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\begin{document}

\section{Fourier transforms, finite spectra and convex
separation}\label{fourier-transforms-finite-spectra-and-convex-separation}

The Gaussian estimates in
\href{gauss-transform-estimates.pdf}{Quantitative estimates for quadratic
Fourier multipliers} use a precise Fourier convention, a real spectral
theorem, and finite-dimensional convex separation. This chapter proves
those facts with their constants and endpoint cases.
\href{metric-foundation-bridges.pdf}{Metric and topological foundations}
supplies the measure and compactness background. Basic references are
{[}Melrose{]} for Fourier analysis and {[}Beezer{]} for
finite-dimensional linear algebra, but the arguments below do not
require either book.

Section 13 of
\href{metric-foundation-bridges.pdf}{Metric
and topological foundations} proves the full original scalar and
finite-coordinate calculus. Its Section 13.12 gives the exact Gaussian,
Rayleigh, logarithmic-contour, matrix and half-line receiving maps,
retaining every original factor, endpoint and norm.

The complete original real-number, compactness, extrema and finite-norm
proofs are in Section 12 of
\href{metric-foundation-bridges.pdf}{Metric
and topological foundations}. Its Section 12.10 gives the exact
receiving minimum and spectral-contour maps; the original coordinates,
norms and constants are retained.

We assume multivariable differentiation and Taylor's formula, smooth
compact cutoffs, change of variables and Fubini for absolutely
integrable functions, dominated convergence and differentiation,
Cauchy--Schwarz, finite-dimensional compactness, and completeness of the
real numbers. We also use elementary matrix algebra, determinants,
Gram--Schmidt, and the usual monomial Schwartz seminorms. The complete
finite basis, Gram, matrix and adjoint proofs are given in Section 10 of
\href{stable-prerequisite-bridges.pdf}{Polynomial
and contour interfaces for stable boundary models}. Each further step is
proved here.

\subsection{1. Fourier inversion on the Schwartz
space}\label{fourier-inversion-on-the-schwartz-space}

For \(d\geq1\), let \(\mathcal S(\mathbb R^d)\) have the seminorms \[
p_{\alpha,\beta}(f)=\sup_x|x^\alpha\partial^\beta f(x)|.
\] Use \(D_j=-i\partial_j\) and \[
Ff(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx,
\qquad
Gv(x)=(2\pi)^{-d}\int e^{ix\cdot\xi}v(\xi)\,d\xi.
\] \textbf{Theorem 1.1 (Fourier inversion).} The map \(F\) is a
continuous automorphism of the Schwartz space with inverse \(G\), and \[
F(D_jf)=\xi_jFf,\qquad F(x_jf)=i\partial_{\xi_j}Ff.
\] For the complex-linear distribution convention,
\(\langle Fu,\phi\rangle=\langle u,F\phi\rangle\); transposition of the
continuous Schwartz maps supplies the transform on \(\mathcal S'\), its
inverse with the corresponding factor, and the differentiated
identities. This convention agrees with the integral for regular
distributions induced by Schwartz functions, by Fubini.

\textbf{Proof of Theorem 1.1.} Differentiation under the integral and
integration by parts give the displayed identities. More precisely,
\(\xi^\alpha\partial_\xi^\beta Ff(\xi)=F[D^\alpha((-ix)^\beta f)](\xi)\).
By the finite Leibniz rule, \(D^\alpha((-ix)^\beta f)\) is a finite
linear combination of derivatives of \(x^\beta f(x)\), with each
coefficient fixed by \(\alpha,\beta\). Its Fourier transform is bounded
in supremum norm by the \(L^1\) norm of that full expression. If
\(N>d\), then \[
\|h\|_1\leq\left(\int_{\mathbb R^d}\langle x\rangle^{-N}\,dx\right)
\sup_x\langle x\rangle^N|h(x)|,
\] so finitely many Schwartz seminorms bound each output seminorm. Thus
\(F:\mathcal S\to\mathcal S\) is continuous.

For \(\varepsilon>0\), put \[
k_\varepsilon(x)=(4\pi\varepsilon)^{-d/2}e^{-|x|^2/(4\varepsilon)}.
\] The one-dimensional Gaussian integral is
\(\int_{\mathbb R}e^{-t^2}\,dt=\sqrt{\pi}\): square the integral, use
Fubini, and in \(\mathbb R^2\) make the polar change of variables with
Jacobian \(r\), obtaining \(2\pi\int_0^\infty e^{-r^2}r\,dr=\pi\). Hence
\(\int k_\varepsilon=1\). Let \[
I_\varepsilon(x)=\int_{\mathbb R^d}e^{ix\cdot\xi}e^{-\varepsilon|\xi|^2}\,d\xi.
\] Integration of a total \(\xi_j\)-derivative gives \[
0=ix_jI_\varepsilon(x)-2\varepsilon
\int_{\mathbb R^d}\xi_j e^{ix\cdot\xi}e^{-\varepsilon|\xi|^2}\,d\xi,
\qquad
\partial_{x_j}I_\varepsilon(x)=-\frac{x_j}{2\varepsilon}I_\varepsilon(x).
\] Its value at zero is \((\pi/\varepsilon)^{d/2}\). Solving these
coordinate equations yields the exact inverse-Gaussian identity \[
(2\pi)^{-d}I_\varepsilon(x)=k_\varepsilon(x).
\]

For \(f\in\mathcal S\), Fubini is legitimate after inserting
\(e^{-\varepsilon|\xi|^2}\). It gives \[
(2\pi)^{-d}\int e^{ix\cdot\xi}e^{-\varepsilon|\xi|^2}Ff(\xi)\,d\xi
=\int_{\mathbb R^d}k_\varepsilon(x-y)f(y)\,dy.
\] On the left, dominated convergence gives \(GFf(x)\), since
\(Ff\in L^1\). On the right, write the difference from \(f(x)\) as
\(\int k_\varepsilon(y)(f(x-y)-f(x))\,dy\). On \(|y|<\delta\), the
mean-value formula bounds the difference by
\(\|\nabla f\|_\infty\delta\). On its complement, boundedness of \(f\)
gives at most \(2\|f\|_\infty\) times the Gaussian mass
\(\int_{|y|\geq\delta}k_\varepsilon(y)\,dy=\pi^{-d/2}\int_{|z|\geq\delta/(2\sqrt{\varepsilon})}e^{-|z|^2}\,dz\),
which tends to zero for fixed \(\delta>0\). First let
\(\varepsilon\downarrow0\), then \(\delta\downarrow0\). Thus the right
side tends to \(f(x)\), uniformly in \(x\). Hence \(GF=I\) on
\(\mathcal S\). The calculation of the other inverse follows below
without changing any Fourier factor. \(\square\)

Here is the two-sided inverse bridge. Define \(Rf(x)=f(-x)\) and
\(a=(2\pi)^{-d}\). A change of variables gives \(FR=RF\), and the
candidate inverse is \(G=aRF\). The proof above gives \(GF=I\).
Therefore \[
FG=aFRF=aRF^2=GF=I.
\] Reflection preserves the monomial seminorms, so \(G\) is continuous.
No equivalence with an unstated alternative topology is required. In
dimension zero the integral convention is the identity on \(\mathbb C\);
this includes that harmless endpoint whenever a zero-dimensional factor
occurs.

\subsection{2. Plancherel and integer Sobolev
weights}\label{plancherel-and-integer-sobolev-weights}

\textbf{Theorem 2.1 (Plancherel on Schwartz functions).} For
\(u\in\mathcal S(\mathbb R^d)\), \[
\|Fu\|_2^2=(2\pi)^d\|u\|_2^2.
\] For \(f,g\in\mathcal S\), substitute \(g=GFg\) from Theorem 1.1 into
\(\int f\overline g\). Fubini is valid because \(f\) and \(Fg\) are
integrable. The inner integral is \(Ff\), so
\(\int f\overline g=(2\pi)^{-d}\int Ff\,\overline{Fg}\). Taking \(g=f\)
proves the displayed identity, including its full factor. No extension
to arbitrary \(L^2\) functions is needed here.

For every integer \(s\geq0\), the multinomial identity and
differentiation under \(F\) give the more precise formula \[
\|\langle\xi\rangle^s Fu\|_2^2
=(2\pi)^d\sum_{|\alpha|\leq s}
\frac{s!}{(s-|\alpha|)!\,\alpha!}\|D^\alpha u\|_2^2.
\] Indeed expand \((1+\xi_1^2+\cdots+\xi_d^2)^s\), multiply by
\(|Fu|^2\), integrate the finite sum and apply Theorem 2.1 to each
derivative. All coefficients are positive, so this also proves both norm
comparisons, with constants depending only on \(d,s\). For \(s=0\) the
sum has one term.

When \(s>d/2\), Cauchy--Schwarz yields \[
\|Fu\|_1\leq
\|\langle\xi\rangle^{-s}\|_2
\|\langle\xi\rangle^sFu\|_2.
\] The first norm is finite: the integral inside the unit ball is
bounded; the annulus \(2^k\leq|\xi|<2^{k+1}\) contributes at most a
constant times \(2^{k(d-2s)}\), whose sum converges. This is the precise
integer-derivative control used in the local multiplier estimate.

\subsection{3. Real spectral
decomposition}\label{real-spectral-decomposition}

\textbf{Theorem 3.1 (Real spectral decomposition).} Every self-adjoint
endomorphism of a finite-dimensional real inner-product space has a real
orthonormal eigenbasis, including repeated and zero eigenvalues.

\textbf{Proof.} In dimension zero the empty basis suffices. Otherwise
maximize \(\langle Av,v\rangle\) over the compact unit sphere, and call
a maximizer \(v\). For each \(w\perp v\), differentiate
\(\langle A(v+tw),v+tw\rangle/\langle v+tw,v+tw\rangle\) at \(t=0\). Its
derivative vanishes, so \(\langle Av,w\rangle=0\); hence
\(Av=\lambda v\) for \(\lambda=\langle Av,v\rangle\). Self-adjointness
makes \(v^\perp\) invariant, since
\(\langle Aw,v\rangle=\langle w,Av\rangle=0\) for \(w\perp v\).
Induction on dimension gives an orthonormal eigenbasis of \(v^\perp\).
Together with \(v\) it gives the asserted basis, with no
distinct-eigenvalue assumption. \(\square\)

The complexification gives a second check of the real eigenspaces.

In dimension zero take the empty basis. Otherwise use Gram--Schmidt to
choose real orthonormal coordinates. The real symmetric matrix \(A\),
regarded as a complex matrix, is Hermitian. The real basis just proved
complexifies to an eigenbasis of \(\mathbb C^d\), and every eigenvalue
remains real. For real \(\lambda\), taking real and imaginary parts
proves \[
\ker_{\mathbb C}(A-\lambda I)
=\ker_{\mathbb R}(A-\lambda I)
 +i\ker_{\mathbb R}(A-\lambda I).
\] Decompose a real vector into complex eigenvectors and take real
parts. Thus the real eigenspaces span \(\mathbb R^d\). If
\(Av=\lambda v\) and \(Aw=\mu w\), symmetry gives \[
(\lambda-\mu)\langle v,w\rangle
=\langle Av,w\rangle-\langle v,Aw\rangle=0.
\] Distinct eigenspaces are orthogonal. Choosing an orthonormal basis in
each gives the required real basis, with all multiplicities retained.

The following triangular-matrix facts retain the zero-dimensional case
and the order of multiplication. Its induction includes dimension zero
with the empty matrix. Products of upper triangular matrices are upper
triangular: when \(i>j\), in each term \(a_{ik}b_{kj}\), either \(k<i\)
or \(k\geq i>j\), so one factor vanishes. An invertible upper triangular
matrix preserves every coordinate flag space \(E_j\); its restriction to
that finite-dimensional space is injective and therefore onto. Its
inverse preserves the same flag and is upper triangular. The diagonal
equation in \(AA^{-1}=I\) gives \((A^{-1})_{jj}=a_{jj}^{-1}\).

\subsubsection{3.1. Real skew-adjoint operators and their full
coordinate
blocks}\label{real-skew-adjoint-operators-and-their-full-coordinate-blocks}

The scalar positivity argument uses both real self-adjoint and real
skew-adjoint finite-dimensional spectral decompositions. The
self-adjoint theorem is proved in Section3 of
\href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
convex separation}, with the original inner product. We now prove the
skew-adjoint theorem and every coordinate comparison needed to use it.

Let \(V\) be a finite-dimensional real vector space with its given
positive-definite inner product \(h\), and let \(A:V\to V\) satisfy \[
h(Au,v)=-h(u,Av)\qquad(u,v\in V).
\tag{SS1}
\] Keep an original ordered coordinate basis \(b\), its full positive
Gram matrix \(H\), and the original coordinate matrix \(M\) of \(A\).
Thus \[
h(J_bx,J_by)=x^{\mathsf T}Hy,\qquad
AJ_bx=J_bMx,\qquad M^{\mathsf T}H+HM=0.
\tag{SS2}
\] The last equality follows by substituting each original pair of
coordinate vectors into SS1. No coefficient, Gram factor or coordinate
of the working operator is replaced.

Put \(B=-A^2\). Two applications of SS1 give \[
h(Bu,v)=h(Au,Av)=h(u,Bv),\qquad
h(Bu,u)=h(Au,Au)\geq0.
\tag{SS3}
\] The proved real self-adjoint theorem applies to this actual \(B\):
its real eigenspaces give an orthogonal direct sum of \(V\), with every
multiplicity retained. If \(Bu=\tau u\), \(u\ne0\), SS3 gives
\(\tau h(u,u)=h(Au,Au)\geq0\), hence \(\tau\geq0\). Moreover \[
\ker B=\ker A,\qquad AB=BA,\qquad
A\ker(B-\tau I)\subset\ker(B-\tau I).
\tag{SS4}
\] For the kernel equality one direction follows from \(B=-A^2\); in the
other direction SS3 gives \(h(Au,Au)=0\), so \(Au=0\). The commuting
identity is the full equality \(-A^3=-A^3\), and substitution proves the
invariant-eigenspace assertion.

For each positive eigenvalue \(\tau\), let \(\lambda=\sqrt{\tau}>0\),
the positive real root constructed in the real-number chapter. Choose a
vector \(u\) of original \(h\)-length one in its eigenspace; finite
Gram--Schmidt provides such a vector. Define \(v=Au/\lambda\). Then
SS1--SS3 give the complete identities \[
h(u,v)=\lambda^{-1}h(u,Au)=0,\qquad
h(v,v)=\lambda^{-2}h(Au,Au)
=\lambda^{-2}\tau h(u,u)=1,
\qquad Au=\lambda v,\quad Av=-\lambda u.
\tag{SS5}
\] Here \(h(u,Au)=-h(Au,u)=-h(u,Au)\), so it is zero over the original
real field. The last identity uses the actual equation \(A^2u=-\tau u\),
retaining both \(\lambda\) factors.

The original \(h\)-orthogonal complement of
\(\operatorname{span}\{u,v\}\) inside that eigenspace is
\(A\)-invariant: for a vector \(w\) in the complement, \[
h(Aw,u)=-h(w,Au)=-\lambda h(w,v)=0,\qquad
h(Aw,v)=-h(w,Av)=\lambda h(w,u)=0.
\tag{SS6}
\] Induction on its finite dimension constructs all pairs there. It
cannot terminate in a positive-eigenvalue line, because SS5 would
construct two orthogonal unit vectors in that line. Thus each positive
eigenspace has even dimension, with one pair per two dimensions. Choose
an \(h\)-orthonormal basis of \(\ker A\) by the proved Gram
construction. Altogether there are \(k\) orthonormal kernel vectors and
\(l\) ordered pairs, with \(k+2l=\dim V\). The actual operator is zero
on those \(k\) vectors and has the full ordered block \[
C_j=\begin{pmatrix}0&-\lambda_j\\ \lambda_j&0\end{pmatrix}
\quad\hbox{on the ordered pair }(u_j,v_j),\qquad
\lambda_j>0.
\tag{SS7}
\] Equal positive numbers are allowed and all their pairs remain.

Let \(S\) have, in the original coordinates \(b\), the columns of this
entire constructed basis, and put
\(C=\operatorname{diag}(0_k,C_1,\ldots,C_l)\). Orthogonality and the
actual images in SS5 prove \[
S^{\mathsf T}HS=I,\qquad MS=SC,\qquad
S^{-1}=S^{\mathsf T}H,\qquad
M=SCS^{-1}=SC S^{\mathsf T}H.
\tag{SS8}
\] The columns form a basis, so \(S\) is invertible; multiplying the
first equation by \(S^{-1}\) proves the displayed inverse formula. These
are exact morphisms between the original coordinates and the displayed
blocks, preserving \(H\) and every coordinate coefficient. They do not
identify \(M\) numerically with \(C\).

Complexify the original maps and inner product. In each pair, \[
A(u_j+i v_j)=-i\lambda_j(u_j+i v_j),\qquad
A(u_j-i v_j)=i\lambda_j(u_j-i v_j).
\tag{SS9}
\] Both eigenvectors are nonzero and independent: their coefficients on
the original real independent pair give determinant \(-2i\ne0\). They
span its complexification, while each chosen kernel vector remains an
eigenvector with eigenvalue zero. Each is nonzero, since its original
squared norm is one. Thus the full complex spectrum is zero with
multiplicity \(k\) and both signs \(i\lambda_j,-i\lambda_j\) for every
pair. The real characteristic polynomial comparison retains the entire
determinant factors: \[
\begin{split}
\det(tI-M)
&=\det S\,\det(tI-C)\,\det(S^{-1})\\
&=\det S\,t^k\prod_{j=1}^l(t^2+\lambda_j^2)\,\det(S^{-1}),\\
\det S\,\det(S^{-1})&=\det(SS^{-1})=1.
\end{split}
\tag{SS10}
\] Every zero and repeated pair is accounted for. In dimension zero the
basis is empty, both coordinate maps are the unique empty bijection,
each determinant and empty product is one, and \(k=l=0\). This proves
the entire real skew-adjoint decomposition used by the scalar positivity
argument at its original finite-dimensional inner product.

\subsection{4. Strict separation of an open convex
set}\label{strict-separation-of-an-open-convex-set}

\textbf{Theorem 4.1 (Strict separation).} Let \(V\) be a
finite-dimensional real vector space, let \(C\subset V\) be nonempty,
open and convex, and let \(x\notin C\). There is a nonzero real covector
\(\eta\) such that \[
\eta(y)<\eta(x)\quad(y\in C),
\qquad \sup_{y\in C}\eta(y)\leq\eta(x).
\] The second inequality includes \(x\) on the boundary. The supremum
need not be attained. The proof uses only finite-dimensional compactness
and the inner product.

Choose an auxiliary Euclidean norm, put \(K=\overline C\), and fix
\(c\in C\) with \(B(c,r)\subset C\), \(r>0\). First, if \(z\in K\) and
\(0\leq t<1\), then \[
(1-t)c+tz\in C.
\] For \(t=0\) this is immediate. For \(0<t<1\), choose \(z_j\in C\)
tending to \(z\). Convexity gives a ball of radius \((1-t)r\) centered
at \((1-t)c+tz_j\) inside \(C\). For sufficiently large \(j\), this ball
contains \((1-t)c+tz\), proving the assertion.

For \(\varepsilon>0\), put \(z_\varepsilon=x+\varepsilon(x-c)\). This
point lies outside \(K\): if it lay in \(K\), then \[
x=\frac{\varepsilon}{1+\varepsilon}c
 +\frac{1}{1+\varepsilon}z_\varepsilon
\] would lie in \(C\) by the preceding assertion. The nonempty closed
set \(K\) has a point \(q_\varepsilon\) nearest to \(z_\varepsilon\). To
see existence without a compactness assumption on \(K\), take a
minimizing sequence. Its distance to \(z_\varepsilon\) is bounded, so it
has a convergent subsequence in finite dimensions; closedness places its
limit in \(K\).

Write \(v_\varepsilon=z_\varepsilon-q_\varepsilon\ne0\). For \(y\in K\),
the segment \(q_\varepsilon+t(y-q_\varepsilon)\), \(0\leq t\leq1\), lies
in \(K\). Differentiating the squared distance at its minimum \(t=0\)
gives \[
\langle v_\varepsilon,y-q_\varepsilon\rangle\leq0.
\] Consequently, for \(u_\varepsilon=v_\varepsilon/|v_\varepsilon|\), \[
\langle u_\varepsilon,y\rangle
\leq\langle u_\varepsilon,z_\varepsilon\rangle
-|v_\varepsilon|
\leq\langle u_\varepsilon,z_\varepsilon\rangle.
\] Choose \(\varepsilon=1/k\) and a convergent subsequence of the unit
vectors, with limit \(u\), \(|u|=1\). Since \(z_\varepsilon\to x\), the
last inequality implies \(\langle u,y\rangle\leq\langle u,x\rangle\) for
every \(y\in K\). The same subsequence works for every \(y\): it was
selected using only the unit vectors, and the inequality holds for every
\(y\) at every index.

Set \(\eta(y)=\langle u,y\rangle\). It is nonzero. For \(y\in C\),
openness gives \(y+\delta u\in C\) for some \(\delta>0\), whence \[
\eta(y)+\delta\leq\eta(x).
\] This proves strict pointwise separation. The supremum statement
follows because \(\eta(C)\) is nonempty and bounded above. In dimension
zero the hypotheses cannot occur, since the only nonempty open set is
all of \(V\).

\subsection{5. Support functions of Minkowski
sums}\label{support-functions-of-minkowski-sums}

\textbf{Proposition 5.1.} For nonempty \(A,B\subset V\), put \[
h_A(\eta)=\sup_{a\in A}\eta(a)\in\mathbb R\cup\{+\infty\}.
\] Then \(h_{A+B}(\eta)=h_A(\eta)+h_B(\eta)\), where
\(A+B=\{a+b:a\in A,b\in B\}\). Nonemptiness excludes \(-\infty\), so the
right side is unambiguous.

Linearity first gives the inequality \(\leq\). If both right-hand
suprema are finite, choose \(a,b\) within \(\varepsilon/2\) of the two
suprema. Their sum gives the reverse inequality up to \(\varepsilon\);
let \(\varepsilon\downarrow0\). If \(h_A(\eta)=+\infty\), fix
\(b_0\in B\). The values \(\eta(a+b_0)\) are unbounded above, so
\(h_{A+B}(\eta)=+\infty\). Interchange \(A,B\) for the other case. No
boundedness, compactness or convexity is needed for this identity.

\subsection{6. Phase checks and
exercises}\label{phase-checks-and-exercises}

The exact Fourier phases used above can be checked without changing
conventions: the segment integral in the fundamental theorem runs from
\(0\) to \(1\); frequency continuity compares
\(e^{-ix\cdot\xi}-e^{-ix\cdot\xi'}\); the squared one-dimensional
Gaussian integral is an integral over \(\mathbb R^2\); Parseval uses the
conjugate transform of \(\psi\) as its second factor; and an \(H^m\)
condition uses \(\langle\xi\rangle^{m}Fu\in L^2\). These checks preserve
the signs and the factor \((2\pi)^{-d}\) in Theorems 1.1 and 2.1.

Exercise. For the open strip \(C=\{(a,b)\in\mathbb R^2:|a|<2\}\) and
boundary point \(x=(2,7)\), determine all separating covectors
satisfying the strict inequality in Theorem 4.1. Explain why the support
value need not be attained and why its finiteness constrains the
covector.

Solution. Write \(\eta(a,b)=\alpha a+\beta b\). Since \(b\) is
unrestricted, finite upper support forces \(\beta=0\). Nonzero
separation at the right boundary then forces \(\alpha>0\), and every
such \(\alpha\) works: \(\alpha a<2\alpha=\eta(x)\). The support equals
\(2\alpha\) but is not attained because \(a=2\) is excluded. This
distinguishes strict pointwise separation from a positive uniform gap at
a boundary point.

\textbf{Exercise 6.2 (A Gaussian transform).} For \(t>0\) and
\(f_t(x)=e^{-t|x|^2}\) on \(\mathbb R^d\), compute \(Ff_t(\xi)\) with
the exact convention of Section 1.

\textbf{Solution.} Apply the Gaussian identity from the proof of Theorem
1.1 after exchanging the roles of \(x\) and \(\xi\). Equivalently, the
same integration-by-parts equation gives
\(\partial_{\xi_j}Ff_t=-\xi_jFf_t/(2t)\), while
\(Ff_t(0)=(\pi/t)^{d/2}\). Hence \[
Ff_t(\xi)=(\pi/t)^{d/2}e^{-|\xi|^2/(4t)}.
\] The positive value at zero fixes the constant and the negative
exponent fixes the phase.

The polar and Gaussian integrations in Section 1 are proved in Sections
15.6--15.7 of
\href{banach-foundation-bridges.pdf}{Banach
estimates, quotient spaces and compact parameter arguments}. These
proofs include completed measurability, both determinant terms, the
radial endpoints and the full Fourier heat-kernel factors.

\subsection{\texorpdfstring{7. Fourier transforms on the complete
\(L^2\)
space}{7. Fourier transforms on the complete L\^{}2 space}}\label{fourier-transforms-on-the-complete-l2-space}

The Schwartz identities above extend to the original completed Lebesgue
\(L^2(\mathbb R^d)\), with both Fourier factors unchanged. The measure,
density, translation and completeness proofs used here are Sections
15.0--15.4 of \href{banach-foundation-bridges.pdf}{Banach estimates,
quotient spaces and compact parameter arguments}. We use their original
measure and scalar field.

\subsubsection{7.1. Constructing both inverse
maps}\label{constructing-both-inverse-maps}

Suppose first that \(d\geq1\). Compact smooth functions are dense in
\(L^2\), by the truncation and mollification proof there, and belong to
the Schwartz space. For \(f\in L^2\), choose Schwartz \(f_j\to f\) in
\(L^2\). Theorem 2.1 gives the exact difference identity \[
 \|Ff_j-Ff_k\|_2=(2\pi)^{d/2}\|f_j-f_k\|_2.
                                                               \tag{FL1}
\] Completeness makes this a convergent sequence. If
\(\widetilde f_j\to f\) is another such sequence, its difference from
\(f_j\) tends to zero in \(L^2\), so (FL1) gives the same output. Define
\(Ff\) as that limit. Linearity follows by approximating each summand
and taking limits; the norm identity follows from continuity of the
norm.

For Schwartz \(h\), Theorem 1.1 gives \(F(Gh)=h\), and Theorem 2.1
applied to \(Gh\) gives \(\|Gh\|_2=(2\pi)^{-d/2}\|h\|_2\). Thus the same
construction extends \(G\) to \(L^2\). Applying continuity to
\(GFf_j=f_j\) and \(FGh_j=h_j\) proves, on the original entire spaces,
\[
 \begin{split}
 F,G&:L^2(\mathbb R^d)\longrightarrow L^2(\mathbb R^d),\\
 GF&=I,\qquad FG=I,\\
 \|Ff\|_2&=(2\pi)^{d/2}\|f\|_2,\qquad
 \|Gh\|_2=(2\pi)^{-d/2}\|h\|_2 .
 \end{split}                                                   \tag{FL2}
\] These are bijections with their original norms. In dimension zero the
measure is the point mass on the one point \(\mathbb R^0\), the space is
\(\mathbb C\), and both maps are the identity with \((2\pi)^0=1\). No
assertion that this point is null is used in that case.

\subsubsection{7.2. Ordinary integrals and simultaneous
approximation}\label{ordinary-integrals-and-simultaneous-approximation}

For any finite exponents \(a,c\geq1\) and \(f\in L^a\cap L^c\), put
\(f_j=1_{\{|x|_\infty\leq j,\ |f(x)|\leq j\}}f\). Dominated convergence
of \(|f|^a\) and \(|f|^c\) proves convergence in both original norms.
Each \(f_j\) is bounded and supported in its displayed coordinate cube.
Fix the compact smooth nonnegative approximate identity \(\rho\), with
\(\int\rho=1\), from Section 15.4 of the linked chapter. For
\(\rho_\varepsilon(x)=\varepsilon^{-d}\rho(x/\varepsilon)\), its proved
translation estimate is \[
 \|\rho_\varepsilon*f_j-f_j\|_t
 \leq\int|\rho(z)|\,\|f_j(\,\cdot-\varepsilon z)-f_j\|_t\,dz
 \longrightarrow0,\qquad t=a,c.                                \tag{FL3}
\] \textbf{The full integral triangle estimate in (FL3).} Fix an
original finite exponent \(t\geq1\), the actual \(f_j\), and
\(\varepsilon>0\). Put \(h_z(x)=f_j(x-\varepsilon z)-f_j(x)\) and
\(M_\varepsilon(x)=\int|\rho(z)||h_z(x)|\,dz\). These are the original
translated differences and their nonnegative integral envelope. The
joint integrand is measurable for completed Lebesgue measure: first take
a Borel representative of the given \(f_j\) class. Its coordinate
compositions are Borel. Changing that representative on a null set \(N\)
changes the composed functions only on the inverse images of
\(N\times\mathbb R^d\) under the identity product map and the invertible
shear \((x,z)\mapsto(x-\varepsilon z,z)\). Its inverse is
\((w,z)\mapsto(w+\varepsilon z,z)\), its full block determinant is one,
and completed product integration and affine substitution make both
inverse images null. Thus the calculation respects the original
completed classes. For each fixed \(x\), the section in \(z\) is also
measurable by the invertible affine substitution
\(z\mapsto x-\varepsilon z\).

The original truncation gives \(|f_j|\leq j\) and support in
\(Q_j=[-j,j]^d\). Consequently \(0\leq M_\varepsilon\leq2j\|\rho\|_1\),
with support contained in the actual compact set
\(Q_j\cup(Q_j+\varepsilon\operatorname{supp}\rho)\). It follows that
\(M_\varepsilon\in L^t\). At \(t=1\), nonnegative Tonelli gives
\(\|M_\varepsilon\|_1=\int|\rho(z)|\|h_z\|_1\,dz\). If \(t>1\), apply
the proved scalar Hölder inequality with its actual conjugate exponent
\(t'=t/(t-1)\). The full power satisfies
\(\|M_\varepsilon^{t-1}\|_{t'}=\|M_\varepsilon\|_t^{t-1}\), since
\((t-1)t'=t\). Nonnegative Tonelli then gives \[
 \begin{aligned}
 \|M_\varepsilon\|_t^t
 &=\int M_\varepsilon(x)^{t-1}
              \left(\int|\rho(z)||h_z(x)|\,dz\right)dx\\
 &=\int|\rho(z)|
              \left(\int M_\varepsilon(x)^{t-1}|h_z(x)|\,dx\right)dz\\
 &\leq\|M_\varepsilon\|_t^{t-1}
                 \int|\rho(z)|\|h_z\|_t\,dz.
 \end{aligned}\tag{FL3a}
\] The last integral is finite, because translation is an isometry and
\(\|h_z\|_t\leq2\|f_j\|_t\). If \(\|M_\varepsilon\|_t>0\), division by
its displayed positive power proves the integral triangle bound. If that
norm is zero, the same bound is immediate without division. The exact
difference formula (LP13), with both original scaling Jacobians
retained, gives \(|\rho_\varepsilon*f_j-f_j|\leq M_\varepsilon\).
Combining these two inequalities proves exactly (FL3), rather than
replacing it by the weaker power-integral bound (LP14). Translation
continuity makes \(\|h_z\|_t\to0\) for each fixed \(z\), and its
original bound \(2|\rho(z)|\|f_j\|_t\) is integrable. Its dependence on
\(z\) is continuous by that same translation theorem and the reverse
norm inequality. Dominated convergence therefore proves the limit in
(FL3) for each of the original exponents \(a,c\), without an
infinite-dimensional integration theorem.

Its bound \(2|\rho(z)|\|f_j\|_t\) is integrable. Choose a positive
\(\varepsilon_j<1/j\) making both errors at most \(1/j\). Then
\(v_j=\rho_{\varepsilon_j}*f_j\) is compact smooth and tends to \(f\) in
both norms. Differentiating this absolutely convergent convolution
proves smoothness, and its support is contained in the sum of the two
original compact supports. This constructs a single approximation,
rather than assuming that two independently chosen approximations
coincide.

Apply this with \(a=1,c=2\). The ordinary Fourier and inverse integrals
satisfy \[
 \begin{split}
 \sup_\xi|Fv_j(\xi)-Ff(\xi)|&\leq\|v_j-f\|_1,\\
 \sup_x|Gv_j(x)-Gf(x)|&\leq(2\pi)^{-d}\|v_j-f\|_1 .
 \end{split}                                                   \tag{FL4}
\] Here the symbols on the right side of each difference denote their
ordinary absolutely defined integrals. Their uniform limits therefore
agree almost everywhere with the \(L^2\) limits in (FL2). Indeed, the
summable-difference argument in the completeness proof supplies an
almost-everywhere convergent subsequence of either \(L^2\) sequence. The
uniform and almost-everywhere limits of that subsequence are equal. Thus
the constructed \(L^2\) classes coincide with the original integrals on
\(L^1\cap L^2\), with the exact inverse factor retained.

The same simultaneous approximation identifies any two finite-exponent
continuous extensions of an operator from Schwartz functions when both
original norm bounds are available. For completeness, if functions
converge in \(L^a\) and \(L^c\) to two limits, select a common
subsequence with summable \(a\)-th and \(c\)-th powers of its errors.
Chebyshev's integral inequality and the measure of the countable tail
unions make both errors tend to zero almost everywhere; the limits must
agree. This is the identification used for the multiplier below.

\subsubsection{7.3. Parseval pairing, multiplication and the actual
adjoint}\label{parseval-pairing-multiplication-and-the-actual-adjoint}

Approximating both inputs by Schwartz functions in \(L^2\), Theorem 2.1
and Cauchy--Schwarz prove \[
 \langle u,v\rangle
   =(2\pi)^{-d}\int_{\mathbb R^d}
                     Fu(\xi)\overline{Fv(\xi)}\,d\xi .
                                                               \tag{FL5}
\] The inner product is linear in the first variable. Each pairing
difference tends to zero by the sum of the two Cauchy--Schwarz bounds,
so the original factor persists.

If \(b\) is a measurable scalar frequency function with
\(\|b\|_\infty\leq B\), multiplication by \(b\) respects the completed
null classes and maps \(L^2\) to itself with norm at most \(B\). Hence
\[
 \begin{split}
 T_b&=G\,M_b\,F,\qquad M_bh=bh,\\
 \|T_bf\|_2
 &\leq(2\pi)^{-d/2}B(2\pi)^{d/2}\|f\|_2=B\|f\|_2,\\
 \langle T_bu,v\rangle
 &=(2\pi)^{-d}\int b(\xi)Fu(\xi)\overline{Fv(\xi)}\,d\xi
   =\langle u,T_{\overline b}v\rangle .
 \end{split}                                                   \tag{FL6}
\] This proves the actual adjoint \(T_b^*=T_{\overline b}\). It uses the
constructed inverse maps, not an assumed representation of an \(L^p\)
dual space. For Schwartz \(f\), \(bFf\in L^1\cap L^2\), so \(T_bf\) is
exactly the original inverse integral. These statements apply in the
elliptic chapter with \(d=n\geq1\); its arbitrary value of the symbol at
zero changes no frequency integral.

\subsection{8. Partial Fourier transforms in the original product
coordinates}\label{partial-fourier-transforms-in-the-original-product-coordinates}

\href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments} supplies the full original integration,
monomial Schwartz and lattice-cutoff proofs used here.
\href{metric-foundation-bridges.pdf}{Metric and topological foundations}
supplies the exact derivative and segment-integration maps.

These proofs extend the course's existing scalar Fourier calculation to
its actual product spaces. The original transform, Lebesgue densities,
complex-linear distribution convention, coordinate orders and every
inverse factor remain. This is a standard foundation and carries no
novelty claim. The independent inputs are the scalar Fourier inversion
and Schwartz Plancherel proofs in Sections 1 and 2 of the Fourier
chapter, the two-sided original L2 extensions in its Section 7, the
scalar measure and density proofs in Banach Sections 15--16, the full
monomial Schwartz proof in Banach Section 18, and the actual cutoff in
Banach Section 21. The distribution, symbol, conormal and operator
receivers do not prove these inputs.

\subsubsection{8.1. The actual transform and every monomial
derivative}\label{the-actual-transform-and-every-monomial-derivative}

Let \(p,q\geq0\) be integers. Write the original coordinates as
\((x,y)\in\mathbb R^p\times\mathbb R^q\), with dual coordinate
\(\xi\in\mathbb R^p\). The monomial seminorms are

\[
\begin{aligned}
 P_{a,b;c,d}(f)=\sup_{x,y}
       |x^a y^b\partial_x^c\partial_y^d f(x,y)|,
 \qquad\\
 (T_xf)(\xi,y)=\int_{\mathbb R^p}e^{-ix\cdot\xi}f(x,y)\,dx,
 \qquad\\
 (G_\xi h)(x,y)=(2\pi)^{-p}
           \int_{\mathbb R^p}e^{ix\cdot\xi}h(\xi,y)\,d\xi.
 \end{aligned}
\tag{PF1}
\]

Dimension zero means a singleton carrying measure one; its empty
monomials and derivative products equal one. Thus these formulas include
the identity when \(p=0\), and the full scalar transform when \(q=0\).

For \(p\geq1\), choose an integer \(M\) with \(2M>p\) and retain the
actual finite integral

\[
 I_{p,M}=\int_{\mathbb R^p}(1+|x|^2)^{-M}\,dx,
 \qquad
 (1+|x|^2)^M=
    \sum_{|r|\leq M}{M!\over(M-|r|)!\,r!}x^{2r}.
 \tag{PF2}
\]

Its finiteness, with the entire integral and all shell bounds retained,
was proved by SC3. This also bounds the integral of any Schwartz
function in the \(x\) variables uniformly in all original \(y\)
variables and in each prescribed \(y\) monomial. Derivatives under the
integral follow from the original dominated derivative theorem: on a
compact parameter segment, every derivative of the integrand is bounded
by a finite sum of original monomial Schwartz seminorms times the
integrable \((1+|x|^2)^{-M}\). The exact fundamental theorem on that
segment identifies each derivative of the integral with the integral of
the derivative. Iterating gives all joint derivatives, with the original
\((-ix)^\gamma\) factors.

Here are the integration-by-parts limits, including their cutoff terms.
Take the actual Banach Section 21 function \(\vartheta\), equal to one
on \([-1,1]^p\) and supported in \([-7/4,7/4]^p\subset(-2,2)^p\), and
set \(\vartheta_R(x)=\vartheta(x/R)\), \(R\geq1\). For any Schwartz
\(g(x,y)\) and multiindex \(a\), compact integration by parts gives

\[
\begin{aligned}
 \xi^a\int e^{-ix\cdot\xi}\vartheta_R(x)g(x,y)\,dx
 \\
 =(-i)^{|a|}\sum_{v\leq a}{a\choose v}R^{-|v|}
       \int e^{-ix\cdot\xi}
          (\partial^v\vartheta)(x/R)
          \partial_x^{a-v}g(x,y)\,dx.
 \end{aligned}
\tag{PF3}
\]

The \(v=0\) term converges by dominated convergence to the full original
derivative integral. Every \(v\ne0\) term is supported where
\(R\leq\max_j|x_j|\leq2R\). Its absolute value is at most its original
coefficient \({a\choose v}R^{-|v|}\|\partial^v\vartheta\|_\infty\) times
the integral of \(|\partial_x^{a-v}g(x,y)|\) over that set, and tends to
zero. Multiplying by any requested \(y\) monomial and taking any \(y\)
derivative gives the same conclusion with its corresponding original
seminorm bound. The left side tends to \(\xi^aT_xg\). Consequently
\(T_xD_x^ag=\xi^aT_xg\), with \(D_x=-i\partial_x\). No boundary
contribution or cutoff derivative has been omitted before its limiting
estimate.

Apply this identity to \(g=(-ix)^\gamma y^\beta\partial_y^\delta f\).
The full product rule gives

\[
\begin{aligned}
 \xi^\alpha y^\beta\partial_\xi^\gamma\partial_y^\delta T_xf
 \\
 =(-i)^{|\alpha|+|\gamma|}
   \sum_{\substack{v\leq\alpha\\v\leq\gamma}}
       {\alpha\choose v}{\gamma!\over(\gamma-v)!}
       \int e^{-ix\cdot\xi}x^{\gamma-v}y^\beta
                \partial_x^{\alpha-v}\partial_y^\delta f(x,y)\,dx.
 \end{aligned}
\tag{PF4}
\]

Taking absolute values and expanding the entire weight in PF2 proves the
finite, explicit seminorm estimate

\[
\begin{aligned}
 \sup_{\xi,y}
  |\xi^\alpha y^\beta\partial_\xi^\gamma\partial_y^\delta T_xf|
 \\
 \leq I_{p,M}
   \sum_{\substack{v\leq\alpha\\v\leq\gamma}}
       {\alpha\choose v}{\gamma!\over(\gamma-v)!}
 \\
   \sum_{|r|\leq M}{M!\over(M-|r|)!\,r!}
       P_{2r+\gamma-v,\beta;\alpha-v,\delta}(f).
 \end{aligned}
\tag{PF5}
\]

Every factorial, multiindex and coefficient is retained. For \(p=0\),
this is the equality of the original \(y\) seminorms, with \(I_{0,M}=1\)
and the singleton empty-index sums. PF5 proves that \(T_x\) maps the
original full product Schwartz space continuously into itself. It does
not rely on a tensor-density assertion, a Schwartz kernel theorem, or a
coordinate pullback theorem.

\subsubsection{8.2. Both inverse maps, with the original
factors}\label{both-inverse-maps-with-the-original-factors}

Reflection in the transformed variables is \(R_xf(x,y)=f(-x,y)\). Its
original monomial seminorms equal those of \(f\). Direct change of the
original integration variable gives \(T_xR_x=R_\xi T_x\), and the actual
inverse in PF1 is \(G_\xi=(2\pi)^{-p}R_xT_\xi\), with the input
coordinates named accordingly. Thus PF5 also proves continuity of
\(G_\xi\), retaining the factor \((2\pi)^{-p}\) on each output bound.

For every fixed original \(y\), the slice \(f(\cdot,y)\) is Schwartz by
PF2 and the original monomial bounds. The proved scalar Fourier
inversion applies to this slice and gives

\[
 G_\xi T_x f(x,y)=f(x,y),\qquad
 T_xG_\xi h(\xi,y)=h(\xi,y),\qquad
 T_\xi T_x f(x,y)=(2\pi)^p f(-x,y).
 \tag{PF6}
\]

Both compositions lie in the original full Schwartz space by PF5, so
pointwise equality is equality of its actual elements. The third
identity follows from the first and the full reflection/inverse formula.
It retains the entire power of \(2\pi\); the original transform has not
been replaced by a unitary convention.

For each transformed coordinate \(j\) and unchanged coordinate \(k\),
PF3, differentiation under PF1 and the original product rule give all
four original maps

\[
 T_x(D_{x_j}f)=\xi_jT_xf,\qquad
 T_x(x_jf)=i\partial_{\xi_j}T_xf,\qquad
 T_x(D_{y_k}f)=D_{y_k}T_xf,\qquad
 T_x(y_kf)=y_kT_xf.
 \tag{PF7}
\]

Their arbitrary finite iterates follow by composition in the actual
indicated order. Formula PF4 supplies the full simultaneous derivative
formula, rather than suppressing its product terms.

\subsubsection{8.3. Tempered distributions and both original dual
topologies}\label{tempered-distributions-and-both-original-dual-topologies}

Let \(\mathcal S'\) denote the complex-linear continuous dual of the
original monomial Schwartz space. In the transposed formula below,
\(T_\xi\phi(x,y)\) is the same original negative-sign transform applied
to the test function's \(\xi\) variables, with its output named \(x\).
Define

\[
 \langle T_xu,\phi(\xi,y)\rangle
       =\langle u,T_\xi\phi(x,y)\rangle,
 \qquad
 \langle G_\xi v,\psi(x,y)\rangle
       =\langle v,G_x\psi(\xi,y)\rangle.
 \tag{PF8}
\]

The test transforms are continuous by PF5 and its full inverse bound, so
these are actual tempered distributions. Transposing the two equations
in PF6 proves that these distribution maps are mutual inverses. On
regular distributions induced by Schwartz functions, Fubini identifies
PF8 with the original integrals in PF1. Here is its full
absolute-integrability bound: choose integers \(M,N\) with \(2M>p\),
\(2N>q\), retain \(I_{p,M}\), \(I_{q,N}\), and put

\[
\begin{aligned}
 A_f=\sum_{\substack{|r|\leq M\\|s|\leq N}}
       {M!\over(M-|r|)!\,r!}{N!\over(N-|s|)!\,s!}
       P_{2r,2s;0,0}(f),\qquad\\
 B_\phi=\sum_{|r|\leq M}{M!\over(M-|r|)!\,r!}
       P_{2r,0;0,0}(\phi),
 \qquad\\
 \iiint |f(x,y)\phi(\xi,y)|\,dx\,d\xi\,dy
       \leq I_{p,M}^2 I_{q,N} A_f B_\phi<\infty.
 \end{aligned}
\tag{PF8a}
\]

The full weight expansions give
\(|f(x,y)|\leq A_f(1+|x|^2)^{-M}(1+|y|^2)^{-N}\) and
\(|\phi(\xi,y)|\leq B_\phi(1+|\xi|^2)^{-M}\). Their product and positive
Tonelli prove PF8a. Empty-dimensional factors retain their unit mass and
empty-index coefficient one. Consequently the distribution convention
agrees with the original function convention, including its inverse
coefficient.

For the strong dual topology use its actual bounded-set seminorm
\(q_C(u)=\sup_{\phi\in C}|\langle u,\phi\rangle|\). PF5 proves that the
image of every bounded Schwartz set \(C\) under \(T_\xi\) is bounded.
Directly from PF8,

\[
 q_C(T_xu)=q_{T_\xi C}(u),\qquad
 q_C(G_\xi v)=q_{G_x C}(v).
 \tag{PF9}
\]

Thus both distribution maps are continuous for the original strong dual
topology; singletons give the weak dual continuity too. No unidentified
distribution topology enters the argument.

The identities in PF7 hold for these distributions with precisely the
same factors and signs. For example,

\[
 \langle T_xD_{x_j}u,\phi\rangle
  =i\langle u,\partial_{x_j}T_\xi\phi\rangle
  =\langle u,T_\xi(\xi_j\phi)\rangle,
 \qquad
 \langle i\partial_{\xi_j}T_xu,\phi\rangle
  =-i\langle u,T_\xi\partial_{\xi_j}\phi\rangle
  =\langle u,x_jT_\xi\phi\rangle.
 \tag{PF10}
\]

Here \(\partial_{x_j}T_\xi\phi=-iT_\xi(\xi_j\phi)\) and
\(T_\xi\partial_{\xi_j}\phi=i x_jT_\xi\phi\), with the latter proved by
PF3 in the test variable. Differentiating \(y\) commutes with the test
transform by the proved dominated derivative step, and multiplication by
\(y\) factors out of that integral. The original distributional signs in
the unchanged variables therefore give the last two identities in PF7.
Their finite iterates keep their actual multiplication order.

\subsubsection{8.4. Original L2 maps and the exact unitary
comparison}\label{original-l2-maps-and-the-exact-unitary-comparison}

For \(f,g\in\mathcal S(\mathbb R^{p+q})\), apply the original scalar
Schwartz Parseval identity at each fixed \(y\). Positive Tonelli, or
absolute Fubini for the pairing, then gives

\[
\begin{aligned}
 \|T_xf\|_{L^2(d\xi\,dy)}^2
       =(2\pi)^p\|f\|_{L^2(dx\,dy)}^2,
 \qquad\\
 \int f(x,y)\overline{g(x,y)}\,dx\,dy
       =(2\pi)^{-p}
         \int T_xf(\xi,y)\overline{T_xg(\xi,y)}\,d\xi\,dy.
 \end{aligned}
\tag{PF11}
\]

The pairing is absolutely integrable on either side by the already
proved integral Cauchy--Schwarz inequality. All densities and the full
power of \(2\pi\) remain.

Compact smooth density in the original \(L^2(dx\,dy)\), proved by
LP10--LP12, supplies Schwartz approximants \(f_j\to f\). PF11 makes
\(T_xf_j\) Cauchy with its exact norm factor \((2\pi)^{p/2}\).
Completeness of the original target \(L^2(d\xi\,dy)\) gives a unique
limit, independent of the approximants. This defines the bounded
original \(T_x\) on the whole original L2 space, with precisely that
norm factor. The inverse \(G_\xi\) extends by its exact
\((2\pi)^{-p/2}\) norm bound. Taking limits in both identities in PF6
shows that these two whole-space maps are mutual inverses. Taking limits
in PF11 gives both original norm and pairing identities for all L2
inputs.

The distribution and L2 maps agree on L2 functions. Indeed for every
fixed Schwartz test, the original integral pairing against that test is
continuous in L2 by Cauchy--Schwarz. Pair PF8 with the same approximants
on both sides and pass to their L2 limits. This identifies the two exact
maps without a pointwise Fourier-integrability assumption for a general
L2 input.

If the receiving statement uses a unitary transform, its exact
comparison is

\[
\begin{aligned}
 U_x=(2\pi)^{-p/2}T_x,\qquad\\
 U_x^{-1}=(2\pi)^{p/2}G_\xi,\qquad\\
 \|U_xf\|_2^2=(2\pi)^{-p}\|T_xf\|_2^2=\|f\|_2^2.
 \end{aligned}
\tag{PF12}
\]

Both inverse maps and their surjectivity have just been proved, so this
receiving comparison is unitary. It leaves the original \(T_x\),
\(G_\xi\), measures and constants explicit throughout the calculation.

\subsubsection{8.5. Exact receiving scope}\label{exact-receiving-scope}

For the conormal coordinates
\((t,z)\in\mathbb R^k\times\mathbb R^{n-k}\), take the actual
substitutions \(p=k\), \(q=n-k\), \(x=t\), \(y=z\), \(\xi=\tau\).
PF1--PF12 then supply the entire partial normal Fourier, full
tempered-distribution, monomial derivative and L2 extension clause with
inverse coefficient \((2\pi)^{-k}\). The codimension-zero factor has
already been proved as the identity; it requires no nonexistent normal
coordinate. The complete original scalar Fourier proof supplies the full
transform when every coordinate is transformed. These are exact maps on
the specified spaces, with every original coordinate and factor
retained.

This supplies the elementary Fourier interface itself. Distribution
test-function tensor operations, nonlinear coordinate pullbacks, bundle
covariance, intrinsic wavefront statements and complex contour
continuation are distinct receiving calculations that still require
their complete actual proofs. No conclusion about those calculations,
whole-course closure, independent review or original-primary-source
comparison is inferred from this Euclidean map.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  {[}Melrose{]} Richard Melrose, \emph{Differential Analysis}, MIT
  18.155 lecture notes, 2004.
\item
  {[}Beezer{]} Robert Beezer, \emph{A First Course in Linear Algebra},
  \href{https://github.com/rbeezer/fcla}{author's source repository}.
\end{itemize}

\end{document}