omega\) be bounded and open. Assume \(A=(a_{jk})\) is real,
symmetric, continuous on \(\overline\Omega\), and positive definite at
every point there. Compactness supplies constants
\(0<\lambda\leq\Lambda<\infty\) with \[
 \lambda|\xi|^2\leq \sum a_{jk}(x)\xi_k\overline{\xi_j}
 \leq\Lambda|\xi|^2\quad(x\in\overline\Omega,\ \xi\in\mathbb C^n). \tag{B8}
\] For complex \(\xi\), apply the real inequality to real and imaginary
parts; symmetry cancels their imaginary cross terms. Define \[
 Q(u,v)=\int_\Omega\sum a_{jk}D_ku\,\overline{D_jv},
 \qquad P=\sum_{j,k}D_j(a_{jk}D_k). \tag{B9}
\] Then for every \(f\in L^2(\Omega)\) there is exactly one
\(u\in V_\Omega\) satisfying \[
 Q(u,v)=(f,v)\quad(v\in V_\Omega),\qquad
 \|u\|_{H^1(\Omega)}\leq C\|f\|_2. \tag{B10}
\] The equation holds distributionally, and no boundary regularity is
assumed.

\textbf{Proof.} Translate coordinates so that \(|x_1|\leq R\) on
\(\Omega\). For a compactly supported test function, integrate
\(\partial_1(x_1|u|^2)\) over the whole space. Cauchy-Schwarz gives \[
 \|u\|_2^2\leq2R\|u\|_2\|\partial_1u\|_2,
 \qquad \|u\|_2\leq2R\|\partial_1u\|_2. \tag{B11}
\] Equivalently, before bounding \(|x_1|\), one has
\(\|u\|_2^2\leq4\|x_1\partial_1u\|_2^2\). Density extends these bounds
to \(V_\Omega\). Hence \(Q(u,u)^{1/2}\) is a norm equivalent to its
complete \(H^1\) norm, and \(|(f,v)|\leq C\|f\|_2 Q(v,v)^{1/2}\).

We give the minimization argument, including existence, instead of
leaving a representation theorem implicit. Set
\(J(v)=Q(v,v)-2\operatorname{Re}(f,v)\). Its infimum \(m\) is finite
because \(r^2-2Cr\|f\|_2\) is bounded below. A minimizing sequence
\(v_j\) is bounded in the \(Q\) norm. The parallelogram identity gives
\[
 Q(v_j-v_k,v_j-v_k)
 =2J(v_j)+2J(v_k)-4J((v_j+v_k)/2)
 \leq2J(v_j)+2J(v_k)-4m. \tag{B12}
\] It is therefore Cauchy and converges in \(V_\Omega\) to a minimizer
\(u\). For real \(t\), differentiating the quadratic polynomial
\(J(u+tv)\) at zero gives the real part of (B10). Applying it to \(iv\)
gives the imaginary part. Set \(v=u\) and use the functional bound to
obtain \(Q(u,u)^{1/2}\leq C\|f\|_2\), hence (B10). The difference of two
solutions has zero \(Q\) norm. Testing (B10) on \(C_c^\infty\) is
exactly the distributional equation. Conversely that equation for an
element of \(V_\Omega\) extends to every \(v\in V_\Omega\) by density
and the continuity of both sides. This proves the equivalence as well as
uniqueness. \(\square\)

The proof also applies to any continuous conjugate-linear forcing
functional on \(V_\Omega\), with its dual norm replacing \(\|f\|_2\).
Thus (B7) gives weak existence for lifted boundary values on a regular
boundary, and a different lift gives the same solution because their
difference belongs to \(V_\Omega\).

\textbf{The complete dual norm and measurable coefficients.} The same
proof needs only an essentially bounded measurable real symmetric matrix
satisfying every bound in (B8) almost everywhere, with the stated
constants \(\lambda,\Lambda\). Pointwise real symmetry gives the complex
inequality; diagonalizing the actual finite symmetric matrix shows
\(|A(x)\xi|\leq\Lambda|\xi|\). Thus
\(|Q(u,v)|\leq\Lambda\|\nabla u\|_2\|\nabla v\|_2\), and
\(\lambda\|\nabla v\|_2^2\leq Q(v,v)\). These estimates, (B11),
completeness of (B6), and the exact identity (B12) prove existence and
uniqueness for every bounded conjugate-linear \(F\) by replacing
\((f,v)\) by \(F(v)\) in the entire minimization argument. This extends
weak existence; the later regularity theorems retain their additional
coefficient hypotheses.

Retain both norms of the forcing on this same space. Set
\(C_R=(1+4R^2)^{1/2}\), using the actual coordinate bound in (B11). Its
proof gives \(\|\nabla v\|_2\leq\|v\|_{H^1}\leq C_R\|\nabla v\|_2\). The
two unit-ball inclusions and homogeneity then give \[
 \begin{split}
 \|F\|_{1,*}&=\sup_{\|v\|_{H^1}\leq1}|F(v)|,\qquad
 \|F\|_{\nabla,*}=\sup_{\|\nabla v\|_2\leq1}|F(v)|,\\
 \|F\|_{1,*}&\leq\|F\|_{\nabla,*}\leq C_R\|F\|_{1,*}.
 \end{split}
 \tag{BO7}
\] Testing \(Q(u,u)=F(u)\) proves, for \(u\ne0\),
\(\lambda\|\nabla u\|_2^2\leq\|F\|_{\nabla,*}\|\nabla u\|_2\); division
is legitimate by (B11). For \(u=0\) the same bounds hold directly.
Consequently the full constants are \[
 \|\nabla u\|_2\leq\lambda^{-1}\|F\|_{\nabla,*}
            \leq\lambda^{-1}C_R\|F\|_{1,*},\qquad
 \|u\|_{H^1}\leq\lambda^{-1}C_R^2\|F\|_{1,*}.
 \tag{BO8}
\] For an actual \(H^1\) lift \(\Phi\), the forcing for \(w=u-\Phi\) is
exactly \(F(v)-Q(\Phi,v)\). It is bounded by
\((\|F\|_{1,*}+\Lambda\|\nabla\Phi\|_2)\|v\|_{H^1}\). If another lift is
\(\Psi=\Phi+\chi\), with \(\chi\in V_\Omega\), its solution is
\(w_\Psi=w_\Phi-\chi\), since substitution gives
\(Q(w_\Psi,v)=F(v)-Q(\Psi,v)\). Adding the respective lifts proves
equality of the resulting original \(u\).

Sergio Vessella's freely accessible
\href{https://arxiv.org/abs/2305.04765v1}{original-author notes,
arXiv:2305.04765v1}, give the variational construction in Chapter 4,
Sections 4.1--4.3. The exact comparison in
\href{../supplements/vessella-dirichlet-comparison.html}{Dirichlet
lifting and the full dual norm}, (VE1)--(VE26), proves the isometry of
its closure space with (B6), compares its original constants with (B8),
and supplies complete counterexamples for the printed lifting sign and
the printed estimate stab when its specified full dual norm is used. The
source correction is identified there as editorial material. Here (B7)
and (BO7)--(BO8) retain the negative lifted form and the required norm
factors.

\textbf{Example: an elliptic coefficient without classical first
derivatives everywhere.} In a ball take
\(a_{jk}(x)=(2+|x_1|)\delta_{jk}\). It is continuous, Lipschitz, and
uniformly positive. Formula (B10) defines a unique energy solution.
Expanding its operator produces the first-order coefficient
\(-i\,\operatorname{sgn}(x_1)\) in front of \(D_1u\), defined almost
everywhere. A theorem requiring continuous lower-order coefficients
would not apply to this expansion. The bounded-lower-coefficient
regularity theorem below does.

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\subsection{4. A reflected solver and its two inverse
identities}\label{a-reflected-solver-and-its-two-inverse-identities}

Let \(p_{\mathrm{orig}}\) be a real homogeneous elliptic quadratic
polynomial, and write \(p\) for its positive signed comparison. Its
quadratic form is definite: on the connected unit sphere for \(n\geq2\),
a nonzero continuous real function has constant sign; for \(n=1\) the
assertion is immediate. Choose \(\sigma\in\{1,-1\}\) so that
\(p=\sigma p_{\mathrm{orig}}\) is positive. It can be reduced to
\(|\eta|^2\) by a linear change preserving the plane \(t=0\) and its
positive side. Indeed write
\(p(\zeta,\tau)=\zeta^TB\zeta+2\tau c^T\zeta+a\tau^2\), where \(a>0\).
Completing the square leaves the positive Schur complement \(B-cc^T/a\).
A tangential shear removes \(c\); a positive normal dilation and an
invertible tangential change normalize the remaining blocks. The dual
physical transformation has last coordinate a positive multiple of
\(t\). All derivative and measure norms are equivalent under this fixed
change. The reflected calculation below uses coordinates in which
\(p(D)=-\Delta\). The exact signed comparison with the original form
follows.

Here is the exact map back to the original form, including its sign and
volume factor. Retain the original \(p_{\mathrm{orig}}\), its sign
\(\sigma\), the positive comparison \(p\), and the displayed \(a,B,c\).
Put \[
S=B-\frac{cc^T}{a},\qquad
F(w,\rho)=\left(S^{1/2}w+\frac{c}{\sqrt a}\rho,\ \sqrt a\,\rho\right),
\qquad
J_F=\sqrt a\,(\det S)^{1/2}.
\] The matrix \(S\) is positive definite: for nonzero \(\zeta\),
evaluate the positive form at \(\tau=-a^{-1}c^T\zeta\) to get
\(\zeta^TS\zeta>0\). The exact square and transformed full symbol are \[
p(\zeta,\tau)=\zeta^TS\zeta+a(\tau+a^{-1}c^T\zeta)^2,\qquad
\sigma p_{\mathrm{orig}}\!\left(S^{-1/2}\eta,\ a^{-1/2}\theta-a^{-1}c^TS^{-1/2}\eta\right)
=|\eta|^2+\theta^2.
\] For \(n=1\), \(S\) is the empty matrix and \(\det S=1\). The inverse
coordinates are \(w=S^{-1/2}(z-ct/a)\) and \(\rho=t/\sqrt a\). Thus
\(F\) carries the upper half-space onto itself and the boundary plane
onto itself. The chain rule gives \[
(p_{\mathrm{orig}}(D_x)u)\circ F
=\sigma\left[-\Delta_{w,\rho}(u\circ F)\right].
\] Indeed the exact derivative maps are \(D_z=S^{-1/2}D_w\) and
\(D_t=a^{-1/2}D_\rho-a^{-1}c^TS^{-1/2}D_w\). Substituting both into the
displayed full square gives \(D_w^TD_w+D_\rho^2\), including the normal
dilation and every mixed term. If \(G_{-\Delta}\) is the reflected
construction's whole-space fundamental solution, the original
fundamental solution is \[
G_{p_{\mathrm{orig}}}(x)
=\sigma J_F^{-1}G_{-\Delta}(F^{-1}x).
\] To verify the factor, change variables \(x=F(w,\rho)\) in a test
pairing: the Jacobian \(J_F\) cancels \(J_F^{-1}\), the two factors of
\(\sigma\) multiply to one, and \(-\Delta G_{-\Delta}=\delta_0\). On a
transformed local patch, the original right or left inverse is
\(f\mapsto [T_{-\Delta}(\sigma f\circ F)]\circ F^{-1}\), with the same
support restrictions transported by \(F\). All constants in its Sobolev
bounds retain their dependence on this fixed map. This conjugation
carries every original coefficient through the Laplacian calculation.

Fix a bounded ball \(X\) centered on the plane and invariant under
reflection. For \(f\in L^2(X_+)\), let \(Tf\) equal \(f\) on \(X_+\),
its negative reflection on \(X_-\), and zero elsewhere. Choose an exact
fundamental solution \(G\) of \(-\Delta\) that is even in \(t\). Such a
solution is obtained by averaging the solution of Section 2 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates} with its reflected distribution.
Convolution with \(Tf\), restricted to \(X_+\), defines \[
 E_0f=(G*Tf)|_{X_+}. \tag{B13}
\] The local \(L^2\) convolution bounds of Section 2 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates} give \[
 \sum_{|\alpha|\leq2}\|D^\alpha E_0f\|_{L^2(X_+)}\leq C_X\|f\|_2,
 \quad p(D)E_0f=f,\quad\gamma E_0f=0\text{ on }X_0. \tag{B14}
\] For the trace, convolution is odd because \(G\) is even and \(Tf\) is
odd; it is locally \(H^2\), so the two one-sided traces agree and
oddness forces them to vanish. The convolution bounds are local; a
global \(L^2\) inverse of the homogeneous Laplacian has not been
asserted.

There is also the distinct left identity \[
 E_0p(D)u_+=u_+,\qquad
 u\in C_c^\infty(X),\quad u|_{X_0}=0. \tag{B15}
\] To verify it, reflect \(u_+\) oddly. The function is continuous at
the plane. Its first normal derivative is the even reflection of
\(\partial_tu_+\), so it too is continuous there. Thus the second normal
derivative has no interface delta. Tangential differentiation does not
differentiate the sign of \(t\), and tangential derivatives of the zero
trace vanish. Therefore \(p(D)Tu_+=T(p(D)u_+)\); the compared Laplacian
has no mixed normal-tangential monomials, and the full original operator
is recovered by the proved conjugation. Compact support permits
transferring \(p(D)\) in the convolution, yielding
\(G*p(D)Tu_+=(p(D)G)*Tu_+=Tu_+\). This proves (B15).

The same reasoning proves a useful density fact. If
\(v\in H^2(\mathbb R^n_+)\) has zero trace and compact support up to the
plane, its odd extension is \(H^2\). The only possible interface term
would be the jump of a first derivative. The normal derivative is even,
and the tangential derivatives have zero trace by Section 2. Convolving
this extension with an even smooth mollifier produces smooth odd
functions converging in \(H^2\); a symmetric cutoff keeps their supports
in any prescribed slightly larger neighborhood. Consequently (B15) and
its norm estimate extend to such \(H^2\) functions whenever their
support lies in the fixed patch.

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\subsection{5. Continuous principal coefficients and a local
inverse}\label{continuous-principal-coefficients-and-a-local-inverse}

Let \[
 P(x,D)=\sum_{|\alpha|\leq2}a_\alpha(x)D^\alpha \tag{B16}
\] near zero. Suppose the principal coefficients are continuous, their
frozen quadratic polynomial is real elliptic, and the lower-order
coefficients are locally essentially bounded. Then a sufficiently small
neighborhood \(X\) has a bounded linear map \(E:L^2(X_+)\to H^2(X_+)\)
with \[
 PEf=f,\quad\gamma Ef=0\text{ on }X_0,\qquad
 EPu_+=u_+\quad(u\in C_c^\infty(X),\ u|_{X_0}=0). \tag{B17}
\] The original continuous-coefficient situation is included. Products
in \(PEf\) are \(L^2\) products, since \(Ef\in H^2\).

\textbf{Proof.} Make the boundary-preserving normalization above and
work on a fixed unit patch in the rescaled variable \(y=x/r\).
Multiplying the equation by \(r^2\) gives a frozen principal operator
\(p(D_y)\) and perturbation \[
 R_r=\sum_{|\alpha|=2}(a_\alpha(ry)-a_\alpha(0))D_y^\alpha
       +\sum_{|\alpha|<2}r^{2-|\alpha|}a_\alpha(ry)D_y^\alpha. \tag{B18}
\] By (B14), \(\|R_rE_0\|_{L^2\to L^2}\to0\). Choose \(r\) making this
norm at most \(1/2\), put \(A_r=R_rE_0\), and set \[
 E_r=E_0(I+A_r)^{-1},\qquad (I+A_r)^{-1}=\sum_{j=0}^\infty(-A_r)^j. \tag{B19}
\] The series converges in operator norm, gives a two-sided inverse, and
has norm at most two. Since \(P_rE_0=I+A_r\), the right identity and
trace assertion follow. For a compact smooth zero-trace \(u\), put
\(g=p(D)u\). Identity (B15) gives \(E_0g=u\), and hence
\(P_ru=(I+A_r)g\). Applying (B19) yields the left identity. Rescaling
back gives (B17) and boundedness of every derivative through order two.
Explicitly the scaled bound is
\(\sum_{j=0}^2r^j\|\nabla^j Ef\|_2\leq Cr^2\|f\|_2\). No smallness of
unscaled lower-order coefficients was assumed. \(\square\)

In particular compact smooth zero-trace functions satisfy \[
 \|w\|_{H^2(X_+)}\leq C\|Pw\|_{L^2(X_+)}. \tag{B20}
\] The odd-density result after (B15) extends (B20) to compactly
supported \(H^2\) zero-trace functions in a slightly smaller patch. This
extension will be used only after the regularized function has been
proved to be \(H^2\).

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\subsection{6. Recovering second derivatives at a flat
boundary}\label{recovering-second-derivatives-at-a-flat-boundary}

Assume the principal coefficients in (B16) are Lipschitz, the frozen
principal polynomial is real elliptic, and the lower-order coefficients
are locally bounded. If \[
 u\in H^1(X_+),\qquad Pu=f\in L^2(X_+),\qquad\gamma u=0\text{ on }X_0, \tag{B21}
\] then for every \(Y\Subset X\), \[
 u\in H^2(Y_+),\qquad
 \sum_{|\alpha|=2}\|D^\alpha u\|_{L^2(Y_+)}
 \leq C_Y\bigl(\|f\|_{L^2(X_+)}+\|u\|_{H^1(X_+)}\bigr). \tag{B22}
\] Here \(Y\) may meet the boundary plane. Merely knowing the interior
theorem would not establish this assertion.

\textbf{Proof.} Choose \(\chi\in C_c^\infty(X)\) equal to one near
\(\overline Y\), put \(v=\chi u\), and extend tangentially by zero
beyond the patch. The full product rule gives \[
 Pv=\chi f+
 \sum_{|\alpha|\leq2}\sum_{0<\beta\leq\alpha}
 \binom\alpha\beta a_\alpha(D^\beta\chi)D^{\alpha-\beta}u
 =:g\in L^2.
 \tag{BO1}
\] Every derivative on \(u\) in the displayed sum has order at most one.
Its norm is bounded by \(C(\|f\|_2+\|u\|_{H^1})\).

Keep \(a=a_{2e_t}\), the original coefficient of \(D_t^2\), and shrink
the patch so that \(|a|\geq a_*>0\). For each other original multi-index
put \(b_\alpha=a^{-1}a_\alpha\). These are comparison coefficient
functions: the equation and estimate will continue to use the original
\(P\). The principal \(b_\alpha\) are Lipschitz; the lower ones are
bounded. All bounds retain their dependence on \(a_*^{-1}\),
\(\|a\|_\infty\), its Lipschitz bound and the original coefficients.

Here is the weak multiplication identity needed before second
derivatives are known. For \(h\in L^2\) and Lipschitz \(c,d\), define
\(cD_jh=D_j(ch)-(D_jc)h\). The weak product rule, tested against
compactly supported \(H^1\) functions, gives \[
 d(cD_jh)=D_j(dch)-\big((D_jd)c+d(D_jc)\big)h
          =(dc)D_jh.
 \tag{BO2}
\] Indeed multiplication by a Lipschitz function is bounded on the test
space \(H^1_0\), so it acts by duality on \(H^{-1}\); applying the
product rule twice gives the displayed terms in their actual order. For
\(d=a^{-1}\), \(c=a\), differentiating \(a^{-1}a=1\) makes the entire
derivative bracket zero. Thus applying multiplication by \(a^{-1}\) to
the original weak equation is an exact identity in \(H^{-1}\); no second
derivative has been assumed.

Let \(J_\varepsilon\) be convolution in \(z\) alone with a compactly
supported smooth approximate identity, and put
\(v_\varepsilon=J_\varepsilon v\). First derivatives converge in
\(L^2\). Every second derivative with a tangential factor already
belongs to \(L^2\), since that factor may differentiate the smooth
convolution kernel and the remaining derivative falls on \(v\in H^1\).
Tangential convolution commutes with \(D_t^2\) distributionally.
Consequently (BO2) and the complete original equation yield \[
 \begin{split}
 aD_t^2v_\varepsilon
 &=aJ_\varepsilon(a^{-1}g)
   -a\sum_{\alpha\ne2e_t}J_\varepsilon(b_\alpha D^\alpha v),\\
 Pv_\varepsilon
 &=aJ_\varepsilon(a^{-1}g)
   +a\sum_{\alpha\ne2e_t}
      \big(b_\alpha D^\alpha J_\varepsilon v
                  -J_\varepsilon(b_\alpha D^\alpha v)\big).
 \end{split}
 \tag{BO3}
\] All sums in this proof range over the original \(|\alpha|\leq2\). For
a principal \(\alpha\ne2e_t\), write \(D^\alpha=D_jD_k\) with \(j\)
tangential. Apply Section 7 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates} to \(D_kv\) at fixed \(t\). It
gives \[
 b_\alpha D^\alpha J_\varepsilon v
       -J_\varepsilon(b_\alpha D^\alpha v)
 \longrightarrow0\quad\hbox{in }L^2.
 \tag{B23}
\] The Lipschitz and convolution bounds are uniform in \(t\). More
explicitly the commutator is bounded by a fixed constant times
\(\|D_kv(\cdot,t)\|_{L^2_z}\); for almost every \(t\) it tends to zero.
The square of this dominating function is integrable, so dominated
convergence proves the displayed full-space convergence. For a
lower-order term put \(h=D^\alpha v\in L^2\). The exact difference
\(b_\alpha J_\varepsilon h-J_\varepsilon(b_\alpha h)
=b_\alpha(J_\varepsilon h-h)+(b_\alpha h-J_\varepsilon(b_\alpha h))\)
tends to zero in \(L^2\) using only boundedness of \(b_\alpha\). Define
\[
 T_\varepsilon g=aJ_\varepsilon(a^{-1}g),\qquad
 \|T_\varepsilon g-g\|_2
 \leq\|a\|_\infty
       \|J_\varepsilon(a^{-1}g)-a^{-1}g\|_2\longrightarrow0.
 \tag{BO4}
\] Thus the actual original operator, with its full \(D_t^2\)
coefficient, satisfies \[
 Pv_\varepsilon=T_\varepsilon g+o_{L^2}(1).
 \tag{B24}
\] This identity and the already known nonnormal second derivatives
express \(aD_t^2v_\varepsilon\) as an \(L^2\) function. Multiplication
by the retained \(a^{-1}\) proves \(D_t^2v_\varepsilon\in L^2\), and
hence \(v_\varepsilon\in H^2\), before (B20) is invoked. Its trace is
zero because tangential convolution commutes with the continuous trace;
its support lies in a fixed compact subpatch for small \(\varepsilon\).

Apply the original estimate (B20) to \(v_\varepsilon-v_\delta\), using
the odd-extension density after (B15). Formula (B24), with (BO4) and
every commutator retained, makes the right side tend to zero.
Completeness gives an \(H^2\) limit, and the \(H^1\) convergence
identifies it with \(v\). Testing against compact smooth functions
identifies all second-derivative limits with its distributional
derivatives. The same estimate gives (B22), with constants containing
the coefficient and reciprocal bounds already stated. For \(n=1\), there
is no tangential convolution: the original equation is
\(aD_t^2v=g-\sum_{\alpha\ne2e_t}a_\alpha D^\alpha v\), whose right side
is \(L^2\). Its actual nonzero coefficient gives \(D_t^2v\in L^2\), and
(B20) applies. \(\square\)

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\subsection{7. Smooth bootstrap and a single inverse on all integer
scales}\label{smooth-bootstrap-and-a-single-inverse-on-all-integer-scales}

Suppose the coefficients are smooth on a neighborhood of the closed flat
patch and retain the ellipticity condition above. For every integer
\(r\geq0\), a zero-trace solution initially in \(H^1\), with
\(Pu\in H^r\), satisfies on a smaller patch \[
 \|u\|_{H^{r+2}(Y_+)}\leq C_{r,Y}\bigl(\|Pu\|_{H^r(X_+)}+\|u\|_{H^1(X_+)}\bigr). \tag{B25}
\] The nested patches used in the proof may depend on \(r\); the
solution does not.

\textbf{Proof.} The case \(r=0\) is (B22). Inductively suppose
\(u\in H^{r+1}\) on the required local patches. For a tangential
\(\beta\) with \(|\beta|=r\), the original differentiated equation is \[
 PD_z^\beta u
 =D_z^\beta f-
   \sum_{|\alpha|\leq2}\sum_{0<\gamma\leq\beta}
       \binom\beta\gamma(D_z^\gamma a_\alpha)
                 D^\alpha D_z^{\beta-\gamma}u.
 \tag{BO5}
\] The right side is \(L^2\): at least one derivative in each commutator
term falls on the original coefficient, leaving at most \(r+1\)
derivatives on \(u\). The function \(D_z^\beta u\) belongs to \(H^1\),
and its trace is zero by (B4) and approximation. Applying (B22) on
nested patches gives all derivatives of total order \(r+2\) having at
most two normal factors.

Retain \(a=a_{2e_t}\) and the complete original equation in the form \[
 aD_t^2u=f-\sum_{\substack{|\alpha|\leq2\\\alpha\ne2e_t}}
                         a_\alpha D^\alpha u.
 \tag{B26}
\] For a target derivative with \(q\geq3\) normal factors, write it as
\(D^{\beta+2e_t}u\), where \(|\beta|=r\) and \(\beta_t=q-2\). The full
differentiated identity is \[
 \begin{split}
 aD^{\beta+2e_t}u
 ={}&D^\beta f
 -\sum_{\substack{|\alpha|\leq2\\\alpha\ne2e_t}}
      \sum_{\gamma\leq\beta}
       \binom\beta\gamma(D^\gamma a_\alpha)
                           D^{\beta-\gamma+\alpha}u\\
 &-\sum_{0<\gamma\leq\beta}
       \binom\beta\gamma(D^\gamma a)
                           D^{\beta-\gamma+2e_t}u.
 \end{split}
 \tag{BO6}
\] In the first sum the terms with \(\gamma=0\) have at most \(q-1\)
normal factors. Every other term on \(u\), including every
differentiated original normal coefficient in the last sum, has total
order at most \(r+1\). Increasing induction on \(q\), beginning with the
tangentially obtained orders, therefore makes the entire right side
\(L^2\). The weak identity (BO2), or its smooth coefficient version,
allows multiplication by the actual \(a^{-1}\), which supplies precisely
the target derivative and its norm bound. None of the derivatives of
\(a\) in (BO6) has been removed. Cutoffs between \(Y\) and \(X\), the
full product formula (BO1), and preceding-order estimates control all
localization terms. There are finitely many multi-indices at each order,
proving (B25) with its stated finite constant. \(\square\)

There is, in addition, \textbf{one} local inverse acting on the whole
half-space, whose higher-regularity property holds for all such
integers. Choose a patch \(X_1\) on which (B17) is constructed, a
smaller \(X\), and smooth cutoffs \(\eta_0,\eta_1\) with compact support
in \(X_1\), where \(\eta_0=1\) near \(\overline X\) and \(\eta_1=1\)
near \(\operatorname{supp}\eta_0\). With \(E_1\) the inverse on
\((X_1)_+\), define \[
 \mathcal E f=\bigl[\eta_0 E_1(\eta_1f|_{(X_1)_+})\bigr]^0_{
 \text{outside }X_1\text{ within the half-space}}. \tag{B27}
\] The cutoff vanishes near the artificial boundary, so the indicated
extension is bounded into \(H^2(\mathbb R^n_+)\). On \(X_+\),
\(P\mathcal Ef=f\) and its trace is zero. If \(u\in C_c^\infty(X)\) has
zero trace, then \(\eta_1Pu=Pu\), so (B17) gives \(\mathcal EPu=u\) on
\(X_+\). Finally (B25), applied on a neighborhood of
\(\operatorname{supp}\eta_0\), gives \[
 \mathcal E:H^r(\mathbb R^n_+)\longrightarrow H^{r+2}(\mathbb R^n_+)
 \quad\text{bounded for every integer }r\geq0. \tag{B28}
\] The operator in (B27) is fixed before \(r\) is chosen. Global
half-space boundedness and the local inverse identities are separate
statements.

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\subsection{8. Global regularity of the energy
solution}\label{global-regularity-of-the-energy-solution}

Under the assumptions of Section 3, suppose now that \(\partial\Omega\)
is \(C^{1,1}\) and the coefficients \(a_{jk}\) are Lipschitz on
\(\overline\Omega\). Then the solution of (B10) satisfies \[
 u\in H^2(\Omega),\qquad \|u\|_{H^2(\Omega)}\leq C\|f\|_2. \tag{B29}
\] If the coefficients, boundary, and forcing are smooth up to the
boundary, then \(u\in C^\infty(\overline\Omega)\).

\textbf{Proof.} On an interior patch, expansion of divergence form gives
\(Pu=\sum a_{jk}D_jD_ku+\sum(D_ja_{jk})D_ku\). The second sum is
\(L^2\), since its coefficients are bounded almost everywhere. The
interior theorem in Section 8 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates} applies to the principal part with
\(p=2,m=2\), giving a local \(H^2\) bound in terms of
\(\|f\|_2+\|u\|_{H^1}\).

In a boundary patch use a \(C^{1,1}\) flattening \(x=F(y)\). Weak
\(H^1\) chain rules were proved in Section 2. The transformed principal
coefficient matrix is obtained by multiplying \(A\) on its two sides by
the inverse derivative of \(F\); it is real positive definite, uniformly
so on a smaller compact patch. Its entries are Lipschitz: \(A\circ F\)
is Lipschitz, and the first derivative of the \(C^{1,1}\) diffeomorphism
and its inverse are Lipschitz on that patch. The additional first-order
coefficients involve first weak derivatives of \(A\) and weak second
derivatives of \(F\), and are bounded almost everywhere. One can first
transform the divergence-form weak identity by change of variables and
then expand it; this justifies the computation for \(u\in H^1\) without
assuming second derivatives. The Jacobian and its inverse are bounded.
The trace is zero by the characterization after (B6). Thus (B22),
including its bounded-lower-coefficient case, gives the desired local
estimate. Here are the weak chart details needed at this lower boundary
regularity. Locally the boundary is a graph \(x_n=h(x')\) with
\(h\in C^{1,1}\); take \[
 F(y',t)=(y',t+h(y')),\qquad
 F^{-1}(x)=(x',x_n-h(x')),\qquad \det DF=1.
 \tag{BR1}
\] On each fixed compact patch \(DF,DF^{-1}\) are bounded and Lipschitz,
and every weak second derivative of either map is bounded. A general
fixed rigid change before this graph map retains these properties and
its actual Jacobian. The \(H^1\) composition result in Section 2
applies, since these are \(C^1\) diffeomorphisms. For a smooth function
\(v\), its first composed derivatives are products of \(Dv\circ F\) with
entries of \(DF\). Applying the Lipschitz weak product rule to these
products gives the full formula, almost everywhere, \[
 \partial_i\partial_j(v\circ F)
 =\sum_{a,b}(\partial_a\partial_bv)\circ F\,
       \partial_iF_a\,\partial_jF_b
   +\sum_a(\partial_av)\circ F\,\partial_i\partial_jF_a.
 \tag{BR2}
\] Every term is in \(L^2\), with its norm bounded by a fixed multiple
of the \(H^2\) norm of \(v\) on the corresponding patch. Approximate a
localized half-space \(H^2\) function by the extension and convolution
in Section 2. The zeroth, first and second expressions in (BR2) are
Cauchy in \(L^2\); testing against compactly supported smooth functions
identifies their limits with the weak derivatives of the composed
function. This proves the bounded \(H^2\) composition map by \(F^{-1}\)
as well. The input energy solution needs only the already proved \(H^1\)
composition until the flattened boundary theorem supplies its \(H^2\)
regularity.

For clarity, the original divergence-form weak identity transforms with
the full coefficient \[
 \mathcal A(y)=|\det DF(y)|\,DF(y)^{-1}
                    A(F(y))\,DF(y)^{-T}.
 \tag{BR3}
\] Change of variables in the two first-derivative integrals proves this
identity for \(H^1\) inputs and tests. Divide by the positive Jacobian
to write the transformed distributional equation on coordinate volume.
Expanding the divergence uses weak first derivatives of the Lipschitz
entries of \(\mathcal A\); they are bounded. The transformed principal
matrix is \(DF^{-1}A(F)DF^{-T}\), and is uniformly positive on the
retained compact patch. All remaining terms involve bounded coefficient
derivatives, the retained Jacobian derivatives and first derivatives of
the solution. The right side is \(f\circ F\), with its actual volume
norm. This proves the nondivergence-form interface to (B22) under
\(C^{1,1}\), without assuming classical second chart derivatives. After
(B22), (BR2) transfers the resulting \(H^2\) function back. Thus (B29)
includes \(C^2\) boundaries and the larger \(C^{1,1}\) class with the
same original operator and energy domain.

A finite cover of the compact closure by these boundary patches and
interior patches, with cutoffs, yields
\(\|u\|_{H^2}\leq C(\|f\|_2+\|u\|_{H^1})\). Bound (B10) removes the
final term and proves (B29). For smooth data use (B25) in every boundary
chart and its interior version, or the same differentiated-equation
argument without a boundary. This gives every integer Sobolev order. To
conclude classical smoothness, extend each localized function by the
integer extension after (B4); for \(k>j+n/2\), Cauchy-Schwarz makes
\(\xi^\alpha\widehat u\in L^1\), \(|\alpha|\leq j\). Fourier inversion
and dominated convergence give continuous derivatives through order
\(j\). Since \(j\) is arbitrary, smoothness holds up to the boundary.
\(\square\)

\textbf{A boundary with a bounded second weak derivative and no
classical second derivative.} In a two-dimensional chart take
\(h(z)=z|z|\), with the domain above its graph. Then \(h'(z)=2|z|\) is
Lipschitz, and \(h''=2\operatorname{sgn}z\) almost everywhere; the
second derivative has no value at zero. The graph flattening (BR1) has
determinant one and the transformed full negative Laplacian is \[
 -\partial_z^2+4|z|\partial_z\partial_t
       -(1+4z^2)\partial_t^2
       +2\operatorname{sgn}z\,\partial_t
 =-\operatorname{div}\!
   \left(\begin{pmatrix}1&-2|z|\\-2|z|&1+4z^2\end{pmatrix}
                      \nabla\right).
 \tag{BR7}
\] Both formulas act weakly. The matrix determinant is one and its
quadratic form is \((\xi_z-2|z|\xi_t)^2+\xi_t^2\); on compact patches it
is uniformly positive. Its entries are Lipschitz, and its displayed
lower-order coefficient is bounded. Thus this graph satisfies the exact
boundary regularity interface although it is not \(C^2\). It is a local
chart example; the global theorem still requires the full bounded domain
and its complete boundary to be \(C^{1,1}\).

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio]{../figures/boundary-weak-chart.png}}
\caption{The exact graph h(z)=z\textbar z\textbar, its flattening, and
the full transformed coefficient matrix.}
\end{figure}

The two graph pieces and the transported grid are numerical samples of
the exact formulas (BR1) and (BR7). The weak derivative has values \(2\)
and \(-2\) on the two open sides; its value at zero does not affect the
operator. The \(H^2\) receiving map is proved in (BR2)--(BR3) and (B29).
The weaker chart hypothesis is established here from the displayed weak
derivative calculation, without a novelty claim.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-SBD-009}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-SBD-009}\label{AN03-SBD-009}\endgroup\ReaderOriginalAnchorStart{an03-sbd-009-complex-quadratic-symbols-and-the-low-frequency-repair}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an03-sbd-009-complex-quadratic-symbols-and-the-low-frequency-repair}\label{an03-sbd-009-complex-quadratic-symbols-and-the-low-frequency-repair}\endgroup\ReaderOriginalAnchorStart{9-complex-quadratic-symbols-and-the-low-frequency-repair}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{9-complex-quadratic-symbols-and-the-low-frequency-repair}\label{9-complex-quadratic-symbols-and-the-low-frequency-repair}\endgroup\ReaderOriginalAnchorStart{complex-quadratic-symbols-and-the-low-frequency-repair}\ReaderOriginalAnchorEnd}
\subsection{9. Complex quadratic symbols and the low-frequency
repair}\label{complex-quadratic-symbols-and-the-low-frequency-repair}

The construction below supplies the complex-coefficient extension of the
local Dirichlet inverse and, with smooth coefficients, its simultaneous
integer Sobolev improvements. An auxiliary boundary at a fixed positive
height removes the low-frequency obstruction. It disappears from the
final local assertions.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-SBD-ROOTS-001}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-SBD-ROOTS-001}\label{AN03-SBD-ROOTS-001}\endgroup\ReaderOriginalAnchorStart{unordered-roots-simple-root-coordinates-and-the-tangential-sphere}\ReaderOriginalAnchorEnd}
\subsubsection{9.0. Unordered roots, simple-root coordinates and the
tangential
sphere}\label{unordered-roots-simple-root-coordinates-and-the-tangential-sphere}

We prove the finite-dimensional root and sphere facts before using them
in the normal-mode construction. The full factorization theorem, with
the original leading coefficient and every multiplicity, is proved in
Sections9.1--9.4 of
\href{stable-prerequisite-bridges.pdf}{Polynomial and
contour tools}. Closed bounded finite-coordinate sets are compact by
Sections12.6--12.9 of \href{metric-foundation-bridges.pdf}{Metric
foundations}. The actual inverse and implicit maps, including their full
remainders and all higher derivatives, are proved in Sections16.4--16.5
of
\href{geometric-microlocal-calculus.pdf}{Geometric
microlocal calculus}. Below we identify the exact polynomial and real
derivative matrices to which those proofs apply. No half-plane root
count is used to prove its own continuity prerequisite.

\paragraph{The original quadratic, including a repeated
root}\label{the-original-quadratic-including-a-repeated-root}

Let \[
 Q(z)=az^2+bz+c,\qquad a\ne0,\qquad
 \mathcal D=b^2-4ac,\qquad d^2=\mathcal D.
 \tag{RC1}
\] The complex-root theorem supplies a square root of the actual number
\(\mathcal D\). When \(\mathcal D=0\), its square root is zero. If
\(\mathcal D\ne0\), its two square roots are \(d,-d\): any other square
root \(e\) obeys \((e-d)(e+d)=0\), so one factor is zero. Put \[
 z_1=\frac{-b+d}{2a},\qquad z_2=\frac{-b-d}{2a},\qquad
 Q(z)=a(z-z_1)(z-z_2).
 \tag{RC2}
\] To prove the last identity, add and multiply the first two
expressions: their sum is \(-b/a\), and their product is
\((b^2-d^2)/(4a^2)=c/a\). Expanding the full last product therefore
gives exactly \(az^2+bz+c\). The original leading factor \(a\), both
occurrences of a repeated root, and the entire discriminant remain.
Changing \(d\) to \(-d\) exchanges the two entries and changes no
unordered pair. When \(\mathcal D=0\), both entries are the original
double root \(-b/(2a)\).

For two unordered root lists \(R=(z_1,z_2)\), \(R'=(w_1,w_2)\), define
their distance by \[
 d_2(R,R')=
 \min\{\max(|z_1-w_1|,|z_2-w_2|),
          \max(|z_1-w_2|,|z_2-w_1|)\}.
 \tag{RC3}
\] Permuting either list exchanges the two candidates, so this is
well-defined for unordered pairs with multiplicity. It is symmetric and
nonnegative, and it vanishes exactly when the lists agree after a
permutation. The triangle inequality follows by composing minimizing
permutations for two consecutive comparisons and applying the ordinary
complex triangle inequality to each matched entry, then taking the
maximum. Thus RC3 is a metric on these unordered pairs. It does not
choose a continuous square-root sign or discard repeated roots.

Suppose the original coefficient triples \((a_k,b_k,c_k)\) tend to
\((a,b,c)\), with \(a\ne0\). For all sufficiently large \(k\),
\(|a_k|\geq|a|/2\), and \(|b_k|+|c_k|\) has a finite bound \(M\). Every
root \(z\) of the actual \(Q_k\) with \(|z|\geq1\) satisfies \[
 |a_k|\,|z|^2\leq |b_k|\,|z|+|c_k|
       \leq (|b_k|+|c_k|)|z|,\qquad
 |z|\leq\frac{2M}{|a|}.
 \tag{RC4}
\] Consequently all roots lie in the fixed disk of radius \(1+2M/|a|\).
List the two roots in nondecreasing order of their original real
coordinates and then their imaginary coordinates, retaining repeats.
This selects representatives of the unordered pairs; no continuity of
the ordering is asserted. Finite-coordinate compactness gives a
subsequence on which both entries converge, say to \(u,v\). Passing to
the limit in the complete factorization, or in each of its three
coefficients, gives \[
 az^2+bz+c=a(z-u)(z-v).
 \tag{RC5}
\] Uniqueness of multiplicities in the full factorization theorem says
that \((u,v)\) is exactly the original unordered root pair of \(Q\). If
convergence in RC3 failed, a subsequence would have distance at least
some \(\varepsilon>0\) from that pair. Applying the preceding
compactness argument to this subsequence gives a further subsequence
whose distance tends to zero, a contradiction. This also proves
epsilon-delta continuity: its failure would provide coefficient triples
within \(1/k\) of the original triple whose root distances stay at least
one fixed positive epsilon, contradicting the just-proved sequence
result. Thus continuity holds at every original coefficient triple with
nonzero leading coefficient, including the discriminant-zero locus. An
ordered continuous choice was neither assumed nor needed.

The following finite-degree extension is separate from the quadratic
entry, and uses the same full factorization proof. For original
polynomials \[
 P_k(z)=\sum_{j=0}^m a_{j,k}z^j,\quad
 a_{j,k}\longrightarrow a_j,\quad a_m\ne0,\qquad
 P(z)=\sum_{j=0}^m a_jz^j,
 \tag{RC6}
\] keep the degree \(m\), leading coefficient and all \(m\) root
occurrences. For \(|z|\geq1\), the original root equation gives
\( |a_{m,k}||z|^m\leq\sum_{j=0}^{m-1}|a_{j,k}||z|^j
\leq |z|^{m-1}\sum_{j=0}^{m-1}|a_{j,k}|\). The nonvanishing leading
coefficient and bounded complete lower-coefficient sum give a fixed disk
containing every root. Any sequence of full root lists has a
coordinatewise convergent subsequence. In the limit the original
identity is \(P(z)=a_m\prod_{j=1}^m(z-u_j)\). Full factorization
uniqueness identifies every limit occurrence with its correct
multiplicity. The same contradiction proves convergence in \[
 d_m(R,R')=\min_{\sigma\in\mathfrak S_m}
                \max_{1\leq j\leq m}|z_j-w_{\sigma(j)}|.
 \tag{RC7}
\] The proof that this is an unordered-multiset metric is exactly the
composition-of-permutations argument, with all \(m\) matched terms
retained. A nonzero constant has the empty root list and distance zero;
the zero polynomial and a vanishing degree-\(m\) leading coefficient are
outside this theorem. No polynomial is replaced by a monic one.

\paragraph{The exact real implicit map at a simple complex
root}\label{the-exact-real-implicit-map-at-a-simple-complex-root}

Let the coefficients of the original \(P(s,z)=\sum_{j=0}^m a_j(s)z^j\)
be \(C^r\) in the original real parameter coordinates \(s\),
\(1\leq r\leq\infty\). Suppose \(P(s_0,z_0)=0\) and \[
 w=\partial_zP(s_0,z_0)
    =\sum_{j=1}^m j a_j(s_0)z_0^{j-1}\ne0.
 \tag{RC8}
\] Write \(z=x+iy\), and apply the real implicit theorem to the actual
map
\(F(s,x,y)=(\operatorname{Re}P(s,x+iy),\operatorname{Im}P(s,x+iy))\).
Finite polynomial expansion of the increment in \(z\) gives its first
term \(w(\Delta x+i\Delta y)\) and its full higher-order remainder. Thus
its actual two by two derivative and inverse at the original point are
\[
 A=\begin{pmatrix}\operatorname{Re}w&-\operatorname{Im}w\\
                   \operatorname{Im}w&\operatorname{Re}w\end{pmatrix},
 \qquad \det A=|w|^2,\qquad
 A^{-1}=\frac1{|w|^2}
       \begin{pmatrix}\operatorname{Re}w&\operatorname{Im}w\\
                       -\operatorname{Im}w&\operatorname{Re}w\end{pmatrix}.
 \tag{RC9}
\] Both matrix products are the identity by direct multiplication. The
denominator is positive by RC8. The unchanged augmented map
\((s,x,y)\mapsto(s,F(s,x,y))\) therefore has precisely the invertible
block derivative of IV9 in the independent implicit proof. That proof
supplies original neighborhoods and a unique \(C^r\) root \(z(s)\) in
the chosen root neighborhood, with \(z(s_0)=z_0\). It retains the
original coefficient functions, parameter coordinates and Euclidean
complex norm; no replacement of \(A\) by the identity is used.

Differentiating the actual equation gives, in every original parameter
direction \(v\), \[
 D_sz(s)[v]=-
    \frac{\displaystyle\sum_{j=0}^m(D_sa_j(s)[v])z(s)^j}
         {\displaystyle\sum_{j=1}^m j a_j(s)z(s)^{j-1}}.
 \tag{RC10}
\] This complex formula is the real matrix expression
\(-A(s)^{-1}D_sF(s,z(s))[v]\), since RC9 is multiplication by the
reciprocal of the actual complex derivative. The full numerator,
denominator, signs and all coefficients remain. For the original
quadratic the denominator is \(2a(s)z(s)+b(s)\), and the numerator is
\(D_sa(s)[v]z(s)^2+D_sb(s)[v]z(s)+D_sc(s)[v]\).

For clarity, every higher derivative is also governed by the complete
original implicit identity. Put
\(Z(s)=(s,\operatorname{Re}z(s),\operatorname{Im}z(s))\). For a finite
set of direction labels and \(2\leq N\leq r\), let \(D_BZ\) retain
exactly the labels in the nonempty block \(B\). In the full
labeled-partition chain rule for \(F(Z(s))=0\), the one-block term is
\(A(s)D^Nz(s)\), with the complex derivative read in its two real
coordinates: the parameter component of \(D^NZ\) is zero for \(N\geq2\).
Hence \[
 D^Nz(s)[v_1,\ldots,v_N]
 =-A(s)^{-1}
  \sum_{\substack{\Pi\in\mathfrak P(\{1,\ldots,N\})\\|\Pi|\geq2}}
    D^{|\Pi|}F(Z(s))[D_BZ(s):B\in\Pi].
 \tag{RC11}
\] Both sides of RC11 are read as their original real and imaginary
coordinate columns; the inverse coordinate map returns the complex
derivative. Every real parameter block, coefficient derivative and
repeated-direction multiplicity remains. The chain rule is proved by
inserting the next label in an existing block or adding its singleton
block; those two possibilities give every new partition once. IV7--IV10
prove the needed smooth inverse and preserve every ordered matrix
factor. Thus RC11 is an exact identity at every available order,
including coefficients that are only \(C^r\).

\paragraph{Continuous paths on the original tangential
sphere}\label{continuous-paths-on-the-original-tangential-sphere}

Let \(d\geq1\) and keep the original sphere
\(S^d=\{u\in\mathbb R^{d+1}:|u|=1\}\). For \(u,v\in S^d\) with
\(v\ne-u\), the original segment \((1-t)u+tv\) never vanishes on
\([0,1]\): a zero in the interior would make \(v\) a negative multiple
of \(u\), and their original unit lengths would force \(v=-u\). The
explicit radial map from \(\mathbb R^{d+1}\setminus\{0\}\) to the sphere
gives the path \[
 q_{u,v}(t)=\frac{(1-t)u+tv}{|(1-t)u+tv|},\qquad
 q_{u,v}(0)=u,\quad q_{u,v}(1)=v.
 \tag{RC12}
\] Its denominator is continuous and positive, so the path is continuous
and belongs to the original sphere. If \(v=-u\), choose an original
coordinate \(j\) with \(|u_j|<1\). Such a coordinate exists because
there are at least two coordinates and \(\sum u_j^2=1\). The exact
vector and its squared length are \[
 w_j=e_j-u_ju,\qquad u\cdot w_j=0,\qquad
 |w_j|^2=1-u_j^2>0,\qquad
 v_j=\frac{e_j-u_ju}{\sqrt{1-u_j^2}}\in S^d.
 \tag{RC13}
\] The vector \(v_j\) is orthogonal to \(u\), so neither pair
\((u,v_j)\), \((v_j,-u)\) is antipodal. Concatenate RC12 for those
pairs: use \(q_{u,v_j}(2t)\) on \([0,1/2]\) and \(q_{v_j,-u}(2t-1)\) on
\([1/2,1]\). Both values at the join are \(v_j\). This constructs a
continuous path between every two original sphere points, retaining
every coordinate and denominator.

To conclude connectedness, a separation into two nonempty relatively
open sets would give a continuous indicator with values zero and one. A
path joining a point of each set would give a continuous real function
on \([0,1]\) taking both endpoint values, contradicting the intermediate
value theorem at \(1/2\). Thus \(S^d\) is connected. The
zero-dimensional sphere \(S^0=\{-1,1\}\) has its two singleton open
components and is not connected; no positive-dimensional argument is
applied to it.

\paragraph{Exact use by the original homogeneous
quadratic}\label{exact-use-by-the-original-homogeneous-quadratic}

Return to the original \[
 p(\xi',\tau)=a\tau^2+b(\xi')\tau+c(\xi'),\quad
 a=p(0,1)\ne0,\quad p(\xi)\ne0\ (\xi\in\mathbb R^n\setminus\{0\}).
 \tag{RC14}
\] For \(\omega\in S^{n-2}\), neither normal root is real: a real root
would give the forbidden nonzero real vector \((\omega,\tau)\). At a
fixed \(\omega\), put
\(\varepsilon=\frac12\min_j|\operatorname{Im}z_j|>0\), retaining both
occurrences. RC3 continuity on an original coefficient neighborhood
pairs both perturbed roots within \(\varepsilon\) of the fixed ones.
Their imaginary signs cannot change. Consequently the number
\(N_+(\omega)\) of upper roots, including multiplicity, is locally
constant. Composition with any RC12--RC13 path is a continuous
integer-valued function on an interval, hence constant by the
intermediate value theorem. When \(n>2\), connectedness makes the upper
count constant on the entire tangential sphere.

The original odd linear coefficient and even quadratic coefficient give,
without discarding any term, \[
 p(-\omega,\tau)=a\tau^2-b(\omega)\tau+c(\omega)
                =p(\omega,-\tau),\qquad
 N_+(-\omega)=2-N_+(\omega).
 \tag{RC15}
\] The constant count therefore satisfies \(N_+=2-N_+\), so \(N_+=1\).
Each root is simple because one is in each open half-plane. RC8--RC11
apply to the original polynomial at each nonzero tangential frequency.
Uniqueness in the upper and lower half-planes makes the smooth local
labels agree on every overlap, producing the actual global
\(\lambda_+,\lambda_-\) there.

For \(r>0\), full homogeneity gives \[
 p(r\xi',r\tau)=r^2p(\xi',\tau),\qquad
 p(r\xi',\tau)=r^2p(\xi',\tau/r),\qquad
 \lambda_\pm(r\xi')=r\lambda_\pm(\xi').
 \tag{RC16}
\] The first two identities retain the original \(a\), both homogeneous
coefficient terms and all powers of \(r\). The last follows from their
exact root correspondence and the signs of the imaginary parts. The
antipodal identities also give \(\lambda_+(-\xi')=-\lambda_-(\xi')\) and
\(\lambda_-(-\xi')=-\lambda_+(\xi')\); the labels are exchanged as
required.

On the original compact sphere, define the actual constants \[
 c_0=\min\{\min_\omega\operatorname{Im}\lambda_+(\omega),
            \min_\omega[-\operatorname{Im}\lambda_-(\omega)]\}>0,
 \qquad
 C_0=\max_\omega(|\lambda_+(\omega)|+|\lambda_-(\omega)|)>0.
 \tag{RC17}
\] Continuity and attained extrema prove the positivity and finiteness:
each imaginary quantity is strictly positive everywhere and has an
attained minimum. RC16 gives exactly CM1 with these constants, and also
\(|\lambda_+-\lambda_-|\geq2c_0|\xi'|\). At every multi-index
\(\alpha\), differentiating the original scaling identity retains the
factor \(r^{|\alpha|}\), and yields \[
 r^{|\alpha|}(\partial_{\xi'}^\alpha\lambda_\pm)(r\xi')
     =r(\partial_{\xi'}^\alpha\lambda_\pm)(\xi'),\qquad
 |\partial_{\xi'}^\alpha\lambda_\pm(\xi')|
 \leq C_{\alpha,\pm}|\xi'|^{1-|\alpha|},
 \quad
 C_{\alpha,\pm}=\max_\omega
       |\partial_{\xi'}^\alpha\lambda_\pm(\omega)|.
 \tag{RC18}
\] This supplies every original smooth-label derivative with its full
frequency power. It makes no assertion of smooth labels through a root
collision or through zero frequency.

When \(n=2\), the sphere is \(S^0\). Keep the additional original
hypothesis of one root in each half-plane at \(\xi'=1\). RC15 then gives
one in each half-plane at \(-1\), and RC16 handles every nonzero
tangential frequency. RC17--RC18 hold on the finite two-point sphere.
Ellipticity alone does not supply that hypothesis; the repeated-root
example below retains its exact counterexample. When \(n=1\), there is
no nonzero tangential frequency and no sphere argument: the original
operator is \(aD_t^2\), whose interval inverse is calculated below with
the same original \(a\). These are precisely the dimensions used by the
scalar normal-mode construction.

\ReaderArmAliases{\ReaderOriginalAnchorStart{precise-prerequisites}\ReaderOriginalAnchorEnd}
\subsubsection{Precise prerequisites}\label{precise-prerequisites}

The finite-dimensional ingredients now have full independent proofs:
original complex factorization in Polynomial and contour tools
Sections9.1--9.4; unordered root continuity, including collisions and
every multiplicity, and exact simple-root implicit coordinates in
Section9.0; original finite-coordinate compactness in Metric foundations
Sections12.5--12.9; and positive-dimensional sphere paths in Section9.0.
The complete root-count and quantitative-separation receiving argument
is proved in RC14--RC18 and CM1. The planar extra root condition and the
one-dimensional interval case remain explicit.

Use Plancherel and the derivative rule for the Fourier transform on
\(\mathbb R^{n-1}\), Fubini and Fatou's lemma, elementary weak
derivatives, and the characterization of integer \(H^m\) by
square-integrable derivatives. The tangential Fourier normalization may
be unitary; using the course convention
\(\widehat f(\xi')=\int e^{-ix'\cdot\xi'}f(x')\,dx'\) merely inserts the
same factor \((2\pi)^{1-n}\) in both Plancherel integrals. Also use the
one-dimensional fundamental theorem for \(H^1\), the trace map proved in
Section 2, completeness of \(L^2\)/\(H^2\), density of smooth functions
for identifying weak derivatives, and the elementary bounded inverse
series \((I+A)^{-1}=\sum_{k\geq0}(-A)^k\) when \(\|A\|<1\). The weak
difference-quotient implication used in the smooth refinement is proved
below, so it is not an unnamed elliptic regularity import.

Only elementary scalar ODE algebra is needed: a solution of
\((D-\lambda_+)(D-\lambda_-)v=0\), with distinct roots, is a linear
combination of \(e^{it\lambda_+}\) and \(e^{it\lambda_-}\). This follows
either by integrating each first-order equation, or by Sections 1--2 of
\href{stable-boundary-models.pdf}{Stable modes and the algebra of
boundary data}. No general boundary pseudodifferential calculus,
parameter ellipticity, coercive variational theorem, or unproved global
inverse for a homogeneous symbol is used.

\ReaderArmAliases{\ReaderOriginalAnchorStart{cm1-root-separation-at-every-nonzero-tangential-frequency}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{cm1-root-separation-at-every-nonzero-tangential-frequency}\label{cm1-root-separation-at-every-nonzero-tangential-frequency}\endgroup\ReaderOriginalAnchorStart{root-separation-at-every-nonzero-tangential-frequency}\ReaderOriginalAnchorEnd}
\subsubsection{Root separation at every nonzero tangential
frequency}\label{root-separation-at-every-nonzero-tangential-frequency}

Let \(p\) be a complex homogeneous polynomial of degree two on
\(\mathbb R^n\), elliptic in the sense

\[
p(\xi)\ne0\qquad(\xi\in\mathbb R^n\setminus\{0\}).
\]

Write

\[
p(\xi',\tau)=a\tau^2+b(\xi')\tau+c(\xi'),\qquad a=p(0,1)\ne0,
\]

where \(b\) and \(c\) are homogeneous of degrees one and two. Assume
that, for every \(\xi'\ne0\), there is one normal root in each open
half-plane. Denote the roots by \(\lambda_+(\xi')\) and
\(\lambda_-(\xi')\), their signs referring to their imaginary parts.
They are distinct, depend smoothly on \(\xi'\ne0\), are positively
homogeneous of degree one, and for constants \(c_0,C_0>0\),

\[
\operatorname{Im}\lambda_+\geq c_0|\xi'|,
\quad-\operatorname{Im}\lambda_-\geq c_0|\xi'|,
\quad |\lambda_+|+|\lambda_-|\leq C_0|\xi'|.
\tag{CM1}
\]

Here is why the assumption on root counts is automatic when \(n>2\). On
the tangential unit sphere, no root can meet the real axis, by
ellipticity. The number of roots in the upper half-plane, counted with
multiplicity, is consequently locally constant. One can see this without
any chosen labeling: the two roots depend continuously as an unordered
pair on the coefficients, and a small perturbation keeps each root away
from the real axis. The sphere \(S^{n-2}\) is connected for \(n>2\), so
this count is constant. Homogeneity gives

\[
p(-\xi',\tau)=p(\xi',-\tau),
\]

which exchanges upper and lower root counts. The constant upper count
must therefore be one. For \(n=2\), it suffices to check \(\xi'=1\); the
displayed identity gives the same conclusion at \(\xi'=-1\), and
positive homogeneity handles all other nonzero values. The
opposite-half-plane roots are simple, so the implicit function theorem
gives their smooth local labels; the sign of the imaginary part makes
those labels consistent. Compactness of the unit sphere, continuity of
the roots, and the absence of real roots give (CM1).

In dimension one no root-count assumption is needed: \(p(D)=aD_t^2\),
\(a\ne0\), and CM3 below is simply the ordinary interval Dirichlet
inverse with its explicit kernel at \(\xi'=0\).

\ReaderArmAliases{\ReaderOriginalAnchorStart{cm2-the-half-line-kernel-and-its-high-frequency-bounds}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{cm2-the-half-line-kernel-and-its-high-frequency-bounds}\label{cm2-the-half-line-kernel-and-its-high-frequency-bounds}\endgroup\ReaderOriginalAnchorStart{the-half-line-kernel-and-its-high-frequency-bounds}\ReaderOriginalAnchorEnd}
\subsubsection{The half-line kernel and its high-frequency
bounds}\label{the-half-line-kernel-and-its-high-frequency-bounds}

Fix \(\xi'\ne0\), put \(r=|\xi'|\), \(\Delta=\lambda_+-\lambda_-\), and
suppress \(\xi'\) in formulas. For \(t,s>0\), define

\[
H(t,s)=\frac{i}{a\Delta}
\left[
\begin{cases}
e^{i\lambda_+(t-s)},&t\geq s,\\
e^{i\lambda_-(t-s)},&t<s
\end{cases}
-e^{i\lambda_+t-i\lambda_-s}
\right].
\tag{CM2}
\]

For each fixed \(s\), this is continuous at \(t=s\), vanishes at
\(t=0\), and solves the homogeneous ODE away from \(s\). Its ordinary
first-derivative jump is

\[
\partial_tH(s+,s)-\partial_tH(s-,s)
=\frac{i}{a\Delta}\,i\Delta=-\frac1a.
\]

Since the coefficient of \(\partial_t^2\) in \(p(\xi',D_t)\) is \(-a\),
the jump contributes exactly \(\delta_s\). There is no delta derivative,
because \(H\) is continuous. Thus

\[
p(\xi',D_t)H(t,s)=\delta_s(t).
\tag{CM3}
\]

The signs in (CM2) are fixed by this jump calculation. For example, if
\(p=\tau^2+r^2\), it becomes

\[
H(t,s)=\frac{e^{-r|t-s|}-e^{-r(t+s)}}{2r},
\]

the positive kernel for \(-\partial_t^2+r^2\) with zero initial boundary
value.

For \(k=0,1,2\), differentiating the regular pieces gives

\[
|\partial_t^kH(t,s)|
\leq C_k r^{k-1}
\bigl(e^{-c_0r|t-s|}+e^{-c_0r(t+s)}\bigr),\qquad t\ne s.
\tag{CM4}
\]

For \(k=2\), this is the bound on the ordinary piecewise derivative; the
distributional derivative also contains \(-a^{-1}\delta_s\). Integration
in either variable of the right-hand side is at most \(C_kr^{k-2}\). The
elementary Schur estimate therefore gives

\[
\left\|\partial_t^k\int_0^\infty H(t,s)g(s)\,ds\right\|_{L^2_t}
\leq C_k r^{k-2}\|g\|_{L^2_s},\qquad k=0,1,2,
\tag{CM5}
\]

where the delta term is included in the constant when \(k=2\). For
completeness, if a kernel \(K\) satisfies \(\sup_t\int|K(t,s)|ds\leq M\)
and \(\sup_s\int|K(t,s)|dt\leq N\), Cauchy--Schwarz with measure
\(|K(t,s)|ds\), followed by Fubini, gives
\(\|Kg\|_2^2\leq MN\|g\|_2^2\). This is the estimate just used.

Formula (CM5) is a parameter estimate, not an \(L^2\to H^2\) inverse at
all frequencies: its zeroth-order bound is \(Cr^{-2}\). The next step
keeps the exact ODE inverse and repairs this low-frequency behavior.

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\subsubsection{\texorpdfstring{A full \(L^2\to H^2\) inverse on an
auxiliary
slab}{A full L\^{}2\textbackslash to H\^{}2 inverse on an auxiliary slab}}\label{a-full-l2to-h2-inverse-on-an-auxiliary-slab}

Fix \(L>0\) and let \(S_L=\mathbb R^{n-1}\times(0,L)\). Define

\[
\phi_{\xi'}(t)=\frac{e^{i\lambda_+t}-e^{i\lambda_-t}}{i\Delta},
\qquad h_{\xi'}(t)=\frac{\phi_{\xi'}(t)}{\phi_{\xi'}(L)}.
\tag{CM6}
\]

For \(\xi'\ne0\), the denominator is nonzero: the two exponentials at
\(L\) have different absolute values. At \(\xi'=0\), set \(\phi_0(t)=t\)
and \(h_0(t)=t/L\). These are the continuous limits as both roots tend
to zero. An explicit nonsingular representation is

\[
\phi_{\xi'}(t)
=t\int_0^1 e^{it(\lambda_-+\theta\Delta)}\,d\theta.
\tag{CM7}
\]

For \(0<t,s<L\), set

\[
G_{\xi'}(t,s)=H(t,s)-h_{\xi'}(t)H(L,s),\qquad \xi'\ne0.
\tag{CM8}
\]

This kernel is zero at both \(t=0\) and \(t=L\), and (CM3) still holds
with \(G\) in place of \(H\), since the subtracted term solves the
homogeneous equation. Its value at zero tangential frequency is

\[
G_0(t,s)=\frac{\min(t,s)}a-\frac{ts}{aL}
=\frac{\min(t,s)(L-\max(t,s))}{aL}.
\tag{CM9}
\]

We prove estimates uniform in \(\xi'\). If \(rL\geq1\), rewrite

\[
h_{\xi'}(t)=
e^{i\lambda_-(t-L)}
\frac{1-e^{i\Delta t}}{1-e^{i\Delta L}}.
\tag{CM10}
\]

The denominator has absolute value at least \(1-e^{-2c_0}\), and
differentiation gives

\[
|\partial_t^kh_{\xi'}(t)|\leq C_kr^k e^{-c_0r(L-t)},\qquad k=0,1,2.
\]

Furthermore, (CM4) at \(t=L\), for \(s<L\), gives

\[
|H(L,s)|\leq Cr^{-1}e^{-c_0r(L-s)}.
\]

Consequently the \(k\)-th derivative of the correction in (CM8) is
bounded by

\[
C_kr^{k-1}e^{-c_0r(2L-t-s)}.
\]

Its integrals in either variable are at most \(C_kr^{k-2}\); the same
holds for the corresponding derivatives of \(H\), restricted to the
slab. Including the second-derivative delta term yields

\[
\|\partial_t^kG_{\xi'}g\|_{L^2(0,L)}
\leq C_kr^{k-2}\|g\|_{L^2(0,L)},\qquad rL\geq1,\quad 0\leq k\leq2.
\tag{CM11}
\]

For \(rL\leq1\), singular-looking factors \(\Delta^{-1}\) can be removed
before estimating. On the two triangles in the square, formula (CM2)
becomes

\[
H(t,s)=
\begin{cases}
a^{-1}\phi_{\xi'}(t)e^{-i\lambda_-s},&t\leq s,\\
a^{-1}e^{i\lambda_+t-i(\lambda_++\lambda_-)s}\phi_{\xi'}(s),&t\geq s.
\end{cases}
\tag{CM12}
\]

Equations (CM7) and (CM12) show that \(H\) and its ordinary derivatives
in \(t\) through order two, separately on the closed triangles, are
uniformly bounded for \(r\leq L^{-1}\). They extend continuously to zero
frequency and give \(H_0(t,s)=a^{-1}\min(t,s)\). The function
\(\phi_{\xi'}(L)\) is continuous and nonzero on the compact tangential
ball \(r\leq L^{-1}\), including its value \(L\) at zero. Its modulus
has a positive minimum. Thus \(h\), its first two derivatives, and the
regular derivatives of \(G\) are uniformly bounded there. Schur's
estimate on the finite square, together with the same fixed delta term
in \(\partial_t^2G\), proves

\[
\|G_{\xi'}g\|_{H^2(0,L)}\leq C_L\|g\|_{L^2(0,L)},\qquad rL\leq1.
\tag{CM13}
\]

Combining (CM11) and (CM13), with
\(\langle\xi'\rangle=(1+|\xi'|^2)^{1/2}\), gives

\[
\sum_{k=0}^2
\langle\xi'\rangle^{2-k}\|\partial_t^kG_{\xi'}g\|_{L^2(0,L)}
\leq C\|g\|_{L^2(0,L)}.
\tag{CM14}
\]

For \(f\in L^2(S_L)\), define \(R_Lf\) by its tangential Fourier
transform:

\[
\widehat{R_Lf}(\xi',t)=\int_0^L G_{\xi'}(t,s)\widehat f(\xi',s)\,ds.
\tag{CM15}
\]

The parameter kernels are measurable, and (CM14) and Plancherel show

\[
R_L:L^2(S_L)\longrightarrow H^2(S_L),\qquad
\|R_Lf\|_{H^2(S_L)}\leq C\|f\|_{L^2(S_L)}.
\tag{CM16}
\]

Indeed, the three terms in (CM14), squared and integrated in \(\xi'\),
control all tangential derivatives through order two, all mixed
tangential/normal derivatives of order two, and the second normal
derivative. The Fourier derivatives agree with weak derivatives by
testing against smooth compact functions and using Plancherel; no
differentiability of the kernel with respect to \(\xi'\) is needed. The
vanishing endpoint values of the kernels imply zero traces at \(t=0,L\).
This can be checked first after a bounded tangential Fourier cutoff with
smooth data in \(t\), where the integral formulas are classical, and
then by (CM16) and continuity of the trace map. Equation (CM3), applied
under the integral first to such data and then by continuity, proves
\(p(D)R_Lf=f\).

Put

\[
\mathcal D_L=\{u\in H^2(S_L):\gamma_0u=\gamma_Lu=0\}.
\]

The homogeneous equation has no nonzero element of \(\mathcal D_L\). In
fact, the partial Fourier transform of such a solution belongs to
\(H^2(0,L)\) for almost every \(\xi'\), satisfies the scalar ODE
distributionally and has both endpoint values zero. At nonzero \(\xi'\),
the general homogeneous solution and the nonzero determinant in (CM6)
force it to be zero. At \(\xi'=0\), it is affine in \(t\), and its two
boundary values again force it to be zero. The scalar ODE assertion for
an \(H^2\) solution follows by applying the first-order
integrating-factor identity successively; the result is classical and
requires no regularity theorem for PDE. Consequently

\[
p(D)R_L=I\quad\hbox{on }L^2(S_L),\qquad
R_Lp(D)=I\quad\hbox{on }\mathcal D_L.
\tag{CM17}
\]

In dimension one, (CM9), the derivative jump, and ordinary interval
integration prove (CM16)--(CM17) directly. Thus the construction also
includes arbitrary nonzero complex \(aD_t^2\).

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\subsubsection{Exact local inverse identities and small variable
coefficients}\label{exact-local-inverse-identities-and-small-variable-coefficients}

Let \(X\) be a bounded neighborhood of a point of \(t=0\), with
\(\overline{X_+}\) below height \(L\), where \(X_+=X\cap\{t>0\}\) and
\(X_0=X\cap\{t=0\}\). Extension by zero embeds \(L^2(X_+)\)
isometrically in \(L^2(S_L)\). Define

\[
E_0 f=(R_Le_{X_+}f)|_{X_+}.
\]

Then \(p(D)E_0f=f\) on \(X_+\), \(\gamma_0E_0f=0\) on \(X_0\), and
\(E_0:L^2(X_+)\to H^2(X_+)\) is bounded. If \(u\) is the restriction to
\(X_+\) of some \(\widetilde u\in C_c^\infty(X)\) with
\(\widetilde u|_{X_0}=0\), its zero extension from \(X_+\) to the slab
is in \(\mathcal D_L\). There are no artificial-side or top-boundary
derivative terms: the support is compact in \(X\). Nothing is extended
through the physical boundary \(t=0\). Hence (CM17) gives the exact left
identity

\[
E_0p(D)u=u\quad\hbox{on }X_+.
\tag{CM18}
\]

The same model works for the original nondivergence-form variable
operator

\[
P=p(D)+\sum_{|\alpha|\leq2}b_\alpha(x)D^\alpha,
\]

provided the coefficient suprema on the patch are small enough. One may
either use the local Neumann-series argument of Section 5 with \(E_0\),
or use the following global slab version, which is useful for the smooth
refinement. Extend the \(b_\alpha\) to bounded functions on the slab,
retaining sufficiently small suprema, and put
\(B=\sum b_\alpha D^\alpha\). If

\[
C_R\sum_{|\alpha|\leq2}\|b_\alpha\|_\infty<\tfrac12,
\tag{CM19}
\]

where \(C_R\) is large enough for (CM16) and the component derivative
estimates, then \(\|BR_L\|_{L^2\to L^2}<1/2\). Thus

\[
T=R_L(I+BR_L)^{-1}:L^2(S_L)\longrightarrow\mathcal D_L
\tag{CM20}
\]

is a right inverse of \(\widetilde P=p(D)+B\). It is also the left
inverse on \(\mathcal D_L\). To see this directly, (CM17) and (CM19)
imply

\[
\|v\|_{H^2}
\leq C_R\|p(D)v\|_2
\leq C_R\|\widetilde Pv\|_2+\tfrac12\|v\|_{H^2},
\qquad v\in\mathcal D_L.
\tag{CM21}
\]

This proves injectivity and the estimate
\(\|v\|_{H^2}\leq2C_R\|\widetilde Pv\|_2\). The difference
\(T\widetilde Pv-v\) lies in the nullspace, proving the left identity.

Smallness does not impose extra assumptions on the local theorem. In
original coordinates let \(a_\alpha\) be bounded near the origin, with
the second-order coefficients continuous there, and let
\(p(\xi)=\sum_{|\alpha|=2}a_\alpha(0)\xi^\alpha\) satisfy CM1. Set
\(x=\varepsilon y\), multiply the equation by \(\varepsilon^2\), and
work in a fixed-size \(y\)-patch. The coefficient differences are

\[
a_\alpha(\varepsilon y)-a_\alpha(0)\quad(|\alpha|=2),
\qquad \varepsilon^{2-|\alpha|}a_\alpha(\varepsilon y)\quad(|\alpha|<2).
\tag{CM22}
\]

Their suprema tend to zero on that fixed patch. Multiplying by a fixed
cutoff, equal to one on a smaller patch, extends them to the slab with
the same small-supremum property. Applying (CM20) and rescaling back
proves the original local inverse statements, with constants allowed to
depend on the chosen neighborhood. If the coefficients are merely
bounded/continuous, this argument gives exactly the \(L^2\to H^2\)
statements; it does not claim higher regularity from those assumptions.

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\subsubsection{One inverse with every integer smoothness
improvement}\label{one-inverse-with-every-integer-smoothness-improvement}

Assume now that the coefficients are smooth on a neighborhood of the
closed local patch. In (CM22) take the cutoff inside that neighborhood.
The extended coefficients are smooth on the closed slab and all their
derivatives are bounded. They continue to satisfy (CM19). The same \(T\)
from (CM20), chosen once at the \(L^2\) level, satisfies

\[
T:H^s(S_L)\longrightarrow H^{s+2}(S_L)
\quad\hbox{boundedly for every integer }s\geq0.
\tag{CM23}
\]

Here are the details, including why no new inverse or new smallness
condition is required when \(s\) increases. Write \(u=Tf\). For a
tangential coordinate \(j<n\), let \(\tau_hu(x',t)=u(x'+he_j,t)\) and
\(\delta_hu=(\tau_hu-u)/h\). Tangential translations preserve
\(\mathcal D_L\). If \(f\) has one tangential derivative in \(L^2\),

\[
\widetilde P\delta_hu
=\delta_hf-\sum_{|\alpha|\leq2}(\delta_h b_\alpha)D^\alpha\tau_hu.
\tag{CM24}
\]

This is simply the difference rule for each product; the constant
coefficients of \(p\) have no difference term. For each fixed nonzero
\(h\), \(\delta_hu\in\mathcal D_L\), so (CM21) applies. The fundamental
theorem of calculus bounds \(\|\delta_hf\|_2\) by \(\|\partial_jf\|_2\),
and bounds \(\|\delta_hb_\alpha\|_\infty\) by
\(\|\partial_jb_\alpha\|_\infty\). Consequently \(\delta_hu\) is
uniformly bounded in \(H^2\).

To justify the passage to a weak derivative without inserting a
compactness theorem, put \(g=D^\beta u\) for \(|\beta|\leq2\). The
uniform \(H^2\) bound just proved gives a uniform \(L^2\) bound on
\(\delta_hg\). Tangential Plancherel and Fatou's lemma imply

\[
\int_{\mathbb R^{n-1}}\int_0^L
|\xi_j|^2|\widehat g(\xi',t)|^2\,dt\,d\xi'
\leq\liminf_{h\to0}
\int_{\mathbb R^{n-1}}\int_0^L
\left|\frac{e^{ih\xi_j}-1}{h}\right|^2
|\widehat g(\xi',t)|^2\,dt\,d\xi'<\infty.
\]

The inverse tangential Fourier transform of \(i\xi_j\widehat g\) is
therefore an \(L^2\) function; testing the Fourier derivative rule
against smooth compact functions identifies it with the weak derivative
\(\partial_jD^\beta u\). Applying this for all \(\beta\) proves
\(\partial_ju\in H^2\) and the same norm bound. Its two boundary traces
vanish: in the partial Fourier description, tangential differentiation
multiplies the endpoint trace by \(i\xi_j\); the original trace is zero.
Equivalently, extend the distributional tangential derivative identity
to the continuous trace map. Thus \(\partial_ju\in\mathcal D_L\).

Repeat this argument for tangential derivatives. For a tangential
multi-index \(\beta\), differentiating the equation in the sense of
distributions gives

\[
\widetilde P\partial'^\beta u
=\partial'^\beta f
-\sum_{|\alpha|\leq2}\sum_{0<\gamma\leq\beta}
\binom\beta\gamma
(\partial'^\gamma b_\alpha)D^\alpha\partial'^{\beta-\gamma}u.
\tag{CM25}
\]

Suppose all tangential derivatives through order \(q\) already belong to
\(\mathcal D_L\), and \(q+1\leq s\). For \(|\beta|=q\), the right side
of (CM25) has each first tangential difference quotient uniformly
bounded in \(L^2\): the derivative of \(\partial'^\beta f\) exists
because \(f\in H^s\); when a difference falls on
\(D^\alpha\partial'^{\beta-\gamma}u\), its required tangential order is
\(|\beta-\gamma|+1\leq q\), already known. Coefficient derivatives are
bounded. Apply (CM24) to the equation for \(\partial'^\beta u\), then
the same weak-derivative argument. Induction proves

\[
\sum_{|\beta|\leq s}\|\partial'^\beta u\|_{H^2(S_L)}
\leq C_s\|f\|_{H^s(S_L)}.
\tag{CM26}
\]

This controls every derivative with at most one normal differentiation
and total order at most \(s+2\), as well as the derivatives with two
normal differentiations and at most \(s\) tangential differentiations.
To obtain the remaining derivatives, write the equation as

\[
D_t^2u=A(x)^{-1}
\left[f-\sum_{\substack{|\alpha|\leq2\\\alpha_n\leq1}}
c_\alpha(x)D^\alpha u\right],
\tag{CM27}
\]

where \(A(x)\) is the coefficient of \(D_t^2\) in \(\widetilde P\).
Shrinking the initial coefficient bound if needed ensures
\(|A(x)|\geq|a|/2\); this is one fixed bound at the original
construction stage. All derivatives of \(A^{-1}\) and \(c_\alpha\) are
bounded. Apply a mixed derivative of total order at most \(s\) to (CM27)
and expand products. Any derivative of \(f\) then has order at most
\(s\). In every term containing \(u\), the normal order is strictly
smaller than the target normal order, and total order is at most
\(s+2\). Induction on normal order, starting with (CM26), proves that
all these weak derivatives exist in \(L^2\) and obey (CM23). Product
differentiation is valid distributionally and all products obtained are
\(L^2\), so this also verifies each induction step without presupposing
the sought regularity. For \(n=1\), the same induction starts directly
from (CM21), with no tangential step.

Finally convert the slab inverse into the globally bounded half-space
operator used in the local theorem. Choose
\(\psi,\chi\in C_c^\infty(\mathbb R^n)\) equal to one on a neighborhood
of the chosen local patch, with support below height \(L\). Restrict
their products to \(t>0\), and define

\[
Ef=\chi\,T(\psi f),
\tag{CM28}
\]

extending the result by zero from the slab across its top into the rest
of \(\mathbb R^n_+\). Because \(\chi\) vanishes in a neighborhood of the
top, this extension preserves every integer Sobolev order.
Multiplication by these fixed smooth cutoffs is bounded on each \(H^s\).
Hence

\[
E:L^2(\mathbb R^n_+)\longrightarrow H^2(\mathbb R^n_+),
\qquad E:H^s(\mathbb R^n_+)\longrightarrow H^{s+2}(\mathbb R^n_+)
\quad(s\in\mathbb N_0).
\tag{CM29}
\]

The trace of \(Ef\) at the physical boundary is zero. On the local patch
where both cutoffs are one and \(\widetilde P=P\), equation (CM20) gives
\(PEf=f\). For a smooth compactly supported local input \(u\) with zero
physical trace, its slab extension lies in \(\mathcal D_L\),
\(\psi Pu=Pu\), and \(T\widetilde Pu=u\); therefore \(EPu=u\) on the
patch. These are local inverse identities, while (CM29) is global
half-space boundedness. If the coefficients were first rescaled, undo
the dilation as in (CM22); the same assertions hold in the resulting
smaller original patch.

No arbitrary global right inverse for the original variable-coefficient
operator is claimed. The coefficients away from the patch and the
auxiliary top boundary serve only to construct one bounded local inverse
with the stated support and regularity properties.

\ReaderArmAliases{\ReaderOriginalAnchorStart{an-original-diagnostic-example}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an-original-diagnostic-example}\label{an-original-diagnostic-example}\endgroup\ReaderOriginalAnchorStart{a-diagnostic-example}\ReaderOriginalAnchorEnd}
\subsubsection{A diagnostic example}\label{a-diagnostic-example}

The exceptional planar root condition is necessary for the decaying
half-space Dirichlet model. Take

\[
p(\xi,\tau)=(\tau-i\xi)^2.
\]

This is elliptic on real nonzero \((\xi,\tau)\): \(\tau-i\xi\) can
vanish there only at the origin. At positive \(\xi\), however, both
roots lie in the upper half-plane. Choose a nonzero
\(F\in C_c^\infty((1,2))\) and define a half-space function by

\[
\widehat u(\xi,t)=F(\xi)t e^{-\xi t}.
\]

Every derivative through every finite order is square-integrable, the
trace at \(t=0\) is zero, and \(p(\xi,D_t)\widehat u=0\). The last
identity follows because
\((D_t-i\xi)\bigl(t e^{-\xi t}\bigr)=-i e^{-\xi t}\), which the same
first-order factor kills. Thus the half-space nullspace with zero
Dirichlet data is nontrivial. The root count in CM1 cannot be silently
removed when \(n=2\).

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\subsubsection{Scope of the complex quadratic
model}\label{scope-of-the-complex-quadratic-model}

Here the auxiliary slab inverse controls both low and high frequencies.
Difference quotients then prove smooth regularity for that same fixed
inverse, so the construction and its regularity estimates use one
operator.

Gerd Grubb's \href{https://web.math.ku.dk/~grubb/dist9n.pdf}{Chapter 9,
\emph{Boundary value problems in a constant-coefficient case}} treats
half-space Fourier methods and the \(I-\Delta\) Dirichlet model. That
\(I-\Delta\) inverse is narrower than the arbitrary homogeneous complex
quadratic polynomial proved here and does not supply the zero-frequency
slab estimate. The Fourier and Sobolev prerequisites are proved in
Sections 1--2.

The formulas (CM1)--(CM29) prove the scalar quadratic assertion and the
smooth inverse refinement. They do not prove the higher-order system or
boundary-operator cases, which require separate arguments.

The proof in Section 6 also applies when the frozen quadratic is complex
and satisfies the root condition of (CM1): replace the reflected model
estimate by (CM16)--(CM21), and use the same odd-extension density for
compact zero-trace \(H^2\) functions. That density concerns the function
space, not commutation of a complex mixed-term operator with reflection.
Keep the actual nonzero \(D_t^2\) coefficient: CM27 retains its original
inverse \(A(x)^{-1}\) multiplying the complete bracket, including every
original coefficient and source term. The full reciprocal derivative
formula in Section9.1 of
\href{euclidean-symbol-calculus.pdf}{Symbols,
operators and Sobolev scales} supplies all derivative factors; no
leading coefficient is discarded. Equations (BO1)--(BO6) give the weak
multiplication map, the full original-operator commutators and every
normal coefficient derivative; the same reconstruction applies here.
Thus no real-coefficient assumption is reintroduced into this extension.

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\subsection{10. Setting, conventions, and the exact analytic
inputs}\label{setting-conventions-and-the-exact-analytic-inputs}

Let \(\Omega\subset\mathbb R^n\), \(n\ge2\), be bounded with smooth
boundary \(\Gamma\). Neither \(\Omega\) nor its complement is assumed
connected. A smooth boundary means a compact embedded hypersurface with
a smooth collar, with \(\Omega\) on one side. Compactness provides
finitely many charts and finitely many connected components. Let \(\nu\)
always point out of \(\Omega\). A superscript minus denotes the trace
from \(\Omega\), a superscript plus the trace from its complement; both
normal derivatives use this same \(\nu\). Write \(\omega_n=|S^{n-1}|\),
and fix \[
 E(x)=\frac{|x|^{2-n}}{(2-n)\omega_n}\quad(n\ge3),
 \qquad E(x)=\frac1{2\pi}\log\frac{|x|}{a}\quad(n=2),\quad a>0.
 \tag{LP1}
\] Then \(\Delta E=\delta_0\). Indeed \(E\) is harmonic off zero and its
outward flux on every centered sphere is one. Integration by parts
outside a shrinking ball, with the test function replaced there by its
value at zero plus an error of size \(O(|x|)\), proves the
distributional assertion. The flux-one normalization is the sign
convention used throughout.

The following inputs are precise dependencies, not layer-potential
assertions assumed in advance.

\begin{itemize}
\tightlist
\item
  \textbf{Traces and lifts.} On a bounded smooth domain,
  \(\gamma:H^1(\Omega)\to H^{1/2}(\Gamma)\) is bounded and onto with a
  bounded right inverse, \(\ker\gamma=H^1_0(\Omega)\), smooth functions
  are dense in these spaces, and localization in smooth collars
  preserves these facts. Surface Sobolev spaces have the chart norms of
  Section 11 of \href{geometric-microlocal-calculus.pdf}{Detecting
  regularity without choosing coordinates}. The trace on both sides of a
  hypersurface agrees for a function in ambient \(H^1\).
\item
  \textbf{The harmonic Dirichlet solution.} The variational harmonic
  Dirichlet solution \(\mathcal H f\in H^1(\Omega)\) exists uniquely for
  each \(f\in H^{1/2}(\Gamma)\), with
  \(\|\mathcal H f\|_{H^1}\le C\|f\|_{H^{1/2}}\). Smooth data produce a
  smooth solution up to the boundary, continuously in every smooth
  seminorm. The local version holds for smooth inhomogeneous equations
  in a collar, including collars in a complementary component. The
  preceding Sections 2--3 and Sections 7--8 supply the exact proofs;
  their use is detailed immediately below.
\item
  \textbf{The energy identity.} Weak Green integration by parts, the
  divergence theorem for smooth functions, and the fact that a function
  with zero weak gradient on a connected domain is constant. For a
  harmonic \(u\in H^1(\Omega)\), its weak outward derivative is the
  functional
  \(\langle\gamma_1u,h\rangle=\int_\Omega\nabla u\cdot\overline{\nabla W}\),
  where \(\gamma W=h\). The functional is independent of the extension
  by harmonicity and the trace facts, and belongs to
  \(H^{-1/2}(\Gamma)\).
\item
  Section 5 of \href{fredholm-stability.pdf}{Finite defects under
  perturbation} supplies the compact-perturbation Fredholm theorem.
  Sections 1--6 of \href{geometric-microlocal-calculus.pdf}{Detecting
  regularity without choosing coordinates} and Section 11 of
  \href{geometric-microlocal-calculus.pdf}{Detecting regularity without
  choosing coordinates} supply the ordinary classical calculus, its
  coordinate transport, elliptic parametrices, and all real Sobolev
  mapping estimates. Section 11 of
  \href{conormal-transmission.pdf}{Singularities along a submanifold and
  smooth boundary passage} and Section 13 of
  \href{conormal-transmission.pdf}{Singularities along a submanifold and
  smooth boundary passage} supply coordinate-invariant ordinary
  transmission and the boundary symbol of a simple or differentiated
  layer of an integer-order classical operator. We use these only for
  the Euclidean Laplace inverse and its smooth coordinate transports;
  this part asserts no result for an arbitrary symbol family or system.
\end{itemize}

The analytic dependencies have concrete proofs earlier in this lesson.
Section 2 proves the trace facts, including a bounded lift in charts and
the zero-trace characterization. For the harmonic Dirichlet solution,
lift the datum to \(F\in H^1\), and apply Section 3 to the zero-trace
equation \(-\Delta v=\Delta F\in H^{-1}\), that is,
\(\int\nabla v\cdot\overline{\nabla w}=-\int\nabla F\cdot\overline{\nabla w}\);
the resulting \(u=F+v\) is harmonic, unique by the energy identity, and
obeys the stated norm bound. For smooth data choose the smooth lift
supplied by the same construction; Sections 7--8 then give all integer
estimates for \(v\), hence smoothness and continuity in smooth
seminorms. Their localized versions give the collar assertion. The
energy identity uses the weak integration identities in Sections 2--3.
Thus these are exact adapters to those earlier proofs, not an imported
harmonic Dirichlet theorem. A locally integrable function with zero weak
gradient is constant on each connected component: convolution on a ball
makes it a smooth function with zero gradient, hence a constant there;
convergence of the convolutions identifies its distributional value, and
overlapping balls equate these constants throughout a connected
component.

The elementary distribution operations, Fourier inversion/Plancherel,
finite-dimensional calculus, integration, and smooth manifold
assumptions are those already declared in
\href{conormal-transmission.pdf}{Singularities along a submanifold and
smooth boundary passage} and
\href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates}. The uses of compactness in this part are on the
compact hypersurface \(\Gamma\), not on an arbitrary noncompact
boundary.

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\subsection{11. Boundary kernels gain one
derivative}\label{boundary-kernels-gain-one-derivative}

For smooth densities set, at points off \(\Gamma\), \[
 S\sigma(x)=\int_\Gamma E(x-y)\sigma(y)\,dS_y,
 \qquad D\phi(x)=\int_\Gamma\partial_{\nu_y}E(x-y)\phi(y)\,dS_y.
 \tag{LP2}
\] They are harmonic off \(\Gamma\), by differentiating the smooth
kernel on every compact set separated from \(\Gamma\). Their boundary
operators are \[
 V\sigma(x)=\int_\Gamma E(x-y)\sigma(y)\,dS_y,
 \quad K\phi(x)=\int_\Gamma\partial_{\nu_y}E(x-y)\phi(y)\,dS_y,
 \quad K'\sigma(x)=\int_\Gamma\partial_{\nu_x}E(x-y)\sigma(y)\,dS_y.
 \tag{LP3}
\] These integrals are ordinary locally integrable integrals for smooth
densities, not unspecified principal values. In particular \(K'\) is the
adjoint of \(K\) for surface measure; \(V\) is selfadjoint. This follows
first by Fubini after absolute integrability, then by duality on Sobolev
spaces.

Here is an explicit regularity argument. In a boundary chart
\(F:U\subset\mathbb R^d\to\Gamma\), \(d=n-1\), put \(z=x-y\). Taylor
expansion gives \[
 |F(x)-F(x-z)|\asymp|z|,\qquad
 [F(x)-F(x-z)]\cdot\nu(F(x-z))=O(|z|^2).
 \tag{LP4}
\] The second estimate also holds with the normal evaluated at \(F(x)\).
Both are uniform with differentiated Taylor remainders on compact chart
subsets. For \(n\ge3\), each localized kernel in (LP3), including the
surface Jacobian, therefore satisfies \[
 |\partial_x^\alpha\partial_z^\beta k(x,z)|
 \le C_{\alpha\beta}|z|^{-d+1-|\beta|},\quad0<|z|<1.
 \tag{LP5}
\] To see that this is a symbol estimate of order \(-1\), split a cutoff
kernel into annuli \(|z|\asymp2^{-j}\), \(j\ge0\). Its Fourier transform
\(a_j(x,\xi)\), differentiated \(\gamma\) times in \(\xi\), has the
bound \[
 |\partial_x^\alpha\partial_\xi^\gamma a_j(x,\xi)|
 \le C_{\alpha\gamma N}2^{-j(1+|\gamma|)}
 (1+2^{-j}|\xi|)^{-N}.
 \tag{LP6}
\] Indeed its support has volume \(O(2^{-jd})\); each factor \(z\) gives
\(2^{-j}\), and integration by parts in \(z\) supplies the last factor
after rescaling to a fixed annulus. Summing separately over
\(2^j\ge\langle\xi\rangle\) and its complement, choosing
\(N>1+|\gamma|\), gives \(C\langle\xi\rangle^{-1-|\gamma|}\). The
off-diagonal part is smooth. Thus all three operators belong to
\(\Psi^{-1}_{1,0}(\Gamma)\).

There is a small dimensional detail when \(n=2\). In a curve chart,
\(F(x)-F(x-z)=z\int_0^1F'(x-tz)dt\). The squared norm of the last
integral is smooth and positive. Hence the single-layer kernel is a
smooth coefficient times \(\log|z|\), plus a smooth function. Its
leading coefficient is the boundary arclength Jacobian. The Fourier
transform of a cutoff \(\log|z|\) has leading term \(-\pi/|\xi|\):
differentiate once, use the principal-value transform
\(\mathcal F(1/z)=-i\pi\operatorname{sgn}\xi\), and divide by \(i\xi\)
away from zero. That transform follows directly by regularizing the odd
integral with \(e^{-\varepsilon|z|}\), integrating its sine part, and
letting \(\varepsilon\downarrow0\). Taylor expanding the smooth
coefficient at \(z=0\), or differentiating this Fourier formula, gives
the successive lower orders; the Taylor remainder is controlled by the
annular argument with its extra powers of \(|z|\). For \(K,K'\), the
numerator in (LP4) is \(z^2\) times a smooth function and the
denominator is \(z^2\) times a positive smooth function. Those localized
kernels are smooth. Consequently \(K,K'\) are in fact smoothing on a
smooth planar boundary. The weaker order \(-1\) conclusion remains valid
in every \(n\ge2\).

We also need the precise leading symbol of \(V\), not just its order.
Write the Laplace inverse near the diagonal as the classical Fourier
operator with high-frequency symbol \(-\chi(\xi)/|\xi|^2\), plus a
smooth kernel. The difference is smooth because applying \(\Delta\) to
it gives the inverse Fourier transform of a compactly supported smooth
function; equivalently its Fourier transform is supported at low
frequencies up to the fundamental-solution ambiguity, and its
derivatives are smooth. In normal coordinates its principal symbol is
\(-[\tau^2+|\eta|_{g_\Gamma}^2]^{-1}\). The integer parity gives
ordinary transmission in both directions, and Section 11 of
\href{conormal-transmission.pdf}{Singularities along a submanifold and
smooth boundary passage} preserves this after a smooth chart change,
including the surface-density factor in the layer. Section 13 of
\href{conormal-transmission.pdf}{Singularities along a submanifold and
smooth boundary passage}, or its absolutely convergent normal integral
in this case, yields \[
 v_{-1}(y,\eta)=\frac1{2\pi}\int_{\mathbb R}
 \frac{-d\tau}{\tau^2+|\eta|_{g_\Gamma}^2}
 =-\frac1{2|\eta|_{g_\Gamma}},\qquad\eta\ne0.
 \tag{LP7}
\] The same theorem gives all homogeneous remainders, so \(V\) is
classical and elliptic of order \(-1\). It also gives classical order
zero for the one-sided double-layer trace; after subtracting the jump in
Section 12 its order-zero symbol vanishes, consistently with the
directly proved order \(-1\) bound for \(K\). Smooth low-frequency
choices do not change any of these symbols.

For every real \(s\), therefore, \[
 V,K,K':H^s(\Gamma)\longrightarrow H^{s+1}(\Gamma)
 \quad\text{continuously},
 \qquad K,K':H^s(\Gamma)\longrightarrow H^s(\Gamma)
 \quad\text{compactly}.
 \tag{LP8}
\] The continuity is precisely the order \(-1\) case of Section 11 of
\href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates}. For compactness, localize to finitely many charts
supported inside coordinate cubes and regard their zero extensions in a
larger torus. Fourier truncation to \(|k|\le N\) has finite rank, while
its tail has \(H^{s+1}\to H^s\) norm at most \(C/N\). Multiplication and
coordinate transport are bounded. Combining these finite-rank
approximations through a finite partition proves the compact embedding
used in (LP8), for negative as well as positive \(s\).

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\subsection{12. Which half of the identity
occurs}\label{which-half-of-the-identity-occurs}

For smooth densities the four traces are \[
 \gamma^-D=\tfrac12 I+K,\qquad\gamma^+D=-\tfrac12 I+K,
 \qquad\gamma^-S=\gamma^+S=V,
 \tag{LP9}
\] \[
 \gamma_1^-S=-\tfrac12 I+K',\qquad
 \gamma_1^+S=\tfrac12 I+K',\qquad
 \gamma_1^-D=\gamma_1^+D.
 \tag{LP10}
\] The last equality uses the same vector \(\nu\) on both sides.

We give the limiting details. Fix a boundary point, rotate its tangent
plane to \(\{x_n=0\}\), and write the boundary as \(y_n=h(y')\), with
\(h(0)=dh(0)=0\), the domain locally below the graph, and outward normal
upward. Approach along \(x=-t e_n\), \(t>0\). The leading part of either
the double-layer kernel or the normal derivative of the single layer is,
respectively, \[
 \frac{t}{\omega_n(|y'|^2+t^2)^{n/2}},\qquad
 -\frac{t}{\omega_n(|y'|^2+t^2)^{n/2}}.
 \tag{LP11}
\] Their integrals over the entire tangent plane are \(1/2\) and
\(-1/2\). One way to calculate the constant is to map the plane radially
to the corresponding hemisphere: \(t(|y'|^2+t^2)^{-n/2}dy'\) is its
solid-angle measure. The kernels concentrate at zero, and their
integrals outside a fixed disk tend to zero. Replacing a smooth density
by its value at zero therefore gives precisely the indicated half jump.

Curvature has an integrable limit. With \(r=|y'|\), the boundary height
is \(O(r^2)\) and its derivatives have the corresponding Taylor bounds.
Subtracting (LP11), the errors due to numerator, Jacobian, and
denominator are bounded, on a small disk, by \(C r^2(r^2+t^2)^{-n/2}\).
For example the denominator difference has numerator bounded by
\(C(|t|r^2+r^4)\); after multiplication by the flat numerator \(t\),
division by one additional \(r^2+t^2\) reduces it to the same bound on a
sufficiently small disk. Its integral on \(r<\varepsilon\) is at most
\(C\varepsilon\), uniformly in \(t\), since \(d=n-1\). Terms multiplying
the difference of density from its value at zero have flat-kernel
integral bounded by \(C t[1+|\log t|]\) on a fixed disk and hence tend
to zero. Outside that disk the kernels converge with every derivative.
Let \(t\to0\) first and then \(\varepsilon\to0\). What remains is
exactly the integrable boundary kernel \(K\) or \(K'\). Approaching from
the other side reverses the sign of (LP11). This proves the double-layer
and single-normal formulas.

For \(S\) itself, the integral on a boundary disk of radius
\(\varepsilon\) is uniformly \(O(\varepsilon)\) if \(n\ge3\), and
\(O(\varepsilon(1+|\log\varepsilon|))\) if \(n=2\). The complementary
integral converges smoothly. Thus its trace is continuous across the
boundary and equals \(V\). The smooth one-sided boundary regularity
follows either from the integer transmission input or, once the smooth
traces have been identified, from local smooth Dirichlet regularity.
This also justifies the distribution calculation for the remaining
normal trace.

Explicitly, for a piecewise smooth function \(w\) harmonic on both
sides, put \([w]=\gamma^+w-\gamma^-w\) and
\([\partial_\nu w]=\gamma_1^+w-\gamma_1^-w\). Integrating twice by parts
gives \[
 \Delta w=[\partial_\nu w]\,\delta_\Gamma
             +\operatorname{div}([w]\nu\delta_\Gamma).
 \tag{LP12}
\] But \(D\phi=-E*\operatorname{div}(\phi\nu\delta_\Gamma)\), so
\(\Delta D\phi=-\operatorname{div}(\phi\nu\delta_\Gamma)\). By (LP9),
\([D\phi]=-\phi\). Comparing (LP12) proves \([\partial_\nu D\phi]=0\).
It also confirms \([\partial_\nu S\sigma]=\sigma\), independently
checking the signs in (LP10).

As a useful normalization check, the divergence theorem applied to (LP2)
gives \(D1=1\) in \(\Omega\) and \(D1=0\) in its complement. Thus
\(K1=1/2\). On a flat boundary the curvature kernels \(K,K'\) vanish.
Accordingly \(2D\) has Dirichlet trace the identity, while \(-2S\) has
outward Neumann trace the identity. These are the two normalized
half-space constructions.

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\subsection{13. Energy traces and Green
representation}\label{energy-traces-and-green-representation}

All formulas needed for the boundary problems extend at the energy
exponents: \[
 S:H^{-1/2}(\Gamma)\to H^1_{\mathrm{loc}}(\mathbb R^n),\qquad
 D:H^{1/2}(\Gamma)\to
 H^1(\Omega)\oplus H^1_{\mathrm{loc}}(\mathbb R^n\setminus\overline\Omega),
 \tag{LP13}
\] where the second local space includes bounded collars on the exterior
side. No decay or global \(H^1\) assertion at infinity is included for
\(S\).

For the first map, the trace facts and duality give
\(\|\sigma\delta_\Gamma\|_{H^{-1}(\mathbb R^n)}\le C\|\sigma\|_{H^{-1/2}(\Gamma)}\):
pair it with an ambient \(H^1\) function and take its trace. The
high-frequency multiplier \(-\chi(\xi)/|\xi|^2\) maps \(H^{-1}\) to
\(H^1\), directly from its weighted Fourier bound. Its difference from
\(E\) is smooth, and convolution with a distribution supported on
compact \(\Gamma\) gives a smooth function on every fixed compact set,
with seminorms bounded by the density norm. This proves the first
assertion. Its trace is \(V\), by smooth approximation and the trace
facts; its weak normal traces are (LP10), by the bounded weak Green
functional. This argument works at \(-1/2\) itself, with no hidden
\(\varepsilon\) loss.

For smooth harmonic \(u\), Green's formula on
\(\Omega\setminus\overline{B(x,\varepsilon)}\) proves \[
 u(x)=D(\gamma u)(x)-S(\partial_\nu u)(x),\quad x\in\Omega.
 \tag{LP14}
\] Indeed the integrand on the outer boundary is
\(u\partial_\nu E-E\partial_\nu u\); its integral equals the value
\(u(x)\), since the inward-facing small sphere contributes
\(-u(x)+o(1)\). The term with \(E\partial_\nu u\) on that small sphere
tends to zero: its bound is \(O(\varepsilon)\) in \(n\ge3\) and
\(O(\varepsilon|\log\varepsilon|)\) in \(n=2\). More invariantly the
corresponding identity in the whole space is \[
 \mathbf1_\Omega u=D(\gamma u)-S(\partial_\nu u).
 \tag{LP15}
\] It follows from
\(\Delta(\mathbf1_\Omega u)=-(\partial_\nu u)\delta_\Gamma-\operatorname{div}((\gamma u)\nu\delta_\Gamma)\)
and the identity \(E*\Delta T=T\) for compactly supported distributions
\(T\). The latter follows by moving the derivatives in convolution onto
\(E\), where \(\Delta E=\delta\).

Define the Dirichlet-to-Neumann map \[
 A:H^{1/2}(\Gamma)\to H^{-1/2}(\Gamma),\qquad Af=\gamma_1(\mathcal H f).
 \tag{LP16}
\] the harmonic Dirichlet solution and the weak normal estimate prove
boundedness. For smooth \(f\), (LP15) applied to \(\mathcal H f\) says
\(Df=\mathbf1_\Omega \mathcal H f+SAf\). The already proved single-layer
bound consequently gives the second assertion of (LP13) for smooth
\(f\), uniformly in its \(H^{1/2}\) norm. Smooth approximation extends
\(D\) uniquely and proves (LP15) for every harmonic
\(u\in H^1(\Omega)\), because such a \(u\) equals
\(\mathcal H\gamma u\). In particular the jump relations now hold as
bounded maps into \(H^{1/2}\) or \(H^{-1/2}\), at exactly the indicated
energy spaces.

For \(f,h\in H^{1/2}(\Gamma)\), \[
 \langle Af,h\rangle
 =\int_\Omega\nabla \mathcal H f\cdot\overline{\nabla \mathcal H h},\qquad
 \langle Af,f\rangle\ge0.
 \tag{LP17}
\] Thus \(A\) is symmetric in the Sobolev duality, and its kernel
consists of traces of functions constant on each connected component of
\(\Omega\). This is not a claim that the layer density kernels are
always identified injectively with those constants.

The compatible energy Cauchy pairs are exactly \[
 \mathcal C_\Omega
 =\{(f,Af):f\in H^{1/2}(\Gamma)\}
 \subset H^{1/2}(\Gamma)\oplus H^{-1/2}(\Gamma).
 \tag{LP18}
\] Necessity is Dirichlet uniqueness; sufficiency is the solution
\(\mathcal H f\). For an arbitrary pair \((f,g)\) the potential
\(Df-Sg\) is harmonic and has interior trace
\(T(f,g)=(\tfrac12+K)f-Vg\). Its interior normal trace is \(A T(f,g)\),
which need not be \(g\). The bounded map \(C(f,g)=(T(f,g),AT(f,g))\) is
a projection onto (LP18): (LP14) makes it the identity on that graph. A
pair is therefore compatible exactly when it is fixed by \(C\). Merely
substituting arbitrary two boundary functions into a representation
formula does not prescribe both traces.

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\subsection{14. Fredholm equations with their actual
defects}\label{fredholm-equations-with-their-actual-defects}

For the normalized Dirichlet ansatz \(u=2D\phi\), the density equation
is \[
 (I+2K)\phi=f\quad\text{on }H^{1/2}(\Gamma).
 \tag{LP19}
\] For the normalized Neumann ansatz \(u=-2S\psi\), it is \[
 (I-2K')\psi=g\quad\text{on }H^{-1/2}(\Gamma).
 \tag{LP20}
\] Both operators are identity plus compact, hence Fredholm of index
zero. The same statement holds on \(H^s(\Gamma)\) for every real \(s\).
A kernel vector is smooth: its equation expresses it as an
order-minus-one operator applied to itself, and iteration raises its
Sobolev order arbitrarily. The identical argument makes every cokernel
distribution smooth. Fredholm solvability is therefore an actual finite
list of smooth dual orthogonality conditions, independent of the
exponent, not an assumption of invertibility.

The defects can be identified topologically. Let
\(\Omega_1,\ldots,\Omega_m\) be the connected components of \(\Omega\).
Then \[
 \ker(I-2K)=\operatorname{span}\{1_{\partial\Omega_1},\ldots,
 1_{\partial\Omega_m}\}.
 \tag{LP21}
\] Here each indicated function is constant on every boundary piece
belonging to that component, and zero on the others. For the inclusion
from right to left, the divergence theorem gives
\(D1_{\partial\Omega_j}=1_{\Omega_j}\). Conversely, if \(K\phi=\phi/2\),
the exterior trace of \(D\phi\) vanishes. On each bounded complementary
component, Dirichlet uniqueness makes the potential zero. On the
unbounded component it is zero as well: \(D\phi=O(|x|^{1-n})\),
\(\nabla D\phi=O(|x|^{-n})\), as follows directly by differentiating
(LP2) for \(|x|\) larger than twice the diameter of \(\Gamma\).
Integrating its energy in a large truncation gives an outer boundary
term \(O(R^{-n})\to0\). Its normal derivative is continuous by (LP10),
so the interior derivative is zero. Applying the interior energy
identity makes \(D\phi\) constant on each \(\Omega_j\), and its interior
trace is \(\phi\). This proves (LP21).

Adjointness, closed range, and index zero now imply \[
 \operatorname{ran}(I-2K')
 =\{g\in H^s(\Gamma):\langle g,1_{\partial\Omega_j}\rangle=0,
       \ 1\le j\le m\},
 \qquad \dim\ker(I-2K')=m.
 \tag{LP22}
\] The pairing is distributional when \(s<0\). At \(s=-1/2\), (LP20) and
(LP13) therefore solve every compatible harmonic Neumann problem. Every
two solutions differ by a constant on each \(\Omega_j\), by the energy
identity. Conversely those compatibility conditions are necessary by
testing the weak Green identity against \(1_{\Omega_j}\). The statement
applies even when one component of \(\Omega\) surrounds another; no
connected-complement assumption was used.

The double-layer defect is different. Let \(G_1,\ldots,G_h\) be the
bounded connected components of \(\mathbb R^n\setminus\overline\Omega\),
and let \(\theta_\ell\) equal one on all parts of \(\Gamma\) adjacent to
\(G_\ell\), zero elsewhere. A complementary component can itself have
several boundary components, for example when there are nested islands.
Then \[
 \ker(I+2K)=\operatorname{span}\{\theta_1,\ldots,\theta_h\}.
 \tag{LP23}
\] Indeed the normal out of \(G_\ell\) is \(-\nu\), so
\(D\theta_\ell=-1_{G_\ell}\) away from the surface. For the converse, a
kernel vector gives a double layer zero in \(\Omega\). Normal continuity
gives zero Neumann data in every complementary component. Its energy
makes the potential constant on each bounded \(G_\ell\) and zero in the
unbounded component, using the same decay estimate as above. The value
jump \(\gamma^-D\phi-\gamma^+D\phi=\phi\) then gives exactly (LP23).

Thus (LP19) is invertible if the complement has no bounded component. If
there are holes, it has kernel and cokernel dimension \(h\), and its
exact range is \[
 \{f\in H^s(\Gamma):\langle f,\eta\rangle=0
       \text{ for every }\eta\in\ker(I+2K')\}.
 \tag{LP24}
\] This is a finite-dimensional obstruction of the pure double-layer
representation; it does not obstruct the Dirichlet problem itself. The
harmonic Dirichlet solution and (LP14) always give the harmonic solution
and its combined representation \(Df-SAf\). If a single density equation
without compatibility conditions is desired, one must add a
finite-dimensional correction, rather than silently discard (LP24).

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\subsection{15. The logarithmic constant and two diagnostic
calculations}\label{the-logarithmic-constant-and-two-diagnostic-calculations}

Changing \(a\) in (LP1) adds a constant to \(E\). It changes \(V\) and
\(S\) by the rank-one operation which multiplies \(\int_\Gamma\sigma\)
by that constant; it changes neither \(D\), \(K\), \(K'\), nor the two
density equations. In dimension two, \[
 S\sigma(x)=\frac{\langle\sigma,1\rangle}{2\pi}\log|x|
       -\frac{\log a}{2\pi}\langle\sigma,1\rangle+O(|x|^{-1})
 \tag{LP25}
\] for smooth \(\sigma\), with the differentiated expansion obtained
directly from (LP2). Thus decay at infinity cannot be assumed for a
general single layer. No such assumption entered (LP22).

For a circle of radius \(R\), the constant density has \(S1=R\log(R/a)\)
in the disk. To compute this, average \(\log|Re^{i\theta}-x|\) for
\(|x|<R\): the real part of the convergent series for
\(\log(1-xe^{-i\theta}/R)\) has average zero. In particular \(a=R\)
makes a nonzero density have identically zero interior potential. The
interior normal derivative is zero for every \(a\), so
\(1\in\ker(I-2K')\) regardless. The proof of (LP22) used the adjoint
equation, and did not make the false inference that the map from kernel
densities to interior constants is injective. Ellipticity of \(V\) in
(LP7) also does not mean that \(V\) itself is invertible.

For the annulus \(a<|x|<b\), a density equal to constants \(c_a,c_b\) on
the two spheres has \(D\phi=c_b\) in the annulus. The inner sphere is
oriented into the hole, and its double layer is zero outside that
sphere. In the ordered constant-density basis (inner, outer), \[
 K=\begin{pmatrix}-1/2&1\\0&1/2\end{pmatrix},\quad
 \ker(I+2K)=\operatorname{span}\{(1,0)\},\quad
 \ker(I-2K)=\operatorname{span}\{(1,1)\}.
 \tag{LP26}
\] The harmonic function taking different constant values on the two
boundary spheres is not a pure double layer. To see this without
restricting the unknown density to constants, average any proposed
representation over rotations: the kernel and the domain commute with
rotations, while those boundary data are invariant. Its averaged density
is constant on each sphere, contradicting the first calculation. The
harmonic solution is an affine function of \(r^{2-n}\) for \(n\ge3\), or
of \(\log r\) for \(n=2\). This example separates failure of an ansatz
from failure of a boundary problem.

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\subsection{16. The Dirichlet-to-Neumann symbol without an assumed layer
inverse}\label{the-dirichlet-to-neumann-symbol-without-an-assumed-layer-inverse}

The map \(A\) in (LP16) is a classical pseudodifferential operator of
order one on \(\Gamma\), and \[
 \sigma_1(A)(y,\eta)=|\eta|_{g_\Gamma},\qquad\eta\ne0.
 \tag{LP27}
\] Here is a proof which retains possible kernels of \(V\). Trace (LP14)
to obtain \[
 VA=K-\tfrac12 I.
 \tag{LP28}
\] On smooth data, symmetry of \(A,V\) and adjointness of \(K,K'\) give
\(AV=K'-\tfrac12 I\). By (LP7) and Section 6 of
\href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates} choose a right parametrix \(Q\in\Psi^1(\Gamma)\)
such that \(VQ=I-R\) with \(R\) smoothing. Then, on smooth data, \[
 A=(K'-\tfrac12 I)Q+AR.
 \tag{LP29}
\] The error \(AR\) is a smoothing operator: \(R\) maps distributions
continuously to smooth functions, and the harmonic Dirichlet solution
together with normal differentiation makes
\(A:C^\infty(\Gamma)\to C^\infty(\Gamma)\) continuous. More explicitly,
apply \(A\) in the output variable of the smooth kernel of \(R\);
continuous parameter differentiation shows that the resulting kernel is
smooth in both variables. Thus (LP29) also defines an extension to all
distributions. The product term is classical by the previously specified
calculus, and its principal symbol is \((-1/2)(-2|\eta|)=|\eta|\). It
agrees with the energy-defined \(A\) by density. This proves (LP27), all
mappings \(A:H^s\to H^{s-1}\), and its elliptic regularity at every real
\(s\), without assuming invertibility of \(V\) or imposing a
normalization in dimension two.

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\subsection{17. Removing every higher normal
derivative}\label{removing-every-higher-normal-derivative}

Use inward distance \(t\ge0\) in a collar of \(\Gamma\), so that
\(\partial_\nu=-\partial_t\) at \(t=0\). The Euclidean metric has the
form \(dt^2+g_t\), and the Laplace equation reads \[
 \partial_t^2u+b(y,t)\partial_tu+L_tu=0,
 \quad b=\partial_t\log\sqrt{\det g_t},\quad
 L_t=\frac1{\sqrt{\det g_t}}\partial_{y_i}
       (\sqrt{\det g_t}\,g_t^{ij}\partial_{y_j}).
 \tag{LP30}
\] This follows directly by applying the divergence formula for the
metric to the gradient; the mixed terms vanish because distance curves
meet parallel surfaces orthogonally.

Set \(T_j=\gamma\partial_t^ju\), \(f=T_0\), \(h=T_1=-g\), where
\(g=\partial_\nu u\). There are tangential differential operators
\(P_j,Q_j\), with \(\operatorname{ord}P_j\le j\),
\(\operatorname{ord}Q_j\le j-1\), such that \[
 T_j=P_j f+Q_jh.
 \tag{LP31}
\] The zero operators are assigned every upper order bound. Start with
\((P_0,Q_0)=(I,0)\) and \((P_1,Q_1)=(0,I)\). Differentiating (LP30)
\(r\) times in \(t\) gives the exact recurrence \[
 T_{r+2}=-\sum_{\ell=0}^r\binom r\ell
 \big[b^{(\ell)}(0)T_{r-\ell+1}
       +L^{(\ell)}(0)T_{r-\ell}\big].
 \tag{LP32}
\] Substitution of the earlier \(P,Q\) proves (LP31) by induction.
Multiplication by \(b^{(\ell)}\) has order zero; \(L^{(\ell)}\) has
order at most two. The displayed order bounds follow term by term.
Formula (LP32) retains all curvature and normal coefficient derivatives;
replacing it by its flat principal part would not prove the exact
reduction.

Let \(B\) be a scalar differential boundary operator of order at most
\(q\), \(q\ge0\), with arbitrary smooth complex coefficients. In a
collar, express it with coefficients to the left as
\(\sum_{|\alpha|+j\le q}c_{\alpha j}(y)D_y^\alpha\gamma\partial_t^j\),
where \(D_y=-i\partial_y\). Equations (LP31)--(LP32) produce the exact
identity on smooth harmonic functions \[
 Bu=B_0f+B_1g,\qquad
 \operatorname{ord}B_0\le q,\quad
 \operatorname{ord}B_1\le q-1.
 \tag{LP33}
\] For \(q=0\), \(B_1=0\). On overlaps the formulas describe the same
operator on the boundary jets because they were obtained by equality
with the given differential operator and the intrinsic equation. A
partition of unity consequently yields global tangential operators. For
arbitrary energy harmonic functions, the right side of (LP33) defines a
distribution in \(H^{1/2-q}(\Gamma)\); it agrees with the classical
boundary operator whenever the original traces exist, in particular for
\(u\in H^r(\Omega)\), \(r>q+1/2\), by the approximation argument below
using the following trace bound. For an ambient \(H^r\) extension,
Fourier inversion in the normal variable and Cauchy--Schwarz bound its
normal-derivative trace of order \(j\le q\) by the square root of
\(\int |\tau|^{2j}(\lambda^2+\tau^2)^{-r}d\tau=C_{r,j}\lambda^{2j+1-2r}\),
where \(\lambda=\langle\eta\rangle\). This integral is finite exactly
when \(r>j+1/2\). Integrating with the boundary weight proves the
\(H^{r-j-1/2}\) bound. The corresponding slice maps are continuous down
to the boundary by the same estimate and density; an extension zero on
the interior side consequently has zero trace, proving independence of
extension. Quotient norms, smooth coordinate transport from Section 11
of \href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates}, and a finite chart partition give the asserted
domain trace bound. It does not assert a classical high normal trace of
every \(H^1\) function.

To justify the approximation for \(q\ge2\), take restrictions \(u_k\) of
smooth approximations to a fixed ambient \(H^r\) extension of \(u\).
Then \(u_k\to u\) in \(H^r(\Omega)\), while \(F_k=\Delta u_k\to0\) in
\(H^{r-2}(\Omega)\). For \(u_k\), the differentiated equation has the
same right side as (LP32) plus \(\gamma\partial_t^jF_k\) in the
recurrence for \(T_{j+2}\). Induction therefore expresses the
discrepancy in (LP33) as a finite sum of smooth tangential differential
operators applied to normal traces of derivatives of \(F_k\), with
combined differentiation order at most \(q-2\). The trace bound makes
each term tend to zero in \(H^{r-q-1/2}(\Gamma)\), because
\(r-2>(q-2)+1/2\). The traces of \(u_k\) converge in their corresponding
spaces as well, so passage to the limit proves (LP33) for the harmonic
\(u\). For \(q\le1\), the identity involves only \(f\) and \(g\), and no
residual term occurs. This argument does not require the smooth
approximants themselves to be harmonic.

The harmonic boundary problem is therefore precisely \[
 \Lambda f=\varphi,\qquad \Lambda=B_0+B_1A\in\Psi^q(\Gamma),
 \tag{LP34}
\] with solution \(u=\mathcal H f\) for \(f\in H^{1/2}\). Its leading
symbol is \[
 \lambda_q(y,\eta)=b_{0,q}(y,\eta)
                   +b_{1,q-1}(y,\eta)|\eta|_{g_\Gamma}.
 \tag{LP35}
\] Equivalently, freeze the highest-order coefficients of the original
\(B\) at \((y,0)\) and apply them to
\(e^{iy\cdot\eta}e^{-t|\eta|_{g_\Gamma}}\): the result at \(t=0\), with
the exponential removed, is (LP35). In the convention
\(D_y=-i\partial_y\), this is
\(\sum_{|\alpha|+j=q}c_{\alpha j}(y)\eta^\alpha(-|\eta|_{g_\Gamma})^j\).
The recurrence has this leading behavior because the frozen equation is
\(\partial_t^2-|\eta|^2=0\). Lower-order terms in (LP32) do not affect
it.

For this scalar Laplace equation and one scalar boundary condition, the
complementing condition is exactly \[
 \lambda_q(y,\eta)\ne0\quad(y\in\Gamma,\ \eta\ne0).
 \tag{LP36}
\] Indeed the space of decaying solutions of the frozen normal equation
at nonzero tangential frequency is one dimensional, spanned by
\(e^{-t|\eta|}\). The boundary map on that space is multiplication by
\(\lambda_q\), so bijectivity is precisely (LP36). Compactness of the
cosphere makes this equivalent to ellipticity of \(\Lambda\) at the
stated order \(q\). If the leading symbol cancels, one cannot claim a
complementing order-\(q\) problem merely because the reduced operator
happens to have a lower-order elliptic term.

Under (LP36), the parametrix and compact Sobolev embedding imply that
\(\Lambda:H^s(\Gamma)\to H^{s-q}(\Gamma)\) is Fredholm for every real
\(s\). Its kernel and the kernel of its formal adjoint are smooth by the
parametrix. Thus its range consists exactly of those data pairing to
zero with \(\ker\Lambda^\dagger\), and solutions are unique modulo
\(\ker\Lambda\). These finite defects are not asserted to vanish or to
have a prescribed index. For \(s\ge1/2\) each such \(f\) determines the
energy harmonic solution \(\mathcal H f\); Sections 7--8 gives its
stronger interior/boundary regularity when applicable. The all-real
\(s\) Fredholm statement concerns the boundary operator and does not
assert an unproved low-regularity Poisson mapping theorem.

This argument treats every differential order of a scalar boundary
condition for the Laplacian. A system, several boundary conditions,
another interior operator, a nonsmooth boundary, or a noncompact
boundary requires its own normal solution space, symbol hypotheses, and
analytic estimates. Those obligations are not closed by (LP36).

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\subsection{17A. The inward-normal layer system and its effective
operator}\label{a.-the-inward-normal-layer-system-and-its-effective-operator}

The introduction to the general boundary theory writes the same Laplace
example with the positive Laplacian and the \textbf{inward} normal. We
now translate that convention exactly and prove the single-layer inverse
used there. This matters because changing the Laplacian and changing the
normal each contribute a sign.

Assume in this section that \(n>2\). Keep the exterior normal \(\nu\),
the operator \(\Delta_-=\sum_j\partial_{x_j}^2\), and the fundamental
solution \(E_-\) from (LP1). Put \[
 \Delta_+=-\Delta_-,\qquad E_+=-E_-,\qquad n=-\nu,
 \qquad \Delta_+E_+=\delta_0.
 \tag{LP39}
\] Let \(S_+,D_+\) be the single- and double-layer potentials built from
\(E_+\) and the inward normal \(n\). Directly from (LP39), \[
 S_+=-S,\qquad D_+=D.
 \tag{LP40}
\] Indeed the double layer receives one minus sign from \(E_+=-E_-\) and
a second from \(\partial_n=-\partial_\nu\). If \(u_0=\gamma u\),
\(g=\partial_\nu u\), and \(u_1=\partial_nu=-g\), then (LP14) becomes \[
 u=D_+u_0-S_+u_1\qquad\text{in }\Omega.
 \tag{LP41}
\]

Taking the interior trace and using (LP9) gives \[
 u_0=k_0u_0+k_1u_1,\qquad
 k_0=\tfrac12I+K,\qquad k_1=V,
 \tag{LP42}
\] or, equivalently, \[
 (I-k_0)u_0-k_1u_1=0.
 \tag{LP43}
\] This identifies the chapter notation without hiding a sign in the
density. Equation (LP7) gives \(k_1\in\Psi^{-1}_{\mathrm{cl}}(\Gamma)\)
and \(\sigma_{-1}(k_1)=-1/(2|\eta|_{g_\Gamma})\).

We still have to prove that \(k_1\) is invertible. Suppose first that
\(V\sigma=0\). Elliptic regularity makes every distributional kernel
vector smooth. The function \(w=S\sigma\) is harmonic on both sides of
\(\Gamma\), is continuous across it, and has zero boundary value.
Dirichlet uniqueness makes it zero in every bounded component. On the
unbounded component, (LP1) gives \[
 w(x)=O(|x|^{2-n}),\qquad \nabla w(x)=O(|x|^{1-n}).
 \tag{LP44}
\] The decay suffices for a direct energy proof on the unbounded
component, with no maximum-principle input. On its truncation by a large
sphere, Green integration gives \[
 \int_{G_\infty\cap B_R}|\nabla w|^2
  =\operatorname{Re}\int_{G_\infty\cap\partial B_R}
                   \overline w\,\partial_r w
  =O(R^{2-n}).
 \tag{BR4}
\] The boundary pieces on \(\Gamma\) contribute zero because \(w=0\)
there. The last exponent is the full product \(R^{2-n}R^{1-n}R^{n-1}\)
from (LP44) and sphere area. For \(n>2\) it tends to zero. The left
integrals are nonnegative and increase to the integral over the whole
component, so that integral is zero. Its weak gradient therefore
vanishes, and connectedness makes \(w\) constant; its decay makes this
constant zero. The exact jump in (LP10) is
\(\gamma_1^+S\sigma-\gamma_1^-S\sigma=\sigma\). Both one-sided functions
are zero, so both derivatives are zero and \(\sigma=0\). Thus \(V\) is
injective.

The kernel \(E_-(x-y)\) is symmetric. Hence \[
 V:H^{-1/2}(\Gamma)\longrightarrow H^{1/2}(\Gamma)
 \tag{LP45}
\] is its own transpose in the boundary duality. It is Fredholm by
ellipticity, and its cokernel is the kernel of the same transposed map.
Its index is therefore zero. Injectivity now gives surjectivity.
Elliptic regularity identifies the kernel and cokernel at every Sobolev
exponent with these same smooth spaces, so \[
 k_1=V:H^{s-1}(\Gamma)\xrightarrow{\;\sim\;}H^s(\Gamma),\qquad
 k_1^{-1}\in\Psi^1_{\mathrm{cl}}(\Gamma)\quad(s\in\mathbb R).
 \tag{LP46}
\]

To justify the operator assertion in (LP46), retain the right parametrix
\(Q\) of Section 16 with \(VQ=I-R\). The actual inverse is bounded at
every exponent. Indeed let \(Q_{\mathrm l}V=I-R_{\mathrm l}\) be a left
parametrix. If no bound \(\|\sigma\|_{H^{s-1}}\le C_s\|V\sigma\|_{H^s}\)
held, choose \(\|\sigma_j\|_{H^{s-1}}=1\) with \(V\sigma_j\to0\) in
\(H^s\). The smoothing remainder \(R_{\mathrm l}\) is compact on
\(H^{s-1}\), by the compact embedding argument in (LP8). A subsequence
of \(R_{\mathrm l}\sigma_j\) converges, and the exact identity
\(\sigma_j=Q_{\mathrm l}V\sigma_j+R_{\mathrm l}\sigma_j\) makes the same
subsequence of \(\sigma_j\) converge to a norm-one vector annihilated by
\(V\), contradicting injectivity. This proves the bound, and
surjectivity gives the bounded actual inverse. Multiplying the right
parametrix identity on the left by that inverse gives the exact ordered
formula \[
 V^{-1}=Q+V^{-1}R.
 \tag{BR5}
\] These inverses agree on their common domains because distributional
injectivity was proved first. The all-exponent bounds and Sobolev
embedding make \(V^{-1}:C^\infty(\Gamma)\to C^\infty(\Gamma)\)
continuous. Applying it to the output variable of the smooth kernel of
\(R\) commutes with every parameter derivative and produces a jointly
smooth kernel, by those bounds. Thus \(V^{-1}R\) is smoothing, and the
actual inverse is classical of order one. No inverse of a symbolic
formula was substituted for this operator.

This also completes the converse trace argument. If a smooth pair
\((u_0,u_1)\) satisfies (LP43), define \(u\) by (LP41). Its boundary
value is \(u_0\). Green's formula for this actual harmonic function
gives \(u=D_+u_0-S_+\partial_nu\). Subtraction shows that
\(S_+(u_1-\partial_nu)=0\) in \(\Omega\). Its boundary trace is
\(-V(u_1-\partial_nu)\), so injectivity proves \(u_1=\partial_nu\).

Let \(A\) be the outward Dirichlet-to-Neumann operator in (LP16), and
set \(A_{\mathrm{in}}=-A\). Equation (LP28) says \(VA=K-\tfrac12I\).
Therefore, with the factors in their actual order, \[
 k_1^{-1}(I-k_0)
 =V^{-1}(\tfrac12I-K)
 =-A=A_{\mathrm{in}}.
 \tag{LP47}
\] For a boundary equation \(b_0u_0+b_1u_1=f\), where both terms have
the same target and Sobolev order, (LP43), (LP46), and (LP47) give the
exact reduction \[
 \bigl(b_0+b_1k_1^{-1}(I-k_0)\bigr)u_0
 =(b_0+b_1A_{\mathrm{in}})u_0=f,\qquad
 u_1=A_{\mathrm{in}}u_0.
 \tag{LP48}
\] No factor has been commuted. Since (LP27) gives
\(\sigma_1(A)=|\eta|_{g_\Gamma}\), \[
 \sigma_1(A_{\mathrm{in}})=-|\eta|_{g_\Gamma},\qquad
 \sigma\!\left(k_1^{-1}(I-k_0)\right)
 =(-2|\eta|_{g_\Gamma})(\tfrac12)=-|\eta|_{g_\Gamma}.
 \tag{LP49}
\] If \(b_0\) has order \(q\) and \(b_1\) has order at most \(q-1\), the
effective principal symbol is \[
 b_{0,q}(y,\eta)-b_{1,q-1}(y,\eta)|\eta|_{g_\Gamma}.
 \tag{LP50}
\] The decaying frozen mode in inward distance \(t\ge0\) is
\(e^{iy\cdot\eta}e^{-t|\eta|}\), whose inward derivative at zero is
\(-|\eta|\). Applying the principal boundary row to that mode gives
exactly (LP50). Thus ellipticity of the effective operator is the scalar
complementing condition.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio]{../figures/laplace_layer_effective_operator.png}}
\caption{The inward and outward normals, the single-layer jump, and the
exact operator chain from compatible Cauchy data to the effective
boundary equation.}
\end{figure}

The normal and jump signs are (LP39)--(LP43) and (LP10). The actual
inverse and its operator class are proved in (LP44)--(LP46), (BR4) and
(BR5); the ordered receiving map is (LP47)--(LP50). The ellipse
represents a schematic smooth domain.

The restriction \(n>2\) is essential to this particular inverse proof.
In dimension two the single layer has the logarithmic term (LP25),
changing the fundamental-solution constant changes \(V\) by rank one,
and the circle example can put a nonzero constant density in its kernel.
The Dirichlet-to-Neumann construction in Section 16 remains valid there
because it used an elliptic parametrix for \(V\), not an assumed global
inverse.

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\subsection{18. Problems and completed
solutions}\label{problems-and-completed-solutions}

\textbf{Problem 1. A compatible pair need not be chosen freely.} On the
unit disk take \(f=0\) and \(g=1\). Can these be the Cauchy data of a
harmonic energy solution? What does the potential \(Df-Sg\) do when the
logarithmic normalization is \(a=1\)?

\textbf{Solution.} Dirichlet uniqueness gives \(\mathcal H 0=0\), hence
\(A0=0\); the pair \((0,1)\) is not in (LP18). Independently its Neumann
integral is \(2\pi\ne0\). By the circle calculation \(S1=0\) inside the
unit disk for \(a=1\), so the potential built from the incompatible pair
is zero. Its actual Cauchy pair is \((0,0)\). Harmonicity of the
representation with arbitrary inputs does not make those inputs its
traces.

\textbf{Problem 2. Curvature in a second normal derivative.} Let the
dimension be \(n\) and let the domain be a ball of radius \(R\). Express
\(\gamma\partial_t^2u\) for a harmonic function in terms of
\(f=\gamma u\), \(g=\partial_\nu u\), using inward distance \(t\).

\textbf{Solution.} Parallel spheres have radius \(R-t\), so
\(b(0)=-(n-1)/R\), while \(L_0=\Delta_\Gamma\). Formula (LP30) gives
\(\gamma\partial_t^2u=-b(0)(-g)-L_0f=-(n-1)g/R-\Delta_\Gamma f\). For
\(u(x)=x_1\), \(g=f/R\) and \(\Delta_\Gamma f=-(n-1)f/R^2\), so the
right side is zero, as it must be because \(x_1\) is affine along radial
distance curves. Omitting the curvature term would give the wrong
answer.

\textbf{Problem 3. A tangential second-order boundary condition.}
Consider \(Bu=-\Delta_\Gamma(\gamma u)+c\partial_\nu u\), with constant
\(c\in\mathbb R\). Determine its reduced operator, principal order, and
whether complementing alone proves uniqueness.

\textbf{Solution.} Here \(B_0=-\Delta_\Gamma\), \(B_1=c\), so
\(\Lambda=-\Delta_\Gamma+cA\) has order two and principal symbol
\(|\eta|^2\), regardless of \(c\). Thus (LP36) holds. Constants on each
component of \(\Omega\) lie in both terms' kernels, so uniqueness fails.
If \(c\ge0\), its quadratic form is
\(\|\nabla_\Gamma f\|_2^2+c\|\nabla \mathcal H f\|_2^2\); the explicit
nonnegative form still has those constants. Ellipticity gives finite
defects and regularity, not automatic invertibility.

\textbf{Problem 4. Count holes, not names of boundary components.}
Suppose a connected annular region surrounds a second disjoint ball,
with positive separation. Count the dimensions of the two density
kernels.

\textbf{Solution.} The domain has two connected components, so (LP22)
gives \(\dim\ker(I-2K')=2\), and the Neumann data require two separate
zero integrals. Its complement has one bounded component, the region
between the surrounding annulus and the inner ball; that complementary
region has two boundary components. Formula (LP23) nevertheless gives
\(\dim\ker(I+2K)=1\). Its generator equals one on both boundary surfaces
adjacent to this one complementary region. Counting each such surface as
an independent hole would overcount the kernel.

For the jump and compatibility comparison, see the Stanford Math 220B
\href{https://web.stanford.edu/class/math220b/handouts/potential.pdf}{potential-theory
handout}; for a broader Dirichlet-to-Neumann framework, see Gerd Grubb's
\href{https://web.math.ku.dk/~grubb/dist11n.pdf}{Chapter 11}, Definition
11.15 and Theorem 11.17. The Stanford notation uses the opposite-sign
fundamental solution \(\Phi=-E\), then inserts a minus sign in its
layers, producing the operators in (LP2). The all-Sobolev proof required
here is given above.

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\subsection{19. The zero-dimensional boundary
case}\label{the-zero-dimensional-boundary-case}

If the ambient dimension is one, a bounded smooth domain is a finite
disjoint union of intervals \((a_j,b_j)\). Surface measure is counting
measure on their endpoints; use \(E(x)=|x|/2\), so again \(E''=\delta\).
Every boundary Sobolev space is the same finite-dimensional vector space
\(\mathbb C^{2m}\). For distinct endpoints \(x_i,x_j\), the layer
matrices are \[
 V_{ij}=|x_i-x_j|/2,\quad
 K_{ij}=-\nu_j\operatorname{sgn}(x_i-x_j)/2,\quad
 K'_{ij}=\nu_i\operatorname{sgn}(x_i-x_j)/2,
 \qquad V_{ii}=K_{ii}=K'_{ii}=0.
 \tag{LP37}
\] Direct one-sided evaluation of the two sums (LP2) gives exactly
(LP9)--(LP10); for example
\(\partial_{y}E(x-y)=-\operatorname{sgn}(x-y)/2\), and its one-sided
diagonal contributions are \(\pm1/2\). Green's formula (LP14) is
ordinary integration by parts on each interval. The harmonic extension
is affine and the Dirichlet-to-Neumann map is the block matrix \[
 A_j=\frac1{b_j-a_j}\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
 \tag{LP38}
\] This follows by taking its slope \((f(b_j)-f(a_j))/(b_j-a_j)\), whose
outward derivatives at the two endpoints have opposite signs. Neumann
data are solvable exactly when their two values sum to zero on each
interval; solutions differ by one constant on each interval.

There are no nonzero tangential covectors, so the symbol argument (LP36)
is not the appropriate assertion here. Also the higher-dimensional decay
used for (LP23) fails: a double layer tends to constants at the two ends
of the line. Its interior values are constant on every interval. The
densities \(1_{\partial\Omega_j}\) produce those independent constants,
so the Dirichlet trace matrix \(I+2K\) has rank \(m\), kernel dimension
\(m\), and range the endpoint data with equal values on each interval.
Thus the pure double layer again need not represent arbitrary Dirichlet
data; the combined representation does. Similarly \(I-2K'\) has rank
\(m\) and range the endpoint pairs of sum zero, by its actual affine
single-layer potentials, or by adjointness and
\(\ker(I-2K)=\operatorname{span}\{1_{\partial\Omega_j}\}\).

Every derivative \(u^{(j)}\), \(j\ge2\), of an affine harmonic function
vanishes. Hence an arbitrary differential boundary condition reduces
exactly to the endpoint values and first outward derivatives.
Substitution of (LP38) is the full finite-dimensional reduced problem;
its range is the orthogonal complement of the kernel of the
conjugate-transpose matrix. No nonempty-cosphere ellipticity condition
or higher-dimensional hole count is substituted for this explicit matrix
test.

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\subsection{20. Worked diagnostics and problems with
solutions}\label{worked-diagnostics-and-problems-with-solutions}

\textbf{Worked diagnostic: the boundary is part of the functional
space.} Let \(\Omega=(-1,0)\cup(0,1)\). A smooth function supported in
\((-1,1)\) and equal to one near zero is an \(H^1(\mathbb R)\) function
supported in \(\overline\Omega\). It does not belong to \(V_\Omega\):
every approximating test function would vanish at zero, and the
one-dimensional estimate (B2) on a bounded interval makes point
evaluation continuous in \(H^1\). Thus the support description proved
for a \(C^1\) domain cannot be transferred to arbitrary open sets. The
closure definition (B6) continues to make sense.

\textbf{Worked diagnostic: removing a mixed term before reflection.} In
two dimensions consider \(p(\xi_1,\xi_2)=2\xi_1^2+2\xi_1\xi_2+\xi_2^2\).
Reflection across \(x_2=0\) changes the parity of the mixed derivative
\(D_1D_2\); a direct odd-reflection inverse for this unnormalized
polynomial would therefore be invalid. Set \(y_1=x_1-x_2\), \(t=x_2\).
Then \(D_{x_1}=D_{y_1}\), \(D_{x_2}=D_t-D_{y_1}\), and substitution
gives \(p(D)=D_{y_1}^2+D_t^2\). The boundary plane and its upper side
are preserved, and reflection now applies. This calculation also
explains why an arbitrary diagonalizing rotation is not the required
coordinate change.

\textbf{Problem 1.} Take nonzero
\(\varphi\in\mathcal S(\mathbb R^{n-1})\) and
\(u(z,t)=e^{-t}\varphi(z)\). Determine which of its zero, even, and odd
extensions belong to \(H^1(\mathbb R^n)\).

\textbf{Solution.} All three are \(L^2\), and the even extension has
derivative \(-\operatorname{sgn}(t)e^{-|t|}\varphi\), together with the
even tangential derivatives, in \(L^2\). Thus it is \(H^1\). Formula
(B5) puts \(\varphi\otimes\delta_0\) in the normal derivative of the
zero extension, so that extension is not \(H^1\). The odd extension has
jump \(2\varphi\) at zero and hence derivative
\(2\varphi\otimes\delta_0-e^{-|t|}\varphi\); it too fails to be \(H^1\).
Odd extension is useful for a zero-trace function, not for an arbitrary
boundary value.

\textbf{Problem 2.} On the unit ball in \(\mathbb R^n\), put
\(a(x)=2+|x_1|\), \(P=-\operatorname{div}(a\nabla)\), and
\(u(x)=1-|x|^2\). Compute \(Pu\), and identify precisely which global
theorem applies.

\textbf{Solution.} Almost everywhere
\(\partial_1a=\operatorname{sgn}(x_1)\), while all its other weak
derivatives vanish. Since \(\nabla u=-2x\), the weak product rule gives
\(Pu=2na+2x\cdot\nabla a=2n(2+|x_1|)+2|x_1|\). There is no interface
delta: the coefficient itself is continuous and Lipschitz. The ball is
smooth, \(a\geq2\), and the matrix \(aI\) is Lipschitz, so (B29) applies
with this bounded forcing. The function is already an explicit smooth
zero-trace solution and is therefore the unique energy solution by
(B10). Its expansion contains a discontinuous but bounded first-order
coefficient. This example checks why bounded lower terms were retained
in (B22).

\textbf{Problem 3.} Explain why the local left identity in (B17) cannot
be replaced by uniqueness of every homogeneous solution in \(X_+\) with
zero trace on \(X_0\).

\textbf{Solution.} For \(P=-\Delta\), the function \(u(z,t)=t\) is
harmonic and vanishes on the flat boundary. Its restriction to a bounded
patch belongs to \(H^2\), but it is not zero. It has no compact support
in the patch. In (B17), the left identity is asserted only for inputs
compactly supported away from the artificial sides of \(X\); those
support conditions are exactly what justify the convolution identity
(B15). Energy uniqueness, when invoked on a bounded domain, uses zero
trace on the entire boundary through \(V_\Omega\).

\textbf{Problem 4.} Show that the half-order loss in the trace
\(H^2\to H^{3/2}\) is quantitatively necessary. Use boundary Fourier
transforms supported in a fixed-radius ball around \(Ne_1\),
\(N\to\infty\), and the lift (B4).

\textbf{Solution.} Choose a nonzero smooth Fourier profile \(b\)
supported in the unit ball and put
\(\widehat\varphi_N(\zeta)=N^{-3/2}b(\zeta-Ne_1)\). On this support
\(\lambda\asymp N\), so \(\|\varphi_N\|_{H^{3/2}}\asymp1\) and
\(\|\varphi_N\|_2\asymp N^{-3/2}\). The exact integral
\(\int_0^\infty e^{-2t\lambda}dt=(2\lambda)^{-1}\) gives
\(\|L\varphi_N\|_{H^2}\asymp1\), because any tangential derivative of
order two in the first direction has squared integral comparable to
\(N^3\|\varphi_N\|_2^2\). Hence no bound
\(\|L\varphi\|_{H^2}\leq C\|\varphi\|_{H^s}\) with \(s<3/2\) can hold:
the right side tends to zero for this sequence while the left side does
not. The same sequence proves the stated sharp trace loss itself: for
every \(\epsilon>0\), \[
 \|L\varphi_N\|_{H^2}\asymp1,\qquad
 \|\gamma L\varphi_N\|_{H^{3/2+\epsilon}}\asymp N^\epsilon.
 \tag{BR6}
\] Hence no bounded trace \(H^2\to H^{3/2+\epsilon}\) exists. The
preceding calculation about lifts proves the separate obstruction to
supplying \(H^2\) solutions from boundary data of lower regularity. Both
statements keep the original profiles, powers and inhomogeneous weights.
For \(n=1\), the boundary is zero dimensional and this high tangential
frequency example is absent.

\textbf{Problem 5.} Let \(u\) be harmonic in the unit ball. Can
\(\partial_\nu u=1\) hold on the entire sphere? What is the normal
derivative for the Dirichlet value \(\varphi(x)=x_1\) on that sphere?

\textbf{Solution.} Integrating the divergence of \(\nabla u\) gives
\(\int_{\partial\Omega}\partial_\nu u=\int_\Omega\Delta u=0\); for an
energy solution this is the same weak identity tested on the constant
function. The integral of the proposed datum one is the positive area of
the sphere, so no such harmonic solution exists. For the second datum,
\(u(x)=x_1\) is harmonic and has the required boundary value. The
exterior normal on the unit sphere is \(x\), so \(\partial_\nu u=x_1\).
Thus the Dirichlet-to-Neumann operator sends this boundary value to
itself. Adding a constant to \(u\) would change its Dirichlet value and
leave its Neumann value unchanged; that is the kernel relevant to the
Neumann problem.

\textbf{Problem 6.} In flat coordinates, reduce the third-order boundary
condition
\(\gamma\partial_t^3u+\gamma\partial_{z_1}\partial_tu=\varphi\) for a
harmonic function on \(t>0\) to its two Cauchy traces. Then specialize
to a decaying tangential Fourier mode.

\textbf{Solution.} Write \(u_0=\gamma u\) and
\(u_1=\partial_\nu u|_{t=0}=-\gamma\partial_tu\). The equation gives
\(\partial_t^2u=-\Delta_z u\), so \(\gamma\partial_t^3u=\Delta_z u_1\)
and \(\gamma\partial_{z_1}\partial_tu=-\partial_{z_1}u_1\). The boundary
equation is \((\Delta_z-\partial_{z_1})u_1=\varphi\). For
\(\widehat u(\zeta,t)=e^{-t|\zeta|}\widehat u_0(\zeta)\), one has
\(u_1=|D_z|u_0\), and the resulting multiplier is
\((-|\zeta|^2-i\zeta_1)|\zeta|\). Its principal homogeneous symbol is
\(-|\zeta|^3\), nonzero for \(\zeta\ne0\). This is a principal-symbol
calculation; it does not by itself decide low-frequency solvability or
remove finite-dimensional obstructions on a compact boundary.

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

The arguments and worked examples needed here are developed above.

John K. Hunter,
\href{https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf}{\emph{Notes
on Partial Differential Equations}, revised 18 June 2014}, Theorems 4.9
and 4.11, gives comparison proofs of Poincaré and weak existence.
Theorem 4.30 treats boundary regularity with \(C^1\) coefficients; the
result in (B29) retains Lipschitz leading coefficients. Theorems 4.31
and 4.32 concern higher and smooth boundary regularity.

Three further study routes have precise starting points.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \textbf{Beyond the proved \(C^{1,1}\) boundary class.} The weak chart
  argument (BR1)--(BR3) proves (B29) for \(C^{1,1}\) boundaries. Going
  below that class requires controlling the term \((Dv)\circ F\,D^2F\)
  and the first derivatives of the full transformed coefficient in
  (BR3). A bi-Lipschitz map alone does not provide either \(L^\infty\)
  bound. Any further extension must replace those two estimates and
  retain the actual trace and energy spaces.
\item
  \textbf{General elliptic boundary systems.} The scalar normal root
  calculation selects one decaying mode. For an operator or system of
  higher order one must construct the entire decaying solution space,
  map every boundary operator on it, verify the complementing condition
  for all nonzero tangential covectors, and solve the
  constant-coefficient problem with inhomogeneous boundary data in its
  correct Sobolev orders. The decaying solution space, its boundary maps
  and the complementing condition are developed in
  \href{stable-boundary-models.pdf}{Stable modes and the algebra of
  boundary data}. That lesson retains its stated system and boundary
  orders. The scalar argument here supplies a concrete model for those
  constructions.
\item
  \textbf{Boundary symbols and global defects.} Use the harmonic
  Dirichlet-to-Neumann symbol and the finite-dimensional layer
  obstructions above as complementary examples. An elliptic principal
  symbol controls high-frequency inversion; topology and nullspaces
  still enter global solvability. For the general reduction, first use
  the coordinate and Sobolev maps in
  \href{geometric-microlocal-calculus.pdf}{Detecting regularity without
  choosing coordinates}, then the Cauchy spaces in
  \href{calderon-cauchy-data.pdf}{Cauchy data and the Calderón
  projector}, and finally the general operator and data constructions in
  \href{arbitrary-boundary-data-reduction.pdf}{Reducing arbitrary
  boundary data to a boundary system}.
\end{enumerate}

\end{document}