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\ReaderArmAliases{\ReaderOriginalAnchorStart{singularities-along-a-submanifold-and-smooth-boundary-passage}\ReaderOriginalAnchorEnd}
\section{Singularities along a submanifold and smooth boundary
passage}\label{singularities-along-a-submanifold-and-smooth-boundary-passage}

A distribution can be rough across a submanifold while remaining stable
under every derivative tangent to it. That stability identifies its
normal-frequency amplitude, including its exact limiting regularity. At
a boundary, the positive and negative normal frequencies carry
additional information: a relation between their homogeneous
coefficients decides whether applying an operator leaves a smooth
function on the interior side. We develop that relation for every
complex type, and then compute the boundary operators produced by simple
and multiple layers.

We use the Fourier convention and quadratic multiplier estimates of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale}, the local symbol tools of
\href{metric-localization.pdf}{Localizing symbols with moving metrics},
and the distributional quantizations and classical symbol changes of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
The Fourier transform is
\(\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx\), its inverse has
coefficient \((2\pi)^{-n}\), and \(D=-i\partial\). Schwartz Plancherel
and its unitary \(L^2\) extension are used with these constants. Basic
prerequisites are distributional differentiation and tensor products,
finite-dimensional calculus, smooth charts and bundles, partitions of
unity, dominated convergence, and the elementary Cauchy integral and
maximum principles. Coordinate changes of classical operator symbols are
derived below by amplitude reduction, using the stated
quadratic-multiplier prerequisite. The normal-form, endpoint,
transmission and boundary-symbol assertions are proved in this lesson.

In local coordinates write
\(x=(t,z)\in\mathbb R^k\times\mathbb R^{n-k}\), with dual variables
\((\tau,\eta)\), and let \(Y=\{t=0\}\). For the nontrivial
normal-frequency discussion \(1\leq k\leq n\). Codimension zero gives
only smooth distributions and is treated by the same tangent-regularity
definition with no normal frequency. An ordinary symbol \(a(z,\tau)\) of
real order \(r\) satisfies \[
|\partial_z^\beta\partial_\tau^\alpha a(z,\tau)|
\leq C_{K,\alpha,\beta}\langle\tau\rangle^{r-|\alpha|}
\quad(z\in K),\qquad \langle\tau\rangle=(1+|\tau|^2)^{1/2}.
\tag{C1}
\] The global symbol class uses constants uniform in the base variables;
the local class uses each compact set \(K\). Additional normal base
variables \(t\) count as base variables in this definition.
Finite-dimensional vector and matrix norms give the corresponding
bundle-chart versions. Complex polyhomogeneous orders always refer to
ordinary estimates of the real part of the order.

Throughout Sections 1--7, \(\Psi^d\) denotes operators with ordinary
\(S^d_{1,0}\) symbols; no polyhomogeneous expansion is assumed. A
polyhomogeneity hypothesis is always stated expressly. The transmission
theory uses the classical expansions specified in Section 10.

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\subsection{1. The endpoint is a Besov
endpoint}\label{the-endpoint-is-a-besov-endpoint}

Partition frequency space into a low-frequency ball and dyadic annuli,
and let \(\Pi_j\) be their Fourier projections, \(j\geq0\). For
\(1\leq p\leq\infty\), define \[
\|u\|_{B^s_{2,p}}
=\left\|\big(2^{js}\|\Pi_j u\|_2\big)_{j\geq0}\right\|_{\ell^p}.
\tag{C2}
\] Changing the first radius or using a smooth finite-overlap dyadic
partition gives an equivalent norm. For the latter statement, each new
frequency block is a sum of a fixed finite number of neighboring old
blocks, with uniformly bounded multipliers; apply the triangle
inequality in \(L^2\) and in \(\ell^p\), and reverse the two partitions.
Sharp annuli give the same conclusion by their overlap with the smooth
blocks. Plancherel and \(\langle\xi\rangle\asymp2^j\) on an annulus show
that \(B^s_{2,2}=H^s\). The superscript \(p\) in the notation
\({}^pH_{(s)}\) used in some accounts means this dyadic \(\ell^p\)
index: it is \(B^s_{2,p}\), not an \(L^p\)-based Sobolev space.

We need three elementary continuity facts, including their endpoint
versions.

First, a localized smooth change of coordinates or multiplication by a
smooth function is bounded on \(B^s_{2,\infty}\) for every real \(s\).
Here localized means that all relevant functions and derivatives are
considered between fixed compact coordinate sets. For nonnegative
integers \(a\), the chain and product rules prove boundedness on
\(H^a\). The adjoint of a localized coordinate pullback is its inverse
pullback times a smooth Jacobian factor; duality therefore proves the
corresponding negative-integer bounds. To pass to the Besov index \(s\),
choose integers \(a<s<b\). For such an operator \(T\), the two Sobolev
bounds give \[
\|\Pi_lT\Pi_j\|_{2\to2}
\leq C\min\big(2^{(j-l)a},2^{(j-l)b}\big).
\] After multiplication by \(2^{ls-js}\), the right side decays
geometrically in \(|l-j|\): use \(a\) if \(j\geq l\) and \(b\) if
\(j<l\). Summation in \(j\) proves the assertion. The series represents
the original distribution, since
\(B^s_{2,\infty}\subset H^{s-\epsilon}\) for every \(\epsilon>0\),
directly by summing \(2^{-2j\epsilon}\). This also proves that the local
Besov spaces are well defined in charts and frames.

The same geometric matrix estimate proves boundedness for every
\(1\leq p\leq\infty\), since convolution with an \(\ell^1\) sequence is
bounded on each \(\ell^p\): apply the triangle inequality to its sum of
shifted sequences. In particular both sequence endpoints are retained.
The operator estimate in the next paragraph has the same summable matrix
form and likewise holds for all these \(p\).

Second, a properly supported pseudodifferential operator with an
ordinary \(S^d_{1,0}\) symbol of real order \(d\) maps
\(B^s_{2,\infty,\mathrm{loc}}\) continuously to
\(B^{s-d}_{2,\infty,\mathrm{loc}}\). We give the estimate rather than
infer it from a Sobolev theorem at an endpoint. After localizing the
output, its left symbol has compact base support. Integration by parts
in that base variable gives the Fourier kernel bound \[
|\widehat p(\xi-\theta,\theta)|
\leq C_M\langle\theta\rangle^d\langle\xi-\theta\rangle^{-M}.
\tag{C3}
\] On neighboring dyadic blocks, integrating the last factor in either
variable and applying the \(L^2\) Schur estimate gives
\(\|\Pi_lP\Pi_j\|\leq C2^{jd}\). On separated blocks, \(|\xi-\theta|\)
is comparable to \(2^{\max(j,l)}\). The same Schur estimate, now using
the two block volumes, gives \(C_M2^{jd}2^{-M\max(j,l)}2^{n(j+l)/2}\).
Choosing \(M\) larger than the dimension and the fixed weights gives a
summable bound after multiplying by \(2^{l(s-d)-js}\). Summing the
blocks proves the assertion for compact input localization. Proper
support permits that localization on any fixed output compact set. The
omitted input is separated from the diagonal; integration by parts in
\(\xi\) in the kernel then gives a smooth function, with all
derivatives, on that output set. This completes the local proof. Matrix
entries obey the same estimates, so arbitrary finite bundle ranks are
allowed.

For reference, the Schur estimate just used follows from Cauchy--Schwarz
in the form \(|Tf(\xi)|^2\leq (\int|K(\xi,\theta)|d\theta)
\int|K(\xi,\theta)||f(\theta)|^2d\theta\), followed by integration in
\(\xi\). No pointwise positivity of the operator is assumed.

Third, local Sobolev estimates in frequency space will be used below. On
a fixed pair of nested balls, any derivative of a smooth function at a
point of the smaller ball is bounded by the \(L^2\) norms of finitely
many derivatives on the larger ball. Multiply by a cutoff equal to one
on the smaller ball and apply Fourier inversion, Cauchy--Schwarz with
\(\langle\zeta\rangle^{-s}\), \(s>n/2\), and Plancherel, exactly as in
Section 2 of \href{gauss-transform-estimates.pdf}{Quadratic Fourier
multipliers at a moving scale}. Translation gives constants independent
of the center. This proves the local estimate from the stated Fourier
prerequisites.

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\subsection{2. Oscillatory amplitudes and removal of normal base
variables}\label{oscillatory-amplitudes-and-removal-of-normal-base-variables}

For a symbol \(a(t,z,\tau)\) of order \(r\), define \[
u(t,z)=\int e^{it\cdot\tau}a(t,z,\tau)\,d\tau.
\tag{C4}
\] This is a distribution for every real \(r\). To define its pairing
with a compactly supported test function, use
\((1-\Delta_t)^N e^{it\cdot\tau}=\langle\tau\rangle^{2N}e^{it\cdot\tau}\),
integrate by parts, and choose \(2N>r+k\). Derivatives of the amplitude
in base variables do not raise its order. The resulting integral is
absolutely convergent and bounded by finitely many test-function
seminorms. Different sufficiently large \(N\)'s agree by integration by
parts after a frequency cutoff and dominated convergence. The same
argument proves independence of that cutoff and continuity under bounded
symbol approximation.

For a symbol with \textbf{global} bounds in all base variables, there is
a reduced amplitude depending only on \((z,\tau)\): \[
\widetilde a(z,\tau)
=\left[e^{-i\langle D_t,D_\tau\rangle}a(t,z,\tau)\right]_{t=0},
\qquad u(t,z)=\int e^{it\cdot\tau}\widetilde a(z,\tau)\,d\tau.
\tag{C5}
\] It belongs to \(S^r\), and truncation of the exponential before total
degree \(N\) leaves \(S^{r-N}\), with every target seminorm controlled
by finitely many input seminorms. The map is also continuous on bounded
symbol sets with their local smooth topologies.

Here is a direct verification of all parts. For Schwartz \(a\), the
partial Fourier transform of (C4) in \(t\), divided by \((2\pi)^k\), is
\((2\pi)^{-k}\iint e^{it\cdot(\theta-\tau)}a(t,z,\theta)\,dt\,d\theta\).
Fourier transformation in the two variables shows that this equals the
first expression in (C5). In particular its sign is negative: a factor
\(e^{ip\cdot t+iq\cdot\tau}\) acquires \(e^{-ip\cdot q}\).

For the estimates apply Section 8 of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} to the phase \(-\langle D_t,D_\tau\rangle\), treating
\(z\) as a passive variable. Use the metric
\(|dt|^2+|dz|^2+\langle\tau\rangle^{-2}|d\tau|^2\) and weight
\(\langle\tau\rangle^r\). This metric is slowly varying. Its phase-dual
distance is finite only for equal passive variables, and there it is a
fixed factor times \(\langle\tau\rangle^2|\Delta t|^2+|\Delta\tau|^2\).
Polynomial frequency-ratio bounds prove the required temperateness; they
are the \((\rho,\delta)=(1,0)\) calculation in Section 8 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
The actual Gauss parameter at the observation point is
\(1/(2\langle\tau\rangle)\). Restriction to \(t=0\) consequently gives
exactly the order \(r-N\) remainder, including tangential derivatives.
Bounded compactly supported approximation now extends the identity of
distributions in (C5), using the defining estimate for (C4). The full
quadratic multiplier acts on tempered distributions; its restriction in
(C5) is asserted on these global symbol spaces, not on arbitrary
tempered distributions for which restriction might be undefined.

For a local symbol, fix a compact working set inside its coordinate
domain and a cutoff \(\chi(t,z)\) supported compactly in that domain and
equal to one on a neighborhood of the working set. Extend
\(a_\chi=\chi a\) by zero in the base variables. It is a global symbol
of order \(r\), so (C5) applies to \(a_\chi\), and its inverse integral
equals \(\chi u\) globally. It therefore equals the original \(u\) on
the working neighborhood. Every remainder estimate is controlled by
finitely many input seminorms on the fixed support of \(\chi\). At
points \((0,z)\) where \(\chi\) is identically one nearby, all
coefficients of the expansion are the derivatives of the original \(a\)
at \(t=0\). Different such cutoffs give reduced amplitudes differing by
\(S^{-\infty}\) locally in those \(z\)'s, since the expansion of their
difference vanishes to every order. No global growth condition or global
reduced representation is asserted for an uncut local symbol. We use
this compact cutoff and zero-extension convention whenever (C5) is
applied to local coordinates below.

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\subsection{3. Energy on an annulus and regularity away from the
submanifold}\label{energy-on-an-annulus-and-regularity-away-from-the-submanifold}

Assume the reduced amplitude \(a(z,\tau)\) has compact \(z\)-support and
global symbol bounds of order \(r\). Then
\(\widehat u(\tau,\eta)=(2\pi)^k\widehat a(\eta,\tau)\), where the
second Fourier transform is only in \(z\). Integration by parts in \(z\)
gives, for every \(M\), \[
|\widehat a(\eta,\tau)|\leq C_M\langle\eta\rangle^{-M}\langle\tau\rangle^r.
\tag{C6}
\] Thus \(\widehat u\) is locally square integrable. If
\(\varphi\in C_c^\infty(\mathbb R^n)\) vanishes near zero, then for
\(R>1\) \[
\int |\widehat u(\xi)|^2|\varphi(\xi/R)|\,d\xi
\leq C_\varphi R^{k+2r}.
\tag{C7}
\] To prove it, split the support of \(\varphi\) into a region where
\(|\tau|\) is bounded below and one where \(|\eta|\) is bounded below.
After scaling \(\tau=R\theta\), the first contribution is bounded by
\(CR^{k+2r}\int\langle\eta\rangle^{-2M}d\eta\), since \(\theta\) lies in
a compact set away from zero. In the second region \(|\eta|\geq cR\),
and (C6) makes the contribution smaller than every prescribed negative
power of \(R\) after choosing \(M\) large, despite the fixed polynomial
growth in \(\tau\). This also works when \(r<0\).

If \(a\) is polyhomogeneous of complex degree \(r\), with leading
homogeneous coefficient \(a_0\), then the exact leading energy limit is
\[
\lim_{R\to\infty}R^{-k-2\operatorname{Re}r}(2\pi)^{-n-k}
\int|\widehat u(\xi)|^2\varphi(\xi/R)\,d\xi
=\iint|a_0(z,\theta)|^2\varphi(\theta,0)\,dz\,d\theta.
\tag{C8}
\] On the first region above, \(R^{-r}\widehat a(\eta,R\theta)\)
converges to \(\widehat {a_0}(\eta,\theta)\), with an integrable
majorant from the symbol bounds and compact base support. The second
region still tends to zero. Dominated convergence and tangential
Plancherel give the right side. The constants are \((2\pi)^{2k}\) from
\(\widehat u\), \((2\pi)^{n-k}\) from tangential Plancherel, and the
factor \((2\pi)^{-n-k}\) displayed on the left; their product is one. A
complex test function is allowed by the same domination argument.

Formula (C7) is exactly \(u\in B^{-r-k/2}_{2,\infty}\). This endpoint
cannot in general be replaced by \(H^{-r-k/2}\). For example, a delta
distribution in \(k\) normal variables has constant normal Fourier
transform: its weighted dyadic block norms at index \(-k/2\) are
bounded, while their squared sum diverges.

If \(\chi(t)\) is compactly supported and equals one near zero, then
\((1-\chi(t))u\) is Schwartz. Indeed, \[
t^\alpha u=\int e^{it\cdot\tau}(-D_\tau)^\alpha a(z,\tau)\,d\tau.
\] Any prescribed number of \((t,z)\)-derivatives of this integral is
absolutely convergent after \(|\alpha|\) is chosen sufficiently large.
On the support of \(1-\chi\), division by a sufficiently large power of
\(|t|\) is smooth, with all derivatives controlled. Compact
\(z\)-support and arbitrary choices of \(\alpha\) give all Schwartz
seminorms. The same Fourier estimates after localization show
\(\operatorname{WF}(u)\subset N^*Y\setminus0\): away from \(Y\) there is
smoothness, and when the tangential covector dominates a fixed positive
fraction of the full covector, (C6) gives rapid decay.

The generators tangent to the plane preserve the amplitude order.
Tangential differentiation acts on \(a\); for \(i,j\leq k\), \[
t_iD_{t_j}u
=\int e^{it\cdot\tau}(-D_{\tau_i})(\tau_j a(z,\tau))\,d\tau.
\tag{C9}
\] A frequency derivative lowers the order raised by the factor
\(\tau_j\). Thus every product of \(D_{z_j}\) and \(t_iD_{t_j}\)
preserves the endpoint just proved. Multiplication by smooth local
coefficients preserves it by Section 1.

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\subsection{4. Recovering the amplitude from tangent
regularity}\label{recovering-the-amplitude-from-tangent-regularity}

\textbf{The compact converse.} Suppose \(u\in\mathcal E'(\mathbb R^n)\),
and for every pair of multiindices satisfying
\(|\alpha_t|\geq|\beta_t|\), \[
x^\alpha D^\beta u\in B^{-r-k/2}_{2,\infty}.
\tag{C10}
\] Then \(u\) has the reduced representation (C4), with a global \(S^r\)
amplitude of compact tangential support.

\textbf{Proof.} Compact support makes \(\widehat u\) smooth with
polynomially bounded derivatives. Fourier transformation, (C2), and
induction using the product rule give \[
\int_{R/2<|\xi|<2R}
|\xi^\beta\partial_\xi^\alpha\widehat u(\xi)|^2d\xi
\leq C_{\alpha\beta}R^{2r+k},\qquad R\geq1,
\quad |\alpha_t|\geq|\beta_t|.
\tag{C11}
\] The product rule first produces derivatives of
\(\xi^\beta\widehat u\). Terms in which a derivative hits the monomial
have smaller indices and preserve the difference
\(|\alpha_t|-|\beta_t|\), so induction isolates (C11).

On the cone \(|\eta|\geq|\tau|\), use arbitrarily large tangential
powers \(\eta^\beta\), with no normal powers. A unit ball about a large
point in this cone is contained in a fixed enlargement of its dyadic
annulus; at least one tangential coordinate is comparable to the radius
on a smaller fixed ball. Formula (C11) bounds the \(L^2\) norm of every
Fourier derivative on that ball by \(C R^{r+k/2-M}\), with arbitrary
\(M\). The local Sobolev estimate in Section 1 then gives rapid decay of
every derivative there. If there are no tangential variables, this cone
has no large points and this step is unnecessary.

On the other cone put \(R=|\tau|\geq1\) and \[
U_R(\theta,\eta)=R^{-r}\widehat u(R\theta,\eta).
\] Use the ellipsoidal annulus \(1/4<|\theta|^2+|\eta/R|^2<4\). In (C11)
choose the number of normal monomial factors equal to the number of
normal derivatives. The powers of \(R\) from those two operations
cancel; the change of variables contributes \(R^k\), canceled by the
normalization in (C11). We obtain uniform \(L^2\) bounds on that
ellipsoidal annulus for
\(\theta^\beta\eta^\gamma\partial_\theta^\alpha\partial_\eta^\delta U_R\),
with \(|\beta|=|\alpha|\), for every \(\alpha,\gamma,\delta\). The sum
of \(|\theta^\beta|^2\) over \(|\beta|=|\alpha|\) is bounded below on a
fixed neighborhood of the unit sphere. Hence these bounds control all
derivatives of \(U_R\), with every tangential frequency weight, on
fixed-size boxes centered at \(|\theta|=1,|\eta|\leq R\). The boxes fit
inside the ellipsoidal annulus with a uniform margin after choosing
their size small; bounded \(R\)'s are absorbed into the local bounds.
The same local Sobolev estimate now gives \[
|\partial_\xi^\alpha\widehat u(\tau,\eta)|
\leq C_{\alpha,M}\langle\tau\rangle^{r-|\alpha_t|}
\langle\eta\rangle^{-M}
\quad\hbox{for every }\alpha,M.
\tag{C12}
\] Together with the first cone this holds globally.

Define \[
a(z,\tau)=(2\pi)^{-n}\int e^{iz\cdot\eta}\widehat u(\tau,\eta)\,d\eta.
\tag{C13}
\] Every derivative of this integral is absolutely convergent by (C12),
including the polynomial \(\eta\)-factors from \(z\)-derivatives. It
gives precisely the estimates (C1). Fourier inversion proves (C4).
Partial Fourier transformation in \(t\) also shows that the amplitude
vanishes outside the projection of \(\operatorname{supp}u\) onto the
\(z\)-space. This proves the support statement and the converse. All
estimates use finitely many of the hypotheses for each output seminorm.
{\ReaderSymbols\char"220E}

Every smooth vector field tangent to \(Y\) is a smooth linear
combination of \(t_i\partial_{t_j}\) and \(\partial_{z_j}\). Indeed, a
normal coefficient \(f(t,z)\) vanishing at \(t=0\) has the exact
division formula \[
f(t,z)=\sum_i t_i\int_0^1(\partial_{t_i}f)(st,z)\,ds.
\tag{C14}
\] The integral is smooth, with all parameter derivatives obtained by
differentiation under the integral. This proves the generator assertion.

Consequently the local amplitude class is exactly the class on which all
products of tangent first-order operators preserve
\(B^{-r-k/2}_{2,\infty,\mathrm{loc}}\). One direction is (C9) and (C14).
For the converse, localize by a compact cutoff. Products of the
generators express the normal balanced monomials in (C10), up to shorter
terms; multiplication by additional powers of \(t\) and tangential
coordinates is harmless on that compact set. Induction gives all of
(C10), so the compact converse applies. Finally choose a locally finite
family \(\psi_j\) with \(\sum_j\psi_j^2=1\). If \(\psi_j u\) has reduced
amplitude \(a_j\), then \(u=\sum_j\psi_j(\psi_j u)\) has full amplitude
\(\sum_j\psi_j(t,z)a_j(z,\tau)\). On each compact base set only finitely
many terms contribute. This proves the local assertion without a support
assumption on \(u\).

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\subsection{5. Intrinsic conormality, bundles and operator
action}\label{intrinsic-conormality-bundles-and-operator-action}

Let \(X\) be a smooth \(n\)-manifold, \(Y\) a closed smooth embedded
submanifold, and \(E\) a smooth complex vector bundle. For real \(m\),
define \(I^m(X,Y;E)\) by requiring \[
L_1\cdots L_Nu\in B^{-m-n/4}_{2,\infty,\mathrm{loc}}(X;E)
\tag{C15}
\] for every \(N\geq0\) and every collection of first-order smooth
operators on \(E\) whose principal symbols vanish on \(N^*Y\). The
topology is generated by the local Besov seminorms of all these images.
Chart and frame changes preserve those spaces by Section 1, and the
principal-symbol vanishing condition is intrinsic, so this definition
and topology are coordinate independent.

In a bundle frame, (C14) expresses every such operator as
\(L=\sum_j A_jM_j+A_0\), where the \(A_j\) are smooth endomorphisms and
the generators \(M_j\) have scalar principal symbols. When a coefficient
is moved to the left across such a generator, the commutator is order
zero: its first-order matrix terms cancel because the principal part is
scalar. Induction on the length of an operator product therefore
expresses it as a finite sum of smooth endomorphisms times words in the
\(M_j\)'s. Thus scalar-principal generators suffice even for systems; no
assumption that the original operator has scalar matrix coefficients is
made.

By Section 4, the local normal form for codimension \(k\) is exactly \[
u(t,z)=\int e^{it\cdot\tau}a(z,\tau)d\tau,
\qquad a\in S^{m+(n-2k)/4}.
\tag{C16}
\] For compactly supported \(u\) in a chart the amplitude can be chosen
globally in these coordinates. Conversely every such representation is
locally conormal. Localization commutes with tangent words up to shorter
words with smooth coefficients, so (C15) holds if and only if it holds
after every member of any partition of unity. These arguments also
compare the normal-form seminorms with the defining topology, using the
finite-seminorm estimates above. For \(k=0\), every derivative is
tangent, so repeated (C15) and local Sobolev estimates imply smoothness;
all orders give the same smooth class.

A properly supported \(P\in\Psi^d(X;E,F)\) acts continuously \[
P:I^m(X,Y;E)\longrightarrow I^{m+d}(X,Y;F).
\tag{C17}
\] For completeness, let \(L\) on \(F\) have scalar tangent principal
symbol, and choose \(L'\) on \(E\) with the same scalar symbol. The
order \(d+1\) terms of \(LP\) and \(PL'\) cancel, so \(LP=PL'+P_0\),
with \(P_0\in\Psi^d(X;E,F)\). This cancellation can be seen directly by
differentiating the local oscillatory kernel; the terms without a
derivative on a coefficient are the identical scalar products, and every
remaining frequency derivative lowers the raised order by one. Apply
this identity successively through a word of tangent generators. For an
empty word, (C17) is the Besov estimate proved from (C3); for a longer
word it reduces to shorter words and the same estimate. Scalar-principal
reduction then gives (C15) for all tangent operators, with
finite-seminorm continuity. This proves (C17) for arbitrary input and
output ranks.

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\subsection{6. The intrinsic principal symbol and its
normalization}\label{the-intrinsic-principal-symbol-and-its-normalization}

Factor out an ambient half-density and write a local conormal section as
\[
u(t,z)=(2\pi)^{-(n+2k)/4}
\int e^{it\cdot\tau}a(z,\tau)d\tau\,|dt\,dz|^{1/2},
\qquad a\in S^{m+(n-2k)/4}.
\tag{C18}
\] Its leading symbol is the class of
\(a(z,\tau)|dz|^{1/2}|d\tau|^{1/2}\), modulo one lower ordinary
amplitude order, on the chart \((z,\tau)\mapsto(0,z;\tau,0)\) of
\(N^*Y\).

We check the full coordinate law. Let \(\kappa=(\kappa_1,\kappa_2)\)
preserve \(t=0\), and pull back the ambient half-density. Formula (C14)
gives \(\kappa_1(t,z)=\Psi(t,z)t\), with \(\Psi(0,z)=\kappa'_{11}(0,z)\)
invertible. Near the submanifold shrink the chart so that \(\Psi\) is
invertible; away from it the distribution is smooth. Changing the
integration variable to \(\tau=\Psi(t,z)^T\theta\) gives the full
amplitude \[
A(t,z,\tau)=a_\kappa(\kappa_2(t,z),\Psi(t,z)^{-T}\tau)
\frac{|\det\kappa'(t,z)|^{1/2}}{|\det\Psi(t,z)|}.
\tag{C19}
\] Every base derivative of the composed symbol introduces a factor
linear in \(\tau\) paired with a frequency derivative, so (C1) retains
its order on compact coordinate sets. Frequency derivatives lower the
order normally. Choose the compact base cutoff \(\chi\) of Section 2
equal to one on the working neighborhood and extend \(\chi A\) by zero.
Reducing this global symbol by (C5) gives the exact amplitude
\(a=[e^{-i\langle D_t,D_\tau\rangle}(\chi A)]_{t=0}\) of the localized
pullback. Its inverse integral agrees with the original pullback on that
neighborhood, and all \(N\)-term remainders have order \(m+(n-2k)/4-N\).
At the working points of \(Y\), the cutoff and all its derivatives in
the expansion are respectively one and zero.

The zeroth term is \[
a_\kappa\big(\kappa_2(0,z),\kappa'_{11}(0,z)^{-T}\tau\big)
|\det\kappa'_{11}(0,z)|^{-1/2}
|\det\kappa'_{22}(0,z)|^{1/2}.
\tag{C20}
\] Indeed, the off-diagonal block \(\kappa'_{12}(0,z)\) vanishes, so the
full determinant is the product of the two diagonal block determinants.
The factor in (C20) is exactly the half-density Jacobian on
\((z,\tau)\). This proves intrinsic invariance modulo one lower order,
including its exact determinant powers. A bundle transition matrix
contributes its value on \(Y\); every additional normal derivative of
that matrix in the reduction is paired with a frequency derivative and
therefore contributes only to lower orders.

On a real rank-\(k\) vector bundle \(V\to Y\), define
\(S^\nu(V;\Omega_V^{1/2})\) by requiring the coefficient of
\(|dz|^{1/2}|d\tau|^{1/2}\) to have ordinary order \(\nu-k/2\). Fiber
dilation multiplies that coordinate half-density by the factor
\(t^{k/2}\); hence its intrinsic homogeneous degree includes \(k/2\).
Base/fiber changes obey the same determinant calculation and preserve
these estimates. In (C18) the intrinsic symbol order is consequently
\(m+n/4\), independent of codimension.

Let \(\widehat E\) be the pullback of \(E|_Y\) to \(N^*Y\). The
construction gives an isomorphism \[
\frac{I^m(X,Y;\Omega_X^{1/2}\otimes E)}{I^{m-1}(X,Y;\Omega_X^{1/2}\otimes E)}
\simeq
\frac{S^{m+n/4}(N^*Y;\Omega_{N^*Y}^{1/2}\otimes\widehat E)}
{S^{m+n/4-1}(N^*Y;\Omega_{N^*Y}^{1/2}\otimes\widehat E)}.
\tag{C21}
\] Surjectivity follows by realizing local amplitudes in (C18),
multiplying by cutoffs in the ambient chart, and summing a locally
finite partition on \(Y\). Formula (C20) makes the resulting principal
symbol the prescribed one. To identify the kernel, localize compactly in
a chart. The reduced amplitude is its partial Fourier transform in the
normal variables, with the fixed factor in (C18), so it is unique. By
the normal-form equivalence, the localized distribution belongs to
\(I^{m-1}\) exactly when this amplitude has one lower order. This proves
injectivity and the kernel assertion; changes off \(Y\) contribute
smooth terms only.

The normalization can be checked in two ways. Before introducing (C18),
put \(r=m+(n-2k)/4\) in (C8); then the power of \(R\) is
\(2\operatorname{Re}m+n/2\), and with the factor \((2\pi)^{-n}\) on the
energy side the limit is \((2\pi)^k\int|a_0|^2\varphi(\theta,0)\).
Inserting the coefficient in (C18) multiplies that limit by
\((2\pi)^{-(n+2k)/2}\). Thus the normalized amplitude has the equivalent
limit with coefficient \((2\pi)^{-n/2}\) on the Fourier-energy side and
coefficient one on the amplitude side.

For a pseudodifferential kernel on a \(d\)-manifold, the ambient
dimension is \(n=2d\) and the diagonal codimension is \(k=d\). Formula
(C18) becomes the usual \((2\pi)^{-d}\) kernel normalization, and the
amplitude order is the operator order. Under \(N^*\Delta\simeq T^*X\),
the cotangent symplectic density supplies \(|dx|^{1/2}|d\xi|^{1/2}\), of
fiber degree \(d/2=n/4\). Multiplying the ordinary operator symbol by it
gives (C21). A general conormal bundle has no such specified canonical
half-density that would turn its symbols into scalar functions.

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\subsection{7. Full action on amplitudes, not only leading
symbols}\label{full-action-on-amplitudes-not-only-leading-symbols}

Let \(P=p(x,D)\) have left symbol of order \(d\), and let \(u\) have the
normalized reduced amplitude \(a(z,\tau)\) of (C18), of order
\(r=m+(n-2k)/4\). Work first with compact input and output localization
in one chart: the exact left symbol \(p\) is extended globally with
compact base support, and \(a\) is the global reduced amplitude of the
compactly localized input from Section 2. Thus both factors have global
symbol bounds. The output amplitude of this localized operator and input
is \[
b(z,\tau)=
\left[
\exp\big(i\langle D_w,D_\eta\rangle-i\langle D_t,D_\tau\rangle\big)
\{p(t,z,\tau,\eta)a(w,\tau)\}
\right]_{w=z,\ t=0,\ \eta=0}.
\tag{C22}
\] The exponential means successive quadratic-multiplier restriction as
below. Every truncation before total degree \(N\) has remainder in
\(S^{d+r-N}\), with all base/frequency derivatives and finite-seminorm
control. In particular \[
\sigma(Pu)=\sigma(P)|_{N^*Y}\,\sigma(u).
\tag{C23}
\]

For Schwartz inputs, insert the Fourier transform of (C18) into the left
quantization integral. Tangential Fourier inversion first gives the full
amplitude \[
a_1(t,z,\tau)=(2\pi)^{-(n-k)}
\iint e^{i(z-w)\cdot\eta}p(t,z,\tau,\eta)a(w,\tau)\,dw\,d\eta.
\] This is the first quadratic multiplier in (C22), evaluated at
\(w=z,\eta=0\). Its estimates follow directly from
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale}, including the passive variables. On the space of
\((x,w,\tau,\eta)\), use \[
G=|dx|^2+|dw|^2+\langle\tau\rangle^{-2}|d\tau|^2
+\langle(\tau,\eta)\rangle^{-2}|d\eta|^2,
\quad M=\langle(\tau,\eta)\rangle^d\langle\tau\rangle^r.
\tag{C24}
\] The product symbol obeys these derivative bounds: derivatives of
\(p\) in \(\tau\) cost at most \(\langle\tau\rangle^{-1}\), and those of
\(a\) have exactly that cost. Small metric displacements preserve both
brackets, so slow variation and local weight comparison hold. For the
phase \(\langle D_w,D_\eta\rangle\), a finite phase distance keeps
\((x,\tau)\) fixed. Relative to an observation point with \(\eta=0\),
the \(\eta\)-part of \(G\) at any input point is no larger, while the
remaining parts agree. The weight ratio is bounded by a power of
\(1+|\eta|^2\), which the phase-dual distance controls. Thus the full
hypotheses of the Gauss restriction theorem hold uniformly along
\(w=z,\eta=0\). When \(n-k>0\), the parameter is
\(1/(2\langle\tau\rangle)\). When \(k=n\), there are no tangential
variables: this first reduction is ordinary multiplication, its phase
and actual Gauss parameter are zero, and the zero-phase branch of
Section 8 of \href{gauss-transform-estimates.pdf}{Quadratic Fourier
multipliers at a moving scale} applies. It gives \(a_1\in S^{d+r}\) and
every remainder order \(d+r-N\), including parameter derivatives.

Now remove the normal base dependence of \(a_1\) by (C5). This gives the
second multiplier and (C22). The constant-coefficient differential
operators commute before restriction. Expanding the two finite Taylor
polynomials and collecting terms of total degree less than \(N\) gives
the stated single exponential expansion; each contraction lowers the
order by one, and the discarded finite terms and both controlled
remainders have order at most \(d+r-N\).

For general symbols, take bounded compactly supported approximants. The
oscillatory definition, the two weak Gauss extensions, and the Schwartz
action of ordinary \(S_{1,0}\) operators and their adjoints pass the
equality to distributions. Explicitly, adjoints applied to a fixed
Schwartz test function converge in Schwartz space: the classical
estimates from Section 8 of \href{weyl-metric-products.pdf}{Two measuring
scales, one Weyl product} bound all output seminorms uniformly, local
smooth convergence follows by dominated oscillatory integration, and a
higher uniform decay seminorm controls the tails. Pair this convergence
with the bounded, distributionally convergent input approximants. For
the original local operator, proper support permits a compact input
cutoff equal to one at all kernel input points over a fixed output
compact set. Use a compact output cutoff equal to one on its working
neighborhood. Splitting into chart pieces near the diagonal then gives
the globally extended symbols just used. The remaining kernel pieces are
separated from the diagonal, so frequency integration by parts makes
their outputs smooth there; after compact output localization their
normal Fourier transforms are \(S^{-\infty}\) amplitudes and are
included in the local representation. This gives equality on the working
neighborhood and every claimed local remainder without imposing growth
at infinity on the original symbols. All estimates are componentwise,
and (C20) glues the leading term, proving (C23) for bundle operators.

If the amplitudes and operator symbols have complex polyhomogeneous
expansions in steps \(h=1/q\), \(q\) a positive integer, every
coordinate and action formula above preserves that expansion. A
frequency derivative lowers degree by an integer, which is \(q\) steps;
multiplying homogeneous components adds their step indices. At each
specified degree only finitely many indices contribute, and the
remainder estimates just proved justify truncation. A common
reciprocal-integer step may be chosen for two different such expansions.
The boundary theory below uses step one, with degrees \(m-j\); its phase
formulas are not assertions for arbitrary noninteger steps.

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\subsection{8. What a normal-frequency tail does on one
side}\label{what-a-normal-frequency-tail-does-on-one-side}

From here on the normal variable is one dimensional, and the interior is
\(t>0\). Smoothness up to the boundary means restriction of a smooth
function on a neighborhood of the closed half-space, locally in the
other variables. We first establish a Fourier test that includes complex
powers and logarithmic exceptions.

The entire family of supported distributions \[
e_\lambda(t)=\frac{t_+^{\lambda-1}}{\Gamma(\lambda)},\qquad
\partial_t e_\lambda=e_{\lambda-1},\qquad
e_0=\delta,
\quad e_{-j}=\delta^{(j)}
\tag{T1}
\] is initially defined by the integral when
\(\operatorname{Re}\lambda>0\). Here is a construction at the remaining
parameters. In its pairing with a test function subtract a Taylor
polynomial at zero inside a cutoff. The subtracted integral converges on
successively larger left half-planes; the integrals of the subtracted
powers give simple poles at the nonpositive integers. These are
precisely canceled by \(1/\Gamma(\lambda)\). Integration by parts proves
the derivative identity on the initial half-plane and then on every
continuation. At \(\lambda=1\) this is the Heaviside distribution, so
differentiation gives the last two identities. This proves the asserted
distributional normalization, rather than assigning a value to a
divergent power integral.

The elementary properties of \(\Gamma\) used here can be obtained from
its Euler integral. Integrating the beta integral repeatedly gives
\(\int_0^1 s^{z-1}(1-s)^Nds=N!/[z(z+1)\cdots(z+N)]\). After the
substitution \(s=y/N\), dominated convergence gives the usual product
limit for \(\Gamma(z)\) when \(\operatorname{Re}z>0\). Taking logarithms
of its factors, and subtracting their linear terms, leaves a uniformly
convergent series on compact sets avoiding the nonpositive integers.
Thus the continued function has no zeros; its only poles are simple, and
the recursion \(\Gamma(z+1)=z\Gamma(z)\) gives residue \((-1)^j/j!\) at
\(-j\). These facts justify the cancellations above.

Laplace damping and the Euler integral give, in our Fourier convention,
\[
\widehat {e_\lambda}(\tau)
=\lim_{\epsilon\downarrow0}(\epsilon+i\tau)^{-\lambda}.
\tag{T2}
\] First prove this for \(\operatorname{Re}\lambda>0\) by integrating
\(e^{-(\epsilon+i\tau)t}t^{\lambda-1}\). Passage to distributions
follows by pairing with a Schwartz function and dominated convergence on
the original \(t\)-integral. Differentiation and the continuation
construction then give all \(\lambda\). On the positive and negative
frequency rays the values in (T2) have ratio \(e^{i\pi\lambda}\). They
are the boundary values of a homogeneous holomorphic function in the
lower half-plane. Reflection in \(t\) gives the corresponding
upper-half-plane function and a distribution supported in \(t\leq0\).

For a more explicit smoothness test, let a frequency function have
homogeneous tails \(c_+\tau^v\) for \(\tau>0\) and \(c_-|\tau|^v\) for
\(\tau<0\), cut off near zero. The singular part on \(t>0\) of its
inverse oscillatory integral \emph{without} the inverse Fourier
coefficient is \[
\Gamma(v+1)
\big[c_+e^{i\pi(v+1)/2}+c_-e^{-i\pi(v+1)/2}\big]
t^{-v-1}.
\tag{T3}
\] For \(\operatorname{Re}v>-1\), insert \(e^{-\epsilon|\tau|}\) on the
two rays, apply the Euler integral, and let \(\epsilon\downarrow0\) at a
fixed \(t>0\). Removing the low-frequency part changes the answer by a
smooth function. Both this cutoff tail and its distributional inverse
depend holomorphically on \(v\) on each vertical strip. Taylor
subtraction at frequency zero therefore continues the displayed
expression; any polar subtractions have inverse transforms that are
polynomials in \(t\), hence smooth.

Away from negative integers, (T3) has a nonsmooth coefficient unless its
bracket vanishes. That vanishing is exactly \[
c_-=e^{i\pi v}c_+.
\tag{T4}
\] At \(v=-\ell\), \(\ell=1,2,\ldots\), put \(v=-\ell+w\) and expand
\(t^{-w}=1-w\log t+O(w^2)\). If \(B(v)\) denotes the bracket in (T3),
the nonsmooth term is \[
-\frac{(-1)^{\ell-1}}{(\ell-1)!}
B(-\ell)t^{\ell-1}\log t.
\tag{T5}
\] It vanishes precisely under (T4) again. Terms arising from
differentiating \(B\), or from the finite part of \(\Gamma\), are
multiples of \(t^{\ell-1}\) and are smooth. Thus the exceptional
integral exponents do not create an exception to the phase test; they
replace a power singularity by a logarithm.

Suppose now that \(b(z,\tau)\) is polyhomogeneous of complex degree
\(v\), with components \(b_j\) of degree \(v-j\). Then \[
\int e^{it\tau}b(z,\tau)d\tau\text{ is smooth up to }t=0+
\quad\Longleftrightarrow\quad
b_j(z,-1)=e^{i\pi(v-j)}b_j(z,1)
\text{ for every }j.
\tag{T6}
\] These conditions also say that every homogeneous component extends
holomorphically to the upper half-plane: use \(b_j(z,1)\zeta^{v-j}\)
with \(0\leq\arg\zeta\leq\pi\).

To verify that the expansion proves the equivalence, a remainder of
order \(\operatorname{Re}v-N\) has a \(C^q\) inverse integral whenever
\(N>\operatorname{Re}v+q+1\), by absolute integration after up to \(q\)
derivatives. If every condition holds, (T3)--(T5) and this estimate give
arbitrary regularity. Conversely take the first component violating
(T4). If its power is \(t^\lambda\), where \(\lambda=-v+j-1\) is not a
nonnegative integer, choose an integer \(q>\operatorname{Re}\lambda\).
Differentiate \(q\) times and multiply by \(t^{q-\lambda}\). Smooth
terms tend to zero; every later singular term tends to zero because it
has an additional positive integer power of \(t\); the first term tends
to its nonzero coefficient times
\(\lambda(\lambda-1)\cdots(\lambda-q+1)\). Choose the truncation
remainder to be \(C^q\). This contradicts smoothness. For a logarithm
\(t^{\ell-1}\log t\), differentiate \(\ell\) times and multiply by
\(t\); exactly the first logarithm has a nonzero limit. The same
argument after tangential differentiation is uniform on compact sets. In
particular a power with nonzero imaginary part and integer real part is
covered: its critical derivative oscillates, and is not silently treated
as a smooth integer power.

We will also use two elementary extension facts. Arbitrary smooth
boundary jets \(f_j(z)\) are realized by a smooth function. Locally, sum
\(\chi(t/\epsilon_j)t^j f_j(z)/j!\), where \(\chi=1\) near zero. Choose
\(\epsilon_j\downarrow0\) so that the \(j\)-th term has every prescribed
compact \(C^N\) seminorm at most \(2^{-j}\) for \(N\leq j/2\); this is
possible because its bound contains \(\epsilon_j^{j-N}\). The finitely
many small indices are harmless, all derivative series converge, and the
desired jets follow at zero. Exhausting tangential compact sets and then
using a partition of unity proves the full assertion. Second, if all
derivatives of a function on \(t>0\) have continuous one-sided limits,
it is smooth up to the boundary. The fundamental theorem of calculus,
using bounded next normal derivatives, proves existence and
compatibility of these jets; the preceding construction supplies
matching jets on \(t<0\). Their difference is flat at zero, so gluing
gives a smooth extension.

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\subsection{9. Supported powers and the spaces of complex
type}\label{supported-powers-and-the-spaces-of-complex-type}

Let \(\overline Y\) be a smooth manifold with boundary inside a smooth
ambient manifold \(X\). In this boundary discussion \(Y\) denotes its
interior. For a complex number \(\mu\), define \[
C_\mu^\infty(\overline Y;E)
=\{u\in I_{\mathrm{phg}}^{\mu-(n-2)/4}(X,\partial Y;E):
\operatorname{supp}u\subset\overline Y\}.
\tag{T7}
\] The order refers to a step-one polyhomogeneous normal form and its
real-part estimates. Smooth terms supported in \(\overline Y\) are
included. In a boundary chart the normal amplitude has degree \(\mu\).
By (T6) applied on the negative side, its homogeneous coefficients
satisfy \[
a_j(z,-1)=e^{-i\pi(\mu-j)}a_j(z,1).
\tag{T8}
\] Conversely these relations allow an expansion by distributions
supported on the positive side; a general amplitude satisfying them has
a smooth inverse on the negative side, and subtracting a smooth
extension of that part produces a supported representative with the same
asymptotic coefficients. Formula (T2) gives a supported representative
of each homogeneous term directly. In particular every smooth leading
coefficient in the bundle fiber can be prescribed independently:
multiply \(e_{-\mu}(t)\) by that coefficient and by a cutoff equal to
one near zero. Its Fourier leading coefficient on the positive ray is
\(e^{i\pi\mu/2}\), which never vanishes, so rescaling gives the desired
normalization.

The space (T7) does not depend on the ambient extension of a boundary
chart. The conormal change-of-coordinate proof in Section 6 gives the
same polyhomogeneous class under any smooth extension, and support is
intrinsic. Equivalently the local description just given glues by the
bundle transition law. This treats arbitrary smooth vector bundles;
finite-order Taylor coefficients and all constructions are taken
componentwise in a frame.

Here is a concrete description of all its members, not only their
leading terms:

\begin{itemize}
\tightlist
\item
  If \(\mu\notin\mathbb Z\), they have expansions
  \(\sum_{j\geq0}u_j(z)t_+^{j-\mu-1}\), with these powers interpreted by
  distributional continuation. After sufficiently many terms the
  remainder has any prescribed finite smoothness.
\item
  If \(\mu\leq-1\) is an integer, they are exactly the zero extensions
  of smooth functions on \(\overline Y\) whose normal jets of orders
  less than \(-\mu-1\) vanish.
\item
  If \(\mu\geq0\) is an integer, they are exactly a zero extension of a
  smooth function plus a finite sum
  \(\sum_{j=0}^{\mu}v_j(z)\delta^{(j)}(t)\), locally at the boundary.
\end{itemize}

To prove these descriptions, match successive Fourier coefficients using
\(e_{j-\mu}\) in (T1)--(T2). Multiplication by a normal cutoff changes
each Fourier expansion by lower terms only, which are matched
successively; equivalently its cutoff is constant near the entire
singular support, so the complementary part has a rapidly decreasing
Fourier transform after compact localization. The difference after \(N\)
terms has amplitude order \(\operatorname{Re}\mu-N\), hence is \(C^q\)
for \(N>\operatorname{Re}\mu+q+1\). When \(\mu\) is not an integer, the
gamma factors are nonzero and are absorbed in the coefficients of the
stated powers. At integer \(\mu\), (T1) gives precisely the listed delta
derivatives and nonnegative powers on the positive side. The smooth jet
construction at the end of Section 8 realizes the latter coefficients by
one smooth function on \(\overline Y\). The remaining amplitude is of
every negative order, so its inverse is smooth; support makes it flat on
the boundary. This proves exact membership statements, including their
converses. One can also realize arbitrary infinite power expansions by
multiplying the \(j\)-th term by a shrinking cutoff: the same \(2^{-j}\)
construction is applied in successively weaker symbol orders, and then
in each fixed finite distributional seminorm. This gives all the
asserted asymptotic remainders.

For clarity, if \(f\) is smooth up to \(t=0\), compactly supported in a
boundary chart, and \(f_0\) is its zero extension, its normal Fourier
transform is \[
\widehat {f_0}(z,\tau)=\int_0^\infty e^{-it\tau}f(t,z)dt
\sim-i\sum_{j\geq0}\tau^{-1-j}D_t^j f(0,z).
\tag{T9}
\] Integrating by parts \(N\) times proves this expansion and its
remainder. To differentiate it in \(\tau\), apply the same integration
by parts to \((-it)^\alpha f\); its vanishing jets give the extra
\(\alpha\) powers of decay. Tangential derivatives commute with the
integral. Consequently (T9) is a full symbol expansion of degree \(-1\).
The reduced amplitude in (C4) is \((2\pi)^{-1}\widehat {f_0}\), so the
conormal order is exactly \(-(n+2)/4\), and these inputs form
\(C_{-1}^\infty\). Arbitrary jets in (T9) can be prescribed by the
smooth extension construction; two such amplitudes with identical jets
differ by \(S^{-\infty}\).

For each integer \(N\geq0\), \(C_{\mu-N}^\infty\subset C_\mu^\infty\).
Smooth multiplication preserves \(C_\mu^\infty\), by (C5) and support; a
first-order derivative maps it into \(C_{\mu+1}^\infty\), by (C22) or
direct differentiation of the power family. Tangential derivatives in a
fixed boundary chart actually preserve its normal order. These
inclusions will let us recover every jet in a transmission condition
from an operator mapping property.

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\subsection{10. All jets are necessary, at every complex
type}\label{all-jets-are-necessary-at-every-complex-type}

Let \(P:E\to F\) be properly supported, with a classical full left
symbol \(p\sim\sum_{j\geq0}p_j\), where \(p_j\) is homogeneous of
complex degree \(m-j\). We say that \(P\) has transmission of type
\(\mu\) into \(Y\) if \(Pu|_Y\) is smooth up to the boundary for every
\(u\in C_\mu^\infty(\overline Y;E)\). The ordinary transmission property
means that zero extensions of arbitrary smooth functions have this
property; it is type \(-1\), or equivalently the phase type zero. Proper
support is the hypothesis ensuring these local distributions can be
acted on without a global compact-support restriction.

The complete criterion is, for every \(j\), every pair of multi-indices
\(\alpha,\beta\), and every boundary point, \[
(\partial_x^\beta\partial_\xi^\alpha p_j)(0,z,-1,0)
=e^{i\pi(m-j-|\alpha|+2\mu)}
(\partial_x^\beta\partial_\xi^\alpha p_j)(0,z,1,0).
\tag{T10}
\] It is an equality of maps between the bundle fibers. In particular it
includes every normal base jet and every tangential frequency jet, not
just the two values of the leading symbol.

We first prove sufficiency. Apply the full amplitude formula (C22) to a
supported input. Each homogeneous output term is a finite sum of
products of a base/frequency derivative of some \(p_j\), of frequency
degree \(d_p\), with a derivative of an input coefficient, of normal
degree \(d_a=\mu-r\), \(r\) an integer. At the two normal rays, (T10)
gives the first factor the ratio \(e^{i\pi(d_p+2\mu)}\); (T8), including
its differentiated version, gives the second the ratio
\(e^{-i\pi d_a}\). Their product is
\(e^{i\pi(d_p+2\mu-d_a)}=e^{i\pi(d_p+d_a)}\), because \(\mu-d_a\) is an
integer. Matrix multiplication has the same scalar phase and preserves
its order. Thus each output coefficient obeys the upper-half-plane
relation in (T6). All remainder orders in (C22) are available, so (T6)
proves the desired smoothness with all derivatives. Localizing and
summing in charts proves it globally.

For necessity, first prescribe an arbitrary leading input coefficient
using (T2). The leading output coefficient is the product
\(p_0(0,z,\tau,0)a_0(z,\tau)\). Its upper-half-plane relation and the
input's lower-half-plane relation give
\(p_0(0,z,-1,0)=e^{i\pi(m+2\mu)}p_0(0,z,1,0)\). Arbitrary input vectors
prove the entire matrix equality.

Next recover every derivative of \(p_0\). A commutator with
multiplication by a coordinate has full symbol a fixed nonzero constant
times the corresponding frequency derivative; a commutator with
\(D_{x_i}\) has full symbol \(D_{x_i}p\). These are exact local
identities obtained by differentiating the Fourier kernel; coordinate
cutoffs equal to the coordinates near the point only add locally smooth
kernels. Therefore an iterated commutator has full symbol a nonzero
constant times \(\partial_x^\beta\partial_\xi^\alpha p\), with leading
degree \(m-|\alpha|\).

Although derivatives need not preserve a particular \(C_\mu^\infty\),
this commutator still has the required mapping property on a
sufficiently smaller such space. Expand its finite nested commutator as
a sum of words with one occurrence of \(P\). If \(N\) is at least the
total number of first-order differential factors, every word to the
right of \(P\) takes \(C_{\mu-N}^\infty\) into \(C_\mu^\infty\); every
word to its left preserves smoothness on the interior up to the
boundary. The preceding leading-coefficient argument therefore applies
to this commutator with input parameter \(\mu-N\). Since
\(e^{2\pi i(\mu-N)}=e^{2\pi i\mu}\), it gives exactly (T10) for \(p_0\)
and all \(\alpha,\beta\).

Finally pass through all lower homogeneous degrees. Choose a frequency
cutoff equal to one near infinity and properly quantize the single
homogeneous symbol \(p_0\). Its sole homogeneous component satisfies
every condition just proved, so the already established sufficiency
shows that this operator has the mapping property. A cutoff of the
kernel near the diagonal makes it proper and changes its local full
symbol only by \(S^{-\infty}\), as follows by integration by parts off
that diagonal. Subtract it from \(P\). The remainder has order \(m-1\)
and the same mapping property. Apply the leading argument and the
commutator argument to this remainder to obtain every jet of \(p_1\).
Induction gives every \(p_j\), proving necessity in full.

The proof also shows why testing only functions with a fixed finite
order of boundary vanishing still detects ordinary transmission. Such
tests form \(C_{-1-N_0}^\infty\) for a fixed nonnegative integer
\(N_0\). The phase is unchanged. In the commutator step choose inputs in
still smaller \(C_{-1-N}^\infty\), so every right-hand word remains
among the permitted tests. The same subtraction induction recovers all
jets of all homogeneous terms. Finite boundary vanishing therefore
cannot remove a failed transmission jet.

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\subsubsection{10.1. An equivalent family of boundary
tests}\label{an-equivalent-family-of-boundary-tests}

The full condition (T10) can be checked using fewer separately
prescribed derivatives, while recovering every jet. Keep the original
homogeneous maps \(p_j\), complex orders \(m,\mu\), coordinates
\(x=(t,z)\), \(\xi=(\tau,\eta)\), and both points \((0,z,\pm1,0)\). Then
(T10) is equivalent to the following identities, for every \(j\geq0\),
normal base order \(b\geq0\), tangential frequency multi-index \(a\),
and every boundary point \(z\): \[
 (\partial_t^b\partial_\eta^a p_j)(0,z,-1,0)
 =e^{i\pi(m-j-|a|+2\mu)}
   (\partial_t^b\partial_\eta^a p_j)(0,z,1,0).
 \tag{TJ1}
\] The equality is between the same bundle fibers in a fixed smooth
frame. It must hold at every boundary point, so it can be differentiated
in every tangential base direction. This does not discard any base jet.

\textbf{Proof of the exact equivalence.} Fix a tangential base
multi-index \(d\), and write \[
 g(\tau)=(\partial_z^d\partial_t^b\partial_\eta^a p_j)(0,z,\tau,0),
 \qquad v=m-j-|a|.
 \tag{TJ2}
\] Differentiating the original homogeneity law in \(\eta\) contributes
exactly the factor \(s^{-|a|}\); base differentiation leaves the degree
unchanged. Hence, for every real \(s>0\), \[
 g(s)=s^v g(1),\qquad g(-s)=s^v g(-1),
 \qquad s^v=\exp(v\log s).
 \tag{TJ3}
\] There is no branch change in this positive real logarithm. Repeated
differentiation on each open ray gives, for every integer \(c\geq0\), \[
 g^{(c)}(1)=(v)_c g(1),\qquad
 g^{(c)}(-1)=(-1)^c(v)_c g(-1),\qquad
 (v)_c=\prod_{r=0}^{c-1}(v-r),\quad(v)_0=1.
 \tag{TJ4}
\] The negative-ray sign follows at each differentiation from
\(s=-\tau\). Differentiating (TJ1) in \(z\), substituting it into (TJ4),
and retaining the entire falling factorial yields \[
 g^{(c)}(-1)=(-1)^c(v)_c e^{i\pi(v+2\mu)}g(1)
           =e^{i\pi(v-c+2\mu)}g^{(c)}(1).
 \tag{TJ5}
\] The scalar phases agree because their quotient is \(e^{2\pi i c}=1\).
If any factor in \((v)_c\) is zero, both sides vanish; no division by
that product has occurred. Set \(\alpha=(c,a)\), \(\beta=(b,d)\). Then
\(|\alpha|=c+|a|\), and (TJ5) is exactly (T10) for those original
indices. They range over every required derivative. Conversely (T10),
restricted to \(c=0,d=0\), gives (TJ1). This proves both implications,
for complex \(m,\mu\), all matrix entries and dimension one, where
\(a,d\) are empty multi-indices. The previously proved mapping property,
invariant reflection and all higher trace consequences therefore receive
precisely the same full jets.

\textbf{Human source and parameter correspondence.}
\href{https://arxiv.org/abs/1310.0951v5}{Gerd Grubb, \emph{Fractional
Laplacians on domains}, arXiv 1310.0951v5, October 1, 2014}, Theorem
2.6, equations (2.11), (2.17), and the paragraph immediately following
its proof, already gives this equivalent family. This is an explicit
proof of that criterion, with no novelty claim. Her supported-power
exponent \(\nu\) corresponds to \(\nu=-\mu-1\) here: her distribution
\(I^\nu=t_+^\nu/\Gamma(\nu+1)\) is exactly \(e_{\nu+1}=e_{-\mu}\) from
(T1), initially where the integral converges and then by the same
derivative continuation. Its integer delta terms and lower-half-plane
branch are retained. The source phase becomes \[
 e^{i\pi(m-2\nu-j-|\alpha|)}
 =e^{i\pi(m+2\mu+2-j-|\alpha|)}
 =e^{i\pi(m-j-|\alpha|+2\mu)}.
 \tag{TJ6}
\] The last equality records the full factor \(e^{2\pi i}=1\). Thus the
comparison identifies the original supported distributions as well as
every phase and index; it does not replace the course's symbols or
proofs.

\textbf{Editorial correction to one source integral.} In the proof of
Grubb's Lemma 2.7, the displayed \(\sigma=-2\) calculation omits \(i\)
and uses the wrong transformed lower endpoint. This source-only error is
already avoided in Problem 3. Retain both boundary values and the entire
original integral. For \(t>0\), \[
 \begin{split}
 \int_{|\tau|>1}e^{it\tau}(\tau^\pm)^{-3}|\tau|\,d\tau
 &=\int_1^\infty\tau^{-2}(e^{it\tau}-e^{-it\tau})\,d\tau\\
 &=2i\int_1^\infty\tau^{-2}\sin(t\tau)\,d\tau
 =2it\int_t^\infty s^{-2}\sin s\,ds.
 \end{split}
 \tag{TJ7}
\] On the negative ray both branches contribute \(-|\tau|^{-2}\), since
\(e^{\pm3\pi i}=-1\). Both original integrals converge absolutely. The
substitution \(s=t\tau\) has \(d\tau=ds/t\), giving the factor \(t\) and
lower endpoint \(t\). For \(0<t<1\), the last integral is \[
 \int_t^\infty s^{-2}\sin s\,ds
 =-\log t+C-\int_0^t\frac{\sin s-s}{s^2}\,ds,
 \quad C=\int_0^1\frac{\sin s-s}{s^2}\,ds+
             \int_1^\infty\frac{\sin s}{s^2}\,ds.
 \tag{TJ8}
\] The first integrand extends analytically to zero, with value zero,
and the second integral converges absolutely. Consequently the complete
singular contribution is \(-2it\log t\), with the remaining term smooth
at zero. The source's conclusion that this example has a nonzero
logarithmic term survives. Neither this correction nor (TJ1) changes the
already valid full-jet theorem.

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\subsection{11. The invariant reflection of a classical
operator}\label{the-invariant-reflection-of-a-classical-operator}

For an operator of fixed complex order \(m\), define, modulo smoothing
operators, \[
\widetilde p(x,\xi)\sim
\sum_{j\geq0}e^{-i\pi(m-j)}p_j(x,-\xi).
\tag{T11}
\] Such a classical symbol exists. To see the asymptotic realization
explicitly, multiply the \(j\)-th homogeneous term by \(\chi(\xi/R_j)\),
where \(\chi\) vanishes near zero and is one near infinity. Choose
\(R_j\to\infty\) so that each later term is at most \(2^{-j}\) in the
first \(j\) compact symbol seminorms of order
\(\operatorname{Re}m-j/2\). Homogeneity makes this possible; a fixed
lower target order only involves finitely many excluded indices. The
series then converges with precisely the required remainders. The
construction is local in the base and glues by a partition. Two
realizations differ by \(S^{-\infty}\). Directly from (T11),
\(\widetilde{\widetilde P}=e^{-2\pi im}P\) modulo smoothing, so
\(e^{i\pi m}\widetilde{\phantom P}\) is an involution when the order
label \(m\) is fixed.

We prove that (T11) is intrinsic. Here are the needed full coordinate
estimates, including their remainder. For a coordinate change
\(\kappa\), write near the diagonal
\(\kappa(x)-\kappa(y)=\Psi(x,y)(x-y)\), using the integral of its
derivative along the segment. The matrix \(\Psi\) is invertible on a
sufficiently small diagonal neighborhood. Choose a compact base cutoff
\(\chi(x,y)\) supported there and equal to one near the working
diagonal; extend the resulting amplitude by zero. Pullback of this
localized operator kernel and the change \(\eta=\Psi(x,y)^T\theta\) give
the global amplitude \[
A(x,y,\eta)=\chi(x,y)p(\kappa(x),\Psi(x,y)^{-T}\eta)
\frac{|\det\kappa'(y)|}{|\det\Psi(x,y)|}
\tag{T12}
\] for functions. For half-densities the numerator is
\(|\det\kappa'(x)\det\kappa'(y)|^{1/2}\); frame changes insert the
corresponding smooth matrices on the two sides. All these extra factors
have frequency degree zero. Each base derivative falling on
\(\Psi^{-T}\eta\) brings a linear frequency factor paired with a symbol
derivative, so \(A\) has order \(m\) with all amplitude estimates.

Its left symbol is the reduction
\([ e^{i\langle D_y,D_\eta\rangle}A(x,y,\eta)]_{y=x}\), with expansion
\(\sum_\alpha\partial_\eta^\alpha D_y^\alpha A(x,x,\eta)/\alpha!\) and
remainder of order \(\operatorname{Re}m-N\) after \(|\alpha|<N\). For
Schwartz amplitudes this follows by inserting the Fourier kernel and
Fourier inversion; the sign also follows by moving \((y-x)^\alpha\) from
a Taylor expansion onto the frequency derivative of the amplitude. For
general amplitudes it is the same Gauss restriction as (C5), with the
opposite sign and passive \(x\), metric
\(|dy|^2+\langle\eta\rangle^{-2}|d\eta|^2\), and parameter
\(1/(2\langle\eta\rangle)\). Thus
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} supplies every finite-seminorm remainder and the weak
extension. A cutoff away from the diagonal gives a smooth kernel by
repeated frequency integration by parts. This proves the full local
coordinate formula needed here: all derivatives of the cutoff vanish on
the working diagonal, and its value there is one. The discarded kernel
pieces are smooth on the working chart and give smoothing symbol terms.

Let \(A_j\) be the homogeneous component of (T12) of degree \(m-j\). A
reduced term with index \(\alpha\) has degree \(m-j-|\alpha|\).
Reflecting the reduced term with (T11) supplies the factor
\(e^{-i\pi(m-j-|\alpha|)}\). Reflecting \(p_j\) first supplies
\(e^{-i\pi(m-j)}\); differentiating \(A_j(x,y,-\eta)\) supplies
\((-1)^{|\alpha|}\). These factors agree. All remainders can be taken
arbitrarily low, so (T11) commutes with coordinates, density choices and
bundle frames modulo smoothing.

It is useful also to check the algebra statement. Apply the same compact
chart cutoffs and global extensions before each reduction, retaining the
smooth off-diagonal pieces as smoothing terms. For local left symbols
the full product and adjoint expansions are \[
p\circ q\sim\sum_\alpha\frac{\partial_\xi^\alpha p\,D_x^\alpha q}{\alpha!},
\qquad
p^{\mathrm{adj}}\sim\sum_\alpha
\frac{\partial_\xi^\alpha D_x^\alpha p^*}{\alpha!}.
\tag{T13}
\] The first formula follows by inserting two kernels: the remaining
integral is
\((2\pi)^{-n}\iint e^{i(x-y)\cdot\eta}p(x,\xi+\eta)q(y,\xi)dy\,d\eta\).
The tangential reduction argument in Section 7, with \(\xi\) as its
retained frequency, gives the expansion and remainder of order
\(\operatorname{Re}(m+m')-N\). The adjoint kernel has amplitude
\(p(y,\xi)^*\); the reduction just proved gives its formula and
remainder of order \(\operatorname{Re}m-N\). Schwartz identities pass to
proper classical operators on distributions by the same Schwartz-adjoint
convergence argument used after (C24). Thus these formulas do not assume
the unresolved general-metric operator-composition extension.

For a product term with component indices \(j,l\), reflection
contributes \(e^{-i\pi(m+m'-j-l-|\alpha|)}\), equal to the two separate
reflection factors times \((-1)^{|\alpha|}\). Hence reflection is
multiplicative with the summed order label. In the order-zero algebra
its factors are the real numbers \((-1)^j\); the same calculation in the
adjoint formula gives
\(\widetilde{P^{\mathrm{adj}}}=(\widetilde P)^{\mathrm{adj}}\).
Therefore it is an involution of that algebra commuting with adjoints,
all modulo smoothing. No such unqualified adjoint assertion is made for
an arbitrary complex order label.

Call a symbol flat on a conic submanifold when every derivative of every
homogeneous component vanishes there. Differentiating (T11) shows that
(T10) is equivalently \[
\widetilde p-e^{2\pi i\mu}p
\text{ is flat on the inward conormal ray of }\partial Y.
\tag{T14}
\] On the outward ray the equivalent expression is
\(e^{-2\pi i(m+\mu)}p-\widetilde p\). This follows either by reflecting
(T14) twice or by reversing the two rays in (T10). In particular type
depends only on \(\mu\) modulo \(\mathbb Z\). For the opposite side, a
type \(\mu'\) is guaranteed by the same relations if \[
m+\mu+\mu'\in\mathbb Z.
\tag{T15}
\] Indeed its desired phase is the inverse of (T10), and their product
is \(e^{2\pi i(m+\mu+\mu')}\). For complex parameters this equals one
exactly when the sum is a real integer. This is a sufficient equivalence
of the two systems of jet conditions; an operator whose relevant jets
all vanish may of course satisfy more than one otherwise distinct type
condition. In particular ordinary transmission is independent of the
side for integer-order operators. Formula (T14) and the intrinsic
mapping characterization prove invariance under boundary charts and
arbitrary bundle trivializations.

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\subsection{12. A normal integral defined by analytic
subtraction}\label{a-normal-integral-defined-by-analytic-subtraction}

Boundary symbols require an integral even when the normal symbol grows.
We define the needed value without treating a divergent real integral as
an ordinary integral.

Let \(q\) be continuous on the real line. Suppose that for some \(R>0\)
there is a function \(Q\), holomorphic in the upper exterior
\(\{\operatorname{Im}\zeta>0,|\zeta|>R\}\), continuous on its boundary,
of at most polynomial growth there, and such that
\(q(t)-Q(t)=O(|t|^{-2})\) on the two tails. Put \[
\begin{split}
\mathcal I_+(q)={}&
\int_{|t|>R}(q(t)-Q(t))dt+\int_{-R}^{R}q(t)dt\\
&-\int_{\pi}^{0}Q(Re^{i\theta})Ri e^{i\theta}d\theta.
\end{split}
\tag{T16}
\] The inner arc in this formula runs clockwise from \(-R\) to \(R\).
Changing \(R\) changes the two real integrals by precisely the integral
over the boundary of an upper half-annulus; Cauchy's theorem cancels the
difference.

To prove independence of \(Q\), their difference \(H\) is holomorphic in
an upper exterior, polynomially bounded, and \(O(|t|^{-2})\) on the real
tails. Apply the maximum principle on this domain to
\(\zeta^2H(\zeta)(1-i\epsilon\zeta)^{-M}\), with \(M\) larger than the
polynomial growth degree plus two. It tends to zero at infinity for each
\(\epsilon>0\). On the real tails it is uniformly bounded, since
\(|1-i\epsilon t|\geq1\), and on the fixed inner arc it has a bound
independent of \(0<\epsilon\leq1\), since \(|1-i\epsilon\zeta|\geq1\)
throughout the upper half-plane. Taking first the outer radius to
infinity and then \(\epsilon\downarrow0\) proves
\(|H(\zeta)|\leq C|\zeta|^{-2}\) in that upper exterior. Cauchy's
theorem now applies with a vanishing outer-arc integral. Its conclusion
is
\(\int_{|t|>R}H(t)dt+\int_\pi^0H(Re^{i\theta})Ri e^{i\theta}d\theta=0\),
exactly the independence required in (T16).

There is a parameter version with no lost endpoint. Let \(F(\zeta,s)\)
be bounded and jointly continuous on the closed upper half-plane times a
parameter space, and holomorphic in \(\zeta\) in the open half-plane.
Then \(QF\) is admissible for \(qF\), and
\(s\mapsto\mathcal I_+(qF(\cdot,s))\) is continuous. The tail of (T16)
is dominated by a fixed integrable multiple of \(|q-Q|\), while the
interval and arc are compact; dominated convergence proves the
statement. Uniform bounds on a compact parameter set suffice, and the
same argument permits derivatives whenever the differentiated tails have
the corresponding common bound.

For a rational function without real poles, take \(R\) beyond every pole
and take \(Q=q\) in that exterior. The two finite pieces of (T16) form
the positively oriented boundary of the upper half-disk. Therefore \[
\mathcal I_+(q)=2\pi i
\sum_{\operatorname{Im}\zeta>0}\operatorname{Res}_{\zeta}q.
\tag{T17}
\] This includes rational functions with polynomial growth. For example
\(\mathcal I_+(t/(t^2+1))=i\pi\), whereas its symmetric real principal
value is zero; the two definitions answer different limiting problems.

Alternatively take \(Q\) to be the finite Laurent sum at infinity
containing precisely the powers of degree at least \(-1\). The omitted
rational tail is \(O(|\zeta|^{-2})\), so this is another admissible
subtraction and gives the same value by the independence proof. This
alternative often makes (T16) convenient to compute.

If \(q\) is in addition a classical normal symbol, its oscillatory
inverse integral at \(t>0\) equals \(\mathcal I_+(e^{it\cdot}q)\). To
verify this carefully, insert
\(F_\epsilon(\zeta)=(1-i\epsilon\zeta)^{-M}\), with \(M\) sufficiently
large. The damped real integral is absolutely convergent, and an upper
contour applied to \(e^{it\zeta}Q(\zeta)F_\epsilon(\zeta)\) gives its
value from (T16). Its outer arc tends to zero, since the damping has
made the polynomial factor decay faster than \(|\zeta|^{-1}\) and the
exponential is bounded in the upper half-plane. The parameter assertion
gives the limit of (T16) as \(\epsilon\downarrow0\). On the other hand,
integrate the damped real integral by parts in frequency enough times.
The derivatives of \(F_\epsilon\) satisfy uniform symbol bounds of the
corresponding negative orders; after sufficiently many integrations all
resulting terms are integrable with a common bound and converge to the
undamped oscillatory definition. The two limits agree. This argument
also explains why the upper half-plane is the one appropriate to the
interior \(t>0\).

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\subsection{13. Boundary traces of every layer
order}\label{boundary-traces-of-every-layer-order}

Assume ordinary transmission, equivalently (T10) with the factor
\(e^{2\pi i\mu}=1\). Write \(\xi=(\tau,\eta)\), with \(\tau\) normal and
\(\eta\) tangential. For every fixed \(z,\eta\), the full symbol
\(p(0,z,\tau,\eta)\) admits an analytic subtraction as in (T16). Indeed
Taylor-expand each homogeneous component at the positive normal ray: \[
p_j(0,z,\tau,\eta)\sim
\sum_\alpha \frac{(\partial_\eta^\alpha p_j)(0,z,1,0)}{\alpha!}
\eta^\alpha\tau^{m-j-|\alpha|}\quad(\tau\to+\infty).
\tag{T18}
\] On the negative ray its coefficients agree with the upper-half-plane
values of these powers by (T10). Choose finitely many terms, for
instance all \(j+|\alpha|<N\) with \(N>\operatorname{Re}m+2\), and use
the branch \(0\leq\arg\zeta\leq\pi\). Their sum is a suitable \(Q\); the
remainder is \(O(|\tau|^{-2})\) on both tails. Taylor's formula and the
classical remainders prove this uniformly with any fixed finite set of
tangential or base derivatives on compact sets of \((z,\eta)\), by
choosing enough terms. The identical statement holds for every normal
base derivative of \(p\) at \(t=0\), because the transmission hypothesis
contains those jets.

The minimal indicated truncation is also valid: retaining exactly
\(j+|\alpha|<2+\operatorname{Re}m\) leaves \(O(|\tau|^{-2})\), since
every omitted integer index has real degree at most \(-2\). The larger
truncations are useful when derivatives of the remainder are required.

Define \[
q(z,\eta)=\frac1{2\pi}\mathcal I_+\big(p(0,z,\cdot,\eta)\big).
\tag{T19}
\] It is a classical tangential symbol of degree \(m+1\), with
homogeneous components \[
q_j(z,\eta)=\frac1{2\pi}\mathcal I_+\big(p_j(0,z,\cdot,\eta)\big),
\qquad \eta\ne0,
\quad \deg q_j=m+1-j.
\tag{T20}
\] For smoothness away from \(\eta=0\), choose an exterior radius larger
than the parameters in a compact \(\eta\)-neighborhood and differentiate
(T16), using the uniform remainders in (T18). Rescaling \(\tau\) and the
radius together proves the asserted homogeneity, including the extra
factor of the scale from \(d\tau\). At small \(\eta\), the full symbol
(T19) remains smooth by the same parameter argument applied to the full
smooth \(p\); the components (T20) need only be defined away from zero.

If the ambient dimension is one, the boundary is zero dimensional.
Formula (T19) then gives the finite-dimensional boundary map directly;
there is no nonzero tangential frequency at which (T20) needs to be
specified.

Here are the full remainder estimates. On \(|\eta|>1\), no low-frequency
cutoff is needed in the homogeneous terms of \(p\), since
\(|(\tau,\eta)|>1\). For any integer \(N>\operatorname{Re}m+2\), the
remainder \(R_N=p-\sum_{j<N}p_j\) has an absolutely convergent normal
integral, and \[
\left|\partial_z^\beta\partial_\eta^\alpha
\int R_N(0,z,\tau,\eta)d\tau\right|
\leq C\langle\eta\rangle^{\operatorname{Re}m+1-N-|\alpha|}.
\tag{T21}
\] This follows by integrating the symbol bound
\(C\langle(\tau,\eta)\rangle^{\operatorname{Re}m-N-|\alpha|}\) and
setting \(\tau=\langle\eta\rangle s\). Its exponent is less than \(-1\).
For this integrable remainder choose \(Q=0\) in (T16), so its plus
integral is its ordinary integral. This proves (T21) for the difference
in (T19)--(T20). Arbitrarily large \(N\) give the expansion at every
requested order; a shorter truncation is that long remainder plus the
intervening homogeneous symbols, with precisely its stated order. Thus
no finite number of symbol seminorms or remainder orders has been
omitted.

We next prove that (T19) actually is the trace of the simple layer. For
\(v\) smooth and compactly supported tangentially, set
\(u=\delta(t)v(z)\). It belongs to \(C_0^\infty\), whose phase is
ordinary transmission, so \(Pu|_{t>0}\) is already known to be smooth up
to the boundary. Its local normal-frequency formula away from zero is \[
Pu(t,z)=(2\pi)^{-n}\int e^{iz\cdot\eta}
\left(\int e^{it\tau}p(t,z,\tau,\eta)d\tau\right)
\widehat v(\eta)d\eta.
\tag{T22}
\] Only the boundary jets of \(p\), rather than its values for positive
\(t\), satisfy (T18). We therefore justify the limit by normal Taylor
expansion. For a chosen integer \(L\), write \[
p(t,z,\xi)=\sum_{l<L}\frac{t^l}{l!}\partial_t^l p(0,z,\xi)
+t^L r_L(t,z,\xi).
\tag{T23}
\] The remainder is a symbol of the same order, locally uniformly in
\(t\). Each frozen coefficient has an analytic subtraction. By Section
12 its oscillatory normal integral is
\(\mathcal I_+(e^{it\cdot}\partial_t^l p(0,z,\cdot,\eta))\), continuous
down to \(t=0\), with polynomial bounds in \(\eta\). Thus all terms with
\(l>0\) tend to zero and the first gives (T19).

For explicit control of the remainder, a symbol \(r\) of fixed order
has, away from \(t=0\), the bound
\(\left|\int e^{it\tau}r(t,z,\tau,\eta)d\tau\right|
\leq Ct^{-K}\langle\eta\rangle^M\), \(0<t<1\), for some \(K,M\). This
follows by a dyadic partition in \(\tau\): the block of size \(R\) is
bounded by a fixed power of \(R\) times a fixed power of
\(\langle\eta\rangle\); after \(a\) frequency integrations by parts it
has the additional factor \((tR)^{-a}\). Sum blocks below and above
\(R=1/t\), choosing \(a\) larger than the growth exponent plus one. A
possible logarithm at zero exponent is bounded by a slightly larger
power of \(t^{-1}\). The same proof treats any fixed number of
derivatives. Crucially these exponents depend on the symbol order and
the requested derivative count, not on the Taylor length \(L\); its
constants may depend on \(L\). Choose \(L>K\). Then the last term of
(T23) tends to zero with a polynomial tangential bound. Taking larger
\(L\) after each desired derivative proves all corresponding boundary
limits. The rapid decrease of \(\widehat v\) justifies their integration
in (T22). Consequently \[
\left.P(\delta(t)v)\right|_{t=0+}=q(z,D_z)v.
\tag{T24}
\]

One can verify the same trace through genuine smooth layers, which also
fixes the distributional meaning of (T22). Choose
\(\varphi\in C_c^\infty(-1,1)\), \(\int\varphi=1\), and replace
\(\delta\) by \(\varphi_\epsilon(t)=\epsilon^{-1}\varphi(t/\epsilon)\).
The inner integral in (T22) acquires \(\widehat\varphi(\epsilon\tau)\).
For \(\operatorname{Im}\zeta\geq0\) and \(t\geq\epsilon\),
\(e^{it\zeta}\widehat\varphi(\epsilon\zeta)=\int\varphi(s)e^{i(t-\epsilon s)\zeta}ds\)
has modulus at most \(\|\varphi\|_1\). It is holomorphic and jointly
continuous, including \((t,\epsilon)=(0,0)\) on compact frequency sets.
The parameter lemma therefore gives the uniform frozen-coefficient
limit. For the Taylor remainder choose \(a<1\) with
\(\operatorname{supp}\varphi\subset[-a,a]\). On the full range
\(t\geq\epsilon\), the preceding off-zero estimate applies at distances
\(t-\epsilon s\geq(1-a)t\), uniformly in \(s\). Thus its contribution
still tends to zero after the chosen \(t^L\) factor. At fixed \(t>0\),
the mollified layers converge distributionally and in the displayed
smooth off-boundary formula to \(P(\delta v)\). This proves the joint
limiting construction and all the tangential derivative bounds used in
(T24), with polynomial tangential growth uniformly for
\(0\leq\epsilon\leq t\) near zero.

There is no restriction to simple layers. For any integers \(k,l\geq0\),
the complete left symbol of \(D_t^kPD_t^l\) is \[
p_{k,l}(t,z,\tau,\eta)=
\sum_{r=0}^k\binom{k}{r}
(D_t^{k-r}p)(t,z,\tau,\eta)\tau^{r+l}.
\tag{T25}
\] This is the ordinary Leibniz rule applied to the kernel, and is
exact. Its order is \(m+k+l\). Each homogeneous term obeys ordinary
transmission: a normal base derivative preserves the phase, and
multiplication by \(\tau^{r+l}\) supplies exactly its integer degree
parity. Thus the preceding theorem applies to it. The boundary operator
\(v\mapsto[D_t^kP(D_t^l\delta\,v)]_{t=0+}\) is a classical tangential
operator of order \(m+k+l+1\), with full symbol
\((2\pi)^{-1}\mathcal I_+(p_{k,l}(0,z,\cdot,\eta))\) and every
homogeneous remainder given as above. Normal coefficient derivatives in
(T25) affect its lower homogeneous terms and cannot be dropped. All
constructions are local and compatible with coordinate and frame changes
because they are traces of the intrinsic operators just proved; a
partition of unity yields the corresponding layer operators for bundles
on a smooth boundary.

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\subsection{14. Three models that distinguish the
hypotheses}\label{three-models-that-distinguish-the-hypotheses}

\textbf{An endpoint measured by a sequence.} Let \(g\) be a nonzero
smooth compactly supported function of \(z\), and set
\(u=\delta(t)g(z)\) with \(k\) normal variables. On large frequency
annuli its squared Fourier mass is comparable to \(R^k\). The upper
estimate follows from (C7); for the lower estimate restrict \(\eta\) to
a fixed ball on which \(\widehat g\) has positive squared integral and
restrict \(\tau\) to a fixed subannulus of radius \(R\). Thus its dyadic
sequence at \(s=-k/2\) is bounded above and below by positive constants.
It belongs to \(B^{-k/2}_{2,\infty}\) and to no \(B^{-k/2}_{2,p}\) with
finite \(p\); it belongs to every \(H^{-k/2-\epsilon}\), \(\epsilon>0\).
The amplitude is order zero, so its intrinsic conormal order is
\((2k-n)/4\). These are three compatible facts, not three different
conventions for a Sobolev exponent.

\textbf{A complex order that distinguishes the two sides.} In one
dimension let a classical symbol have homogeneous tails
\(p_0(\tau)=\tau^i\) for \(\tau>0\) and \(p_0(\tau)=e^{-\pi}|\tau|^i\)
for \(\tau<0\); smooth it near zero and localize its kernel properly. It
satisfies ordinary transmission into \(t>0\): the ratio is
\(e^{i\pi i}=e^{-\pi}\), and differentiation gives every higher
frequency condition. There are no tangential frequencies, and the local
symbol is independent of the base, so the other jets vanish. Into the
opposite side, ordinary transmission would require \(1=e^{-2\pi}\),
which fails. Type \(\mu'=-i\) does work on that side, since
\(m+0+\mu'=0\) in (T15). The imaginary part of the type is
mathematically active.

\textbf{Residues give a boundary operator.} At nonzero tangential
frequency put \(a=|\eta|\) and take \(p(\tau,\eta)=(\tau^2+a^2)^{-1}\).
A smooth low-frequency extension makes a classical symbol of order
\(-2\); proper localization changes only the local smoothing ambiguity.
It has ordinary transmission, since each frequency derivative has the
parity of its integer degree. The pole at \(\tau=ia\) has residue
\((2ia)^{-1}\), so (T20) gives principal boundary symbol
\(1/(2|\eta|)\), of order \(-1\). One normal derivative gives the
principal symbol \(i/2\), and two give \(-|\eta|/2\). Directly, the
inverse normal transform is \(e^{-a t}/(2a)\) for \(t>0\), and applying
\(D_t=-i\partial_t\) confirms each value and sign. The polynomial part
of \(\tau^2/(\tau^2+a^2)\) has plus integral zero, as (T17) requires.

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\subsection{15. Problems with full
solutions}\label{problems-with-full-solutions}

\textbf{Problem 1.} In the first model, explain why replacing the Besov
endpoint by the statement that every tangent derivative lies in
\(H^{-k/2-\epsilon}\) for every \(\epsilon>0\) loses information.
Produce a normal symbol-like amplitude that meets these weakened
estimates but is not of order zero.

\textbf{Solution.} In one normal dimension choose smooth disjoint dyadic
bumps \(\psi(2^{-j}\tau)\), supported where \(1<2^{-j}\tau<3/2\), and
put \(a(z,\tau)=g(z)\sum_{j\geq1}j\psi(2^{-j}\tau)\). It has frequency
derivatives bounded by \(C_\alpha\langle\tau\rangle^{\epsilon-\alpha}\)
for every \(\epsilon>0\), but its zeroth derivative is unbounded on the
bump maxima, so it is not \(S^0\). Form \(u\) by (C4) and compactly
localize near the normal origin if desired. Every tangent generator
\(tD_t\) replaces a bump by a finite linear combination of its scaled
derivatives with the same factor \(j\); tangential derivatives act on
\(g\). Balanced normal monomials have the same bound. The squared dyadic
mass is at most \(C j^2 2^j\), so the weighted Sobolev sum at
\(-1/2-\epsilon\) is bounded by \(C\sum j^2 2^{-2j\epsilon}<\infty\) for
every word. Before localization the endpoint sequence is comparable to
\(j\) along these annuli, and is unbounded. Localization equal to one
near \(t=0\) subtracts an off-plane smooth rapidly decreasing term, by
the proof after (C8) applied at any positive order \(\epsilon\); it
cannot cancel this unbounded dyadic sequence. The exact compact converse
would forbid an \(S^0\) representation. Thus arbitrarily small Sobolev
losses cannot replace the stated Besov endpoint.

\textbf{Problem 2.} In ambient dimension two and codimension one, change
coordinates by \(\kappa(t,z)=(2t,3z)\). If the normalized amplitude in
the new chart is \(a_\kappa(z,\tau)=h(z)\tau^r\) on the positive
frequency ray, what is the old-chart principal amplitude? Check the
intrinsic degree.

\textbf{Solution.} Formula (C20) gives
\(a(z,\tau)=2^{-r}\sqrt{3/2}\,h(3z)\tau^r\) modulo one lower order. The
square-root factor is the Jacobian of the induced map
\((z,\tau)\mapsto(3z,\tau/2)\) on half-densities; replacing it by the
normal integration determinant alone would be incorrect. Since
\(n=2,k=1\), ordinary amplitude order is \(r=m\), while the fiber
half-density adds \(1/2\). The intrinsic order is \(m+1/2=m+n/4\), as
required.

\textbf{Problem 3.} Let \(b(\tau)=\operatorname{sgn}(\tau)|\tau|^{-2}\)
on \(|\tau|>1\), with any smooth extension near the two endpoints. Find
its first nonsmooth inverse-integral term on \(t>0\). Does it pass (T6)?

\textbf{Solution.} Altering bounded frequencies changes only a smooth
function. The two tails give \(2i\int_1^\infty\tau^{-2}\sin(t\tau)d\tau
=2it\int_t^\infty s^{-2}\sin s\,ds\). Near zero,
\(s^{-2}\sin s=s^{-1}+O(s)\), so the singular part is \(-2it\log t\).
Here \(v=-2,c_+=1,c_-=-1\), whereas (T4) requires \(c_-=c_+\). The
logarithmic failure agrees with (T5); neither the factor \(i\) nor the
lower limit \(t\) may be removed in the rescaling.

\textbf{Problem 4.} Why is it insufficient to check only the values of
homogeneous components at the boundary normal rays, without their base
derivatives? Give a one-dimensional order-zero example and identify its
first nonsmooth output term on an ordinary input.

\textbf{Solution.} Let \(a(\tau)\) be smooth, equal to one for large
positive \(\tau\), and zero for large negative \(\tau\). Near the
boundary take \(p(t,\tau)=t a(\tau)\), and arrange compact local support
and proper support. Its order-zero homogeneous term vanishes at \(t=0\)
on both rays, but its first normal base derivative has values one and
zero. It fails (T10). Choose an ordinary smooth input equal to one near
zero on the positive side and compactly supported. Formula (T9) gives
Fourier leading term \(-i\tau^{-1}\); all further boundary jets vanish.
The positive-frequency part of its output is consequently
\((2\pi)^{-1}t\int_1^\infty e^{it\tau}(-i\tau^{-1})d\tau\) modulo smooth
terms. At \(v=-1\), (T5) gives leading term \(i(2\pi)^{-1}t\log t\).
This is not smooth up to zero, although the undifferentiated
boundary-ray test passed.

\textbf{Problem 5.} Put \(P=b(t,z)Q\), where the leading symbol of \(Q\)
is \((\tau^2+|\eta|^2)^{-1}\), and suppose this is its full homogeneous
model near the point. Find the two contributions to the full normal
derivative trace of its simple layer. Which would be missed by using
only \(\tau p\)?

\textbf{Solution.} Formula (T25) with \(k=1,l=0\) is
\(p_{1,0}=\tau p+D_t p\). Its plus integral therefore has homogeneous
terms \(ib(0,z)/2\), of order zero, and \((D_tb)(0,z)/(2|\eta|)\), of
order \(-1\), with any other lower symbol terms treated in the same way.
Direct differentiation of \(b(t,z)e^{-|\eta|t}/(2|\eta|)\) gives both.
The second term is the normal derivative of the coefficient and would be
lost by keeping only \(\tau p\).

\textbf{Problem 6.} Fix \(m=1/2+i\) and inward type \(\mu=1/3-2i\).
Describe every opposite-side type guaranteed solely by the phase
relation. Is there a real such type?

\textbf{Solution.} Equation (T15) gives \(\mu'=N-m-\mu=N-5/6+i\),
\(N\in\mathbb Z\). These differ by integers and hence give the same
opposite-side phase. Every one has imaginary part one, so none is real.
Requiring only the real part of \(m+\mu+\mu'\) to be integral would
leave an exponential of nonunit modulus and would not give the inverse
jet relation. An operator flat at all relevant conormal jets is an
exceptional case satisfying every phase, which is why the question asks
what the phase relation alone guarantees.

\subsection{Prerequisites and scope}\label{prerequisites-and-scope}

The advanced prerequisites are the symbol cutoff/localization bounds of
\href{metric-localization.pdf}{Localizing symbols with moving metrics};
the possibly degenerate Gauss multiplier, its restriction theorem,
finite-seminorm remainders and weak extension in Section 8 of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale}; and the distributional Fourier kernels, quantization
changes, classical Schwartz estimates and \(L^2\) Fourier extension in
Section 4 of \href{weyl-metric-products.pdf}{Two measuring scales, one
Weyl product}, Section 5 of \href{weyl-metric-products.pdf}{Two measuring
scales, one Weyl product} and Section 8 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
The elementary entry base listed at the start includes Schwartz density
and completeness of \(L^2\), so the extension from Schwartz Plancherel
is legitimate. The local symbol coordinate, product and adjoint formulas
used here have been derived with complete remainder orders from these
prerequisites. These classical local formulas do not establish operator
composition for arbitrary compatible metrics.

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\subsection{References}\label{references}

The exact dyadic endpoint is also stated in
\href{https://www.math.purdue.edu/~sabarre/Papers/TripleInt.pdf}{Sá
Barreto and Wang's author draft on triple interactions}, dated 25
October 2018, §2.1; that discussion refers elsewhere for its normal-form
equivalence. It is not used here as a proof of the compact converse or
the bundle symbol theorem.

\href{https://arxiv.org/pdf/1310.0951v5}{Grubb's 2014 version of the
work on fractional Laplacians and transmission spaces}, §2, provides a
useful comparison of the complex-type condition. Its supported-space
parameter \(\nu\) corresponds to \(\nu=-\mu-1\) here; using \(-\mu\)
would give the same phase but a different space.
\href{https://web.math.ku.dk/~grubb/dist10.pdf}{Grubb's chapter on
boundary operators}, §§10.3--10.4, gives a Laurent-symbol comparison for
the upper residue integral. The general analytic subtraction lemma and
the full complex-order remainder estimates required here have been
proved above.

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

A natural next question is how the collection of layer traces combines
into a projection onto boundary data of solutions. The residue model
gives a concrete starting point, while the all-jet transmission
criterion identifies which operators admit those traces in the smooth
category. For noninteger type, the spaces in (T7) suggest the
appropriate weighted boundary data instead of ordinary smooth zero
extensions. Those elliptic projection, solvability and weighted mapping
theorems require additional arguments and are not consequences asserted
by this lesson.

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\subsection{16. Complex contour integrals and their complete
foundations}\label{complex-contour-integrals-and-their-complete-foundations}

The boundary subtraction and wave integrals require actual contour,
continuation and Gamma proofs. This section supplies them with their
original orientations, branches, factors and endpoint domains. The
finite calculus and original circle constant are proved in
\href{metric-foundation-bridges.pdf}{Metric foundations}; measure
construction, product integration, affine substitution and the Gaussian
are proved in \href{banach-foundation-bridges.pdf}{Banach estimates and
measure foundations}. The nonlinear measure substitution and the full
sphere density are proved here before their use in the beta and ball
integrals.

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\subsubsection{16.1. The actual contour integral and triangle
theorem}\label{the-actual-contour-integral-and-triangle-theorem}

Let \(\Omega\subset\mathbb C\) be open. Holomorphic means complex
differentiable at every point of \(\Omega\); continuity of the
derivative is not assumed. A contour is a finite concatenation of
continuously differentiable maps \(\gamma:[a,b]\to\Omega\). Its integral
is the original oriented Riemann integral

\[
\int_\gamma f(\zeta)\,d\zeta
 =\int_a^b f(\gamma(t))\gamma'(t)\,dt,\qquad
\left|\int_\gamma f\,d\zeta\right|
 \leq \sup_{\gamma([a,b])}|f|\int_a^b|\gamma'(t)|\,dt.
\tag{CX1}
\]

This is defined for continuous \(f\). The real and imaginary coordinate
integrals, their triangle inequality and the limit of Riemann sums prove
the bound. Reversing the original parameter endpoints reverses the sign.
Concatenation adds the integrals, including their endpoints. A
continuously differentiable increasing reparameterization retains its
derivative factor by substitution; reversing the orientation changes its
sign. These assertions apply to each piece separately. A zero-length
piece contributes zero.

For an oriented triangle \(T\) whose closed interior lies in \(\Omega\),
let \(I(T)=\int_{\partial T}f\,d\zeta\). Join the side midpoints,
forming four triangles with the inherited positive orientation. Each
interior edge occurs twice with opposite orientation, so \(I(T)\) is the
sum of their four boundary integrals. Choose a child whose integral has
modulus at least \(|I(T)|/4\), then repeat. At each step choose the
first child in a fixed finite ordering satisfying that inequality; no
additional infinite-choice assumption is needed. The resulting closed
triangles \(T_k\) are nested, with diameter \(2^{-k}d\) and perimeter
\(2^{-k}L\), where \(d,L\) are the original diameter and perimeter.
Compactness gives a common point \(z_*\), and the shrinking diameter
makes it unique.

Complex differentiability at \(z_*\) gives

\[
f(\zeta)=f(z_*)+f'(z_*)(\zeta-z_*)
                 +(\zeta-z_*)\epsilon(\zeta),\qquad
\epsilon(\zeta)\longrightarrow0\quad(\zeta\longrightarrow z_*).
\tag{CX2}
\]

The boundary integrals of the first two terms are zero: the fundamental
theorem on each side evaluates the respective primitives \(f(z_*)\zeta\)
and \(f'(z_*)(\zeta-z_*)^2/2\), and the endpoint values cancel.
Therefore CX1 gives

\[
|I(T)|\leq4^k|I(T_k)|
\leq dL\sup_{\zeta\in T_k}|\epsilon(\zeta)|
\longrightarrow0.
\tag{CX3}
\]

Thus every such triangle integral is zero. A degenerate triangle has
zero integral directly by cancellation of its collinear oriented sides.
No real partial derivative or continuity of \(f'\) has been used.

There is a point-removal version needed below. Suppose \(g\) is
continuous on a disk and holomorphic except possibly at one point \(w\).
Its triangle integrals are still zero. For a triangle having \(w\) as a
vertex, remove the homothetic small triangle of ratio \(\delta>0\) at
that vertex. The remaining quadrilateral splits into two triangles
avoiding \(w\), whose integrals vanish by CX3. Interior-edge
cancellation makes the original integral equal the small triangle
integral. On its compact neighborhood \(g\) is bounded, so CX1 bounds
this by \(\delta L\sup|g|\), tending to zero. If \(w\) is inside a
triangle, split it into the three triangles joining \(w\) to its
vertices. If it is on an edge, split into the two triangles with \(w\)
as a vertex. Opposite internal sides cancel in both cases. If \(w\) is
outside, CX3 already applies.

For any continuous \(g\) with zero triangle integrals on a disk centered
at \(a\), define \(F(z)=\int_{[a,z]}g\,d\zeta\) on that disk. The
triangle with vertices \(a,z,z+h\) gives

\[
F(z+h)-F(z)=h\int_0^1g(z+th)\,dt,\qquad F'(z)=g(z).
\tag{CX4}
\]

Continuity makes the quotient tend to \(g(z)\). The complex derivative
is also the real coordinate derivative, since the remainder is
\(o(|h|)\). Along every contour piece the already proved real chain rule
gives \((F\circ\gamma)'=g(\gamma)\gamma'\). Consequently every closed
contour in this disk has integral zero. This applies to the
point-removal function as well; it does not assume in advance that the
derivative of a holomorphic function is holomorphic.

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\subsubsection{16.2. The circle formula, every coefficient and both
continuation
principles}\label{the-circle-formula-every-coefficient-and-both-continuation-principles}

Let the closed disk \(|\zeta-a|\leq R\) lie in \(\Omega\), and let
\(|z-a|<R\). Set

\[
g(\zeta)=
 \begin{cases}
 (f(\zeta)-f(z))/(\zeta-z),&\zeta\ne z,\\
 f'(z),&\zeta=z.
 \end{cases}
\tag{CX5}
\]

It is continuous at \(z\) by complex differentiability and holomorphic
elsewhere by the proved field/product rules. The point-removal argument
and CX4 give zero circle integral for \(g\). Use the original positively
oriented circle \(\zeta=a+Re^{i\theta}\), \(0\leq\theta\leq2\pi\). The
original complex exponential is its everywhere convergent series; the
full binomial product gives \(e^{u+v}=e^ue^v\), and termwise
differentiation gives \((e^u)'=e^u\). Its real and imaginary
restrictions agree with the circle parameter constructed in the metric
companion by their first-order equations and initial values. Thus this
circle uses that companion's actual \(\pi\), rather than a new
convention.

The geometric series for \((\zeta-z)^{-1}\) converges uniformly on the
circle. With its derivative factor \(iRe^{i\theta}\), its term of index
zero integrates to \(2\pi i\); each other term integrates to zero by the
fundamental theorem for \(e^{-im\theta}/(-im)\). It follows that

\[
\int_{|\zeta-a|=R}\frac{d\zeta}{\zeta-z}=2\pi i,\qquad
f(z)=\frac1{2\pi i}\int_{|\zeta-a|=R}
                          \frac{f(\zeta)}{\zeta-z}\,d\zeta .
\tag{CX6}
\]

Let \(M=\sup_{|\zeta-a|=R}|f(\zeta)|\). Expanding the same full
denominator and using CX1 proves

\[
\begin{aligned}
f(z)&=\sum_{m=0}^\infty c_m(z-a)^m,&
c_m&=\frac1{2\pi i}\int_{|\zeta-a|=R}
                     \frac{f(\zeta)}{(\zeta-a)^{m+1}}\,d\zeta,\\
|c_m|&\leq MR^{-m},&
\sup_{|z-a|\leq r}\left|f(z)-\sum_{m=0}^Nc_m(z-a)^m\right|
 &\leq \frac{M(r/R)^{N+1}}{1-r/R}\quad(0\leq r<R).
\end{aligned}
\tag{CX7}
\]

At \(r=0\), the remainder is exactly zero for \(N\geq0\). For every
derivative order \(j\), the differentiated terms on a compact disk of
positive radius \(r<R\) are bounded by the summable series

\[
M\sum_{m\geq j}\frac{m!}{(m-j)!}r^{m-j}R^{-m}.
\tag{CX8}
\]

Its successive ratio tends to \(r/R<1\). The metric companion's uniform
differentiation theorem now proves all derivatives, including at the
center by continuity. Hence

\[
f^{(j)}(a)=j!c_j,\qquad |f^{(j)}(a)|\leq j!MR^{-j}.
\tag{CX9}
\]

In particular complex differentiability implies a convergent local power
series and continuous derivatives of every order; these have been proved
rather than included in the definition.

If the first nonzero coefficient of a nonzero local power series at
\(a\) has index \(m\), factor it as
\((z-a)^m(c_m+\sum_{\ell>m}c_\ell(z-a)^{\ell-m})\). The parenthesis
tends to \(c_m\ne0\), so the zero at \(a\) is isolated and has exactly
that multiplicity. A holomorphic function on a connected open set which
vanishes on a set with an accumulation point in that open set vanishes
everywhere. Indeed the series at that point cannot have a first nonzero
coefficient, so it vanishes on a disk. The set of points with a zero
neighborhood is open. At a limit point inside the domain it has either
an eventual equal point or distinct zeros accumulating there; the same
series argument makes it closed relative to the domain. Connectedness
makes it the whole domain. Applying this to the difference proves the
identity theorem with the actual common connected domain.

Finally, if holomorphic \(f_n\) converge uniformly on every compact
subset of \(\Omega\), their limit \(f\) is holomorphic and every
\(f_n^{(j)}\) converges to \(f^{(j)}\) uniformly on smaller compact
disks. On each fixed circle uniform convergence permits passage through
CX6 and CX7. The limit therefore has the same circle representation and
coefficient formula. The bound in CX8 with the uniformly bounded circle
values proves differentiation and convergence on every smaller disk, for
every fixed \(j\). A finite disk cover gives the result on an arbitrary
compact subset. This proves the required locally uniform continuation
principle without assuming it.

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\subsubsection{16.3. The full dominated holomorphic integral
map}\label{the-full-dominated-holomorphic-integral-map}

Let \((S,\mu)\) be an arbitrary positive measure space. The compact
envelopes below restrict every required product integral to an actual
sigma-finite positive-envelope part; Section16.3.1 proves that reduction
and the completed joint measurability before Fubini is used. Suppose
\(f(z,x)\) is measurable in \(x\) for every \(z\in\Omega\), and
\(f(\,\cdot\,,x)\) is holomorphic on \(\Omega\) for every \(x\) outside
one fixed null set. For every compact \(K\subset\Omega\), assume an
actual integrable \(M_K\geq0\) bounds \(|f(z,x)|\) for all \(z\in K\),
almost everywhere. Define the original integral
\(F(z)=\int_S f(z,x)d\mu(x)\).

Dominated convergence proves continuity of \(F\) on every smaller
compact neighborhood. Take a closed disk
\(|\zeta-a|\leq R\subset\Omega\), and \(|z-a|\leq r<R\). CX6 for the
integrand has denominator of modulus at least \(R-r\) on that circle.
The integral of its absolute value over the original circle and \(S\) is
at most \(2\pi R\|M_K\|_1/(R-r)\). Thus the proved absolute Fubini
theorem gives

\[
\begin{aligned}
F(z)&=\frac1{2\pi i}\int_{|\zeta-a|=R}
                    \frac{F(\zeta)}{\zeta-z}\,d\zeta,\\
F^{(m)}(a)&=\int_S\partial_z^m f(a,x)\,d\mu(x),&
\int_S|\partial_z^m f(a,x)|\,d\mu(x)
 &\leq m!R^{-m}\|M_K\|_1 .
\end{aligned}
\tag{CX10}
\]

The first line gives a convergent power series by the actual geometric
denominator expansion, hence holomorphy of \(F\). For the second line
use CX9 for each integrand and its circle formula for the derivative.
The full absolute bound is integrable, so the same Fubini interchange
identifies the coefficient with the integral of the integrand
derivative. Differentiability under the integral is therefore a
consequence, not an extra assumption.

For \(d\) complex parameters on an actual product of disks, the same
argument applied successively to their circle integrals gives, with
multiindex \(\alpha=(\alpha_1,\ldots,\alpha_d)\) and original radii
\(R_j\),

\[
\partial^\alpha F(a)=\int_S\partial^\alpha f(a,x)d\mu(x),
\qquad
\int_S|\partial^\alpha f(a,x)|d\mu(x)
 \leq\left(\prod_{j=1}^d\alpha_j!R_j^{-\alpha_j}\right)\|M_K\|_1 .
\tag{CX11}
\]

Every circle contributes its own \(1/(2\pi i)\), derivative factorial
and radius. The iterated geometric series converges absolutely on every
smaller product of disks, because its bound is the product of the \(d\)
geometric sums times \(\|M_K\|_1\). It therefore supplies a joint power
series, including every mixed derivative, rather than only separate
differentiability. Additional continuous real parameters can be carried
through this proof when the actual envelope is common on their compact
parameter sets; dominated convergence supplies their continuity. Their
derivatives can be passed through only when their full differentiated
integrands have corresponding integrable envelopes, exactly as in the
real dominated-integral provider.

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\paragraph{16.3.1. The actual circle and product
measurability}\label{the-actual-circle-and-product-measurability}

To interchange the original parameter and circle integrals, first
establish their joint measurability and absolute envelope bounds. It
retains the original measure space \((S,\mu)\), the integrand
\(f(z,x)\), every compact envelope and the original integral
\(F(z)=\int_S f(z,x)\,d\mu(x)\). Its purpose is to prove the product
measurability needed for the stated Fubini steps, rather than assume
that separate measurability has already supplied it.

Fix the original closed disk \(K=\{|z-a|\leq R\}\subset\Omega\),
\(R>0\), and the original circle parametrization \(\zeta(t)=a+Re^{it}\),
\(0\leq t\leq2\pi\). The hypotheses give one fixed null set \(N\)
outside which \(f(\,\cdot\,,x)\) is holomorphic, hence continuous on the
circle. Enlarge it by the null set for this compact envelope, so
\(|f(z,x)|\leq M_K(x)\) for every \(z\in K\), \(x\notin N\), with
\(\int_S M_K\,d\mu<\infty\). A finite modification of the envelope on
its own null exceptional set can be covered by this same \(N\); the
original integral is unchanged.

For \(k\geq1\), partition \([0,2\pi)\) into its \(2^k\) half-open
intervals \(I_{k,j}=[2\pi j/2^k,2\pi(j+1)/2^k)\), and retain the
endpoint \(\{2\pi\}\) separately. Define the actual finite sum

\[
f_k(t,x)=\sum_{j=0}^{2^k-1}
\boldsymbol1_{I_{k,j}}(t)
f\!\left(a+Re^{2\pi i j/2^k},x\right)
\ +\boldsymbol1_{\{2\pi\}}(t)f(a+R,x).
\tag{CMI1}
\]

Each function \(x\mapsto f(a+Re^{2\pi i j/2^k},x)\) is measurable by the
original hypothesis. The corresponding function on the product is
measurable: the inverse image of a Borel set on each strip is that strip
times a measurable subset of \(S\), and the sum is finite. Thus every
\(f_k\) is measurable for the product sigma-algebra. For each original
\(t\) and \(x\notin N\), continuity on the circle gives
\(f_k(t,x)\to f(\zeta(t),x)\). The countable limit is measurable off the
product cylinder \([0,2\pi]\times N\). That cylinder has product measure
zero, and its entire subsets are measurable in the completed product.
Hence the original function \((t,x)\mapsto f(\zeta(t),x)\), with its
actual values on that cylinder retained, is measurable for the completed
product. No pointwise values of the working integrand have been
substituted.

The product argument can be confined to an actual sigma-finite part even
when the whole original measure space is not sigma-finite. Put \[
S_K=\bigcup_{j=1}^\infty\{x:M_K(x)\geq1/j\}.
\tag{CMI2}
\] Every set in this union has measure at most \(j\int_S M_K\,d\mu\), by
the proved integral of its indicator and the pointwise inequality
\(\boldsymbol1_{\{M_K\geq1/j\}}\leq jM_K\). Thus \(\mu|_{S_K}\) is
sigma-finite. On \(S\setminus S_K\) the original \(M_K\) is zero and
\(f(z,x)=0\) for every \(z\in K\), outside \(N\). The original integral
therefore retains the entire decomposition \[
\int_S f(z,x)\,d\mu(x)
=\int_{S_K}f(z,x)\,d\mu(x)
 +\int_{S\setminus S_K}f(z,x)\,d\mu(x),
\qquad
\int_{S\setminus S_K}f(z,x)\,d\mu(x)=0 .
\tag{CMI3}
\] The zero complement is proved from the actual envelope; it is not a
dropped contribution. Product measurability and completed complex Fubini
on the sigma-finite factor \(S_K\) are supplied by the original measure
chapter. The finite circle parameter is the other factor. Null
exceptional cylinders are measurable in its completion, with measure
zero as stated above.

For \(|z-a|\leq r<R\), the exact circle differential is
\(d\zeta=iRe^{it}\,dt\) and its modulus is \(R\,dt\). The original
kernel has denominator of modulus at least \(R-r\). Consequently \[
\int_0^{2\pi}\int_{S_K}
\left|
\frac{f(a+Re^{it},x)}{a+Re^{it}-z}
iRe^{it}
\right|d\mu(x)\,dt
\leq
\frac{2\pi R}{R-r}\int_S M_K(x)\,d\mu(x).
\tag{CMI4}
\] Every circle factor and the zero-complement equality in CMI3 remain.
This is precisely the absolute envelope required to interchange the
original \(S\)-integral and the positively oriented circle in CX10. For
its \(m\)-th coefficient the additional original Cauchy multiplier is
\(m!/(2\pi i)\) and the denominator is \((\zeta-a)^{m+1}\). The absolute
parameter integral is bounded by \[
\frac{m!}{2\pi}2\pi R R^{-m-1}
\int_S M_K\,d\mu
=m!R^{-m}\int_S M_K\,d\mu .
\tag{CMI5}
\] This proves both product measurability and the complete bound before
the derivative/Fubini step. Each derivative
\(x\mapsto\partial_z^m f(a,x)\) is measurable outside \(N\), because it
is a pointwise limit of countably many measurable difference quotients
on the original coordinate disk. Define its extension to the exceptional
set \(N\) to be zero, since a holomorphic derivative was required only
off \(N\). This is a measurable extension on the original measure space,
agrees with every required derivative, and changes no integral.

For the \(d\)-parameter extension, take the original product of closed
disks centered at \(a_j\) with radii \(R_j>0\). The precise hypotheses
are measurability in \(x\) for every parameter tuple, separate
holomorphy in each complex coordinate outside one fixed null set, and
one actual integrable \(M_K\) for the whole compact product. These
suffice; joint holomorphy is the conclusion.

Joint measurability on the product of its circles follows by induction
on their number. For one circle it is CMI1. For \(d>1\), replace only
the first circle parameter by its finite dyadic endpoints. Each
resulting summand is measurable in the remaining \(d-1\) parameters and
\(x\) by the induction hypothesis, because fixing that endpoint
preserves the original measurability and separate continuity.
Multiplication by its first-parameter interval indicator gives product
measurability. As its mesh tends to zero, separate continuity in that
first coordinate gives pointwise convergence to the original function
for every remaining parameter tuple and every \(x\notin N\). The
completed null-cylinder argument proves the claim, with the entire
original tuple retained. There is no assumption that separate continuity
already gives joint continuity.

Apply the original one-variable Cauchy formula successively for the
fixed other coordinates. The absolute completed Fubini bound on the
circle product is \[
\left(\prod_{j=1}^d\frac{2\pi R_j}{R_j-r_j}\right)
\int_S M_K\,d\mu,\qquad 0\leq r_j<R_j .
\tag{CMI6}
\] It comes from the full product of the \(d\) differentials
\(iR_je^{it_j}\,dt_j\), not a unit circle substitute. The resulting
original multiple formula is \[
F(z)=\left(\prod_{j=1}^d\frac1{2\pi i}\right)
\int_{|\zeta_1-a_1|=R_1}\cdots
\int_{|\zeta_d-a_d|=R_d}
\frac{F(\zeta_1,\ldots,\zeta_d)}
     {\prod_{j=1}^d(\zeta_j-z_j)}
\ d\zeta_d\cdots d\zeta_1 .
\tag{CMI7}
\] All factors have their original positive orientations; Fubini
justifies reversing the iterated order as well. In each denominator use
its full convergent geometric expansion with its original numerator
powers. On a smaller product of disks the absolute majorant is the
product of the \(d\) geometric series, each of ratio \(r_j/R_j<1\). Its
sum times \(\int_S M_K\,d\mu\) bounds the whole series, so the actual
multiple Cauchy integral is a joint convergent power series.
Differentiation of this joint series and the full coefficient integrals
give \[
\partial^\alpha F(a)
=\int_S\partial^\alpha f(a,x)\,d\mu(x),\qquad
\int_S|\partial^\alpha f(a,x)|\,d\mu(x)
\leq
\left(\prod_{j=1}^d\alpha_j!R_j^{-\alpha_j}\right)
\int_S M_K\,d\mu .
\tag{CMI8}
\] Mixed derivatives of the integrand itself are supplied by the same
multiple circle representation before integration in \(x\). Each is
measurable by its countable original difference quotients off the fixed
null set, and CMI8 gives its absolute integral bound. This closes the
actual measurability, Fubini, mixed-derivative and joint-power-series
steps used in CX10--CX11, including the full original radius and
factorial factors. Additional real parameters still need their actual
continuity and differentiated envelopes on compact parameter sets, as
the original source states; no real derivative hypothesis is invented.

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\subsubsection{16.4. The original winding
number}\label{the-original-winding-number}

For a closed contour \(\Gamma\) and a point \(w\) off its image define

\[
n_\Gamma(w)=\frac1{2\pi i}\int_\Gamma\frac{d\zeta}{\zeta-w}.
\tag{CX12}
\]

On each contour piece set \(U(t)=\gamma(t)-w\), which never vanishes.
Define the continuous accumulated integral
\(A(t)=\int_{\Gamma|_{[a,t]}}d\zeta/(\zeta-w)\), with the pieces
concatenated in their original order. The fundamental theorem and
product rule give \((U(t)e^{-A(t)})'=0\) on each piece, since
\(A'=U'/U\). Endpoint continuity makes the constant the same on all
pieces. Hence

\[
U(t)=U(a)e^{A(t)},\qquad e^{A(b)}=1 .
\tag{CX13}
\]

The complex exponential has modulus \(e^{\operatorname{Re}z}\), by its
product law and conjugate series. Its purely imaginary restriction is
the original circle parameter. That circle traverses the unit circle
once during \(0\leq\theta<2\pi\), as proved by the arctangent chart and
arc-length calculation in the metric companion. Consequently \(e^z=1\)
holds exactly for \(z=2\pi i k\), \(k\in\mathbb Z\). CX13 therefore
proves that CX12 is an integer, with its sign fixed by the original
contour orientation.

The contour image is compact. On a neighborhood of any \(w\) outside it
the denominator has a positive uniform lower bound. CX1 or dominated
convergence proves continuity of CX12 there. Its integer values make it
locally constant, and therefore constant on every connected component of
the complement of the contour. If the image lies in \(|\zeta|\leq L\)
and \(|w|>L\), the full geometric expansion gives

\[
\frac1{\zeta-w}
 =-\sum_{m=0}^{\infty}\frac{\zeta^m}{w^{m+1}}.
\tag{CX14}
\]

It converges uniformly on the contour. Each numerator \(\zeta^m\) has
the global primitive \(\zeta^{m+1}/(m+1)\), so every term has zero
closed-contour integral. Thus \(n_\Gamma(w)=0\) on the unbounded outside
component. Reversing \(\Gamma\) negates its winding number;
concatenating closed contours adds their winding numbers. These are
exact oriented-integral identities.

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\subsubsection{16.5. The complete contour theorem and every finite-pole
residue}\label{the-complete-contour-theorem-and-every-finite-pole-residue}

A finite integer sum of closed contours is also allowed: add the
original integrals and winding numbers with precisely those integer
coefficients. In an integral bound use the sum of each length times the
absolute value of its integer coefficient. Suppose \(\Gamma\) lies in
\(\Omega\) and \(n_\Gamma(w)=0\) for every \(w\notin\Omega\). For
holomorphic \(f\) on \(\Omega\), define off the contour

\[
\Phi(w)=\frac1{2\pi i}\int_\Gamma\frac{f(\zeta)}{\zeta-w}\,d\zeta .
\tag{CX15}
\]

It is holomorphic there by CX10, with the positive distance to the
compact contour supplying the actual local envelope. For \(w\in\Omega\),
the function

\[
H(w)=\frac1{2\pi i}\int_\Gamma
             \frac{f(\zeta)-f(w)}{\zeta-w}\,d\zeta
\tag{CX16}
\]

is holomorphic even at points on the contour. At \(\zeta=w\) the
quotient is defined as \(f'(w)\). To prove its required holomorphy in
\(w\), expand \(f\) on a small disk at a diagonal point. Each divided
power has the full polynomial \(((\zeta-a)^m-(w-a)^m)/(\zeta-w)
=\sum_{j=0}^{m-1}(\zeta-a)^{m-1-j}(w-a)^j\). The coefficient bounds in
CX7 make this sum locally uniformly convergent, with every derivative on
smaller disks by CX8. Away from the diagonal the quotient is a quotient
of holomorphic functions with nonzero denominator. On a compact
\(w\)-set and the compact contour, a finite disk cover of the diagonal
supplies a uniform bound for those divided differences; on its compact
complement the denominator is bounded away from zero. CX10 therefore
proves the asserted integral holomorphy.

Off the contour, CX12 gives \(H=\Phi-n_\Gamma f\). Let \(V\) be the open
set of points off the contour with \(n_\Gamma=0\). It is open by local
constancy. The hypothesis places every point outside \(\Omega\) in
\(V\). On \(V\), define \(H=\Phi\); this agrees with CX16 on the
overlap. Since \(\Omega\cup V=\mathbb C\), these definitions construct
one entire \(H\), including every contour point.

If the contour lies in \(|\zeta|\leq L\), let \(\ell\) be its total
length with absolute integer multiplicities. For \(|w|>L\), CX14 gives
\(n_\Gamma(w)=0\) and

\[
|H(w)|=|\Phi(w)|
\leq \frac{\ell\,\sup_\Gamma|f|}{2\pi(|w|-L)}
\longrightarrow0.
\tag{CX17}
\]

Continuity bounds \(H\) on the remaining compact disk, so it is bounded
on the whole plane. CX9 at any center and any radius gives
\(|H'(a)|\leq \sup_{\mathbb C}|H|/R\); letting \(R\to\infty\) proves
\(H'=0\). The fundamental theorem on each straight segment makes \(H\)
constant, and CX17 makes that constant zero. This is the full Liouville
argument, including its actual Cauchy factor.

Consequently both original contour conclusions are

\[
\frac1{2\pi i}\int_\Gamma\frac{f(\zeta)}{\zeta-w}\,d\zeta
       =n_\Gamma(w)f(w)\quad(w\in\Omega\setminus\Gamma),
\qquad
\int_\Gamma f(\zeta)\,d\zeta=0 .
\tag{CX18}
\]

For the second conclusion use \(\Phi(w)=0\) at sufficiently large \(w\),
multiply CX15 by \(w\), and pass to the uniform limit
\(w/(\zeta-w)\to-1\) on the compact contour. All prefactors remain in
that calculation. The winding hypothesis is necessary: for
\(f(\zeta)=1/\zeta\) on \(\mathbb C\setminus\{0\}\), a positive circle
has winding one at the omitted point and integral \(2\pi i\). This
example identifies precisely the domain condition; it does not supply
that condition by assuming away a pole.

For a deformation \(\gamma_s\) of closed contours within \(\Omega\),
with a fixed finite piece decomposition and jointly continuous
\(\gamma_s,\partial_t\gamma_s\), the winding at every point outside
\(\Omega\) is continuous in \(s\), by its uniform positive denominator
bound on the compact parameter/contour product. It is integer-valued, so
it is constant. The cycle \(\gamma_1-\gamma_0\) thus has winding zero
outside \(\Omega\). Applying CX18 proves equality of their holomorphic
integrals. This proves the contour-deformation rule with the exact
domains and actual orientation.

Now let \(f\) be meromorphic on \(\Omega\) with exactly the finite poles
\(a_1,\ldots,a_p\), of actual orders \(m_1,\ldots,m_p\geq1\), and let
\(\Gamma\) avoid them. At \(a_j\) the holomorphic function
\(g_j(\zeta)=(\zeta-a_j)^{m_j}f(\zeta)\), with \(g_j(a_j)\ne0\), has the
power series proved in CX7. Its entire principal part and residue are

\[
\begin{aligned}
P_j(\zeta)&=\sum_{\ell=1}^{m_j}
        \frac{g_j^{(m_j-\ell)}(a_j)}{(m_j-\ell)!}
                         (\zeta-a_j)^{-\ell},\\
\operatorname{Res}_{a_j}f
 &=\frac{g_j^{(m_j-1)}(a_j)}{(m_j-1)!}.
\end{aligned}
\tag{CX19}
\]

Subtracting every \(P_j\) gives a holomorphic function on \(\Omega\):
its local residual power series has only nonnegative exponents at each
original pole, and all the other subtracted principal parts are
holomorphic there. CX18 gives zero integral for this difference when the
same winding condition outside \(\Omega\) holds. Every term with
\(\ell\geq2\) has the global primitive
\(-(\zeta-a_j)^{1-\ell}/(\ell-1)\) away from that pole, so its
closed-contour integral is zero by the full chain and fundamental
theorem. Each \(\ell=1\) term has integral \(2\pi i n_\Gamma(a_j)\) by
CX12. Thus

\[
\int_\Gamma f(\zeta)d\zeta
 =2\pi i\sum_{j=1}^p n_\Gamma(a_j)
                  \frac{g_j^{(m_j-1)}(a_j)}{(m_j-1)!}.
\tag{CX20}
\]

Every original pole, order, derivative factorial and winding
multiplicity is retained. Empty pole sets give CX18. A rational function
is covered by \(\Omega=\mathbb C\), where the exterior winding condition
is empty; its polynomial part has a global polynomial primitive and
contributes zero.

For later orientation checks, a positive full circle has winding one
inside and zero outside by the exact geometric integral in CX6 and CX14.
A positive upper half-disk boundary runs along the real interval from
\(-R\) to \(R\), then counterclockwise along the upper arc from \(R\) to
\(-R\). Its winding is zero outside the closed half-disk: such points
can be joined to infinity through the lower half-plane or outside the
radius without crossing the contour. Its winding is constant on the
connected open upper half-disk. At \(w=i\delta\), \(0<\delta<R\), the
real segment integral has imaginary part
\(\int_{-R}^R\delta/(t^2+\delta^2)dt\to\pi\) and zero real part. The
upper arc integral tends to \(i\pi\), uniformly away from its
denominator as \(\delta\downarrow0\). The winding tends to one, so its
integer value is one throughout that half-disk. Reversal negates this
value. This proves the half-plane residue orientation used in T17
directly.

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\subsubsection{16.6. Bounded and exhausted maximum principles in the
original
domains}\label{bounded-and-exhausted-maximum-principles-in-the-original-domains}

Suppose \(|f|\) has a local maximum at an interior point \(a\). If
\(f(a)=0\), it vanishes on that neighborhood. Otherwise, if the local
power series is nonconstant, let \(m\geq1\) be its first nonconstant
coefficient. Choose a complex unit \(u\) so that \(c_m u^m\) is a
positive real multiple of \(f(a)\); existence follows from the actual
circle parameter by dividing that argument by the integer \(m\). For
small positive \(r\), \(f(a+ru)=f(a)+c_m u^m r^m+o(r^m)\). Its component
in the direction of \(f(a)\) is strictly greater than \(|f(a)|\),
contradicting the maximum. Thus \(f\) is constant locally, and CX7's
identity theorem makes it constant on the connected component. This
proves the interior maximum-modulus principle.

If \(\Omega\) is bounded, \(f\) is holomorphic in it and continuous on
its closure, compactness gives a maximum on that closure. If it is
attained in the interior, the preceding argument makes \(f\) constant on
that component. That bounded open component has a boundary point: follow
a ray from one of its points until the first exit from a containing
bounded ball and take the first component-exit limit. Its boundary lies
in \(\partial\Omega\), because any interior ball meeting the component
at a limit point would be connected to it and hence belong to the same
component. Continuity therefore makes the constant value a boundary
value. Consequently

\[
\sup_{\overline\Omega}|f|\leq\sup_{\partial\Omega}|f|.
\tag{CX21}
\]

The reverse inequality is immediate. Disconnected bounded domains cause
no exception, since the argument applies to the component where the
attained maximum lies.

For an upper half-plane or upper exterior, the exhausted form retains
the boundary at infinity. Apply CX21 on its intersection with
\(|\zeta|<L\), removing the prescribed inner disk in the exterior case.
On that bounded domain the boundary consists of its original real and
inner-arc pieces and the outer arc. If the outer-arc supremum tends to
zero, CX21 and \(L\to\infty\) give the bound by the original boundary
pieces. No claim discards an outer contribution without this hypothesis
and calculation.

Here is the exact polynomial-growth application in the original CN
proof. Suppose \(H\) is holomorphic in
\(\{\operatorname{Im}\zeta>0,\ |\zeta|>R\}\), continuous on its
boundary, satisfies \(|H(\zeta)|\leq C_g(1+|\zeta|)^d\) there with
\(d\geq0\), and has \(|t|^2|H(t)|\leq C_t\) on the two real tails.
Choose an integer \(M>d+2\). For \(0<\epsilon\leq1\), define the entire
original damping expression

\[
F_\epsilon(\zeta)=\zeta^2H(\zeta)(1-i\epsilon\zeta)^{-M},
\qquad
|1-i\epsilon\zeta|^2
 =(1+\epsilon\operatorname{Im}\zeta)^2
                     +\epsilon^2(\operatorname{Re}\zeta)^2
 \geq1+\epsilon^2|\zeta|^2 .
\tag{CX22}
\]

Its sole possible damping pole is \(-i/\epsilon\), outside the upper
domain. On the real tails \(|F_\epsilon|\leq C_t\). On the fixed inner
arc it is bounded by
\(C_R=R^2\sup_{\operatorname{Im}\zeta\geq0,|\zeta|=R}|H(\zeta)|\),
independently of \(\epsilon\). On the outer arc \(|\zeta|=L\),

\[
\sup|F_\epsilon|
\leq C_g\epsilon^{-M}L^{2-M}(1+L)^d
\longrightarrow0\quad(L\longrightarrow\infty).
\tag{CX23}
\]

The exponent and the factor \(\epsilon^{-M}\) are retained. First let
\(L\to\infty\) for each fixed \(\epsilon\), using CX21. Then, at each
fixed original \(\zeta\), let \(\epsilon\downarrow0\). This gives

\[
|F_\epsilon(\zeta)|\leq\max(C_t,C_R),\qquad
|H(\zeta)|\leq\max(C_t,C_R)|\zeta|^{-2}.
\tag{CX24}
\]

The order of limits is necessary; CX23 supplies no uniform outer limit
in \(\epsilon\). This proves the actual exterior maximum estimate with
every damping, radius and growth factor still present.

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\subsubsection{16.7. The entire original upper subtraction and its
parameter
map}\label{the-entire-original-upper-subtraction-and-its-parameter-map}

Use exactly the original hypotheses from CN: \(q\) is continuous on
\(\mathbb R\); for some \(R>0\), \(Q\) is holomorphic on the upper
exterior, continuous on its boundary and polynomially bounded there; and
\(q(t)-Q(t)=O(|t|^{-2})\) on both real tails. Its original definition,
with its whole clockwise arc and sign, is

\[
\begin{split}
\mathcal I_+(q)={}&
\int_{|t|>R}(q(t)-Q(t))dt+\int_{-R}^{R}q(t)dt\\
&-\int_{\pi}^{0}Q(Re^{i\theta})Ri e^{i\theta}d\theta.
\end{split}
\tag{CX25}
\]

Every integral in this definition exists: the difference tail is
absolutely integrable, and the interval and arc have compact continuous
integrands.

For \(R<L\), Cauchy's theorem for \(Q\) on the upper half-annulus gives

\[
\int_{R<|t|<L}Q(t)dt
 +\int_{\pi}^{0}Q(Re^{i\theta})Ri e^{i\theta}d\theta
 -\int_{\pi}^{0}Q(Le^{i\theta})Li e^{i\theta}d\theta=0 .
\tag{CX26}
\]

Here is the boundary justification when \(Q\) is only continuous on the
real boundary. First use inner radius \(a>R\), outer radius \(b<L\), and
height \(\delta>0\), on the boundary of
\(\{a<|\zeta|<b,\ \operatorname{Im}\zeta>\delta\}\). Its contour lies
strictly in the original holomorphy domain. Its winding is zero at every
point outside that domain: a point in the excluded inner disk can be
joined to zero within that disk and then to infinity through the lower
half-plane, avoiding the contour; a point in the lower closed half-plane
has the same outside path. CX18 therefore gives zero integral on this
contour. Let \(\delta\downarrow0\). Uniform continuity of \(Q\) on the
relevant compact closure, the explicit circle/segment parameterizations,
and their bounded lengths make each piece converge to the corresponding
horizontal segment or circular arc integral. Then let \(a\downarrow R\),
\(b\uparrow L\), with the same compact continuity bound. The inner arc
retains its clockwise orientation, and the outer arc is
counterclockwise, producing exactly CX26.

Subtracting the two definitions CX25 at radii \(L\) and \(R\) gives
precisely the left side of CX26: the difference of the real pieces is
\(\int_{R<|t|<L}Q(t)dt\). Thus the radius is independent, with neither a
discarded arc nor a sign change.

Suppose \(Q_1,Q_2\) are two admissible subtractions. Increase \(R\) if
necessary so both are defined on the same original exterior, and set
\(H=Q_1-Q_2\). The two real-tail remainders give \(H(t)=O(|t|^{-2})\).
CX22--CX24 prove the full bound \(|H(\zeta)|\leq C|\zeta|^{-2}\)
throughout that exterior. Its outer arc in CX26 has modulus at most
\(\pi L\,CL^{-2}=\pi C/L\), tending to zero. Its real tails are
absolutely integrable, so letting \(L\to\infty\) gives

\[
\int_{|t|>R}H(t)dt
 +\int_\pi^0H(Re^{i\theta})Ri e^{i\theta}d\theta=0 .
\tag{CX27}
\]

The difference of CX25 for \(Q_1,Q_2\) is the negative of this entire
expression, hence zero. This proves independence of the subtraction from
its actual growth and matching hypotheses.

For rational \(q\) with no real pole, choose \(R\) beyond every pole and
set \(Q=q\) on that exterior. The tail difference is exactly zero. The
remaining real interval minus the clockwise arc is the positively
oriented upper half-disk boundary. CX20 and its proved half-disk winding
therefore give the complete original conclusion

\[
\mathcal I_+(q)=2\pi i
 \sum_{\operatorname{Im}a_j>0}
           \frac{g_j^{(m_j-1)}(a_j)}{(m_j-1)!},
\qquad
g_j(\zeta)=(\zeta-a_j)^{m_j}q(\zeta).
\tag{CX28}
\]

All pole orders and the polynomial part are allowed. In the original
example \(q(t)=t/(t^2+1)\), its upper pole at \(i\) has residue
\(i/(2i)=1/2\), so \(\mathcal I_+(q)=i\pi\). Its symmetric real
principal value is zero by oddness on every finite symmetric interval.
The different answers retain the actual arc term; they do not result
from replacing one integral definition with the other.

For the original parameter version let \(F(\zeta,s)\) be jointly
continuous on the closed upper half-plane times its parameter space,
holomorphic in \(\zeta\) in the open half-plane, and bounded there
uniformly for \(s\) in each specified compact parameter set. Then \(QF\)
is an admissible subtraction for \(qF\): \((q-Q)F\) is bounded by the
actual integrable tail \(|q-Q|\) times that parameter bound, and \(QF\)
retains polynomial growth. In CX25 its interval and arc remain compact,
and its tail has this common envelope. Dominated convergence proves
continuity in \(s\), including every endpoint which belongs to the given
closed parameter space. For complex parameters in an open domain with
holomorphic \(F\) and the same compact-parameter envelope, CX10--CX11
prove holomorphic dependence and all allowed differentiated formulas.
For real derivatives their full differentiated integrands need the
corresponding common envelopes; then the proved real dominated-integral
map gives the same result. The radius and all three original integration
pieces remain fixed on that parameter neighborhood, so no parameter
endpoint or contour derivative is silently omitted.

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\subsubsection{16.8. The original Gamma integral, every pole and the
full reciprocal
comparison}\label{the-original-gamma-integral-every-pole-and-the-full-reciprocal-comparison}

For \(\operatorname{Re}z>0\), retain the original Euler integral

\[
\Gamma(z)=\int_0^\infty e^{-u}u^{z-1}du,
\qquad u^{z-1}=\exp((z-1)\log u)\quad(u>0).
\tag{CX29}
\]

On a compact parameter set choose
\(0<\sigma\leq\operatorname{Re}z\leq M\). The absolute integrand is
bounded by \(e^{-u}(u^{\sigma-1}+u^{M-1})\), integrable at both original
endpoints. Every parameter derivative adds its actual factor
\((\log u)^j\). Near zero its absolute integral is bounded by
\(\int_0^1u^{\sigma-1}|\log u|^jdu=j!/\sigma^{j+1}\): put \(u=e^{-v}\)
and integrate \(v^je^{-\sigma v}\) by parts \(j\) times, retaining its
zero endpoint terms. At infinity, \(\log u\leq u\) and an exponential
dominates each original finite power, by the real exponential series.
Thus every differentiated integrand has an integrable compact-parameter
envelope. CX10 proves holomorphy and every derivative of the original
Gamma integral.

For each integer \(N\geq1\), integration by parts \(N\) times gives

\[
\begin{aligned}
B(z,N+1)&=\int_0^1t^{z-1}(1-t)^Ndt
              =\frac{N!}{z(z+1)\cdots(z+N)},\\
N^zB(z,N+1)&=\int_0^N u^{z-1}(1-u/N)^Ndu
                 \longrightarrow\Gamma(z).
\end{aligned}
\tag{CX30}
\]

One integration has zero boundary values because
\(\operatorname{Re}z>0\) and the remaining positive power vanishes at
one; it gives \(N/z\) times the integral with parameters \(z+1,N\). The
last integral is \(1/(z+N)\), with its endpoint at one retained. For
\(N=0\) the initial integral is directly \(1/z\). In the second line the
change \(u=Nt\) retains the entire factor \(N^z\). Extend the integrand
by zero for \(u\geq N\). On \(0<u<N\), the inequality
\((1-u/N)^N\leq e^{-u}\) follows from \(\log(1-v)\leq-v\), proved by
integrating its negative derivative. The preceding envelope therefore
gives convergence uniformly on compact \(z\)-sets, with both integration
endpoints retained.

Let \(H_N=\sum_{k=1}^N1/k\). Its original difference satisfies \[
(H_{N+1}-\log(N+1))-(H_N-\log N)
 =\frac1{N+1}-\log(1+1/N)<0 .
\tag{CX31}
\] The inequality follows by integrating \(1/t>1/(N+1)\) from \(N\) to
\(N+1\). The comparison \(H_N\geq\int_1^{N+1}dt/t=\log(N+1)\) bounds
this decreasing sequence below. Its real limit is the original constant
\(\gamma\). The exact reciprocal of the second line in CX30 is the
entire finite product

\[
P_N(z)=zN^{-z}\prod_{k=1}^N(1+z/k)
 =z\exp\bigl(z(H_N-\log N)\bigr)
              \prod_{k=1}^N(1+z/k)e^{-z/k}.
\tag{CX32}
\]

To prove its whole-plane limit, on \(|z|\leq R\) take an integer
\(K>2R\). For \(k>K\), use the actual local logarithm series at one:

\[
\log(1+z/k)-z/k
 =\sum_{j=2}^\infty\frac{(-1)^{j+1}}j(z/k)^j,\qquad
|\log(1+z/k)-z/k|\leq 2R^2/k^2 .
\tag{CX33}
\]

The series derivative is the geometric sum \(1/(1+z/k)\) times \(1/k\);
its value at zero is zero, so exponentiating gives exactly \(1+z/k\).
Thus no branch is chosen across a zero of a finite initial factor. The
tail logarithms converge uniformly on that disk. Their first derivatives
are \(-z/[k(k+z)]\), bounded by \(2R/k^2\). At order \(j\geq2\), their
derivatives are \((-1)^{j-1}(j-1)!/(k+z)^j\), bounded by
\(2^j(j-1)!/k^j\). All derivative series therefore converge uniformly on
smaller disks, or equivalently by the proved locally uniform holomorphic
limit theorem. Keeping the first \(K\) factors exactly and
exponentiating the convergent tail proves an entire function

\[
G(z)=z e^{\gamma z}\prod_{k=1}^\infty(1+z/k)e^{-z/k},
\qquad G(z)\Gamma(z)=1\quad(\operatorname{Re}z>0).
\tag{CX34}
\]

The second identity follows because \(P_N(z)N^zB(z,N+1)=1\) exactly for
every \(N\), and both original factors converge locally uniformly. This
fixes the full comparison constant; \(G\) is an auxiliary reciprocal,
not a working replacement for \(\Gamma\).

The tail exponential is never zero. The retained finite factors
therefore show that \(G\) has precisely the simple zeros
\(0,-1,-2,\ldots\). It has no other zero anywhere in the plane. Its
reciprocal consequently gives the unique meromorphic continuation of the
original \(\Gamma\), with no zeros, and simple poles precisely at those
points. Directly \(P_N(1)=(N+1)/N\), so \(G(1)=1\). Integration by parts
in CX29, with its actual vanishing endpoints, gives
\(\Gamma(z+1)=z\Gamma(z)\) on the right half-plane. Hence
\(G(z)=zG(z+1)\) there; the identity theorem extends this entire
identity to every \(z\). Iteration at the original pole \(-m\),
\(m\in\mathbb N_0\), gives

\[
\begin{aligned}
G(z)&=z(z+1)\cdots(z+m)G(z+m+1),\\
G'(-m)&=(-1)^m m!,&
\operatorname{Res}_{z=-m}\Gamma(z)&=\frac{(-1)^m}{m!}.
\end{aligned}
\tag{CX35}
\]

For the derivative, exactly the factor \(z+m\) vanishes; its derivative
is one. The product of the others at \(-m\) is \((-1)^m m!\), and the
last factor is \(G(1)=1\). At \(m=0\) this is the empty product one.
Taking the reciprocal of the local series with this nonzero linear
coefficient gives the stated residue. Every original pole, sign and
factorial is retained.

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\subsubsection{16.9. The full beta and duplication
identities}\label{the-full-beta-and-duplication-identities}

We first prove the nonlinear measure substitution actually needed here.
Let \(F:U\to V\) be a \(C^1\) diffeomorphism between open subsets of the
original \(\mathbb R^d\), with its actual \(C^1\) inverse. For
\(d\geq1\), use the original maximum coordinate norm only as an
auxiliary comparison norm; the coordinates and map remain unchanged. The
determinant \(J(x)=|\det DF(x)|\) is positive and continuous. Let
\(Q\subset U\) be a closed coordinate cube with compact closure in
\(U\), subdivide it into \(N^d\) cubes of side \(h\), and denote a child
center by \(a\) and its actual derivative by \(A=DF(a)\). On the fixed
compact cube, let \(M=\sup\|DF(x)^{-1}\|_\infty<\infty\), and let
\(\delta(h)\) bound \(\|DF(x)-DF(a)\|_\infty\) on each child. Uniform
continuity makes \(\delta(h)\to0\). The segment fundamental theorem
gives the entire error, without replacing \(F\) by \(A\):

\[
\begin{aligned}
F(a+v)&=F(a)+Av+R_a(v),&
R_a(v)&=\int_0^1(DF(a+tv)-A)v\,dt,\\
E_a(v)&=A^{-1}R_a(v),&
\|E_a(v)\|_\infty&\leq M\delta(h)h/2
       \quad(\|v\|_\infty\leq h/2).
\end{aligned}
\tag{CX35a}
\]

Take \(N\) large enough that \(M\delta(h)<1\). In the image coordinates
\(w=A^{-1}(y-F(a))\), the image of the child cube lies in the cube of
half-side \((1+M\delta(h))h/2\). It contains the open cube of half-side
\((1-M\delta(h))h/2\). To prove the latter inclusion, follow the segment
\(y(s)=F(a)+sAw\), \(0\leq s\leq1\), for such a \(w\). Its initial point
belongs to the open image of the child interior, because the given
inverse is continuous. If the segment first leaves this open image,
compactness of the closed child and continuity of \(F^{-1}\) place that
first exit on the image of its boundary. At a boundary point some
original coordinate of \(v\) has modulus \(h/2\), so CX35a gives
\(\|v+E_a(v)\|_\infty\geq(1-M\delta(h))h/2\), contrary to the strict
bound on every \(sw\). The image of the child interior has boundary
exactly the image of the child boundary, since \(F\) and its inverse are
continuous on a neighborhood of that compact child. Thus the first-exit
argument uses the actual image, not an assumed surjectivity of a linear
approximation.

The affine measure formula and those two inclusions give

\[
(1-M\delta(h))^d|\det A|h^d
 \leq |F(Q_a)|\leq
(1+M\delta(h))^d|\det A|h^d.
\tag{CX35b}
\]

Every original child face and its image are null. The face itself is a
coordinate hyperplane piece. On the compact original cube, the segment
derivative bound makes \(F\) Lipschitz with a finite constant \(L\).
Cover a face of side \(s\) by at most \(C N^{d-1}\) coordinate pieces of
diameter at most \(ds/N\). Each image lies in a coordinate cube of side
at most \(2Lds/N\). The sum of their volumes is at most
\(C N^{d-1}(2Lds/N)^d\), tending to zero. For \(d=1\), the faces are
finite sets of points and this same estimate has \(N^0\) pieces. The
face constants and original coordinate lengths are retained. Since \(F\)
is injective, distinct child images overlap only on these null image
faces. Sum CX35b over all children. The determinant sums are Riemann
sums for the continuous original \(J\). Their upper and lower bounds
have limits \(\int_QJ(x)\,dx\), proving \(|F(Q)|=\int_QJ(x)\,dx\).

For Borel \(E\subset U\), both \(\lambda(E)=|F(E)|\) and
\(\nu(E)=\int_EJ(x)\,dx\) are measures: \(F\) is a homeomorphism, hence
carries Borel sets to Borel sets, and its injectivity preserves disjoint
unions. Half-open dyadic cubes with closures inside \(U\) form an
intersection-closed countable generating family covering \(U\); cubes in
one dyadic grid are either nested or disjoint. Removing their null faces
from the closed-cube equality proves equality on each such cube and its
intersections with the family. The finite-measure generating-family
theorem on each covering cube, followed by a countable disjoint
localization of that cover, proves \(\lambda=\nu\) on all Borel sets of
\(U\). They are finite on each compact covering cube. Since \(J\) is
positive, \(\int_EJ=0\) is equivalent to \(|E|=0\): intersect with the
countable sets \(J\geq1/k\) for the reverse implication. Equality of the
measures therefore preserves null sets in both directions. The
Borel-set-plus-subset-of-a-Borel-null-set description of completion
proves the same assertions for every completed Lebesgue measurable set
and function. Indicators, increasing simple approximations and the four
real and imaginary parts now prove the full substitution:

\[
\int_V f(y)\,dy
 =\int_U f(F(x))\,|\det DF(x)|\,dx.
\tag{CX35c}
\]

This holds with \(+\infty\) allowed for every nonnegative measurable
\(f\). For complex \(f\), it holds precisely when either, and hence
both, of the corresponding absolute integrals is finite. Applying it to
the actual inverse gives the converse formula with \(|\det DF^{-1}|\);
the determinant chain rule retains both factors and proves their product
one. In dimension zero the spaces have their single point, the empty
determinant and mass are one, and the same formula is immediate. This
proves the needed measure theorem, including completed measurability and
orientation reversal, from the original affine formula and finite
calculus.

For \(\operatorname{Re}z>0,\operatorname{Re}w>0\), define the original
beta integral and retain the whole change of variables

\[
\begin{aligned}
B(z,w)&=\int_0^1t^{z-1}(1-t)^{w-1}dt,\\
x&=ut,\quad y=u(1-t),\quad u=x+y>0,\quad t=x/(x+y)\in(0,1),\\
\left|\det\frac{\partial(x,y)}{\partial(u,t)}\right|&=u,\\
\Gamma(z)\Gamma(w)
 &=\int_0^\infty\int_0^\infty
            e^{-x-y}x^{z-1}y^{w-1}dx\,dy\\
 &=\int_0^\infty\int_0^1
   e^{-u}(ut)^{z-1}(u(1-t))^{w-1}u\,dt\,du
 =\Gamma(z+w)B(z,w).
\end{aligned}
\tag{CX36}
\]

The absolute product integral is finite because its two original
one-variable absolute integrals have positive real exponents. The
diffeomorphism has precisely the displayed positive absolute determinant
and both inverse maps. Apply completed-measure substitution and absolute
complex Fubini, already proved in the integration chapter. Positive real
\(u,t,1-t\) have real logarithms, so the power product is exactly
\(u^{z+w-2}t^{z-1}(1-t)^{w-1}\), with the additional Jacobian \(u\)
still present. This proves CX36, including every factor and endpoint. On
compact parameter sets the corresponding logarithmic derivatives are
integrable at zero and one, by the same power/log integral bounds as for
CX29. CX11 proves joint holomorphy. Since \(\Gamma(z+w)\ne0\), its exact
quotient is \(B(z,w)=\Gamma(z)\Gamma(w)/\Gamma(z+w)\) in this original
convergence domain. The explicit meromorphic function
\(\Gamma(z)\Gamma(w)G(z+w)\) continues that integral. Uniqueness follows
by the one-variable identity theorem first in \(z\), for every \(w\) in
the original right half-plane, then in \(w\) for fixed \(z\) away from
the poles; clearing local denominators extends the equality through
their meromorphic charts. This asserts the continuation, without
suppressing any intersection cancellation or assigning a value to a
divergent endpoint integral.

For \(\operatorname{Re}z>0\), first substitute \(t=(1+v)/2\) in
\(B(z,z)\), retaining \(dt=dv/2\) and \(t(1-t)=(1-v^2)/4\). Then use
evenness and \(s=v^2\):

\[
\begin{aligned}
B(z,z)&=2^{1-2z}\int_{-1}^1(1-v^2)^{z-1}dv
       =2^{1-2z}B(1/2,z),\\
\Gamma(z)^2\Gamma(z+1/2)
       &=2^{1-2z}\Gamma(1/2)\Gamma(z)\Gamma(2z),\\
\Gamma(1/2)
       &=\int_0^\infty e^{-v^2}(v^2)^{-1/2}\,2v\,dv
         =2\int_0^\infty e^{-v^2}dv=\sqrt{\pi},\\
\Gamma(2z)&=2^{2z-1}\pi^{-1/2}\Gamma(z)\Gamma(z+1/2).
\end{aligned}
\tag{CX37}
\]

The half-interval substitution contributes \(ds/(2\sqrt s)\); the two
symmetric intervals cancel exactly that factor two, producing the full
\(B(1/2,z)\) integral. The second line is CX36 with both original Gamma
denominators multiplied out. Nonvanishing of the original \(\Gamma(z)\)
permits the stated comparison to the last line. The Gaussian value uses
the integration chapter's proved original polar integral and original
circle constant. For its \(v=0\) endpoint, perform the substitution on
positive truncated intervals and pass by absolute convergence. The last
line is therefore exactly the original duplication identity, with its
factors of two and pi fixed. Both sides are meromorphic functions of
\(z\), and CX7 extends their equality from the right half-plane to their
actual meromorphic domain.

We also prove the full sphere density before using it in an arbitrary
dimension. Let \(n\geq2\),
\(S^{n-1}=\{\theta\in\mathbb R^n:\sum_{j=1}^n\theta_j^2=1\}\), with its
original Euclidean surface measure \(dS\). On the hemisphere
\(\epsilon\theta_i>0\), \(\epsilon\in\{1,-1\}\), use the actual graph
coordinates \(u=(\theta_j)_{j\ne i}\), \(|u|<1\), and
\(h(u)=\theta_i=\epsilon\sqrt{1-\sum_{j\ne i}u_j^2}\). The coordinate
tangent Gram matrix and radial Jacobian are

\[
\begin{aligned}
\partial_{u_j}h&=-u_j/h,&
\mathcal G_{jk}&=\delta_{jk}+u_ju_k/h^2,\\
\det\mathcal G&=1+\sum_{j\ne i}u_j^2/h^2=1/h^2,&
\sqrt{\det\mathcal G}&=1/|h|,\\
\Phi_{i,\epsilon}(r,u)&=r\theta(u),&
|\det D\Phi_{i,\epsilon}|&=r^{n-1}/|h|
                  =r^{n-1}\sqrt{\det\mathcal G}.
\end{aligned}
\tag{CX37a}
\]

For the Gram determinant, multilinearity in its columns leaves the
identity determinant and the \(n-1\) terms with exactly one column from
\(u u^T/h^2\); terms with two such columns vanish because those columns
are dependent. The remaining terms are \(u_j^2/h^2\). For the radial
determinant, factor \(r\) from each of the \(n-1\) angular columns.
After placing the \(i\)-th row first, its block columns are \((h,u)^T\)
and \((-u_j/h,e_j)^T\). Adding \(\sum u_j/h\) times the respective lower
rows to the first row makes its first entry \(h+\sum u_j^2/h=1/h\) and
its other entries zero. The row permutation affects the sign only, so
the absolute determinant is exactly the displayed value. The inverse on
the cone \(\epsilon x_i>0\) is \(r=|x|\), \(u=(x_j/|x|)_{j\ne i}\); both
maps are smooth with nonzero Jacobian.

The density \(\sqrt{\det\mathcal G}\,du\) is the original Euclidean
surface density, not a radial definition substituted for it. On
overlapping graph charts, the tangent chain rule gives
\(\mathcal G_u=(DT)^T\mathcal G_vDT\), hence
\(\sqrt{\det\mathcal G_u}=\sqrt{\det\mathcal G_v}|\det DT|\). CX35c in
dimension \(n-1\) proves equality of their integrals. This verifies that
the graph densities define one coordinate-independent completed surface
measure.

For a finite smooth partition on these hemispheres, take the already
constructed smooth function \(\chi\geq0\) which vanishes at nonpositive
inputs and is positive at positive inputs. Put \(c=1/(2\sqrt n)\),
\(b_{i,\epsilon}(\theta)=\chi(\epsilon\theta_i-c)\), and
\(b(\theta)=\sum_{i,\epsilon}b_{i,\epsilon}(\theta)\). Some coordinate
has modulus at least \(1/\sqrt n\), since their full squares sum to one,
so \(b>0\). The functions \(\rho_{i,\epsilon}=b_{i,\epsilon}/b\) sum to
one; their supports are inside the respective hemispheres and have
\(|h|\geq c\). For nonnegative measurable \(f\), apply CX35c to each
actual radial chart with the factor \(\rho_{i,\epsilon}(x/|x|)\), then
add all \(2n\) terms. Each cone chart has its own original inverse and
Jacobian. The origin has original \(n\)-dimensional measure zero, and
the partition sums to one at every other point. Thus

{\small
\[
\int_{\mathbb R^n}f(x)\,dx
 =\sum_{i=1}^n\sum_{\epsilon\in\{1,-1\}}
   \int_0^\infty\int_{|u|<1}
      f(r\theta(u))\rho_{i,\epsilon}(\theta(u))
          r^{n-1}|h(u)|^{-1}\,du\,dr
 =\int_0^\infty\int_{S^{n-1}}f(r\theta)r^{n-1}\,dS(\theta)\,dr.
\tag{CX37b}
\]
}

Every term is nonnegative, so Tonelli allows infinite values and either
order. Applying the same formula to \(|f|\) proves its exact
absolute-integrability condition; the four component integrals give the
complex formula in that domain. The support condition \(|h|\geq c\)
bounds every angular density, so the finite chart cover gives finite
surface mass. It is positive because one nonempty open graph patch has a
strictly positive density. The radius-zero endpoint may be added with
weighted integrand zero; it contributes no mass.

For \(n=1\), use the two original maps \(r\mapsto r\) and \(r\mapsto-r\)
from \(r>0\), both with absolute derivative one. The sphere
\(S^0=\{1,-1\}\) has counting measure, so CX37b holds with its two terms
and \(r^{n-1}=1\). No angular chart of negative dimension is used. For
every \(n\geq1\), the original Cartesian Gaussian and CX37b therefore
give

\[
\begin{aligned}
(\sqrt\pi)^n
 &=\int_{\mathbb R^n}\prod_{j=1}^n e^{-x_j^2}\,dx
 =|S^{n-1}|\int_0^\infty e^{-r^2}r^{n-1}\,dr\\
 &=|S^{n-1}|\int_0^\infty e^{-s}
                    (\sqrt s)^{n-1}\frac{ds}{2\sqrt s}
 =\frac{|S^{n-1}|}{2}\Gamma(n/2),\\
|S^{n-1}|&=\frac{2\pi^{n/2}}{\Gamma(n/2)}.
\end{aligned}
\tag{CX37c}
\]

The substitution is first on positive finite intervals; absolute
convergence then passes both endpoints. The Cartesian factors are the
integration chapter's independently proved original Gaussian values. The
radial factor is finite and positive by CX29. This fixes the entire
original sphere coefficient, including its factor two. At \(n=1\),
\(\Gamma(1/2)=\sqrt\pi\) gives the original counting mass two.

Consequently the original ball integral in WHK follows, for \(n\geq1\)
and \(\operatorname{Re}\lambda>(n-1)/2\), from this full polar density
and \(s=r^2\):

\[
\begin{aligned}
\int_{|y|<1}(1-|y|^2)^{\lambda-(n+1)/2}dy
 &=\frac{2\pi^{n/2}}{\Gamma(n/2)}
       \int_0^1 r^{n-1}(1-r^2)^{\lambda-(n+1)/2}dr\\
 &=\frac{\pi^{n/2}}{\Gamma(n/2)}
       B\bigl(n/2,\lambda+(1-n)/2\bigr)\\
 &=\pi^{n/2}
       \frac{\Gamma(\lambda+(1-n)/2)}{\Gamma(\lambda+1/2)}.
\end{aligned}
\tag{CX38}
\]

The original sphere coefficient is CX37c, proved above using every
original radial and angular Jacobian. Both beta parameters have positive
real part, exactly the displayed convergence condition. For \(n=1\), the
sphere has its two original endpoints and coefficient two; the same
calculation includes both half-intervals. In dimension zero the ball is
a single point with mass one; the rightmost Gamma quotient is also one
wherever its original integral comparison is used, so no
zero-dimensional sphere formula or negative-dimensional Jacobian is
introduced.

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\subsubsection{16.10. The original Laplace branch and every dominated
derivative}\label{the-original-laplace-branch-and-every-dominated-derivative}

On the right half-plane \(p=x+iy\), \(x>0\), take
\(\operatorname{Log}p=\frac12\log(x^2+y^2)+i\arctan(y/x)\), using the
actual real logarithm and principal arctangent. Direct differentiation
gives \(\partial_x\operatorname{Log}p=(x-iy)/(x^2+y^2)=1/p\) and
\(\partial_y\operatorname{Log}p=i/p\). The proved real differentiability
remainder therefore gives complex derivative \(1/p\). The branch is
holomorphic, real on \(p>0\), and has argument in \((-\pi/2,\pi/2)\).

For \(\operatorname{Re}p>0,\operatorname{Re}z>0\), the original Laplace
identity is

\[
\int_0^\infty e^{-pt}t^{z-1}dt
       =\Gamma(z)\exp(-z\operatorname{Log}p)
       =\Gamma(z)p^{-z}.
\tag{CX39}
\]

For positive real \(p\), the change \(u=pt\) retains the full factor
\(p^{-z}\) and reduces the integral exactly to CX29. For fixed \(z\),
both sides are holomorphic in the entire original right half-plane. On
compact parameter sets choose \(\operatorname{Re}p\geq\delta>0\) and
\(0<\sigma\leq\operatorname{Re}z\leq M\). Every mixed derivative of the
integrand is its full factor \((-t)^a(\log t)^b e^{-pt}t^{z-1}\). The
power/log estimates at zero and the exponentially decaying bound at
infinity give a common integrable envelope for every fixed pair \(a,b\).
CX11 proves joint holomorphy and all derivative interchanges. The
identity theorem in \(p\), from its positive-real accumulation set
inside the domain, now proves CX39 at every allowed complex \(p\).

In particular, with \(a,b\in\mathbb N_0\), the complete differentiated
receiving identity is

\[
\begin{aligned}
\int_0^\infty(-t)^a(\log t)^b e^{-pt}t^{z-1}dt
 &=(-1)^a p^{-z-a}
       \sum_{c=0}^b\binom bc\Gamma^{(c)}(z+a)
                            (-\operatorname{Log}p)^{b-c},\\
\Gamma(z+a)&=\Gamma(z)\prod_{j=0}^{a-1}(z+j).
\end{aligned}
\tag{CX40}
\]

For the first line, initially differentiate in \(p\) \(a\) times in the
actual integral; it is \((-1)^a\Gamma(z+a)p^{-z-a}\) by CX39.
Differentiate in \(z\) \(b\) times, retaining every Leibniz term, sign
and logarithmic power. The second line is the entire original Gamma
recursion; at \(a=0\) its empty product is one. Thus a shifted Gamma
factor is explicitly compared with the original Gamma, rather than
absorbing its product into a different object.

On the upper half-plane the branch
\(\operatorname{Log}_+\zeta=\operatorname{Log}(-i\zeta)+i\pi/2\) retains
argument \((0,\pi)\) and derivative \(1/\zeta\). Consequently its
continuous boundary powers on the positive and negative real rays have
ratio \(e^{i\pi v}\) for every original complex exponent \(v\), as
required by T6. For \(p=\epsilon+i\tau\), \(\epsilon>0\), the branch in
CX39 tends, at each \(\tau\ne0\), to
\(\log|\tau|+i\pi\operatorname{sgn}\tau/2\). Thus its negative power has
negative-ray/positive-ray ratio \(e^{i\pi z}\), with the full
\(\Gamma(z)\) factor still present in the unnormalized integral. Passage
at \(\tau=0\) is the original distributional damping and
Taylor-subtraction construction, never substitution into a divergent
point value.

\end{document}