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\begin{document}

\ReaderArmAliases{\ReaderOriginalAnchorStart{local-inverses-and-distance-weighted-elliptic-estimates}\ReaderOriginalAnchorEnd}
\section{Local inverses and distance-weighted elliptic
estimates}\label{local-inverses-and-distance-weighted-elliptic-estimates}

Ellipticity gives a local inverse with exactly as many derivatives as
the order of the operator. The useful question here is how little
regularity its coefficients may have. Continuity will suffice for the
highest-order coefficients. Lower-order coefficients can be unbounded;
their admissible integrability depends on how many derivatives separate
their term from the principal part.

We first build a constant-coefficient solver and record its quantitative
bounds. Multiplication estimates then determine the coefficient
assumptions. A separate localization argument gives estimates on
arbitrary open subsets of one fixed neighborhood, with constants
independent of the subset. Finally, a commutator argument supplies the
missing highest derivatives when the equation is initially defined only
as a distribution.

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\subsection{1. Conventions and prerequisite
contracts}\label{conventions-and-prerequisite-contracts}

Let \(n,m\geq1\) be integers and \(1<p<\infty\). The order \(m\) is
arbitrary. Coefficients and functions may be complex-valued. We use the
Fourier convention of Section 1 of
\href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
convex separation}:

\[
D_j=-i\partial_j,\qquad
\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx,\qquad
\mathcal F^{-1}h(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}h(\xi)\,d\xi.
\]

The space \(W^{s,p}(V)\), for a nonnegative integer \(s\), consists of
distributions whose derivatives of orders at most \(s\) belong to
\(L^p(V)\). We write \(\nabla^j u\) for the finite array
\((D^\alpha u)_{|\alpha|=j}\), using its pointwise \(\ell^p\) norm
inside an \(L^p\) norm. Switching to the sum of the component norms
changes only constants depending on \(n,m,p\).

The proofs use the following entry results, each with its stated
hypotheses.

\begin{itemize}
\tightlist
\item
  \textbf{Distribution operations:} Distributions carry the usual
  test-function topology. Multiplication by a smooth function and
  distributional differentiation are continuous operations. A
  distribution supported in a compact set has finite order: after
  inserting a fixed cutoff near its support, its pairing is bounded by a
  constant times finitely many suprema of derivatives of the test
  function on a fixed compact neighborhood. It therefore acts on smooth
  functions through that cutoff and is tempered. Convolution is defined
  when one factor is compactly supported; differentiation transfers
  between its factors, and associativity holds in the uses here where
  all but at most one factor are compactly supported. Convolution of a
  tempered distribution with a compactly supported distribution is
  tempered. A smooth kernel paired with a compactly supported
  distribution gives a smooth function, also locally when all relevant
  kernel arguments stay away from its singular set. Fourier inversion of
  a compactly supported distribution is smooth, with derivatives
  obtained by differentiating its exponential pairing. These
  distribution-calculus entry facts are used in (L4)--(L5), (L12),
  (L14), and the smoothness argument at the end of Section 2. Sections
  13.6--13.10 of \href{geometric-microlocal-calculus.pdf}{Detecting
  regularity without choosing coordinates} give the complete proofs of
  this interface, including the exact applications to the inverse
  identities below.
\item
  \textbf{Integration and weak derivatives:} Lebesgue integration,
  Hölder and Young inequalities, Tonelli, dominated and monotone
  convergence; completeness of \(L^p\); density of
  \(C_c^\infty(\mathbb R^n)\) in \(L^p\) for finite \(p\); translation
  continuity and convergence of smooth approximate identities in these
  spaces. Locally, smooth approximation holds in integer \(W^{s,p}\). A
  distribution represented by an \(L^p\) limit has the corresponding
  distributional derivatives. These are the integration and
  weak-derivative facts used below. Sections 15.1--15.5 of
  \href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
  and compact parameter arguments} prove these operations with their
  full norms, scale factors and derivative signs, and identify their
  exact applications below.
\item
  \textbf{Fourier multiplier estimate:} If
  \(b\in C^{n+2}(\mathbb R^n\setminus\{0\})\) and \[
  \max_{|\beta|\leq n+2}\sup_{\xi\ne0}
  |\xi|^{|\beta|}|\partial_\xi^\beta b(\xi)|\leq B,
  \] then \(b(D)\) extends from Schwartz functions to
  \(L^p(\mathbb R^n)\), with norm at most \(C_{n,p}B\). The source is
  Terence Tao, \emph{Lecture Notes 4 for 247A}, Theorem 4.4, pp.~19--20;
  its singular-integral implication is Corollary 2.10, pp.~8--9. The
  bound uses only finitely many symbol derivatives. Section 13 proves
  this statement with the original operator, finite exponents and full
  constants; its Fourier maps are constructed in Section 7 of
  \href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
  convex separation}.
\item
  \textbf{Fractional-integration estimate:} For \(0<s<n\),
  \(1<p<q<\infty\), and \(1/q=1/p-s/n\), convolution with \(|x|^{s-n}\)
  maps \(L^p(\mathbb R^n)\) to \(L^q(\mathbb R^n)\). The source is Tao,
  \emph{Lecture Notes 2 for 247A}, Corollary 6.3, p.~16, with
  Proposition 6.1 on p.~15. No strong \(L^\infty\) endpoint is included.
  Section 12 proves this original kernel estimate, including its finite
  endpoint, full constants and derivative-array application.
\item
  \textbf{Lipschitz product rule:} A Lipschitz scalar function has weak
  first derivatives in \(L^\infty\), bounded by its Lipschitz constant,
  and satisfies the weak product rule with smooth functions. This local
  fact also permits integration by parts against compactly supported
  smooth functions. It will be used only in the weak-coefficient part.
  Section 11 supplies the complete weak-derivative, product, cutoff and
  commutator proofs for the original complex coefficients.
\end{itemize}

There is also a broader constant-strength comparison. For an operator
with nonzero frozen symbol, continuous coefficients and constant
strength near a point, it provides a local \(L^2\) right inverse, a left
inverse on compactly supported smooth functions, and bounded \(Q(D)E\)
when the constant-coefficient operator \(Q\) is weaker than the frozen
operator. In this terminology, write
\(\widetilde q(\xi)^2=\sum_\beta|\partial_\xi^\beta q(\xi)|^2\);
``weaker'' means \(\widetilde q\leq C\widetilde p\), and constant
strength means pairwise comparability of the frozen \(\widetilde p_x\).
Section 14 proves this full constant-strength assertion with both
inverse identities and every weaker-operator bound. It does not supply
the rough lower-coefficient \(L^p\) theorem below. The elliptic
construction below retains its additional finite-exponent estimates.

The equation in that antecedent has a specific interpretation. The
finite-dimensional weaker-operator representation is
\(P(x,D)=P_0(D)+\sum_{\nu=1}^r c_\nu(x)P_\nu(D)\), with \(P_0\) the
frozen operator, \(P_\nu\) weaker than \(P_0\), and continuous \(c_\nu\)
vanishing at the frozen point. Define
\(PEf=P_0Ef+\sum_\nu c_\nu P_\nu Ef\). Each \(P_\nu Ef\), including
\(P_0Ef\), belongs to \(L^2\) by the stated operator bounds; after
shrinking the neighborhood, its product with \(c_\nu\) is an \(L^2\)
function. Section 14.6 proves this graph-domain interpretation: each
continuous coefficient multiplies an already defined local
square-integrable derivative.

The proofs below establish the fundamental solution, the local Hölder
multiplier bound, the perturbation argument, the distance-weighted
estimate, and the Lipschitz commutator. The Banach-space
geometric-series argument is also available in Section 3 of
\href{fredholm-stability.pdf}{Finite defects under perturbation}; we
recall its short application where it is needed.

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\subsection{2. A frozen solver without a restriction on dimension or
order}\label{a-frozen-solver-without-a-restriction-on-dimension-or-order}

Let \(q(\xi)\) be homogeneous of degree \(m\), with

\[
|q(\xi)|\geq c|\xi|^m\quad(\xi\in\mathbb R^n).                 \tag{L1}
\]

Choose \(\chi\in C_c^\infty(\mathbb R^n)\), equal to one for
\(|\xi|\leq1/2\) and zero for \(|\xi|\geq1\). Set

\[
b_0(\xi)=\frac{1-\chi(\xi)}{q(\xi)},\qquad F_0=\mathcal F^{-1}b_0.
                                                                    \tag{L2}
\]

The quotient is smooth across zero because its numerator vanishes there.
Repeated differentiation of a reciprocal, or induction in the identity
\(q/q=1\), gives

\[
|\partial^\beta b_0(\xi)|\leq C_\beta(1+|\xi|)^{-m-|\beta|}.
                                                                    \tag{L3}
\]

In particular \(F_0\) is tempered and \(q(D)F_0=\delta+\omega\), where
\(\omega=-\mathcal F^{-1}\chi\) is Schwartz.

Here is an exact fundamental solution. When \(m<n\), the locally
integrable function \(1/q\) defines a tempered distribution \(T\). When
\(m\geq n\), put \(L=m-n\) and define, for \(\varphi\in\mathcal S\),

\[
\langle T,\varphi\rangle=
\int_{\mathbb R^n}\frac{\displaystyle\varphi(\xi)-\chi(\xi)
  \sum_{|\beta|\leq L}\frac{\partial^\beta\varphi(0)}{\beta!}\xi^\beta}
 {q(\xi)}\,d\xi.                                                   \tag{L4}
\]

Taylor's theorem bounds the numerator by \(C|\xi|^{L+1}\) near zero. Its
quotient is integrable there, since the radial power after including the
volume element is \(n-1-m+L+1=0\). At infinity, Schwartz decay applies.
These same estimates, in terms of finitely many Schwartz seminorms,
prove that \(T\) is tempered. As \(L<m\), all derivatives of
\(q\varphi\) of orders at most \(L\) vanish at zero. Consequently
\(qT=1\). Thus

\[
G=\mathcal F^{-1}T,\qquad q(D)G=\delta.                            \tag{L5}
\]

Moreover, \(T-b_0\) is supported in \(\operatorname{supp}\chi\). Its
inverse Fourier transform \(H=G-F_0\) is smooth: on a fixed compact
frequency set one may differentiate the exponential in its
distributional pairing any number of times. Those differentiated
pairings depend continuously on \(x\), by the finite-order bound for a
compactly supported distribution. This proves smoothness, without
assuming that \(G\) is an ordinary homogeneous function. In the
dimensions where logarithmic terms occur, (L4) includes them
automatically.

We need two kernel estimates. Fix \(|\alpha|=j\leq m\) and put
\(k=m-j\). The symbol of \(D^\alpha F_0\) is \(\xi^\alpha b_0(\xi)\).
With \(\psi(\xi)=\chi(\xi/2)-\chi(\xi)\), its decomposition into annuli
has inverse transforms

\[
K_\ell(x)=2^{\ell(n-k)}\kappa_\alpha(2^\ell x),\quad
\kappa_\alpha=\mathcal F^{-1}\!\left(\psi(\xi)\frac{\xi^\alpha}{q(\xi)}\right),
\quad \ell=0,1,\ldots.                                             \tag{L6}
\]

The function \(\kappa_\alpha\) is Schwartz. Indeed its Fourier transform
is smooth with compact support away from zero; multiplication by powers
of \(x\) in the inverse transform is integration by parts. The
telescoping identity \(\sum_{\ell\geq0}\psi(2^{-\ell}\xi)=1-\chi(\xi)\)
proves (L6) in tempered distributions. For every integer \(N\),

\[
|K_\ell(x)|\leq C_N2^{\ell(n-k)}(1+2^\ell|x|)^{-N}.              \tag{L7}
\]

If \(k>0\), then
\(\sum_\ell\|K_\ell\|_1\leq C\sum_\ell2^{-\ell k}<\infty\). It therefore
represents an \(L^1\) kernel. Splitting the sum at \(2^\ell|x|=1\)
gives, for \(0<|x|\leq1\),

\[
|D^\alpha F_0(x)|\leq C
\begin{cases}
|x|^{k-n},&0<k<n,\\
1+|\log|x||,&k=n,\\
1,&k>n.
\end{cases}                                                       \tag{L8}
\]

For example the terms below the splitting point form a geometric sum
when \(k\ne n\); when \(k=n\), their number is at most
\(C(1+|\log|x||)\). Terms above the splitting point form a convergent
geometric tail if \(N>n-k\). When \(|x|\geq1\), (L7), with \(N\) as
large as desired, gives rapid decay. These arguments also prove
smoothness away from zero by differentiating (L7). In particular,

\[
D^\alpha F_0\in L^r_{\mathrm{loc}}
\quad\hbox{if}\quad k>n(1-1/r),\qquad 1\leq r\leq\infty.          \tag{L9}
\]

For \(r<\infty\) this follows by integrating \(\rho^{r(k-n)+n-1}\); the
logarithmic case is integrable to every finite power. The stated
condition for \(r=\infty\) is \(k>n\). No assertion that a highest
derivative is an integrable kernel is being made. When \(k=0\), its
boundedness on \(L^p\) follows from Fourier multiplier estimate and
(L3).

For \(0<k<n/p\), (L8), its decay at infinity, and Fractional-integration
estimate give the finite endpoint

\[
\|D^\alpha F_0*g\|_q\leq C\|g\|_p,
\qquad \frac1q=\frac1p-\frac{k}{n}.                              \tag{L10}
\]

Together with the multiplier case \(k=0,q=p\), this is the global
endpoint estimate whenever the displayed \(q\) is finite. For strictly
larger \(1/q\), Young's inequality applies to a localized kernel: choose
\(r\) by \(1+1/q=1/p+1/r\) and use (L9). This includes \(q=\infty\)
precisely when \(k>n/p\). Interpolation with, or the finite-measure
inclusion from, the finite endpoint handles the other
\(p\leq q<\infty\).

Consequently, if \(V\subset B_1\), extend \(g\in L^p(V)\) by zero and
define \(S_Vg=(G*g)|_V\). For every \(j\leq m\),

\[
\|D^\alpha S_Vg\|_{L^q(V)}\leq C\|g\|_{L^p(V)},\qquad
p\leq q\leq\infty,\quad
\frac1q\geq\frac1p-\frac{k}{n},                                  \tag{L11}
\]

where the last inequality must be strict if \(q=\infty\). The constant
is independent of the particular \(V\subset B_1\), with the frozen
polynomial and the chosen exponents fixed. To verify the smooth
correction, all arguments \(x-y\), for \(x,y\in V\), lie in \(B_2\);
hence each derivative of \(H\) is bounded there and its convolution is
bounded by \(C\|g\|_1\leq C|B_1|^{1-1/p}\|g\|_p\). Its \(L^q(V)\) norm
has the required uniform bound. The same reasoning localizes the kernels
in the strict Young estimates.

Both inverse identities are exact:

\[
q(D)S_Vg=g\text{ in }V,\qquad S_Vq(D)v=v\quad(v\in C_c^\infty(V)).
                                                                    \tag{L12}
\]

For the first, convolve (L5) with the compactly supported distribution
given by the zero extension of \(g\). For the second, differentiation
may be transferred to the compactly supported smooth factor:
\(G*q(D)v=(q(D)G)*v=v\).

For completeness, any other fundamental solution differs from \(G\) by a
smooth function. If \(q(D)w=0\), take a cutoff \(\eta\) equal to one
near any prescribed compact set. Then \[
\eta w=F_0*q(D)(\eta w)-\omega*(\eta w).
\] Here \(q(D)(\eta w)=[q(D),\eta]w\) has compact support away from that
set. The first convolution is smooth near the set because \(F_0\) is
smooth away from zero; the second is smooth because \(\omega\) is smooth
and \(\eta w\) has compact support. This proves the assertion locally
everywhere. It also explains the smooth-remainder assertion for a
regularized inverse with a different low-frequency cutoff.

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\subsection{3. A parameter in the constant-coefficient
estimate}\label{a-parameter-in-the-constant-coefficient-estimate}

\textbf{Lemma.} For a homogeneous elliptic \(q\) of degree \(m\),

\[
\sum_{|\alpha|\leq m} A^{m-|\alpha|}\|D^\alpha v\|_p
\leq C\bigl(\|q(D)v\|_p+A^m\|v\|_p\bigr),\qquad A>0,\quad
v\in W^{m,p}(\mathbb R^n).                                        \tag{L13}
\]

The constant is independent of \(A\). It can also be chosen uniformly
when \(q\) ranges over a compact set of homogeneous elliptic polynomials
of degree \(m\).

\textbf{Proof.} At \(A=1\), the Fourier identity
\(q b_0=1+\widehat\omega\) gives \[
D^\alpha v=(D^\alpha F_0)*q(D)v-(D^\alpha\omega)*v.                \tag{L14}
\] It holds first on Schwartz functions. Each first operator is bounded
on \(L^p\) by the multiplier contract for top order, and by the \(L^1\)
kernel estimate for lower orders. The second kernel is Schwartz. Smooth
cutoff and mollification approximate any \(W^{m,p}\) function in that
norm, so the identity and estimate extend to the stated domain. For
\(w(y)=v(y/A)\), the derivative norm is
\(\|D^\alpha w\|_p=A^{n/p-|\alpha|}\|D^\alpha v\|_p\), while homogeneity
gives \(\|q(D)w\|_p=A^{n/p-m}\|q(D)v\|_p\). Apply the estimate at one
and multiply by \(A^{m-n/p}\).

Finally, a compact family of elliptic polynomials has a positive common
minimum of \(|q(\theta)|\) on the unit sphere. Its coefficients are
bounded. The reciprocal differentiation identity bounds every required
derivative in (L3) uniformly; the integrations by parts and the
multiplier theorem use only finitely many of these bounds. The same
\(C\) therefore works for the family. \(\square\)

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\subsection{\texorpdfstring{4. The coefficient table and a local inverse
in
\(L^p\)}{4. The coefficient table and a local inverse in L\^{}p}}\label{the-coefficient-table-and-a-local-inverse-in-lp}

Consider \[
P=\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha,
\qquad P_m=\sum_{|\alpha|=m}a_\alpha(x)D^\alpha                     \tag{L15}
\] near zero. Assume that the coefficients of \(P_m\) are continuous and
its frozen symbol \(q(\xi)=\sum_{|\alpha|=m}a_\alpha(0)\xi^\alpha\) is
elliptic. For every lower-order term set \(k=m-|\alpha|>0\) and require
the following local integrability.

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Derivative gap \(k\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Coefficient space \(L^{r_\alpha}_{\rm loc}\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Exponent used for \(D^\alpha S_Vg\)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(k<n/p\) & \(r_\alpha=n/k\) & \(1/q_\alpha=1/p-k/n\) \\
\(k=n/p\) & \(r_\alpha=p+\varepsilon_\alpha\), some
\(\varepsilon_\alpha>0\) &
\(q_\alpha=p(p+\varepsilon_\alpha)/\varepsilon_\alpha\) \\
\(k>n/p\) & \(r_\alpha=p\) & \(q_\alpha=\infty\) \\
\end{longtable}
}

There are finitely many coefficients, so different positive
\(\varepsilon_\alpha\) cause no difficulty. At the critical gap any
finite \(q_\alpha\) is allowed by (L11). The table gives
\(1/p=1/r_\alpha+1/q_\alpha\) in every row.

\textbf{Theorem.} Every sufficiently small open neighborhood \(V\) of
zero has a linear operator \(E:L^p(V)\to L^p(V)\) satisfying

\[
PEf=f\quad(f\in L^p(V)),\qquad EPv=v\quad(v\in C_c^\infty(V)),     \tag{L16}
\]

and, simultaneously for all \(|\alpha|\leq m\),

\[
D^\alpha E:L^p(V)\longrightarrow L^q(V)\text{ bounded if }
p\leq q\leq\infty,\quad
\frac1q\geq\frac1p-\frac{m-|\alpha|}{n},                           \tag{L17}
\]

with strict inequality when \(q=\infty\). The equation in (L16) is a
distributional equality and every term of \(PEf\) is represented by an
\(L^p\) function. In particular \(Ef\in W^{m,p}(V)\).

\textbf{Proof.} Use \(S_V\) from (L11), with the same frozen polynomial,
and compute \[
PS_V=I+R_V,\qquad
R_Vg=\sum_{|\alpha|=m}(a_\alpha-a_\alpha(0))D^\alpha S_Vg
       +\sum_{|\alpha|<m}a_\alpha D^\alpha S_Vg.                   \tag{L18}
\] The top-order terms have norms at most
\(C\sup_V|a_\alpha-a_\alpha(0)|\|g\|_p\). For a lower-order term,
Hölder's inequality and the corresponding row of the table give \[
\|a_\alpha D^\alpha S_Vg\|_p
\leq C\|a_\alpha\|_{L^{r_\alpha}(V)}\|g\|_p.                      \tag{L19}
\] All \(r_\alpha\) are finite. On shrinking the surrounding ball, the
first coefficients tend to zero by continuity and the norms in (L19)
tend to zero by absolute continuity of their integrals. The constants in
(L11) do not grow as \(V\) shrinks inside \(B_1\). Choose the ball so
that \(\|R_V\|\leq1/2\) for every \(V\) contained in it.

The series \(B_V=\sum_{\nu=0}^\infty(-R_V)^\nu\) converges in operator
norm, since \(L^p(V)\) is Banach. Multiplication of its finite partial
sums by \(I+R_V\) leaves an error of norm at most \(2^{-N}\), so \(B_V\)
is its two-sided inverse and \(\|B_V\|\leq2\). Define \[
E=S_VB_V.                                                        \tag{L20}
\] Equations (L18) and (L11) give the right identity and every bound in
(L17). They also justify all coefficient products used in the equation.

For the left identity let \(g=q(D)v\), extended by zero, where
\(v\in C_c^\infty(V)\). The second identity in (L12) gives \(S_Vg=v\),
and therefore \((I+R_V)g=Pv\). The inverse on \(L^p(V)\) implies
\(B_VPv=g\), and hence \(EPv=v\). This checks the same \(E\) on both
stated domains. It asserts neither a boundary condition for \(Ef\) nor
uniqueness among all local solutions. \(\square\)

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\subsection{5. The Hölder alternative, with its scaling made
explicit}\label{the-huxf6lder-alternative-with-its-scaling-made-explicit}

For \(0<\gamma<1\), let \(\overline C^\gamma(V)\) be the bounded
functions with finite seminorm
\([f]_{\gamma,V}=\sup_{x\ne y\in V}|f(x)-f(y)|/|x-y|^\gamma\). On a ball
\(B_r\) use the equivalent, scaled norm \[
N_r(f)=\|f\|_{\infty,B_r}+r^\gamma[f]_{\gamma,B_r}.                 \tag{L21}
\] This is a Banach norm: a Cauchy sequence converges uniformly, and the
difference quotient bound passes to that limit, including the Cauchy
bound for the seminorm. The product inequality
\(N_r(fg)\leq N_r(f)N_r(g)\) follows by adding and subtracting
\(f(x)g(y)\).

\textbf{Extension.} A function in this space has a unique continuous
extension to \(\overline B_r\), by its Hölder bound. Extend it to all of
\(\mathbb R^n\) by \[
\mathcal A_rf(x)=
\begin{cases}
f(x),&|x|\leq r,\\
(2-|x|/r)f(rx/|x|),&r<|x|<2r,\\
0,&|x|\geq2r.
\end{cases}                                                       \tag{L22}
\] To check the norm including pairs in different regions, let \(\pi_r\)
be radial projection onto \(\overline B_r\), and let
\(\theta_r(x)=\min(1,\max(0,2-|x|/r))\). Then
\(\mathcal A_rf=\theta_r(f\circ\pi_r)\). The projection has Lipschitz
constant at most two: for exterior points expand the difference of their
normalized vectors, ordering their radii; for an interior and an
exterior point insert the point where their connecting segment first
meets the sphere. Also
\(|\theta_r(x)-\theta_r(y)|\leq\min(1,|x-y|/r)\leq(|x-y|/r)^\gamma\).
The product difference estimate now gives \[
\|\mathcal A_rf\|_\infty+r^\gamma[\mathcal A_rf]_{\gamma,\mathbb R^n}
\leq C_\gamma N_r(f),                                             \tag{L23}
\] with a constant independent of \(r\).

\textbf{The frozen Hölder bound.} For \(h\) bounded and globally
\(\gamma\)-Hölder, the top-order kernels in (L6) have zero integral.
Thus \[
K_\ell*h(x)=\int K_\ell(z)(h(x-z)-h(x))\,dz,
\quad
\|K_\ell*h\|_\infty\leq C2^{-\ell\gamma}[h]_\gamma.                \tag{L24}
\] The derivative kernel also has zero integral, and the same
computation gives
\(\|\nabla(K_\ell*h)\|_\infty\leq C2^{\ell(1-\gamma)}[h]_\gamma\). For
points a distance \(t\leq1\) apart, estimate a term either by twice its
supremum or by \(t\) times its derivative supremum. Sum the derivative
bound where \(2^\ell t\leq1\) and the supremum bound where
\(2^\ell t>1\). The geometric sums are at most
\(C_\gamma t^\gamma[h]_\gamma\). For \(t>1\), the convergent supremum
sum suffices. Therefore the sum defines a bounded \(C^\gamma\) function
with norm at most \(C_\gamma(\|h\|_\infty+[h]_\gamma)\). For compactly
supported \(h\), it is the distribution \((D^\alpha F_0)*h\): the symbol
partial sums converge on Schwartz test functions, and (L24) identifies
their locally uniform limit. For lower orders \(k>0\), the sum of the
\(L^1\) kernel norms is finite, so both the supremum and the Hölder
seminorm are bounded directly by convolution. Finally, convolution with
\(D^\alpha H\), for \(h\) supported in \(B_2\) and output restricted to
\(B_1\), is bounded in the full \(C^\gamma\) norm. Use the supremum of
that derivative and its first derivatives on \(B_3\), the mean-value
theorem, and \(\|h\|_1\leq|B_2|\|h\|_\infty\).

It follows that the operator \(T f=(G*\mathcal A_1 f)|_{B_1}\) satisfies
\[
\|D^\alpha Tf\|_{\overline C^\gamma(B_1)}
\leq C\|f\|_{\overline C^\gamma(B_1)},\qquad |\alpha|\leq m.       \tag{L25}
\] This argument works directly for the full Hölder space. It does not
require smooth functions to be dense in the same Hölder norm.

Rescale this single solver by \[
S_rf(x)=r^m\,T(f(r\,\cdot))(x/r),\qquad x\in B_r.
\] Homogeneity of \(q\) gives \[
q(D)S_rf=f,\qquad
N_r(D^\alpha S_rf)\leq C r^{m-|\alpha|}N_r(f).                    \tag{L26}
\] If \(g=q(D)v\) with \(v\in C_c^\infty(B_r)\), its rescaled extension
in (L22) is precisely its zero extension. Equivalently \(S_rg\) is
convolution with the fundamental solution \(r^{m-n}G(\cdot/r)\). Hence
\(S_rq(D)v=v\). This remains true when \(G\) has logarithmic terms; no
homogeneity assertion about \(G\) was used.

\textbf{Theorem.} If every coefficient of \(P\) is \(C^\gamma\) near
zero and its frozen principal symbol is elliptic, then on a sufficiently
small centered ball there is a linear \(E_\gamma\) on
\(\overline C^\gamma\) with \[
PE_\gamma f=f,\qquad E_\gamma Pv=v\ (v\in C_c^\infty),\qquad
D^\alpha E_\gamma:\overline C^\gamma\to\overline C^\gamma
\text{ bounded for }|\alpha|\leq m.                              \tag{L27}
\]

\textbf{Proof.} Compute \(PS_r=I+R_r\) as in (L18). For a highest-order
coefficient, \[
N_r(a_\alpha-a_\alpha(0))\leq2r^\gamma[a_\alpha]_{\gamma,B_r}.
\] For a lower-order coefficient its norm \(N_r(a_\alpha)\) remains
bounded as \(r\downarrow0\), while (L26) supplies the extra factor
\(r^{m-|\alpha|}\). The product inequality therefore gives \[
\|R_r\|_{N_r\to N_r}\leq C\left(
r^\gamma\sum_{|\alpha|=m}[a_\alpha]_{\gamma,B_r}
 +\sum_{|\alpha|<m}r^{m-|\alpha|}N_r(a_\alpha)\right)\longrightarrow0.
                                                                    \tag{L28}
\] Choose this norm below \(1/2\), and set \(E_\gamma=S_r(I+R_r)^{-1}\).
The same geometric-series and test-function argument used for (L20)
proves both identities. Equation (L26) gives the stronger estimate
\(N_r(D^\alpha E_\gamma f)\leq2C r^{m-|\alpha|}N_r(f)\). At fixed \(r\),
scaled and unscaled norms are equivalent, proving (L27). The small
factor in the top terms is \(r^\gamma\); the unscaled Hölder seminorm of
a coefficient need not decrease on smaller balls. \(\square\)

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\subsection{6. An estimate uniform over open
subsets}\label{an-estimate-uniform-over-open-subsets}

In this part the operator is the principal part \(P_m\) alone. Fix a
compact neighborhood \(K\) of zero, contained in the coefficient domain,
on which the coefficients are continuous and \[
\left|\sum_{|\alpha|=m}a_\alpha(x)\xi^\alpha\right|
\geq c_K|\xi|^m\qquad(x\in K,\ \xi\in\mathbb R^n).                \tag{L29}
\] Such a \(K\) is obtained by shrinking around an elliptic point, using
continuity and compactness of the unit sphere. For any open
\(V\subset K\), let
\(d_V(x)=\operatorname{dist}(x,\mathbb R^n\setminus V)\).

\textbf{Theorem.} There is a constant \(C\), depending on
\(K,P_m,n,m,p\) but not on \(V\), such that, for \(u\in W^{m,p}(V)\) and
\(0\leq j\leq m\),

\[
\|d_V^j\nabla^j u\|_{L^p(V)}
\leq C\bigl(\|d_V^mP_mu\|_{L^p(V)}+\|u\|_{L^p(V)}\bigr)^{j/m}
        \|u\|_{L^p(V)}^{1-j/m}.                                  \tag{L30}
\]

For \(j=m\) this is a linear estimate, and for \(j=0\) it is the
identity bound. If \(u=0\) as an \(L^p\) function, every derivative is
zero, and the assertion is interpreted accordingly. Lower-order terms
are not included in \(P_m\) in (L30).

\textbf{Proof.} The frozen polynomials
\(q_y(\xi)=\sum_{|\alpha|=m}a_\alpha(y)\xi^\alpha\), \(y\in K\), form a
compact family satisfying (L29). Thus (L13) has a common constant. Let
\(\eta\in C_c^\infty(B_1)\) equal one on \(B_{1/2}\). For a ball
\(B(y,r)\subset V\), apply (L13) to \(\eta((x-y)/r)u(x)\), extended by
zero, with \(A=M/r\). On \(B(y,r/2)\) its derivatives equal those of
\(u\). Expanding the frozen operator gives three types of terms: \[
q_y(D)(\eta_yu)=\eta_yP_mu+\eta_y(q_y(D)-P_m)u+[q_y(D),\eta_y]u.
\] The middle term is at most \(\omega_a(r)|\nabla^m u|\), up to a fixed
array constant, where \(\omega_a(r)\to0\) is a common modulus of
continuity of the finitely many coefficients on \(K\). Every term in the
last commutator contains a derivative of \(u\) of some order \(j<m\),
multiplied by a coefficient bounded by \(Cr^{j-m}\). Taking \(p\)th
powers, using the finite-sum inequality, and multiplying by \(r^{mp}\),
gives \[
\begin{split}
\sum_{j=0}^m M^{p(m-j)}r^{pj}
       \int_{B(y,r/2)}|\nabla^j u|^p
\leq C\bigg(&\int_{B(y,r)}|r^mP_mu|^p
 +\omega_a(r)^p\int_{B(y,r)}|r^m\nabla^m u|^p\\
 &+\sum_{j<m}\int_{B(y,r)}|r^j\nabla^j u|^p
 +M^{mp}\int_{B(y,r)}|u|^p\bigg).
\end{split}                                                       \tag{L31}
\] All constants in this inequality are independent of \(M,r,y,V\).

Choose a small number \(r_0>0\), to be fixed below, and put \[
\rho(y)=\min(r_0,d_V(y)/2).                                       \tag{L32}
\] The distance function is 1-Lipschitz, by the triangle inequality and
taking an infimum. Thus \(\rho\) is \(1/2\)-Lipschitz. Every ball
\(B(y,\rho(y))\) is contained in \(V\). The following two geometric
bounds ensure uniform overlap when the centers, rather than a discrete
covering, are integrated: \[
\begin{array}{ll}
|x-y|<\rho(y)&\Longrightarrow\quad
       \rho(y)/2<\rho(x)<3\rho(y)/2,\\[2pt]
|x-y|<2\rho(x)/5&\Longrightarrow\quad
       4\rho(x)/5<\rho(y)<6\rho(x)/5
       \quad\hbox{and}\quad |x-y|<\rho(y)/2.
\end{array}                                                       \tag{L33}
\] They follow immediately by subtracting the two radii and using the
Lipschitz bound. The second ball of centers lies in \(V\), since
\(2\rho(x)/5<d_V(x)\). Writing \(v_n=|B_1|\), they imply \[
\int_{\{y:x\in B(y,\rho(y))\}}\frac{dy}{\rho(y)^n}\leq3^n v_n,
\qquad
\int_{\{y:x\in B(y,\rho(y)/2)\}}\frac{dy}{\rho(y)^n}\geq3^{-n}v_n.
                                                                    \tag{L34}
\] For the upper bound, the centers lie in \(B(x,2\rho(x))\) and
\(\rho(y)>2\rho(x)/3\). For the lower bound, integrate just over
\(B(x,2\rho(x)/5)\), where \(\rho(y)<6\rho(x)/5\). The same estimates,
with constants depending also on \(j,p\), hold if the integrands are
multiplied by \(\rho(y)^{pj}\), after factoring out \(\rho(x)^{pj}\).

Integrate (L31), with \(r=\rho(y)\), against \(dy/\rho(y)^n\). Tonelli
and (L34) give \[
\begin{split}
\sum_{j=0}^m M^{p(m-j)}\|\rho^j\nabla^j u\|_p^p
\leq C\bigg(&\|\rho^mP_mu\|_p^p
 +\omega_a(r_0)^p\|\rho^m\nabla^m u\|_p^p\\
 &+\sum_{j<m}\|\rho^j\nabla^j u\|_p^p+M^{mp}\|u\|_p^p\bigg).
\end{split}                                                       \tag{L35}
\] First choose \(r_0\) so the coefficient of the highest-order term on
the right is less than \(1/2\). Next choose \(M_0\geq1\) sufficiently
large that the terms of all lower orders on the right can be absorbed by
half of their terms on the left whenever \(M\geq M_0\). Indeed each
corresponding left coefficient is at least \(M^p\). This proves, with
\(F=\|\rho^mP_mu\|_p\) and \(U=\|u\|_p\), \[
\|\rho^j\nabla^j u\|_p
\leq C\bigl(M^{j-m}F+M^jU\bigr),\qquad M\geq M_0.               \tag{L36}
\]

If \(U>0\) and \(F>M_0^mU\), choose \(M=(F/U)^{1/m}\), obtaining
\(C F^{j/m}U^{1-j/m}\). If \(F\leq M_0^mU\), choose \(M=M_0\), obtaining
\(CU\). In both cases the bound is at most \(C(F+U)^{j/m}U^{1-j/m}\).
Finally \(d_V\) is bounded above by a constant depending only on the
bounded set \(K\); (L32) therefore gives \(c\,d_V\leq\rho\leq d_V/2\),
with \(c>0\) depending only on \(K,r_0\). Replacing \(\rho\) by \(d_V\)
proves (L30), with a constant independent of the shape, connectedness,
or boundary regularity of \(V\). \(\square\)

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\subsection{7. A Lipschitz coefficient through a
mollifier}\label{a-lipschitz-coefficient-through-a-mollifier}

We now give the exact weak product used in the remainder of the lesson.
If \(v\in L^p_{\rm loc}\) and \(a\) is Lipschitz, define \[
aD_jv=D_j(av)-(D_ja)v                                             \tag{L37}
\] as a distribution. The two products on the right are locally
integrable. This definition agrees with the usual product for smooth
\(v\), and depends continuously on \(v\) in \(L^p\) on compact sets,
when tested against a fixed compactly supported smooth function.

If \(u\in W^{m-1,p}_{\rm loc}\), choose \(j\) with \(\alpha_j>0\) and
use (L37) with \(v=D^{\alpha-e_j}u\) to define \(aD^\alpha u\),
\(|\alpha|=m\). This is independent of that choice: smooth approximation
of \(u\) in \(W^{m-1,p}\) on a neighborhood of a test function's support
makes each definition converge to the same classical product on the
approximants. The continuous dependence just proved identifies the
limits.

\textbf{Commutator lemma.} Suppose \(a:\mathbb R^n\to\mathbb C\) has
Lipschitz constant \(M\), \(v\in L^p(\mathbb R^n)\), and
\(\varphi\in C_c^\infty(\mathbb R^n)\). Put
\(\varphi_\varepsilon(x)=\varepsilon^{-n}\varphi(x/\varepsilon)\). Then
\[
\begin{split}
C_\varepsilon^a v&=(aD_jv)*\varphi_\varepsilon
                     -a(D_jv*\varphi_\varepsilon),\\
\|C_\varepsilon^a v\|_p
&\leq M\|v\|_p\int_{\mathbb R^n}
       \bigl(|\varphi(z)|+|z|\,|D_j\varphi(z)|\bigr)\,dz,
\end{split}                                                       \tag{L38}
\] and \(C_\varepsilon^a v\to0\) in \(L^p\) for each fixed \(v\).

\textbf{Proof.} Applying (L37), or integrating by parts first for smooth
\(v\), gives \[
\begin{split}
C_\varepsilon^a v(x)
={}&\int [a(x-y)-a(x)]v(x-y)D_j\varphi_\varepsilon(y)\,dy\\
 &-\int (D_ja)(x-y)v(x-y)\varphi_\varepsilon(y)\,dy.
\end{split}                \tag{L39}
\] The right side makes sense for \(v\in L^p\). Its absolute value is
bounded by convolution of \(M|v|\) with
\(|y||D_j\varphi_\varepsilon(y)|+|\varphi_\varepsilon(y)|\). Young's
inequality yields (L38), because changing \(y=\varepsilon z\) makes that
kernel's \(L^1\) norm independent of \(\varepsilon\). Approximation in
\(L^p\) proves that (L39) represents the distribution in (L38), using
the continuity of the product (L37) on compact sets. No boundedness of
\(a\) on all of \(\mathbb R^n\) is required; its at most linear growth
also makes the original products legitimate distributions.

For \(h\in C_c^\infty\), use the unintegrated expression \[
C_\varepsilon^a h(x)=\int[a(x-y)-a(x)]D_jh(x-y)\varphi_\varepsilon(y)\,dy.
\] Its \(L^p\) norm is at most
\(\varepsilon M\|D_jh\|_p\int|z||\varphi(z)|\,dz\), which tends to zero.
Given \(v\), approximate it by such an \(h\); (L38) bounds
\(C_\varepsilon^a(v-h)\) uniformly in \(\varepsilon\). Let
\(\varepsilon\downarrow0\) first, and then \(\|v-h\|_p\downarrow0\).
This proves strong convergence. \(\square\)

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\subsection{8. Recovering the highest derivatives from the
equation}\label{recovering-the-highest-derivatives-from-the-equation}

\textbf{Weak regularity and weighted estimate.} Retain the fixed compact
neighborhood and ellipticity of Section 6, and suppose that all
coefficients of \(P_m\) are Lipschitz on a neighborhood of \(K\). Let
\(V\subset K\) be open. If \[
u\in W^{m-1,p}(V),\qquad P_mu\in L^p(V),                           \tag{L40}
\] where \(P_mu\) is defined by (L37), then
\(u\in W^{m,p}_{\rm loc}(V)\), all the weighted derivatives in (L30)
belong to \(L^p(V)\), and (L30) holds with the same type of constant.
Global unweighted highest-derivative integrability up to \(\partial V\)
is not asserted.

\textbf{Proof.} Fix a ball \(B_0\) compactly contained in \(V\). Choose
nested slightly larger balls inside \(V\), a cutoff \(\chi_0\) equal to
one near \(\overline B_0\), and a cutoff \(\chi_1\) equal to one on an
open neighborhood \(W\) of \(\operatorname{supp}\chi_0\). Take
\(\overline W\subset V\). Set \(v=\chi_0u\), extend it by zero, and set
\(b_\alpha=\chi_1a_\alpha\), extended by zero to \(\mathbb R^n\). These
\(b_\alpha\) are globally Lipschitz and bounded. Product
differentiation, justified by the smooth approximation used after (L37),
gives \[
P_mv=\chi_0P_mu+
\sum_{|\alpha|=m}\sum_{\substack{\beta\leq\alpha\\|\beta|<m}}
 \binom{\alpha}{\beta}a_\alpha(D^{\alpha-\beta}\chi_0)D^\beta u.
                                                                    \tag{L41}
\] Every term on the right is \(L^p\) and supported in \(V\); its zero
extension is also \(\sum b_\alpha D^\alpha v\) on the whole space.

Choose \(\varphi\in C_c^\infty\) of integral one and let
\(v_\varepsilon=v*\varphi_\varepsilon\). For each top-order \(\alpha\),
choose \(j\) as in (L37). Applying (L38) to \(D^{\alpha-e_j}v\in L^p\)
shows \[
\sum_{|\alpha|=m} b_\alpha D^\alpha v_\varepsilon
=(P_mv)*\varphi_\varepsilon
 -\sum_{|\alpha|=m}C_\varepsilon^{b_\alpha}(D^{\alpha-e_j}v)
\longrightarrow P_mv\quad\hbox{in }L^p(\mathbb R^n).              \tag{L42}
\] For small \(\varepsilon\), the support of \(v_\varepsilon\) is in
\(W\), where \(b_\alpha=a_\alpha\). In particular, (L42) is convergence
of \(P_mv_\varepsilon\) in \(L^p(W)\). Also \(v_\varepsilon\to v\) in
\(L^p\). Apply (L30), with \(j=m\) and open set \(W\), to the smooth
difference \(v_\varepsilon-v_\delta\). Its right side tends to zero.
Since \(d_W\) has a positive lower bound on \(\overline B_0\), every
highest derivative is Cauchy in \(L^p(B_0)\). Its limit is the
corresponding distributional derivative of \(v=u\) there. This proves
local \(W^{m,p}\) regularity.

To pass to the full open set without imposing boundary regularity, put
\(V_t=\{x\in V:d_V(x)>t\}\), \(t>0\). Its closure is compact in \(V\),
since \(V\subset K\) is bounded. Local regularity and a finite compact
cover imply \(u\in W^{m,p}(V_t)\). For \(x\in V_t\), its own distance
function satisfies \[
d_V(x)-t\leq d_{V_t}(x)\leq d_V(x).                               \tag{L43}
\] The first inequality follows because the ball of radius \(d_V(x)-t\)
about \(x\) lies in \(V_t\), by the Lipschitz property of \(d_V\); the
second follows from \(V_t\subset V\). Hence, as \(t\downarrow0\), the
distances, extended by zero off \(V_t\), increase pointwise to \(d_V\).
Apply (L30) on \(V_t\). Its constants are independent of \(t\). Monotone
convergence for the derivative integrals, and dominated convergence for
the \(u\) and \(P_mu\) integrals, give (L30) on \(V\). This also proves
the asserted weighted integrability. \(\square\)

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\subsection{9. Annular decay and infinite-order
vanishing}\label{annular-decay-and-infinite-order-vanishing}

Let \(P_m\) be elliptic with continuous principal coefficients on a
fixed neighborhood of zero. Suppose
\(u\in W^{m,p}_{\rm loc}(K^\circ\setminus\{0\})\), and for some real
\(N\) and all sufficiently small \(r>0\), \[
\int_{r<|x|<2r}|u|^p\,dx=O(r^N),\qquad
|P_mu(x)|\leq C\sum_{|\alpha|<m}|x|^{|\alpha|-m}|D^\alpha u(x)|
\quad\hbox{a.e. off zero}.                                      \tag{L44}
\]

\textbf{Corollary.} For every \(|\alpha|\leq m\), \[
\int_{r<|x|<2r}|r^{|\alpha|}D^\alpha u|^p\,dx=O(r^N).             \tag{L45}
\] The same conclusion holds when the principal coefficients are
Lipschitz and the initial assumption is only
\(u\in W^{m-1,p}_{\rm loc}\) off zero, with (L44) interpreted
distributionally as in (L37). In that case the inequality in (L44) means
that the distribution \(P_mu\) has the indicated locally \(L^p\)
representative.

\textbf{Proof.} On \(A_r=\{r<|x|<2r\}\), let \(d=d_{A_r}\),
\(U=\|u\|_{L^p(A_r)}\), and
\(S=\sum_{j<m}\|d^j\nabla^j u\|_{L^p(A_r)}\). The weak version is first
legitimate on this annulus by Section 8; all lower derivatives and
\(P_mu\) are \(L^p\) on it because its closure lies in the punctured
neighborhood. The same is true in the original strong version. Since
\(d\leq r\leq|x|\), (L44) implies \(F=\|d^mP_mu\|_p\leq C S\). Notice
also that \(U\leq S\). Applying (L30) at every \(j<m\), \[
S\leq C\sum_{j<m}S^{j/m}U^{1-j/m}
\leq C S^{(m-1)/m}U^{1/m}.                                      \tag{L46}
\] If \(S=0\), the desired bound is immediate. Otherwise divide by
\(S^{(m-1)/m}\) and raise to the \(m\)th power, obtaining \(S\leq CU\).
For \(m=1\), the sum is just \(S=U\), consistent with this calculation.
The top-order case of (L30) then gives
\(\|d^m\nabla^m u\|_p\leq C(F+U)\leq CU\).

On \(M_r=\{4r/3<|x|<5r/3\}\) one has \(d\geq r/3\). Thus \[
\int_{M_r}|r^j\nabla^j u|^p\,dx\leq C U^p=O(r^N),\qquad j\leq m.
                                                                    \tag{L47}
\] To obtain the original full annulus, not just its middle, use
\(M_{s r}\) for \(s=3/4,9/10,11/10,13/10\). Their radial intervals
overlap and cover \((r,2r)\): after dividing by \(r\), they are
\((1,5/4)\), \((6/5,3/2)\), \((22/15,11/6)\), and \((26/15,13/6)\).
Their associated larger annuli still lie in the coefficient neighborhood
for small \(r\). Each application of (L47) has \((sr)^N=O(r^N)\), and
\(r^j\) is a fixed multiple of \((sr)^j\). Summing these four bounds
proves (L45). \(\square\)

Under the differential inequality in (L44), the condition that its
annular integral be \(O(r^N)\) for every \(N>0\) is therefore equivalent
to the same condition for every scaled derivative in (L45). It suffices
to include orders below \(m\), because they include \(u\); the top-order
bounds then follow. This is the infinite-order vanishing statement used
later in unique continuation.

The annular and ball formulations agree as well. If an annular bound
holds for a positive exponent \(N\), split \(B_r\setminus\{0\}\) into
the disjoint annuli of inner radii \(2^{-\nu-1}r\), \(\nu\geq0\). Their
bounds sum to at most \(C r^N\sum_{\nu\geq0}2^{-(\nu+1)N}\). The value
at the origin has no effect on the integral. Conversely, an annulus is
contained in \(B_{2r}\). For an unweighted derivative of order \(j\),
replace \(N\) in (L45) by \(N+pj\); this gives every desired positive
decay exponent for that derivative as well. This argument establishes
decay and integrability; a conclusion that the solution vanishes in a
neighborhood requires a unique-continuation theorem in addition.

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\subsection{10. Problems with full
solutions}\label{problems-with-full-solutions}

\textbf{Problem 1: all three coefficient regimes in one operator.} In
dimension six, take \(m=4\), \(p=2\), and \[
P=(1+|x|^{1/3})(D_1^2+\cdots+D_6^2)^2
  +\sum_{|\alpha|\leq3}b_\alpha(x)D^\alpha.
\] Determine sufficient local spaces for each \(b_\alpha\), and all
admissible derivative exponents for its inverse.

\textbf{Solution.} The frozen symbol is \(|\xi|^4\); the principal
coefficients are continuous, although no first derivative at zero is
required. The critical gap is \(n/p=3\). Order-three coefficients have
gap one and may lie in \(L^6_{\rm loc}\); order-two coefficients may lie
in \(L^3_{\rm loc}\); order-one coefficients require
\(L^{2+\varepsilon_\alpha}_{\rm loc}\); and order-zero coefficients
require only \(L^2_{\rm loc}\). These coefficients can be unbounded. For
example, a coefficient of order three equal to \(|x|^{-1/2}\) near zero
is in \(L^6_{\rm loc}(\mathbb R^6)\), because its sixth power has radial
integral proportional to \(\int_0^r t^2\,dt\).

On a sufficiently small neighborhood, (L17) gives order-four derivatives
in \(L^2\), order-three derivatives in every \(L^q\) with
\(2\leq q\leq3\), order-two derivatives for \(2\leq q\leq6\), order-one
derivatives for every finite \(q\geq2\), and the solution itself in
\(L^\infty\) as well as all these finite spaces. The first-order term
uses, in particular, \(q=2(2+\varepsilon_\alpha)/\varepsilon_\alpha\).
There is no \(L^\infty\) assertion for the first derivatives at this
critical gap.

\textbf{Problem 2: why \(L^p\) alone is insufficient in the critical
multiplication step.} In a small disk in \(\mathbb R^2\), construct
\(a\in L^2\) and \(w\in W^{1,2}\) such that \(aw\notin L^2\). Relate
this to the critical row of the table.

\textbf{Solution.} Near zero write \(L(r)=\log(e/r)\) and take \[
w(x)=L(|x|)^{1/4},\qquad
a(x)=|x|^{-1}L(|x|)^{-5/8},
\] with smooth cutoffs away from zero. The integral for \(|a|^2\) near
zero is a constant times \(\int_0^r t^{-1}L(t)^{-5/4}\,dt\), which is
finite. The function \(w\) is square-integrable, and its gradient has
squared radial integral bounded by
\(C\int_0^r t^{-1}L(t)^{-3/2}\,dt<\infty\). These classical derivatives
off zero are also its weak derivatives: integration by parts outside a
disk of radius \(\delta\) produces a boundary term bounded by
\(C\delta L(\delta)^{1/4}\), tending to zero. Thus \(w\in W^{1,2}\). But
\(|aw|^2\) has radial integral
\(\int_0^r t^{-1}L(t)^{-3/4}\,dt=\infty\).

Here \(k=1=n/p\). The example disproves the unrestricted multiplication
estimate \(L^2\cdot W^{1,2}\subset L^2\), which would be needed to use
just \(a\in L^p\) at the critical gap. It does not purport to be a
nonexistence example for every elliptic equation. The stronger
coefficient space \(L^{2+\varepsilon}\), paired with a sufficiently
large finite derivative exponent, is exactly what makes (L19) valid. The
displayed \(a\) belongs to no \(L^{2+\varepsilon}\), as its extra radial
power already forces divergence.

\textbf{Problem 3: check the annular weights on a singular solution
profile.} Let \(P_2=\sum_jD_j^2=-\Delta\), \(n\geq2\), and
\(u(x)=|x|^\lambda\) off zero, with \(\lambda\in\mathbb R\). Find the
annular exponent and verify the differential inequality required in
(L44).

\textbf{Solution.} Direct radial differentiation gives
\(-\Delta u=-\lambda(\lambda+n-2)|x|^{\lambda-2}\). Thus
\(|P_2u|\leq C|x|^{-2}|u|\), one of the allowed terms in (L44). All
derivatives exist and are locally \(L^p\) in the punctured neighborhood.
On an annulus, changing variables \(x=ry\) gives \[
\int_{r<|x|<2r}|u|^p\,dx
=r^{p\lambda+n}\int_{1<|y|<2}|y|^{p\lambda}\,dy.
\] So \(N=p\lambda+n\), which may be negative, zero, or positive. A
derivative of order \(j\leq2\) is homogeneous of degree \(\lambda-j\),
unless it vanishes identically. Multiplication by \(r^j\) restores the
same power \(r^{p\lambda+n}\) after integration. This verifies both the
weight and the exponent in (L45), including profiles that are not
integrable at the origin before imposing a sufficiently positive
exponent.

\textbf{Problem 4: strong commutator convergence need not be operator
norm convergence.} Let \(a(x)=x_j\) in (L38), and choose
\(\varphi\in C_c^\infty\) of integral one. Show why the commutator tends
to zero on each fixed \(L^p\) function even though its \(L^p\) operator
norm does not tend to zero.

\textbf{Solution.} Formula (L39), with \(D_j=-i\partial_j\), gives
convolution with \[
k_\varepsilon(y)=i\partial_{y_j}\bigl(y_j\varphi_\varepsilon(y)\bigr)
=\varepsilon^{-n}k_1(y/\varepsilon).
\] This is a nonzero smooth compactly supported kernel: if
\(\partial_j(y_j\varphi)=0\), then on each line parallel to the \(j\)th
axis the compactly supported function \(y_j\varphi\) is constant, hence
zero, forcing \(\varphi=0\), contrary to its integral. There is
\(h\in C_c^\infty\) with \(k_1*h\ne0\); for instance, convolution with a
smooth approximate identity converges to the nonzero \(k_1\). Define
\(h_\varepsilon(x)=\varepsilon^{-n/p}h(x/\varepsilon)\). A change of
variables gives \[
\|h_\varepsilon\|_p=\|h\|_p,\qquad
\|k_\varepsilon*h_\varepsilon\|_p=\|k_1*h\|_p>0.
\] Thus the operator norm is bounded below independently of
\(\varepsilon\). The lemma applies to a fixed input; this example
changes the input with the smoothing scale. The weak regularity proof in
(L42) uses precisely the strong convergence on its finitely many fixed
lower derivatives.

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\subsection{11. The weak derivative of the original Lipschitz
coefficient}\label{the-weak-derivative-of-the-original-lipschitz-coefficient}

The entry fact about Lipschitz coefficients needs a proof because it is
used inside (L37), rather than only as a bound on a smooth coefficient.
We prove it from the integration results in Sections 15.1--15.5 of
\href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments} and the Hilbert representation proof in
Section 2.3 of \href{lower-bounded-spectral-calculus.pdf}{The full
calculus of a lower-bounded self-adjoint operator}. The proof retains
the actual complex coefficient. No differentiability almost everywhere
or measure-representation theorem for the dual of \(L^1\) is assumed.

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\subsubsection{11.1. Difference quotients and a bounded measurable
derivative}\label{difference-quotients-and-a-bounded-measurable-derivative}

Let \(\Omega\subset\mathbb R^n\) be open and let
\(a:\Omega\to\mathbb C\) satisfy \(|a(x)-a(y)|\leq M|x-y|\) for all
points under consideration, with \(M\geq0\). The local version uses such
a bound on each interior neighborhood. Fix a bounded coordinate cube
\(Q\) with closure in \(\Omega\). For \(\phi\in C_c^\infty(Q)\), its
support and all sufficiently small translates remain in \(Q\). The
original change of variables and the scalar fundamental theorem give \[
 \begin{split}
 -\int_Q a(x)\partial_j\phi(x)\,dx
 &=\lim_{t\to0}\int_Q
       \frac{a(x+t e_j)-a(x)}{t}\phi(x)\,dx,\\
 \left|-\int_Q a\partial_j\phi\right|
 &\leq M\|\phi\|_{L^1(Q)}
 \leq M|Q|^{1/2}\|\phi\|_{L^2(Q)} .
 \end{split}                                                   \tag{LC1}
\] In the change of variables the difference of translated tests is
supported in one fixed compact subset of \(Q\). The function \(a\) is
bounded there: compare its values with a fixed point in that
neighborhood. The difference quotients of the tests converge uniformly
to \(-\partial_j\phi\). Dominated convergence therefore proves the first
line, including its minus sign, while the Lipschitz bound proves the
second.

Here the dense test subspace of \(L^2(Q)\) can also be verified without
a boundary assertion. For an arbitrary \(f\in L^2(Q)\), choose compact
smooth cutoffs \(0\leq\chi_k\leq1\) supported in \(Q\), equal to one on
increasing interior cubes exhausting \(Q\). Dominated convergence gives
\(\chi_k f\to f\) in \(L^2(Q)\). Extend \(\chi_k f\) by zero. Smooth it
by the original compact kernel of integral one from (LP14); choosing its
scale smaller than the distance of \(\operatorname{supp}\chi_k\) to
\(Q^c\), divided by a radius containing the kernel support, keeps the
smoothed support inside \(Q\). Formula (LP14) gives arbitrarily small
\(L^2\) error. Thus \(C_c^\infty(Q)\) is dense. Every original support
and scale restriction is retained.

The linear functional in (LC1) extends uniquely to \(L^2(Q)\).
Completeness of that space is (LP10), and its inner product is the
original integral \(\langle f,h\rangle=\int_Q f\overline h\), linear in
the first variable. The Hilbert representation proof cited above,
applied to the conjugate of this linear functional, gives
\(g_{j,Q}\in L^2(Q)\) such that \[
 -\int_Q a\partial_j\phi=\int_Q g_{j,Q}\phi,
 \qquad
 \left|\int_Q g_{j,Q}h\right|\leq M\|h\|_{L^1(Q)}
                         \quad(h\in L^2(Q)).
                                                               \tag{LC2}
\] The second inequality follows by approximating \(h\) in \(L^2\) by
compact tests; their \(L^1\) errors are at most \(|Q|^{1/2}\) times the
\(L^2\) errors. The complex representative in (LC2) is the conjugate of
the Hilbert representing vector, so no conjugation has been lost.

For measurable \(E\subset Q\), take
\(h=1_E\overline{g_{j,Q}}/|g_{j,Q}|\), with value zero where
\(g_{j,Q}=0\). It belongs to \(L^2(Q)\), and (LC2) gives
\(\int_E|g_{j,Q}|\leq M|E|\). If \(E=\{|g_{j,Q}|>M+\delta\}\) for
\(\delta>0\), this forces \(|E|=0\). Taking a countable union over
positive reciprocal integers proves \[
 g_{j,Q}\in L^\infty(Q),\qquad
 \|g_{j,Q}\|_{L^\infty(Q)}\leq M,\qquad
 \partial_j a=g_{j,Q}\quad\hbox{on }Q.                         \tag{LC3}
\] On overlapping cubes the two representatives agree almost everywhere.
Indeed their difference integrates to zero against every compact smooth
test in the overlap. Exhaust that overlap by bounded interior cubes and
use the density argument just given in each cube. Approximating the
complex conjugate of the difference in \(L^2\) proves that its squared
modulus has zero integral there.

A countable collection of interior cubes covers \(\Omega\). Define
\(g_j\) by taking the first cube containing each point; discard the
countable union of the exceptional null sets on overlaps. These
measurable representatives agree locally and retain every local bound.
To verify the global distribution identity for a compact test, cover its
support by finitely many of these cubes, choose nonnegative smooth bumps
\(\theta_l\) supported in those cubes whose sum is positive on a
neighborhood of the test support, and write
\(\phi=\sum_l\phi\theta_l/(\sum_r\theta_r)\). Each summand is a compact
smooth test in one cube; all original weights remain in this identity.
Summing (LC2) proves \(\partial_j a=g_j\) throughout \(\Omega\). In the
original derivative convention, \(D_j a=-i g_j\); hence
\(\|D_j a\|_\infty\leq M\) wherever the same Lipschitz bound holds. No
global boundedness of \(a\) was required.

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\subsubsection{11.2. The complex gradient bound and the converse on the
actual
domain}\label{the-complex-gradient-bound-and-the-converse-on-the-actual-domain}

There is a sharper form of the preceding bound, with a necessary
distinction for a complex coefficient. Repeat (LC1) in a real direction
\(v\in\mathbb Q^n\), replacing \(te_j\) by \(tv\). The resulting
functional is represented by \(\sum_j v_jg_j\), by the coordinate
identities and uniqueness. Thus \(|\sum_jv_jg_j(x)|\leq M|v|\) almost
everywhere. Remove the null exceptional sets for this countable set of
directions. Continuity in \(v\) then gives the exact pointwise real
linear map \[
 A(x)v=\sum_{j=1}^n v_jg_j(x),\qquad
 \|A(x):\mathbb R^n\to\mathbb C\|_{\rm op}\leq M
                                      \quad\hbox{a.e.}          \tag{LC4}
\] For real \(a\), this operator norm is \((\sum_j|g_j|^2)^{1/2}\). For
complex \(a\), its real matrix has rows \((\operatorname{Re}g_j)_j\) and
\((\operatorname{Im}g_j)_j\). The two row bounds, and the \(n\)
coordinate bounds, give both original estimates \[
 \sum_{j=1}^n|g_j|^2\leq 2M^2,\qquad
 \sum_{j=1}^n|g_j|^2\leq nM^2 .
                                                               \tag{LC5}
\] For each row the Euclidean norm is the supremum of its real
directional pairing, bounded by (LC4). The coordinate estimate is (LC3).
The complex Euclidean gradient bound cannot be replaced by \(M\) in
general: \(a(x_1,x_2)=x_1+i x_2\) has Lipschitz constant one, original
derivative array \((1,i)\), real operator norm one and Euclidean array
norm \(\sqrt2\). These are different norms on the same retained
derivative array.

Conversely, suppose \(a\in L^1_{\rm loc}(\Omega)\) has weak coordinate
derivatives \(g_j\in L^\infty_{\rm loc}\), with the map (LC4) bounded by
\(M\) on a neighborhood of a compact segment. Choose a nonnegative
compact smooth \(r\) with \(\int r=1\), and retain
\(r_\varepsilon(x)=\varepsilon^{-n}r(x/\varepsilon)\). On interior
points its local smoothing satisfies
\(\partial_j a_\varepsilon=r_\varepsilon*g_j\), by (LP19) after
inserting an interior cutoff equal to one near the points and their
kernel translates. Consequently, for every real \(v\), \[
 |\partial_v a_\varepsilon(x)|
 =\left|\int r_\varepsilon(y)A(x-y)v\,dy\right|
 \leq M|v|\int r_\varepsilon(y)\,dy=M|v|.
                                                               \tag{LC6}
\] The nonnegativity in this converse is the chosen kernel's hypothesis,
and its scale and integral remain explicit. Integration along any
retained interior segment gives
\(|a_\varepsilon(x)-a_\varepsilon(y)|\leq M|x-y|\).

These smoothings converge locally uniformly to a representative of the
original \(a\). Here is a direct verification. They are Cauchy in
\(L^1\) on every relatively compact interior neighborhood, by (LP14)
with an interior cutoff. Fix compact \(K\) and \(\rho>0\) such that its
\(\rho\)-neighborhood has compact closure in \(\Omega\). If \(M>0\) and
\(|a_\varepsilon(x)-a_\delta(x)|\geq\eta>0\) at \(x\in K\), both
smoothings have Lipschitz constant at most \(M\) on a small ball about
\(x\). Set \(b=\min(\rho/2,\eta/(4M))\). On the ball of radius \(b\),
their difference has modulus at least \(\eta/2\). The coordinate box
centered at \(x\) with side lengths \(2b/\sqrt n\) lies in that ball, up
to its measure-zero faces. Thus their original \(L^1\) distance on the
neighborhood is at least \[
          \frac{\eta}{2}\left(\frac{2b}{\sqrt n}\right)^n .
                                                               \tag{LC7}
\] This contradicts the Cauchy property for sufficiently small scales,
proving uniform Cauchy convergence on \(K\). If \(M=0\), the smoothings
are constant on each such ball, and their \(L^1\) distance is that
constant difference times its positive original box volume; the same
conclusion follows. Covering \(K\) by finitely many balls handles
locally varying bounds. The uniform limit is continuous and is equal to
\(a\) almost everywhere by the \(L^1\) limit. Passing to the limit in
the segment inequality proves a local Lipschitz representative with
exactly the map bound \(M\). On a convex open \(\Omega\) with a global
bound, every segment between its points has compact image in \(\Omega\),
so the representative is globally \(M\)-Lipschitz. On a general open
set, the conclusion holds along interior segments or paths, retaining
their actual lengths; no inequality across disconnected components is
inferred.

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\subsubsection{11.3. The full weak product, highest derivatives and
cutoff
extension}\label{the-full-weak-product-highest-derivatives-and-cutoff-extension}

Let \(h\) be smooth. Applying (LC2) to \(h\phi\), and using the scalar
product rule on that test, gives \[
 -\int ah\partial_j\phi
       =\int(g_jh+a\partial_jh)\phi,\qquad
 D_j(ah)=(D_j a)h+aD_jh.                                       \tag{LC8}
\] For \(h\in W^{1,p}_{\rm loc}\), \(1\leq p<\infty\), use its smooth
approximations from (LP19) on a neighborhood of the test support. The
locally bounded \(a\) and \(g_j\) multiply the converging \(L^p\)
components continuously, so (LP16) passes (LC8) to that original \(h\).
For \(p=\infty\), take these same local limits in \(L^2\) on
finite-measure test neighborhoods; both \(h\) and its derivatives belong
there to \(L^2\), while the resulting products retain their local
essential bounds. This proves the product identity at that endpoint
without a false strong \(L^\infty\) smoothing assertion.

For arbitrary \(v\in L^p_{\rm loc}\), retain precisely the distribution
definition (L37). Its full pairing and continuity bound are \[
 \begin{split}
 \langle aD_jv,\phi\rangle
 &=-\int v aD_j\phi-\int v(D_j a)\phi,\\
 |\langle aD_jv,\phi\rangle|
 &\leq\|v\|_{L^p(K)}
     \left(\|aD_j\phi\|_{L^{p'}(K)}
                    +\|(D_j a)\phi\|_{L^{p'}(K)}\right),
 \qquad K=\operatorname{supp}\phi .
 \end{split}                                                   \tag{LC9}
\] The conjugate exponents include their two endpoints. Every \(-i\)
factor is still inside the original \(D_j\)'s. Formula (LC8) proves
agreement with the classical product for smooth \(v\). If
\(u\in W^{m-1,p}_{\rm loc}\), \(1\leq p<\infty\), approximate all its
derivatives up to \(m-1\) on a neighborhood of \(K\) by one smooth
sequence. For each \(j\) with \(\alpha_j>0\), apply (LC9) to
\(v=D^{\alpha-e_j}u\). Its approximating products are the same classical
\(aD^\alpha u_k\), because the original smooth coordinate derivatives
commute. Their distributional limits therefore agree for every choice of
\(j\). This proves the claimed independence in (L37), and the same
approximations prove the entire product differentiation (L41), retaining
each \(\binom{\alpha}{\beta}\) and each original ordered derivative.

The coefficient cutoff extension in Section 8 also has an exact global
bound. Let \(\chi\in C_c^\infty(N)\), where \(a\) is Lipschitz with
constant \(M\) on the open \(N\). Choose \(\rho>0\) with the compact
\(\rho\)-neighborhood of \(\operatorname{supp}\chi\) contained in \(N\),
let \(A\) be the supremum of \(|a|\) there, and put
\(L_\chi=\sum_j\|\partial_j\chi\|_\infty\). For \(b=\chi a\) extended by
zero, the original estimates give \[
 \|b\|_\infty\leq A\|\chi\|_\infty,\qquad
 \operatorname{Lip}(b)
 \leq\max\left(M\|\chi\|_\infty+A L_\chi,
                     \frac{2A\|\chi\|_\infty}{\rho}\right).
                                                               \tag{LC10}
\] For points at distance less than \(\rho\), if at least one cutoff
value is nonzero, both points lie in that compact neighborhood. Expand
the original difference as \(\chi(x)(a(x)-a(y))+a(y)(\chi(x)-\chi(y))\)
and use the scalar segment bound for \(\chi\). If neither cutoff value
is nonzero, the difference is zero. For points separated by at least
\(\rho\), use the two separate amplitude bounds, giving the second
constant. This covers points outside \(N\) as well. An identically zero
cutoff gives the zero function directly. The full weak derivative is
\(D_jb=\chi D_ja+aD_j\chi\), extended by zero: an interior test cutoff
equal to one near \(\operatorname{supp}\chi\) proves that no boundary
term is added. Thus every original \(b_\alpha=\chi_1a_\alpha\) used in
(L41)--(L42) is actually bounded and globally Lipschitz.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-LIP-004}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-LIP-004}\label{AN03-LCE-LIP-004}\endgroup\ReaderOriginalAnchorStart{the-commutator-bound-with-every-scale-factor}\ReaderOriginalAnchorEnd}
\subsubsection{11.4. The commutator bound with every scale
factor}\label{the-commutator-bound-with-every-scale-factor}

The commutator argument has a useful precise scope: its bound holds for
\(1\leq p\leq\infty\), and its strong convergence holds for
\(1\leq p<\infty\). This strengthens the commutator itself; the elliptic
inverse theorems above keep their original \(1<p<\infty\) hypotheses.

For \(a:\mathbb R^n\to\mathbb C\) globally \(M\)-Lipschitz and
\(v\in L^p\), insert (LC9) into the original convolution in (L38). The
differentiation signs give exactly (L39); the first integral contains
\([a(x-y)-a(x)]D_j\varphi_\varepsilon(y)\), and the second contains
\(-(D_ja)(x-y)\varphi_\varepsilon(y)\). The convolution kernels are
compact, so each pairing is defined. If \(p\) is finite, the at most
linear growth \(|a(x)|\leq|a(0)|+M|x|\) also gives a tempered product on
the original Schwartz tests by Hölder with the full weight
\((|a(0)|+M|x|)\); the same integral bound applies at infinity. Nothing
requires globally bounded \(a\).

Using (LC3) and the exact Lipschitz difference, the absolute kernel is
bounded by
\(M(|y||D_j\varphi_\varepsilon(y)|+|\varphi_\varepsilon(y)|)\). Its full
change of variables is \[
 \begin{split}
 &\int\left(|y|\,\varepsilon^{-n-1}
                |D_j\varphi(y/\varepsilon)|
        +\varepsilon^{-n}|\varphi(y/\varepsilon)|\right)dy\\
 &=\int\left(\varepsilon|z|\,\varepsilon^{-n-1}|D_j\varphi(z)|
             +\varepsilon^{-n}|\varphi(z)|\right)\varepsilon^n dz\\
 &=\int\left(|z|\,|D_j\varphi(z)|+|\varphi(z)|\right)dz .
 \end{split}                                                   \tag{LC11}
\] Young's (LP9) proves (L38) at every displayed norm endpoint. For
compact smooth \(h\), the unintegrated original commutator has norm at
most \(\varepsilon M\|D_jh\|_p\int|z||\varphi(z)|\,dz\). For finite
\(p\), choose such \(h\) with arbitrarily small \(\|v-h\|_p\), using
Section 15.3 of the integration chapter. The complete estimate is \[
 \|C_\varepsilon^a v\|_p
 \leq M\|v-h\|_p\int
          (|\varphi(z)|+|z||D_j\varphi(z)|)\,dz
       +\varepsilon M\|D_jh\|_p\int|z||\varphi(z)|\,dz .
                                                               \tag{LC12}
\] Let the scale tend to zero and then the approximation error tend to
zero. This proves the original strong limit needed by every term of
(L42), and also the commutator's \(p=1\) extension.

At \(p=\infty\), strong convergence for every input is false. In
dimension one take \(a(x)=|x|\), \(v=1_{(0,\infty)}\), and a nonzero
compact smooth \(\varphi\). The actual distribution products are
\(D v=-i\delta_0\) and \(aD v=0\), as follows either from (LC9) or from
\(D(x_+)=-i1_{(0,\infty)}\). Therefore \[
 C_\varepsilon^a v(x)=i|x|\,\varepsilon^{-1}\varphi(x/\varepsilon),
 \qquad
 \|C_\varepsilon^a v\|_\infty
          =\sup_z |z|\,|\varphi(z)|>0 .                        \tag{LC13}
\] The point \(z=0\) contributes zero; nonzero smooth \(\varphi\) has a
nonzero value at some nonzero point, proving positivity. The original
boundedness estimate survives, while its finite-norm approximation
argument has exactly the finite-\(p\) scope already proved.

These proofs close the Lipschitz entry used in (L37)--(L42). They supply
the actual bounded weak derivative, the product pairing and choice
independence, the global coefficient cutoff bound and the
scale-independent commutator estimate. They preserve the original
complex coefficient, its coordinate derivatives, every mixed derivative
and cutoff contribution, and the finite-\(p\) regularity argument.
Sections 12--13 supply the fractional-integration and multiplier proofs
with their original hypotheses and exact receiving constants.

\ReaderArmAliases{\ReaderOriginalAnchorStart{fractional-integration-with-every-original-scale-retained}\ReaderOriginalAnchorEnd}
\subsection{12. Fractional integration with every original scale
retained}\label{fractional-integration-with-every-original-scale-retained}

We now prove the fractional-integration prerequisite in Section 1. The
kernel is the original \(|x|^{s-n}\); no constant is absorbed into it
and no input norm is set to one. The proof uses the completed coordinate
measure, product integration, Hölder, finite-\(p\) translation
continuity and monotone convergence from Section 15 of
\href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments}. Its finite-cube compactness is proved
in Section 15.0 there. We do not assume an interpolation theorem, a dual
representation theorem or a Lorentz-space theorem in this proof.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-FRAC-001}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-FRAC-001}\label{AN03-LCE-FRAC-001}\endgroup\ReaderOriginalAnchorStart{cube-averages-and-the-full-weak-bound}\ReaderOriginalAnchorEnd}
\subsubsection{12.1. Cube averages and the full weak
bound}\label{cube-averages-and-the-full-weak-bound}

For \(r>0\), let \(Q(x,r)=x+(-r,r)^n\), with its original volume
\((2r)^n\). For a locally integrable measurable \(f\), define \[
 \mathcal Mf(x)=\sup_{r>0}\frac1{(2r)^n}
                         \int_{Q(x,r)}|f(y)|\,dy.              \tag{FI1}
\] Changing the open faces to closed faces does not alter the integral.
The average at a fixed \(r\) is continuous in \(x\). To prove this
locally, truncate \(f\) to a sufficiently large bounded coordinate cube
containing all relevant \(Q(x,r)\), obtaining a global \(L^1\) function
\(g\). Write the average as its convolution with the bounded cube
indicator. Translation by \(h\) changes its numerator by at most
\(\|\tau_hg-g\|_1\), which tends to zero by (LP12). The same argument
applies in every bounded neighborhood of \(x\).

For a fixed \(x\), the numerator is continuous in \(r>0\): on a compact
radius interval the indicators converge almost everywhere as \(r\)
varies, and all integrands are dominated by \(|f|\) on a fixed bounded
cube. The only exceptions are coordinate faces, which are null.
Dominated convergence (LP2) applies. Consequently the supremum in (FI1)
is also the supremum over positive rational \(r\), so it is measurable.
Its superlevel sets are open because it is a supremum of continuous
functions of \(x\).

Suppose \(f\in L^1(\mathbb R^n)\) and \(\lambda>0\). Every \(x\) in
\(E_\lambda=\{\mathcal Mf>\lambda\}\) is the center of an open cube with
average greater than \(\lambda\). Fix a compact \(K\subset E_\lambda\)
and take a finite subcover from these cubes. The finite-cover property
follows from the nested-box proof in Section 15.0: enclose \(K\) in a
closed coordinate cube, add its open complement relative to \(K\) to the
cover, and apply that proved finite-cover result.

Choose among the finite family a cube of largest radius, retain it, and
discard every cube meeting it. Repeat on the undiscarded family. The
retained cubes \(Q_j=Q(x_j,r_j)\) are disjoint. Every discarded cube has
radius at most that of the retained cube it met. If \(Q(x,r)\) meets
\(Q(x_j,r_j)\) and \(r\leq r_j\), then \(|x-x_j|_\infty<r+r_j\); for
every \(y\in Q(x,r)\), \(|y-x_j|_\infty<2r+r_j\leq3r_j\). Thus the
discarded cube lies in \(Q(x_j,3r_j)\), with the exact volume \[
 |Q(x_j,3r_j)|=(6r_j)^n=3^n(2r_j)^n,\qquad
 |K|\leq\sum_j(6r_j)^n
       <\frac{3^n}{\lambda}\sum_j\int_{Q_j}|f|
       \leq\frac{3^n}{\lambda}\|f\|_1.                        \tag{FI2}
\] No contribution from a discarded cube is counted in the last sum.
Disjointness of the retained cubes gives precisely the last inequality.

To pass from compact subsets to the open set, exhaust \(E_\lambda\) by
\(K_k=\{x\in[-k,k]^n:\operatorname{dist}_\infty(x,E_\lambda^c)\geq1/k\}\).
If the complement is empty, take \(K_k=[-k,k]^n\). The distance to a
nonempty closed set is continuous: the triangle inequality gives its
difference bound by \(|x-x'|_\infty\). Thus each \(K_k\) is closed and
bounded and is compact by the same closed-cube proof. The \(K_k\)'s
increase to \(E_\lambda\), since every point of an open set has a small
interior cube. Measure continuity from (LP1) now proves \[
 |\{x:\mathcal Mf(x)>\lambda\}|
              \leq\frac{3^n}{\lambda}\|f\|_1.                 \tag{FI3}
\] In particular \(\mathcal Mf<\infty\) almost everywhere for
\(f\in L^1\), by sending \(\lambda\) to infinity in this bound. The
operator is subadditive, is unchanged when an input is changed on a null
set, and satisfies \(\mathcal Mf\leq\|f\|_\infty\) for essentially
bounded input. These statements follow from the corresponding numerator
inequalities in each original cube.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-FRAC-002}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-FRAC-002}\label{AN03-LCE-FRAC-002}\endgroup\ReaderOriginalAnchorStart{the-strong-maximal-bound-without-interpolation}\ReaderOriginalAnchorEnd}
\subsubsection{12.2. The strong maximal bound without
interpolation}\label{the-strong-maximal-bound-without-interpolation}

Let \(1<p<\infty\), \(f\in L^p\), and \(\lambda>0\). Put
\(f_\lambda=f1_{\{|f|>\lambda/2\}}\). The pointwise bound
\(|f_\lambda|\leq(\lambda/2)^{1-p}|f|^p\) makes it \(L^1\), while
\(|f-f_\lambda|\leq\lambda/2\) almost everywhere. Subadditivity and
(FI3) give \[
 |\{\mathcal Mf>\lambda\}|
 \leq|\{\mathcal Mf_\lambda>\lambda/2\}|
 \leq\frac{2\,3^n}{\lambda}
                     \int_{\{|f|>\lambda/2\}}|f(x)|\,dx.      \tag{FI4}
\] For any measurable \(h\geq0\), possibly infinite, the scalar identity
\(h(x)^p=p\int_0^{h(x)}\lambda^{p-1}\,d\lambda\) and nonnegative (LP4)
give the level-set formula
\(\int h^p=p\int_0^\infty\lambda^{p-1}|\{h>\lambda\}|\,d\lambda\). Use
it for \(h=\mathcal Mf\), insert (FI4), and apply nonnegative product
integration once more. All original cutoff and power factors give \[
 \begin{split}
 \|\mathcal Mf\|_p^p
 &\leq 2\,3^n p\int_{\mathbb R^n}|f(x)|
          \left(\int_0^{2|f(x)|}\lambda^{p-2}\,d\lambda\right)dx\\
 &=\frac{2\,3^n p}{p-1}\int_{\mathbb R^n}
                    |f(x)|(2|f(x)|)^{p-1}\,dx
   =\frac{2^p3^n p}{p-1}\|f\|_p^p .
 \end{split}                                                   \tag{FI5}
\] When \(f(x)=0\), the inner integral is over an empty interval and its
contribution is zero. The condition \(p>1\) is exactly what makes the
lower endpoint integrable. Define the explicit retained constant \[
 C^{\mathcal M}_{n,p}=\left(\frac{2^p3^n p}{p-1}\right)^{1/p};
 \qquad \|\mathcal Mf\|_p\leq C^{\mathcal M}_{n,p}\|f\|_p.                 \tag{FI6}
\] This proof establishes the estimate directly on the complete original
\(L^p\) space, with no interpolation assumption. It also makes
\(\mathcal Mf\) finite almost everywhere for these inputs.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-FRAC-003}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-FRAC-003}\label{AN03-LCE-FRAC-003}\endgroup\ReaderOriginalAnchorStart{the-original-fractional-kernel-and-its-finite-endpoint}\ReaderOriginalAnchorEnd}
\subsubsection{12.3. The original fractional kernel and its finite
endpoint}\label{the-original-fractional-kernel-and-its-finite-endpoint}

Fix \(0<s<n\) and \(1<p<n/s\). Write \(p'=p/(p-1)\), \(F=\|f\|_p\), and
\[
 \theta=1-\frac{sp}{n}>0,\qquad
 q=\frac p\theta=\frac{np}{n-sp},\qquad
 \gamma=(n-s)p'-n>0.                                          \tag{FI7}
\] Thus \(1/q=1/p-s/n\), \(q>p\), and \(\theta q=p\). The sign of
\(\gamma\) follows by multiplying \(p<n/s\) by its positive
denominators. For a measurable \(f\in L^p\), consider the nonnegative
potential \[
 I_s|f|(x)=\int_{\mathbb R^n}|x-y|^{s-n}|f(y)|\,dy .
\] Its integrand at \(y=x\) may be set to zero, as that singleton is
null. The function is measurable by completed nonnegative product
integration. Local integrability of \(f\) follows from Hölder on each
bounded coordinate cube.

For \(R>0\), split the original integral into \(0<|x-y|\leq R\) and
\(|x-y|>R\). Decompose the first part into the annuli
\(2^{-j-1}R<|x-y|\leq2^{-j}R\), \(j\geq0\). Since \(s-n<0\), the kernel
on the \(j\)-th annulus is at most \((2^{-j-1}R)^{s-n}\). Its input
integral is bounded by the integral on \(Q(x,2^{-j}R)\); sphere and cube
boundary choices have no effect here. The defining cube average gives \[
 \begin{split}
 \int_{0<|x-y|\leq R}|x-y|^{s-n}|f(y)|\,dy
 &\leq\sum_{j=0}^\infty
       (2^{-j-1}R)^{s-n}(2\,2^{-j}R)^n\mathcal Mf(x)\\
 &=\frac{2^{2n-s}}{1-2^{-s}}R^s\mathcal Mf(x)
   =A_{n,s}R^s\mathcal Mf(x),\\
 A_{n,s}&=\frac{2^{2n-s}}{1-2^{-s}} .
 \end{split}                                                   \tag{FI8}
\] To justify the boundary comment without a higher-dimensional polar
formula, every fixed sphere is null: for a fixed first \(n-1\)
coordinates its last-coordinate section has at most two points, hence
one-dimensional measure zero. Nonnegative product integration (LP4)
gives zero total measure. When \(n=1\) the sphere is itself at most two
points. All the sets and integrands are Borel apart from the input's
already treated completed null modification.

For the far part use Hölder, and split the kernel's \(p'\)-th power into
annuli \(2^jR<|z|\leq2^{j+1}R\). Each is contained in the coordinate
cube of radius \(2^{j+1}R\), whose full volume is \((2^{j+2}R)^n\).
Therefore \[
 \begin{split}
 \int_{|z|>R}|z|^{(s-n)p'}\,dz
 &\leq\sum_{j=0}^\infty
           (2^jR)^{(s-n)p'}(2^{j+2}R)^n
   =\frac{2^{2n}}{1-2^{-\gamma}}R^{-\gamma},\\
 \int_{|x-y|>R}|x-y|^{s-n}|f(y)|\,dy
 &\leq B_{n,s,p}F R^{s-n/p},\\
 B_{n,s,p}
 &=\left(\frac{2^{2n}}{1-2^{-\gamma}}\right)^{1/p'},
 \qquad -\frac{\gamma}{p'}=s-\frac np .
 \end{split}                                                   \tag{FI9}
\] The last exponent identity retains the calculation
\(-\gamma/p'=-(n-s)+n/p'=s-n/p\). The convergence of the full geometric
series uses the actual \(\gamma>0\).

If \(F=0\), the input vanishes almost everywhere and its potential is
zero. Suppose \(F>0\). At a point where \(0<\mathcal Mf(x)<\infty\),
choose the actual radius \(R=(F/\mathcal Mf(x))^{p/n}\). Combining
(FI8)--(FI9) gives \[
 \begin{split}
 I_s|f|(x)
 &\leq A_{n,s}\mathcal Mf(x)R^s
           +B_{n,s,p}F R^{s-n/p}\\
 &=(A_{n,s}+B_{n,s,p})
                F^{sp/n}(\mathcal Mf(x))^{1-sp/n}.
 \end{split}                                                   \tag{FI10}
\] At a point where \(\mathcal Mf(x)=0\), the near bound is zero for
every \(R\); the far bound tends to zero as \(R\to\infty\), because
\(s-n/p<0\). Thus the same conclusion holds there. The remaining points,
where the maximal function is infinite, form a null set by (FI6).

Raise (FI10) to the \(q\)-th power and use \(\theta q=p\). The complete
bound is \[
 \begin{split}
 \|I_s|f|\|_q
 &\leq(A_{n,s}+B_{n,s,p})
                 F^{sp/n}\|\mathcal Mf\|_p^\theta\\
 &\leq H_{n,s,p}\|f\|_p,\qquad
 H_{n,s,p}=(A_{n,s}+B_{n,s,p})\bigl(C^{\mathcal M}_{n,p}\bigr)^{\,1-sp/n}.
 \end{split}                                                   \tag{FI11}
\] Here \(sp/n+\theta=1\) accounts for the final input norm; both
geometric-series constants and the whole maximal constant remain
explicit.

In particular the original integral for \(I_sf\) converges absolutely
almost everywhere for real or complex \(f\), and \(|I_sf|\leq I_s|f|\).
Inputs equal almost everywhere give equal output classes: translate the
input null set at each fixed \(x\) and use the completed integral, with
the kernel value at its singular singleton as specified. Integral
linearity gives a linear map \(I_s:L^p\to L^q\), and (FI11) proves its
boundedness. This proves the whole original fractional-integration
contract, with its actual kernel and finite endpoint.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-FRAC-004}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-FRAC-004}\label{AN03-LCE-FRAC-004}\endgroup\ReaderOriginalAnchorStart{the-weak-first-endpoint-and-two-proved-strong-endpoint-obstructions}\ReaderOriginalAnchorEnd}
\subsubsection{12.4. The weak first endpoint and two proved
strong-endpoint
obstructions}\label{the-weak-first-endpoint-and-two-proved-strong-endpoint-obstructions}

For \(f\in L^1\), put \(F=\|f\|_1\). The near estimate (FI8) is
unchanged. On \(|x-y|>R\), the original decreasing kernel is at most
\(R^{s-n}\), so the far estimate is \(FR^{s-n}\). If \(F>0\) and
\(0<\mathcal Mf(x)<\infty\), choose \(R=(F/\mathcal Mf(x))^{1/n}\).
Treat zero maximal values as above; its infinite values are null by
(FI3). With \(\sigma=n/(n-s)>1\), the resulting bound and its level
estimate are \[
 \begin{split}
 I_s|f|(x)&\leq(A_{n,s}+1)F^{s/n}
                          (\mathcal Mf(x))^{1-s/n},\\
 |\{x:|I_sf(x)|>\tau\}|
 &\leq3^n(A_{n,s}+1)^\sigma(F/\tau)^\sigma,
                       \qquad \tau>0 .
 \end{split}                                                   \tag{FI12}
\] Indeed the level implication requires
\(\mathcal Mf>(\tau/((A_{n,s}+1)F^{s/n}))^\sigma\). Insert this value
into (FI3); the remaining norm exponent is \(1+s\sigma/n=\sigma\). The
case \(F=0\) gives a zero potential directly. Thus this is a proved weak
\(L^1\)-to-\(L^\sigma\) bound, and also proves that the absolutely
defined potential is finite almost everywhere for \(L^1\) input.

The strong \(L^\sigma\) assertion at this endpoint is false. Take the
original \(f=1_{[-1,1]^n}\), whose integral is \(2^n\). On
\(2^j<|x|_\infty\leq2^{j+1}\), \(j\geq0\), each \(y\) in its support
satisfies \(|x-y|\leq\sqrt n(2^{j+1}+1)\). The negative kernel exponent
consequently gives \[
 \begin{split}
 I_sf(x)&\geq 2^n
                  [\sqrt n(2^{j+1}+1)]^{s-n},\\
 \int_{\{2^j<|x|_\infty\leq2^{j+1}\}}|I_sf(x)|^\sigma\,dx
 &\geq(2^n)^\sigma n^{-n/2}
             \frac{(2^{j+2})^n-(2^{j+1})^n}{(2^{j+1}+1)^n}\\
 &\geq(2^n)^\sigma n^{-n/2}\frac{4^n-2^n}{3^n}>0.
 \end{split}                                                   \tag{FI13}
\] We used \((s-n)\sigma=-n\) and \(2^{j+1}+1\leq3\,2^j\), retaining
both shell volumes. The shells are disjoint, so their infinite sum
diverges. This disproves the strong first endpoint.

The other excluded assertion is strong \(L^{n/s}\)-to-\(L^\infty\). Put
\(p_0=n/s>1\), choose \(1/p_0<a\leq1\), and keep the original shells and
coefficients \[
 \begin{split}
 A_j&=\{y:2^{-j-1}<|y|_\infty\leq2^{-j}\},\qquad j\geq1,\\
 f(y)&=\sum_{j=1}^\infty j^{-a}2^{js}1_{A_j}(y),\\
 \|f\|_{p_0}^{p_0}
 &=\sum_{j=1}^\infty j^{-ap_0}2^{jsp_0}
             \bigl((2^{1-j})^n-(2^{-j})^n\bigr)
   =(2^n-1)\sum_{j=1}^\infty j^{-ap_0}<\infty .
 \end{split}                                                   \tag{FI14}
\] The series converges because \(ap_0>1\), by comparison with the
elementary integral of \(t^{-ap_0}\). If \(|x|_\infty\leq2^{-J}\) and
\(j\leq J\), then \(|x-y|\leq2\sqrt n\,2^{-j}\) for \(y\in A_j\). Using
\(s-n<0\), the full lower bound is \[
 \begin{split}
 I_sf(x)
 &\geq\sum_{j=1}^J
      (2\sqrt n\,2^{-j})^{s-n}j^{-a}2^{js}
          \bigl((2^{1-j})^n-(2^{-j})^n\bigr)\\
 &=(2\sqrt n)^{s-n}(2^n-1)\sum_{j=1}^Jj^{-a}.
 \end{split}                                                   \tag{FI15}
\] The finite sums tend to infinity because \(a\leq1\), again by the
elementary integral comparison. Each lower bound holds on a cube of
positive measure \((2^{1-J})^n\). Hence the output has infinite
essential supremum, rather than merely an infinite value at one
exceptional point. Hölder on the supporting cube gives
\(\|f\|_1\leq(2\,2^{-1})^{n(1-1/p_0)}\|f\|_{p_0}<\infty\). Its far
integral is harmless for this compactly supported \(L^1\) input; local
absolute convergence almost everywhere also follows from the weak
estimate just proved. Thus it gives an actual measurable potential with
unbounded essential supremum. The original elliptic statements still
exclude this strong infinity endpoint.

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\subsubsection{12.5. The exact receiving kernel and derivative-array
estimates}\label{the-exact-receiving-kernel-and-derivative-array-estimates}

For (L10), put \(k=m-|\alpha|\) with \(0<k<n/p\), retaining the original
frozen polynomial and cutoff. Take the finite original bound
\(C_{\alpha,\mathrm{near}}=\sup_{0<|z|\leq1}|z|^{n-k}|D^\alpha F_0(z)|\)
from (L8). For an integer \(N>n-k\), use the actual Schwartz bound of
its full Fourier integral in (L6). The complete annular estimate (L7)
then gives, when \(|z|\geq1\), \[
 \begin{split}
 C_{\alpha,N}&=\sup_z(1+|z|)^N|\kappa_\alpha(z)|,\\
 \kappa_\alpha(z)&=(2\pi)^{-n}
       \int e^{iz\cdot\xi}\psi(\xi)\frac{\xi^\alpha}{q(\xi)}\,d\xi,\\
 \sum_{\ell=0}^\infty|K_\ell(z)|
 &\leq C_{\alpha,N}|z|^{-N}
                  \sum_{\ell=0}^\infty2^{\ell(n-k-N)}
   =\frac{C_{\alpha,N}}{1-2^{n-k-N}}|z|^{-N}\\
 &\leq\frac{C_{\alpha,N}}{1-2^{n-k-N}}|z|^{k-n},\\
 C_\alpha&=C_{\alpha,\mathrm{near}}
                  +\frac{C_{\alpha,N}}{1-2^{n-k-N}} .
 \end{split}                                                   \tag{FI16}
\] Thus the original kernel \(D^\alpha F_0\), already represented by its
\(L^1\) annular sum for \(k>0\), satisfies
\(|D^\alpha F_0(z)|\leq C_\alpha|z|^{k-n}\) almost everywhere in the
whole original space. The Fourier inverse factor in its definition (L6)
remains in \(C_{\alpha,N}\); this comparison does not redefine the
kernel or its quantization.

Apply (FI11) with \(s=k\) to the original input \(g\). Absolute
domination proves \[
 \|D^\alpha F_0*g\|_q
 \leq C_\alpha H_{n,k,p}\|g\|_p,\qquad
       \frac1q=\frac1p-\frac kn,\quad 0<k<n/p.                \tag{FI17}
\] This supplies the finite endpoint in (L10), without interpolation,
Lorentz duality or a Fourier convention change. The compactly supported
zero extension of an input on \(V\subset B_1\) has exactly its original
\(L^p(V)\) norm; the constant is independent of \(V\).

For the finite array \(\nabla^j\) used in Section 1, \(q\geq p\) lets us
apply Minkowski to the sum of the \(p\)-th powers in \(L^{q/p}\). If
\(k=m-j\) satisfies the same gap condition, every component therefore
gives the exact array bound \[
 \begin{split}
 \left\|\left(\sum_{|\alpha|=j}
             |D^\alpha F_0*g|^p\right)^{1/p}\right\|_q
 &\leq\left(\sum_{|\alpha|=j}
                 \|D^\alpha F_0*g\|_q^p\right)^{1/p}\\
 &\leq H_{n,k,p}
         \left(\sum_{|\alpha|=j}C_\alpha^p\right)^{1/p}\|g\|_p .
 \end{split}                                                   \tag{FI18}
\] Every original multi-index and component constant is retained. The
number of terms is \(\binom{n+j-1}{j}\), as proved after (LP17), with
value one when \(j=0\). We do not substitute a single component for this
array. For the smooth correction \(H=G-F_0\) in (L11), all original
differences \(x-y\), \(x,y\in V\subset B_1\), belong to \(B_2\). Hölder
gives the original \(\|g\|_1\leq|B_1|^{1-1/p}\|g\|_{L^p(V)}\); each
output derivative is bounded by its actual supremum on \(B_2\) times
this input integral. Therefore \[
 \begin{split}
 \|\nabla^j(H*g)\|_{L^q(V;\ell^p)}
 &\leq |V|^{1/q}|B_1|^{1-1/p}
      \left(\sum_{|\alpha|=j}
          \sup_{z\in B_2}|D^\alpha H(z)|^p\right)^{1/p}
        \|g\|_{L^p(V)}\\
 &\leq |B_1|^{1/q+1-1/p}
      \left(\sum_{|\alpha|=j}
          \sup_{z\in B_2}|D^\alpha H(z)|^p\right)^{1/p}
        \|g\|_{L^p(V)} .
 \end{split}                                                   \tag{FI19}
\] These suprema are finite because each derivative is continuous on the
compact closure of \(B_2\). The array triangle inequality combines this
with (FI18), retaining both contributions. The strict Young cases still
use (L9). Together these prove the fractional portion of (L11) and its
coefficient-table applications at the finite endpoint. Section 13
supplies the top-order multiplier proof and its full receiving
constants. Section 14 supplies the complete constant-strength inverse
proof.

\ReaderArmAliases{\ReaderOriginalAnchorStart{the-full-finite-exponent-multiplier-proof}\ReaderOriginalAnchorEnd}
\subsection{13. The full finite-exponent multiplier
proof}\label{the-full-finite-exponent-multiplier-proof}

The following proof supplies the original multiplier entry used in
Sections 2--4. It keeps its Fourier phases, inverse factor, original
symbol and exceptional point. All integration and completeness
operations are proved in the linked course chapters.

Fix \(n\geq1\), \(M=n+2\), and the original scalar symbol
\(b\in C^M(\mathbb R^n\setminus\{0\})\). Retain \[
 B=\max_{|\beta|\leq M}\sup_{\xi\ne0}
          |\xi|^{|\beta|}|\partial^\beta b(\xi)|<\infty .
                                                               \tag{MP1}
\] The value at the exceptional point zero can be chosen arbitrarily:
its frequency measure is zero by the completed coordinate measure proof
in Section 15.0 of the linked integration chapter, and every transform
and multiplier below respects the completed null class. We take it to be
zero for the integral representatives. There is no atom at zero in the
original Lebesgue frequency measure. The original inverse factor and
phases are \[
 Ff(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx,\qquad
 Gh(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}h(\xi)\,d\xi,\qquad
 Tf=G(bFf).
                                                               \tag{MP2}
\]

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\subsubsection{\texorpdfstring{13.1. Constructing the actual \(L^2\)
Fourier
maps}{13.1. Constructing the actual L\^{}2 Fourier maps}}\label{constructing-the-actual-l2-fourier-maps}

Theorems 1.1 and 2.1 of \href{prerequisite-bridges.pdf}{Fourier
transforms, finite spectra and convex separation} prove the inverse
identities and \(\|Ff\|_2=(2\pi)^{n/2}\|f\|_2\) on the original Schwartz
space. Compact smooth functions are dense in \(L^2\) in Section 15.3 of
\href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments}, so the Schwartz space is dense too.
For each \(f\in L^2\), choose \(f_j\) in that space tending to \(f\).
The exact norm identity makes \(Ff_j\) Cauchy in the complete \(L^2\)
space; its limit is independent of the approximating sequence by the
same difference identity. Define \(Ff\) by this limit. The inverse map
extends similarly with norm factor \((2\pi)^{-n/2}\), because \(G\) is
the inverse on Schwartz space. The two inverse identities extend by
density and continuity. Section 7.1 of the Fourier chapter constructs
these maps on their original complete spaces; Section 7.2 proves the
simultaneous approximations used next. Thus these are actual bijective
Fourier maps on the original \(L^2\) space, with both factors retained.

If \(f\in L^1\cap L^2\), truncation in space and in value followed by
the mollification proof there gives a single sequence of compact smooth
functions converging in both norms. The ordinary Fourier integrals
converge uniformly by their \(L^1\) difference bound, while their
\(L^2\) classes converge to the constructed Fourier limit. An
almost-everywhere convergent subsequence of the latter, supplied by the
summable-difference argument in its completeness proof, identifies that
class with the ordinary integral. The same argument for \(G\) retains
its extra \((2\pi)^{-n}\) in the uniform difference bound. Polarization
of the original Schwartz Parseval identity, followed by \(L^2\)
convergence, gives the exact pairing \[
 \langle u,v\rangle
   =(2\pi)^{-n}\int Fu(\xi)\overline{Fv(\xi)}\,d\xi,\qquad
 \|Tf\|_2\leq B\|f\|_2,\qquad
 T^*f=G(\overline b\,Ff).
                                                               \tag{MP3}
\] Here inner products are linear in their first variable. The adjoint
identity follows directly by substituting the product into the displayed
pairing; it does not assume an \(L^p\) dual representation. On Schwartz
input, \(bFf\in L^1\cap L^2\), so (MP2) is exactly the absolutely
defined original multiplier integral.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-MULT-002}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-MULT-002}\label{AN03-LCE-MULT-002}\endgroup\ReaderOriginalAnchorStart{complete-annular-kernel-and-gradient-constants}\ReaderOriginalAnchorEnd}
\subsubsection{13.2. Complete annular kernel and gradient
constants}\label{complete-annular-kernel-and-gradient-constants}

Use the original compact smooth cutoff \(\chi\), equal to one for
\(|\eta|\leq1/2\) and zero for \(|\eta|\geq1\), and put \[
 \psi(\eta)=\chi(\eta/2)-\chi(\eta),\qquad
 b_j(\xi)=\psi(2^{-j}\xi)b(\xi),\qquad j\in\mathbb Z .
                                                               \tag{MP4}
\] The support of \(\psi\) is contained in \(1/2\leq|\eta|\leq2\). No
positivity of this cutoff is required. For
\(a_j(\eta)=\psi(\eta)b(2^j\eta)\), the inverse integral is \[
 K_j(x)=(2\pi)^{-n}2^{jn}
                 \int e^{i2^jx\cdot\eta}a_j(\eta)\,d\eta .
                                                               \tag{MP5}
\] These finite integrals define smooth output functions: every output
derivative inserts its actual compact-frequency monomial, which is
integrable. We require only \(M\) frequency derivatives, rather than
incorrectly asserting that the finite-regularity frequency symbol is
Schwartz.

For each coordinate \(i\) and \(0\leq r\leq M\), define the actual
cutoff constant \[
 P_{i,r}=\sum_{t=0}^r\binom rt2^t
                    \|\partial_i^{r-t}\psi\|_1,\qquad
 P_0=\|\psi\|_1 .
                                                               \tag{MP6}
\] The chain derivative of \(b(2^j\eta)\) is
\(2^{jt}(\partial_i^tb)(2^j\eta)\). Its modulus is at most
\(B|\eta|^{-t}\leq B2^t\) on the original support. The full Leibniz sum
therefore gives \(\|\partial_i^ra_j\|_1\leq BP_{i,r}\). Choose a
coordinate \(i\) with \(|x_i|\geq|x|/\sqrt n\). Integrating \(M\) times
in that original coordinate retains the factor \((i2^jx_i)^{-M}\);
compact support eliminates the boundary terms. The direct integral and
this integrated integral prove \[
 \begin{split}
 c_0&=(2\pi)^{-n}
            \max\{P_0,n^{M/2}\max_iP_{i,M}\},\\
 |K_j(x)|&\leq c_0B\,2^{jn}
                     \min\{1,(2^j|x|)^{-M}\}.
 \end{split}                                                   \tag{MP7}
\] All constants are finite and depend only on the actual cutoff and
dimension. On \(|x|\leq2^{-j}\), the original containing cube has volume
\((2^{1-j})^n\). On \(2^{r-j}<|x|\leq2^{r-j+1}\), \(r\geq0\), use its
containing cube of volume \((2^{r-j+2})^n\) and the second bound in
(MP7). The full integral comparison gives \[
 \|K_j\|_1\leq c_0B\,2^{jn}(2^{1-j})^n
   +c_0B\sum_{r=0}^\infty
       2^{jn}2^{-rM}(2^{r-j+2})^n
 =c_0B\left(2^n+\frac{2^{2n}}{1-2^{n-M}}\right)<\infty .
\] No rescaled unit cube or suppressed Jacobian is needed for this
estimate.

For coordinate \(\ell\), differentiation of (MP5) gives the full factor
\(i2^{j(n+1)}\) and the integrand \(\eta_\ell a_j(\eta)\). Its \(M\)-th
derivative in coordinate \(i\) is
\(\eta_\ell\partial_i^Ma_j+M\delta_{i\ell}\partial_i^{M-1}a_j\). Thus,
with every product-rule term retained, \[
 \begin{split}
 c_\ell&=(2\pi)^{-n}
   \max\{2P_0,\ n^{M/2}\max_i(2P_{i,M}
                           +M\delta_{i\ell}P_{i,M-1})\},\\
 |\partial_\ell K_j(x)|&\leq c_\ell B\,2^{j(n+1)}
                     \min\{1,(2^j|x|)^{-M}\}.
 \end{split}                                                   \tag{MP8}
\] This is the original gradient scale \(2^{j(n+1)}\); no \(2\pi\) is
inserted into the gradient under the phase \(e^{ix\cdot\xi}\). The
inverse factor \((2\pi)^{-n}\) remains in each actual integral and
cutoff constant.

For \(x\ne0\), choose the integer \(J\) with \(1/2<\rho=2^J|x|\leq1\).
The low and high sums in (MP7) give, respectively, \[
 \begin{split}
 |K(x)|&\leq c_0B|x|^{-n}
 \left(\frac{\rho^n}{1-2^{-n}}
            +\frac{(2\rho)^{-2}}{1-2^{-2}}\right)
       \leq A_0B|x|^{-n},\\
 |\nabla K(x)|&\leq
 B|x|^{-n-1}\left(\sum_{\ell=1}^nc_\ell^2\right)^{1/2}
 \left(\frac{\rho^{n+1}}{1-2^{-n-1}}
            +\frac{(2\rho)^{-1}}{1-2^{-1}}\right)
       \leq A_1B|x|^{-n-1},\\
 A_0&=c_0\left(\frac1{1-2^{-n}}+\frac1{1-2^{-2}}\right),\\
 A_1&=\left(\sum_{\ell=1}^nc_\ell^2\right)^{1/2}
       \left(\frac1{1-2^{-n-1}}+\frac1{1-2^{-1}}\right).
 \end{split}                                                   \tag{MP9}
\] Here \(K=\sum_{j\in\mathbb Z}K_j\). Both series and the derivative
series converge uniformly on every compact set avoiding zero, by the
same positive geometric bounds; the finite fundamental theorem of
calculus then proves that \(K\) is \(C^1\) there with the displayed
derivative sum. The complete \(\rho\) contributions are retained before
their uniform comparison.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-MULT-003}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-MULT-003}\label{AN03-LCE-MULT-003}\endgroup\ReaderOriginalAnchorStart{identifying-the-original-operator-outside-the-support}\ReaderOriginalAnchorEnd}
\subsubsection{13.3. Identifying the original operator outside the
support}\label{identifying-the-original-operator-outside-the-support}

The full finite telescoping identity is \[
 \sum_{j=-N}^Nb_j(\xi)
 =b(\xi)\bigl(\chi(2^{-N-1}\xi)-\chi(2^N\xi)\bigr)
 \longrightarrow b(\xi)\quad(\xi\ne0).
                                                               \tag{MP10}
\] Its modulus is at most \(2\|\chi\|_\infty B\). Dominated convergence
in the exact \(L^2\) Fourier pairing proves convergence of these finite
multiplier operators to \(T\) in \(L^2\).

The full original distribution kernel is \(\mathcal K=Gb\), not a
substitution of the off-diagonal function \(K\) at its singular point.
For every Schwartz test \(\phi\), its exact complex-linear pairing is \[
 \langle\mathcal K,\phi\rangle
   =(2\pi)^{-n}\int b(\xi)F\phi(-\xi)\,d\xi,\qquad
 \sum_{j=-N}^NK_j\longrightarrow\mathcal K
                         \quad\hbox{in }\mathcal S' .
                                                               \tag{MP10a}
\] The integral is absolutely convergent because \(b\) is bounded and
\(F\phi\) is Schwartz. The telescoping bound and dominated convergence
prove the limit on every test. It also converges in the strong dual
topology, uniformly on every bounded subset \(E\) of the original
Schwartz space. Indeed, put
\(H_L(E)=\sup_{\phi\in E,\xi}(1+|\xi|)^L|F\phi(\xi)|<\infty\), for an
integer \(L>n\). The finiteness follows from the continuous original
Fourier map and the finite expansion of the weight into its monomial
seminorms. The difference of the actual finite symbol and \(b\) is
\(b[\chi(2^{-N-1}\xi)-1-\chi(2^N\xi)]\). The last cutoff is supported in
\(|\xi|\leq2^{-N}\), and the preceding cutoff difference vanishes for
\(|\xi|\leq2^N\). Consequently, \[
 \begin{split}
 \sup_{\phi\in E}\left|
    \left\langle\sum_{j=-N}^NK_j-\mathcal K,\phi\right\rangle\right|
 &\leq(2\pi)^{-n}B H_L(E)
 \left[\|\chi\|_\infty(2^{1-N})^n\right.\\
 &\left.\qquad+(1+\|\chi\|_\infty)
                \int_{|\xi|\geq2^N}(1+|\xi|)^{-L}\,d\xi\right]
 \longrightarrow0 .
 \end{split}                                                   \tag{MP10b}
\] The low term uses its original containing cube; the high term tends
to zero by absolute integrability of the displayed weight, proved by the
original dyadic cube comparison. This verifies the precise distribution
topology as well as the limit on individual tests. For a compact smooth
test supported away from zero, the uniform kernel convergence already
proved identifies this pairing with \(\int K\phi\). Hence the
restriction of the actual full kernel to the complement of zero is
exactly the regular distribution defined by \(K\). Its possible
contribution supported at zero is retained by (MP10a).

The exact convolution map is also retained. For Schwartz \(f,\phi\), put
\(\Theta(y)=\int f(x-y)\phi(x)\,dx\), whose full Schwartz bounds are
proved in Section 13.10, GC19--GC24, of
\href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates}. Absolute product integration gives
\(F\Theta(-\xi)=Ff(\xi)F\phi(-\xi)\). Thus the full distributional
convolution has the actual pairing \[
 \langle\mathcal K*f,\phi\rangle
   =(2\pi)^{-n}\int b(\xi)Ff(\xi)F\phi(-\xi)\,d\xi
   =\langle G(bFf),\phi\rangle .
\] This proves \(\mathcal K*f=Tf\) with its entire distribution kernel.
For example, when \(b(\xi)=c_{\mathrm{const}}\) is constant, Fourier
inversion in this same pairing gives
\(\mathcal K=c_{\mathrm{const}}\delta_0\) and \(T=c_{\mathrm{const}}I\).
The off-diagonal restriction is then zero, while the original operator
remains present in the \(L^2\) and level-set calculation. No diagonal
contribution is dropped.

For \(f\in L^2\) with compact support, Hölder gives \(f\in L^1\). Each
finite term has the identity \[
 G(b_jFf)(x)=\int K_j(x-y)f(y)\,dy .
                                                               \tag{MP11}
\] To verify it, substitute (MP5) before its frequency rescaling, and
apply absolute Fubini; the bound is \((2\pi)^{-n}\|b_j\|_1\|f\|_1\). The
result is the ordinary inverse integral, equal to its \(L^2\) class by
Section13.1 and \href{prerequisite-bridges.pdf}{the
Fourier chapter, Section7.2}. On a compact set separated from the input
support, the series of these convolution integrals converges uniformly
and absolutely, because the positive kernel sum is bounded there by
\(A_0B\) times the fixed negative distance power and \(\|f\|_1\). A
subsequence of the \(L^2\) operator convergence identifies the resulting
function with \(Tf\) almost everywhere there. Exhaust the complement of
the compact support by countably many separated compact sets. This
proves \(Tf(x)=\int K(x-y)f(y)\,dy\) almost everywhere outside the
support, for the original operator rather than a new kernel model.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-MULT-004}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-MULT-004}\label{AN03-LCE-MULT-004}\endgroup\ReaderOriginalAnchorStart{dyadic-differentiation-and-decomposition-proved-from-the-entry-measure}\ReaderOriginalAnchorEnd}
\subsubsection{13.4. Dyadic differentiation and decomposition proved
from the entry
measure}\label{dyadic-differentiation-and-decomposition-proved-from-the-entry-measure}

Keep the actual half-open dyadic cubes \(2^{-j}(k+[0,1)^n)\),
\(j\in\mathbb Z\), \(k\in\mathbb Z^n\). Two such cubes are disjoint or
one contains the other; this follows coordinatewise from the integer
dyadic grid. For \(f\in L^1\), define its dyadic maximal function as the
supremum of the absolute averages on all cubes containing the point. For
each positive height \(a\), cubes whose average of \(|f|\) exceeds \(a\)
have maximal members: their ancestors' volumes tend to infinity while
their input integrals are at most \(\|f\|_1\), so sufficiently coarse
ancestors have average at most \(a\). The maximal cubes are disjoint,
form a countable family, and cover every high cube. Their exact original
volumes give \[
 |\{\mathcal M_df>a\}|\leq a^{-1}\|f\|_1 .
                                                               \tag{MP12}
\]

Let \(A_jf(x)\) be the average over its unique dyadic cube of side
\(2^{-j}\). For compact continuous \(v\),
\(|A_jv(x)-v(x)|\leq\omega_v(\sqrt n\,2^{-j})\to0\). The modulus tends
to zero by uniform continuity on a large compact containing its support;
outside that compact the same continuity follows from zero extension.
For \(f\in L^1\), choose compact smooth \(v\) with arbitrarily small
\(\|f-v\|_1\), by its density proof. The exact difference estimate is \[
 \limsup_{j\to\infty}|A_jf(x)-f(x)|
       \leq\mathcal M_d(f-v)(x)+|f(x)-v(x)|.
                                                               \tag{MP13}
\] For \(\delta>0\), (MP12) and the integral bound for \(|f-v|\) show
that the set where the left side exceeds \(2\delta\) has measure at most
\(2\|f-v\|_1/\delta\). Taking the infimum over these approximations
gives zero. A countable union over positive rational \(\delta\) proves
\(A_jf\to f\) almost everywhere. Apply the same proof to \(|f|\). This
establishes the needed differentiation theorem without importing one.

For \(f\in L^1\cap L^2\) and a height \(a>0\), select its maximal high
cubes \(Q\). Each satisfies \[
 a<|Q|^{-1}\int_Q|f|\leq2^na,
 \qquad \sum_Q|Q|\leq a^{-1}\|f\|_1 .
                                                               \tag{MP14}
\] The upper bound follows from its parent, whose original volume is
\(2^n|Q|\) and whose average is at most \(a\). Write
\(m_Q=|Q|^{-1}\int_Qf\), set \(g=f\) outside the union and \(g=m_Q\) on
each cube, and put \(d_Q=(f-m_Q)1_Q\). Dyadic differentiation gives
\(|f|\leq a\) almost everywhere outside the union. The original complex
averages and disjoint cubes then give \[
 \begin{split}
 f&=g+\sum_Qd_Q,\qquad \int d_Q=0,\qquad
 \|g\|_\infty\leq2^na,\qquad \|g\|_1\leq\|f\|_1,\\
 \|g\|_2^2&\leq2^na\|f\|_1,\qquad
 \sum_Q\|d_Q\|_1\leq2\|f\|_1 .
 \end{split}                                                   \tag{MP15}
\] The sums converge in \(L^1\); they also converge in \(L^2\), because
\(f-g\in L^2\) and its pieces have disjoint supports. In particular
every \(d_Q\) is an actual compactly supported \(L^1\cap L^2\) function,
with its original mean exactly zero.

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\subsubsection{13.5. Complete weak estimate with the actual symbol
bound}\label{complete-weak-estimate-with-the-actual-symbol-bound}

If \(B=0\), (MP1) makes \(b=0\) away from its null exceptional point and
\(T=0\). Otherwise fix the output level \(\lambda>0\) and use the
decomposition at the actual input height \(a=\lambda/B\). This retains
the original \(f\), \(b\), \(B\) and output level; none is replaced by a
unit-norm object.

For a selected cube \(Q\) of side \(\ell_Q\) and center \(c_Q\), set
\(R_Q=2\sqrt n\,\ell_Q\) and \(Q^*=c_Q+(-R_Q,R_Q)^n\). Its original
volume is \((4\sqrt n)^n|Q|\). If \(x\notin Q^*\) and \(y\in Q\), then
\(|x-c_Q|\geq R_Q\) and \(|y-c_Q|\leq\sqrt n\,\ell_Q/2\leq|x-c_Q|/4\).
Every point of the segment between \(x-y\) and \(x-c_Q\) consequently
has length at least \(3|x-c_Q|/4\). The zero mean and (MP9) give \[
 |Td_Q(x)|\leq A_1B(4/3)^{n+1}|x-c_Q|^{-n-1}
                        \int_Q|y-c_Q||d_Q(y)|\,dy .
                                                               \tag{MP16}
\] The original kernel representation in Section13.3, established from
(MP11), applies to each \(d_Q\). On dyadic Euclidean shells beyond
\(R_Q\), use the original containing coordinate cubes to get \[
 \int_{|x-c_Q|\geq R_Q}|x-c_Q|^{-n-1}\,dx
 \leq\sum_{j=0}^\infty
            (2^jR_Q)^{-n-1}(2^{j+2}R_Q)^n
 =\frac{2^{2n+1}}{R_Q}.
                                                               \tag{MP17}
\] Sphere boundaries are null by the coordinate-section proof in FI8.
Therefore \[
 \int_{(Q^*)^c}|Td_Q|
 \leq A_1B(4/3)^{n+1}
        \frac{\sqrt n\,\ell_Q}{2}\frac{2^{2n+1}}{R_Q}\|d_Q\|_1
 =D_nB\|d_Q\|_1,\qquad
 D_n=2^{2n-1}(4/3)^{n+1}A_1 .
                                                               \tag{MP18}
\] Both actual radius factors and all derivative constants remain
explicit.

Let \(\Omega^*=\bigcup_QQ^*\). The series of \(Td_Q\) converges
absolutely in \(L^1((\Omega^*)^c)\) by (MP15)--(MP18). Its \(L^2\)
partial sums also converge to \(T(f-g)\). Almost-everywhere subsequences
identify the two limits, so
\(\int_{(\Omega^*)^c}|T(f-g)|\leq2D_nB\|f\|_1\). The \(L^2\) estimate of
\(Tg\), the enlarged volume sum and the last integral estimate now give
\[
 \begin{split}
 |\{|Tf|>\lambda\}|
 &\leq|\Omega^*|+|\{|Tg|>\lambda/2\}|
       +|\{x\notin\Omega^*:|T(f-g)|>\lambda/2\}|\\
 &\leq\left((4\sqrt n)^n+4\,2^n+4D_n\right)
                   \frac{B}{\lambda}\|f\|_1
   =W_n\frac{B}{\lambda}\|f\|_1 .
 \end{split}                                                   \tag{MP19}
\] This weak estimate is proved for every \(f\in L^1\cap L^2\),
precisely the domain needed in the next argument.

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\subsubsection{13.6. Strong finite exponents without an interpolation
import}\label{strong-finite-exponents-without-an-interpolation-import}

Let \(1<p<2\), initially \(f\in L^p\cap L^2\). At each \(\lambda>0\),
split it at its actual value threshold \(\lambda/B\) into
\(f_{\rm high}\) and \(f_{\rm low}\). The high part is in
\(L^1\cap L^2\), because
\(|f|1_{\{|f|>\lambda/B\}}\leq(\lambda/B)^{1-p}|f|^p\). The low part is
in \(L^2\), including for an arbitrary \(L^p\) input, by
\(|f_{\rm low}|^2\leq(\lambda/B)^{2-p}|f|^p\). Linearity in \(L^2\),
(MP19) and (MP3) give \[
 |\{|Tf|>\lambda\}|
 \leq\frac{2W_nB}{\lambda}
             \int_{\{|f|>\lambda/B\}}|f|
       +\frac{4B^2}{\lambda^2}
             \int_{\{|f|\leq\lambda/B\}}|f|^2 .
                                                               \tag{MP20}
\] Use the level-set identity proved in FI5. Nonnegative Fubini retains
every threshold factor: \[
 \begin{split}
 \|Tf\|_p^p
 &\leq2W_nBp\int|f(x)|
                    \int_0^{B|f(x)|}\lambda^{p-2}\,d\lambda\,dx\\
 &\quad+4B^2p\int_{\{f\ne0\}}|f(x)|^2
                    \int_{B|f(x)|}^{\infty}\lambda^{p-3}\,d\lambda\,dx\\
 &=\left(\frac{2W_np}{p-1}+\frac{4p}{2-p}\right)
                         B^p\|f\|_p^p .
 \end{split}                                                   \tag{MP21}
\] At \(f=0\), the second contribution is zero, so no product of zero
with an infinite scalar integral is used. The conditions \(p>1\) and
\(p<2\) are exactly the two scalar endpoint integrability conditions.
Define \(L_{n,p}=(2W_np/(p-1)+4p/(2-p))^{1/p}\). On Schwartz inputs this
proves the original norm bound. Density and complete \(L^p\) extend
\(T\) uniquely to all \(L^p\), with norm at most \(L_{n,p}B\). Its
\(L^p\) and \(L^2\) realizations agree on the intersection: use
simultaneous smooth approximation in both norms and identify their
limits in measure, or by an almost-everywhere subsequence. At \(p=2\),
use the exact bound \(B\) in (MP3).

For \(p>2\), let \(r=p/(p-1)\in(1,2)\). The original conjugate symbol
satisfies the same complete derivative bounds (MP1), since
real-coordinate derivatives commute with complex conjugation. Hence the
just-proved theorem applies to \(T^*\) with norm at most \(L_{n,r}B\) on
\(L^r\), and it agrees with its \(L^2\) realization on bounded compactly
supported inputs by simultaneous mollification, as proved in Section 7.2
of the Fourier chapter.

For Schwartz \(f\), put \(u=Tf\in L^2\). On
\(E_{R,N}=\{|x|_\infty\leq R,\ |u(x)|\leq N\}\), define the actual test
\(h_{R,N}=1_{E_{R,N}}|u|^{p-2}u\), setting it to zero at \(u=0\). It is
bounded and supported in a finite coordinate cube, so belongs to
\(L^2\cap L^r\). The exact \(L^2\) adjoint identity and Hölder give \[
 \int_{E_{R,N}}|u|^p
   =\langle u,h_{R,N}\rangle
   =\langle f,T^*h_{R,N}\rangle
   \leq L_{n,r}B\|f\|_p
                \left(\int_{E_{R,N}}|u|^p\right)^{1/r}.
                                                               \tag{MP22}
\] Divide only if the finite integral is positive; the zero case is
direct. Increasing \(R,N\) to infinity and applying the proved
monotone-convergence theorem proves \(\|Tf\|_p\leq L_{n,r}B\|f\|_p\).
Density and completeness again extend this original operator to the
entire \(L^p\) space. This uses its actual nonlinear test function,
rather than assuming an \(L^p\) dual representation.

Thus the original multiplier theorem is fully proved with \[
 \|b(D)f\|_p\leq C^{\rm mult}_{n,p,\chi}B\|f\|_p,\qquad
 C^{\rm mult}_{n,p,\chi}=
 \begin{cases}
 L_{n,p},&1<p<2,\\
 1,&p=2,\\
 L_{n,p/(p-1)},&2<p<\infty .
 \end{cases}                                                   \tag{MP23}
\] The displayed cutoff dependence records the actual chosen cutoff.
Fixing one original cutoff for each dimension makes it a
dimension/exponent constant in the original statement; all working
formulas retain that cutoff and its derivative norms.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-MULT-007}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-MULT-007}\label{AN03-LCE-MULT-007}\endgroup\ReaderOriginalAnchorStart{the-proved-derivative-bound-underclaim}\ReaderOriginalAnchorEnd}
\subsubsection{13.7. The proved derivative-bound
underclaim}\label{the-proved-derivative-bound-underclaim}

The proof used, among the given frequency derivative bounds, only \[
 B_{\rm coord}=\max_{\substack{1\leq i\leq n\\0\leq r\leq n+2}}
         \sup_{\xi\ne0}|\xi|^r|\partial_i^rb(\xi)| .
                                                               \tag{MP24}
\] For a symbol still in \(C^{n+2}(\mathbb R^n\setminus\{0\})\), the
finiteness of this smaller displayed maximum suffices, even if no bound
is imposed on its mixed derivatives. Each Leibniz and
integration-by-parts estimate above uses one original coordinate
repeatedly; the output-gradient identity adds only its explicitly
displayed zeroth and first coordinate-monomial terms. Every later step
uses just the resulting original kernel/gradient bounds and the zeroth
symbol bound. Replacing the bound parameter in that exact proof by
\(B_{\rm coord}\) therefore proves the stronger statement with the same
cutoff constants. The original full \(B\) and every one of its
hypotheses remain visible in the comparison, and \(B_{\rm coord}\leq B\)
recovers (MP23). This is a proved consequence of the calculation,
without a novelty claim; it is presented here as an explicit addition
while the original statement remains identifiable.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-LCE-MULT-008}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-LCE-MULT-008}\label{AN03-LCE-MULT-008}\endgroup\ReaderOriginalAnchorStart{full-constants-in-the-original-elliptic-estimates}\ReaderOriginalAnchorEnd}
\subsubsection{13.8. Full constants in the original elliptic
estimates}\label{full-constants-in-the-original-elliptic-estimates}

For the original frozen polynomial \(q\) in (L1), retain
\(b_0=(1-\chi)/q\) and every constant \(C_\nu\) in (L3). At top order
\(|\alpha|=m\), its symbol is \(b_\alpha(\xi)=\xi^\alpha b_0(\xi)\). For
every \(|\beta|\leq n+2\), the exact finite Leibniz formula and its
weighted estimate are \[
 \begin{split}
 \partial^\beta b_\alpha(\xi)
 &=\sum_{\substack{\gamma\leq\beta\\\gamma\leq\alpha}}
       \binom{\beta}{\gamma}\frac{\alpha!}{(\alpha-\gamma)!}
       \xi^{\alpha-\gamma}\partial^{\beta-\gamma}b_0(\xi),\\
 |\xi|^{|\beta|}|\partial^\beta b_\alpha(\xi)|
 &\leq\sum_{\substack{\gamma\leq\beta\\\gamma\leq\alpha}}
       \binom{\beta}{\gamma}\frac{\alpha!}{(\alpha-\gamma)!}
       C_{\beta-\gamma}
       \frac{|\xi|^{m+|\beta|-|\gamma|}}
            {(1+|\xi|)^{m+|\beta|-|\gamma|}}\\
 &\leq B_\alpha,\qquad
 B_\alpha=\max_{|\beta|\leq n+2}
       \sum_{\substack{\gamma\leq\beta\\\gamma\leq\alpha}}
       \binom{\beta}{\gamma}\frac{\alpha!}{(\alpha-\gamma)!}
       C_{\beta-\gamma}.
 \end{split}                                                   \tag{MP25}
\] The exponent in every ratio is nonnegative, since
\(|\gamma|\leq|\beta|\); no polynomial factor or reciprocal derivative
is discarded. The symbol is smooth at zero because the numerator of
\(b_0\) vanishes on its original neighborhood. Thus (MP23) gives \[
 \begin{split}
 \|D^\alpha F_0*g\|_p
 &\leq C^{\rm mult}_{n,p,\chi}B_\alpha\|g\|_p,\\
 \left\|\left(\sum_{|\alpha|=m}
               |D^\alpha F_0*g|^p\right)^{1/p}\right\|_p
 &\leq C^{\rm mult}_{n,p,\chi}
            \left(\sum_{|\alpha|=m}B_\alpha^p\right)^{1/p}\|g\|_p .
 \end{split}                                                   \tag{MP26}
\] There are exactly \(\binom{n+m-1}{m}\) original multiindices in this
array. Its equality with the sum of the \(p\)-th powers of the component
norms follows by integrating the finite sum; no component is suppressed.
On a compactly supported input the distributional convolution agrees
with the multiplier. To prove this for the actual \(L^p\) input, mollify
its zero extension with uniformly bounded compact support. The sequence
converges in \(L^p\), its multipliers converge in \(L^p\), and
convolution with the original tempered distribution converges
distributionally by the compact-factor test-function bounds in the
convolution chapter. Differentiation and uniqueness of the
distributional limit identify the two operators.

For lower orders \(|\alpha|<m\), put \(k=m-|\alpha|>0\). The exact
annular scaling in (L6) gives
\(\|K_\ell\|_1=2^{-\ell k}\|\kappa_\alpha\|_1\). Define the actual
operator constants \[
 J_\alpha=
 \begin{cases}
 C^{\rm mult}_{n,p,\chi}B_\alpha,&|\alpha|=m,\\
 \displaystyle\frac{\|\kappa_\alpha\|_1}{1-2^{-(m-|\alpha|)}},
                                      &|\alpha|<m,
 \end{cases}
 \quad
 \Omega_\alpha=\|D^\alpha\omega\|_1,\quad
 C_q^{\rm est}=
 \max\left\{\sum_{|\alpha|\leq m}J_\alpha,\
            \sum_{|\alpha|\leq m}\Omega_\alpha\right\}.
                                                               \tag{MP27}
\] The inverse Fourier factor is retained in the original
\(\kappa_\alpha=\mathcal F^{-1}(\psi\xi^\alpha/q)\) and
\(\omega=-\mathcal F^{-1}\chi\); their displayed norms are not norms of
substituted kernels. Integrating each original (L14) by the proved Young
inequality and (MP26) gives its estimate with \(C_q^{\rm est}\) at
\(A=1\). For every original \(A>0\), the actual comparison
\(w(y)=v(y/A)\) gives
\(\|D^\alpha w\|_p=A^{n/p-|\alpha|}\|D^\alpha v\|_p\),
\(\|q(D)w\|_p=A^{n/p-m}\|q(D)v\|_p\), and \(\|w\|_p=A^{n/p}\|v\|_p\).
Multiply the obtained inequality by \(A^{m-n/p}\). This proves exactly
(L13), with every displayed \(A^{m-|\alpha|}\) and \(A^m\) term, using
\(C_q^{\rm est}\), for its whole stated \(W^{m,p}\) domain by the
already proved approximation.

The constants can be uniform on an original compact family of elliptic
\(q\)'s. The function \((q,\theta)\mapsto|q(\theta)|\), on that family
times the unit sphere, has a positive minimum by compactness and
ellipticity; the finitely many polynomial coefficients have a finite
maximum. All derivatives of \(q\) on the fixed annulus are therefore
uniformly bounded. The exact reciprocal recurrence \[
 \partial^\nu(1/q)
 =-q^{-1}\sum_{\substack{0<\gamma\leq\nu}}
       \binom{\nu}{\gamma}
       (\partial^\gamma q)\,
       \partial^{\nu-\gamma}(1/q),\qquad |\nu|>0
                                                               \tag{MP28}
\] retains every product and its negative sign. Induction gives uniform
bounds for each derivative needed in (L3), with homogeneity transporting
the original powers from the sphere and the fixed cutoff region treated
by the same recurrence. It also uniformly bounds every derivative of the
compact annular symbol defining \(\kappa_\alpha\). For an integer
\(N>n\), coordinate integration by parts as in (MP7) bounds that kernel
by its original direct integral near zero and a constant times
\(|x|^{-N}\) outside; the full containing-cube sums integrate this
bound. Thus each \(\|\kappa_\alpha\|_1\), \(B_\alpha\) and
\(C_q^{\rm est}\) is uniform. The fixed \(\omega\) is unchanged.

Finally retain the smooth correction \(H=G-F_0\) in the local solver.
For \(V\subset B_1\), the complete top-order array satisfies \[
 \|\nabla^m S_Vg\|_{L^p(V)}
 \leq\left[
 C^{\rm mult}_{n,p,\chi}
       \left(\sum_{|\alpha|=m}B_\alpha^p\right)^{1/p}
 +\left(\sum_{|\alpha|=m}
                 \sup_{B_2}|D^\alpha H|^p\right)^{1/p}
       |V|^{1/p}|B_1|^{1-1/p}
 \right]\|g\|_{L^p(V)} .
                                                               \tag{MP29}
\] Every difference \(x-y\) lies in the original \(B_2\); Hölder bounds
its input integral by the original \(|B_1|^{1-1/p}\), and integrating
its constant output over \(V\) gives \(|V|^{1/p}\). This is the exact
\(k=0,q=p\) case of (L11) and the missing top-order part of (L17). The
original perturbative inverse \(B_V\) there has norm at most \(2\).
Substituting \(g=B_Vf\) in (MP29) consequently bounds the same array of
\(Ef=S_VB_Vf\) by twice its displayed bracket times \(\|f\|_{L^p(V)}\).
Lower orders retain their proved fractional or strict Young estimates
and the original smooth corrections in Section 12.5. Both inverse
identities, each coefficient product and all original distance weights
remain those in Sections3--9.

\ReaderArmAliases{\ReaderOriginalAnchorStart{a-local-inverse-for-every-constant-strength-family}\ReaderOriginalAnchorEnd}
\subsection{14. A local inverse for every constant-strength
family}\label{a-local-inverse-for-every-constant-strength-family}

The broader assertion from Section 1 is proved here without ellipticity.
The polynomial denoted by \(p\) in this section is the full frozen
symbol \(P_0\), rather than the finite exponent used in the earlier
elliptic estimates. The original polynomial family, coordinates and
coefficient multiplication are retained. This construction proves a
local assertion; it supplies neither uniqueness among all local
solutions nor a global \(L^2\) inverse.

\ReaderArmAliases{\ReaderOriginalAnchorStart{conventions-precise-assertion-and-maps}\ReaderOriginalAnchorEnd}
\subsubsection{14.1. Conventions, precise assertion, and
maps}\label{conventions-precise-assertion-and-maps}

Let \(n\geq1\), let \(D_j=-i\partial_{x_j}\), and let
\(p\in\mathbb C[\xi_1,\ldots,\xi_n]\) be nonzero. Write \(m=\deg p\).
All multiindices below belong to \(\mathbb N^n\), with
\(\alpha!=\prod_j\alpha_j!\). Define

\[
 \widetilde p(\xi)^2
 =\sum_{\alpha\in\mathbb N^n}|\partial_\xi^\alpha p(\xi)|^2,
 \qquad \xi\in\mathbb R^n.                                      \tag{CS1}
\]

This sum includes the zeroth derivative and every nonzero higher
derivative. Terms beyond the degree are zero. For every polynomial
\(q\), use the identical full definition of \(\widetilde q\), and define

\[
 V_p=\{q:\ \widetilde q(\xi)\leq C_q\widetilde p(\xi)
       \text{ for every }\xi\in\mathbb R^n\text{ and some finite }C_q\},
 \qquad
 \|q\|_p=\sup_{\xi\in\mathbb R^n}
             \frac{\widetilde q(\xi)}{\widetilde p(\xi)}.           \tag{CS2}
\]

The assertion to be proved is the following. Suppose \(x_0\) is in an
open set \(\Omega\), \(p_\nu\in V_p\) for \(1\leq\nu\leq r\), and
\(c_\nu\in C(\Omega)\) satisfy \(c_\nu(x_0)=0\). Put

\[
 P(x,D)=p(D)+\sum_{\nu=1}^r c_\nu(x)p_\nu(D).                    \tag{CS3}
\]

There is a bounded open neighborhood \(U\) of \(x_0\), with
\(\overline U\subset\Omega\), and a bounded linear map
\(E:L^2(U)\to L^2(U)\), such that:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  For every \(q\in V_p\), the distribution \(q(D)Ef\) is an \(L^2(U)\)
  function and \(q(D)E:L^2(U)\to L^2(U)\) is bounded.
\item
  The equation \(PEf=f\) holds in \(L^2(U)\), hence in
  \(\mathcal D'(U)\), where \(PEf\) means the sum of the already defined
  \(L^2\) functions \(p(D)Ef+\sum c_\nu p_\nu(D)Ef\).
\item
  For every \(u\in C_c^\infty(U)\), \(EPu=u\) in \(L^2(U)\).
\end{enumerate}

We also prove that a continuous polynomial family of constant strength
has precisely the finite representation (CS3), without any assumption on
differentiability of its coefficients. The proof will give explicit
finite constants and all spaces and maps used.

Fourier conventions are

\[
 \widehat f(\xi)=\int_{\mathbb R^n}e^{-iy\cdot\xi}f(y)\,dy,
 \qquad
 f(x)=(2\pi)^{-n}\int_{\mathbb R^n}e^{ix\cdot\xi}\widehat f(\xi)\,d\xi.
                                                                    \tag{CS4}
\]

For \(f\in C_c^\infty\), its Fourier transform is defined at every
\(\eta\in\mathbb C^n\) by the same absolutely convergent integral. It is
entire in each coordinate, with derivatives obtained under the integral.
Distribution pairings here are complex bilinear. Thus the transpose of
\(p(D)\) is \(p(-D)\), with no coefficient conjugation.

The prerequisite proofs are part of this course. Sections 1 and 7 of
\href{prerequisite-bridges.pdf}{Fourier
transforms, finite spectra and convex separation} prove Fourier
inversion, the complete-space Plancherel maps and bounded measurable
multipliers, with exactly (CS4). Sections 9.1--9.4 of
\href{stable-prerequisite-bridges.pdf}{Polynomial
and contour interfaces for stable boundary models} prove complex
polynomial factorization with every leading factor and multiplicity; its
Section 10 proves the finite matrix and basis facts. Sections 15.0--15.4
of
\href{banach-foundation-bridges.pdf}{Banach
estimates, quotient spaces and compact parameter arguments} prove the
completed measure, product integration, convergence, completeness and
smooth approximation operations. Section 13 of
\href{metric-foundation-bridges.pdf}{Metric
and topological foundations} proves the scalar calculus, exponential and
finite-coordinate Taylor rules. Sections 13.6--13.10 of
\href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates} prove the distribution operations. Section 14.5
below calculates every convolution transfer used by this construction.
No PDE solvability theorem enters as a premise.

\ReaderArmAliases{\ReaderOriginalAnchorStart{positivity-finite-dimension-and-the-original-coefficient-representation}\ReaderOriginalAnchorEnd}
\subsubsection{14.2. Positivity, finite dimension, and the original
coefficient
representation}\label{positivity-finite-dimension-and-the-original-coefficient-representation}

Choose a monomial coefficient \(a_{\alpha_0}\neq0\) of \(p\) with
\(|\alpha_0|=m\). Then

\[
 d_p:=\alpha_0!|a_{\alpha_0}|>0,
 \qquad |\partial^{\alpha_0}p(\xi)|=d_p,
 \qquad \widetilde p(\xi)\geq d_p.                              \tag{CS5}
\]

For \(m=0\), use \(d_p=|p|\). No zero or exceptional real point is
removed from (CS2).

Every \(q\in V_p\) has degree at most \(m\). Indeed, polynomial growth
gives \(\widetilde p(\xi)\leq A_p(1+|\xi|)^m\). If \(q\) had degree
\(d>m\), its leading homogeneous part would be nonzero at some real
vector \(v\). To see that such a real vector exists, a polynomial
vanishing on all real vectors has every coefficient zero, by repeated
use of the one-variable fact that a nonzero polynomial has only finitely
many roots. Thus \(q(tv)=t^d q_d(v)+O(t^{d-1})\), contradicting
\(|q(tv)|\leq\|q\|_p A_p(1+t|v|)^m\) as \(t\to+\infty\).

The map \(q\mapsto (\partial^\alpha q(\xi))_\alpha\) is linear, and the
Euclidean triangle inequality at each \(\xi\) proves the triangle
inequality in (CS2). Positive definiteness follows from the zeroth
derivative. Hence \(V_p\) is a complex vector subspace of the
finite-dimensional space

\[
 \mathcal P_m=\left\{\sum_{|\alpha|\leq m}a_\alpha\xi^\alpha\right\},
 \qquad \dim\mathcal P_m=\binom{n+m}{m},                         \tag{CS6}
\]

and \(\|\cdot\|_p\) is a finite norm on that subspace. In particular
\(p,1\in V_p\), with \(\|p\|_p=1\) and \(\|1\|_p\leq d_p^{-1}\).

Now suppose \(x\mapsto p_x\) is an original differential-operator symbol
family with continuous monomial coefficients and \(p_{x_0}=p\), and that
\(p_x\in V_p\) for every \(x\) near \(x_0\). Pairwise constant strength
implies this assumption by comparison with the frozen point. The
preceding degree argument puts every \(p_x\) in \(\mathcal P_m\),
without discarding any original nonzero coefficient.

Choose a basis \(p_1,\ldots,p_r\) of \(V_p\). List the monomials of
\(\mathcal P_m\) in a fixed order, and let \(B\) be the
\(\binom{n+m}{m}\times r\) matrix whose \(\nu\)-th column is the full
monomial coefficient vector of \(p_\nu\). It has independent columns, so
\(B^*B\) is invertible: \(z^*B^*Bz=\|Bz\|_2^2>0\) for \(z\neq0\). If
\(a(x)\) denotes the full coefficient vector of \(p_x\), define

\[
 c(x)=(B^*B)^{-1}B^*\big(a(x)-a(x_0)\big).                     \tag{CS7}
\]

Because \(p_x-p\in V_p\), its coefficient vector lies in
\(\operatorname{ran}B\), so (CS7) gives the exact equality

\[
 p_x(\xi)=p(\xi)+\sum_{\nu=1}^r c_\nu(x)p_\nu(\xi).            \tag{CS8}
\]

Every \(c_\nu\) is continuous, and \(c_\nu(x_0)=0\). This proves the
entire finite representation. It retains the original polynomial family
rather than replacing it by a principal part, a completion, or another
symbol.

For completeness the strength norm also has exact finite-dimensional
comparison maps. If \(q=\sum_\nu z_\nu p_\nu\), the triangle inequality
gives

\[
 \|q\|_p\leq\sum_\nu|z_\nu|\|p_\nu\|_p
       \leq\left(\sum_\nu\|p_\nu\|_p^2\right)^{1/2}\|z\|_2.
                                                                    \tag{CS8a}
\]

Let \(a(q)\) be the original full monomial coefficient vector. Because
its entries are \(\partial^\alpha q(0)/\alpha!\),

\[
 \|a(q)\|_2\leq\widetilde q(0)
                  \leq\widetilde p(0)\|q\|_p,
 \qquad
 \|z\|_2\leq\|(B^*B)^{-1}B^*\|_{2\to2}
                       \widetilde p(0)\|q\|_p.                \tag{CS8b}
\]

Thus the strength norm and the basis-coordinate norm compare in both
directions with the displayed finite constants; continuity of the
original coefficient map already proved continuity of (CS7).

\ReaderArmAliases{\ReaderOriginalAnchorStart{a-finite-measurable-circle-selection-with-a-full-strength-lower-bound}\ReaderOriginalAnchorEnd}
\subsubsection{14.3. A finite measurable circle selection with a full
strength lower
bound}\label{a-finite-measurable-circle-selection-with-a-full-strength-lower-bound}

We construct Borel functions \(\theta:\mathbb R^n\to[0,1]^n\) and
\(\rho:\mathbb R^n\to[1,2]\), each taking only finitely many values, and
a constant \(\kappa_{m,n}>0\), such that

\[
 |p(\xi+\rho(\xi)e^{it}\theta(\xi))|
       \geq\kappa_{m,n}\widetilde p(\xi)
       \quad(\xi\in\mathbb R^n,\ 0\leq t\leq2\pi).              \tag{CS9}
\]

For \(m=0\), take \(\theta=0\), \(\rho=1\), and \(\kappa_{0,n}=1\). In
the rest of this section \(m\geq1\).

Let

\[
 W=\{(j_1/m,\ldots,j_n/m):0\leq j_\ell\leq m\},\qquad M=(m+1)^n.
                                                                    \tag{CS10}
\]

Index a square matrix \(A\) by \(w\in W\) and
\(\alpha\in\{0,\ldots,m\}^n\), with \(A_{w,\alpha}=w^\alpha\). This
matrix is invertible. In one variable, evaluation at the \(m+1\)
distinct points has zero kernel, because a polynomial of degree at most
\(m\) with those zeros is zero. In several variables, apply that
one-variable statement successively in each coordinate; a polynomial
with degree at most \(m\) separately in each variable and zero at the
full product grid has every coefficient zero. Domain and codomain both
have dimension \(M\), proving invertibility. Put

\[
 L=\|A^{-1}\|_{\ell^2\to\ell^2},\qquad H=m!L\sqrt M.          \tag{CS11}
\]

Taylor's exact polynomial identity gives

\[
 p(\xi+w)=\sum_{|\alpha|\leq m}
                  \frac{\partial^\alpha p(\xi)}{\alpha!}w^\alpha.
                                                                    \tag{CS12}
\]

Extend its coefficient array by zero to the product index set in (CS10).
Since \(\alpha!\leq m!\), (CS11) and (CS12) give

\[
 \widetilde p(\xi)
 \leq m!\left(\sum_\alpha
                |\partial^\alpha p(\xi)/\alpha!|^2\right)^{1/2}
 \leq H\max_{w\in W}|p(\xi+w)|.                               \tag{CS13}
\]

Choose \(\theta(\xi)\) to be the first vector in a fixed ordering of
\(W\) maximizing \(|p(\xi+w)|\). The choice is Borel, as it is made
among finitely many continuous functions with a fixed rule for ties. For

\[
 r_\xi(z)=p(\xi+z\theta(\xi))=\sum_{k=0}^{d(\xi)}a_k(\xi)z^k,
 \qquad d(\xi)\leq m,
\]

we have \(r_\xi(1)\neq0\) and

\[
 \max_k|a_k(\xi)|
       \geq\frac{|r_\xi(1)|}{m+1}
       \geq\frac{\widetilde p(\xi)}{H(m+1)}.                    \tag{CS14}
\]

Here is the required one-variable circle calculation in full. For any
nonzero \(r(z)=a_d\prod_{j=1}^d(z-\lambda_j)\), \(d\leq m\), set

\[
 R_\ell=1+\ell/m\quad(0\leq\ell\leq m),\qquad
 \varepsilon=1/(3m).                                         \tag{CS15}
\]

A fixed root modulus excludes at most one of the \(m+1\) radii by the
condition \(\big||\lambda_j|-R_\ell\big|<\varepsilon\), since the radii
are separated by \(1/m>2\varepsilon\). There are at most \(m\) roots
counted with multiplicity. Consequently some \(R_\ell\) has
\(\big||\lambda_j|-R_\ell\big|\geq\varepsilon\) for every root. For
\(|z|=R_\ell\): if \(|\lambda_j|\leq4\), then

\[
 |z-\lambda_j|\geq\varepsilon
             \geq(\varepsilon/4)\max(1,|\lambda_j|);
\]

if \(|\lambda_j|>4\), then

\[
 |z-\lambda_j|\geq|\lambda_j|-2
             >|\lambda_j|/2
             \geq(\varepsilon/4)\max(1,|\lambda_j|).
\]

Thus

\[
 \min_{|z|=R_\ell}|r(z)|
 \geq(\varepsilon/4)^d|a_d|\prod_j\max(1,|\lambda_j|).
                                                                    \tag{CS16}
\]

The elementary-symmetric coefficient formulas give, for each \(k\),

\[
 |a_k|\leq|a_d|\prod_j(1+|\lambda_j|)
          \leq2^d|a_d|\prod_j\max(1,|\lambda_j|).
\]

Combining with (CS16), and using \(\varepsilon/8<1\), yields

\[
 \min_{|z|=R_\ell}|r(z)|
       \geq(\varepsilon/8)^d\max_k|a_k|
       \geq(\varepsilon/8)^m\max_k|a_k|.                        \tag{CS17}
\]

When \(d=0\), the same inequality follows directly from \(|r|=|a_0|\);
no roots are present.

Define the explicit constant

\[
 \kappa_{m,n}=\frac{(1/(24m))^m}{H(m+1)}
             \quad(m\geq1).                                  \tag{CS18}
\]

For each \(\xi\), at least one radius in (CS15) satisfies (CS9), by
(CS14) and (CS17). Choose the first such radius, calling it
\(\rho(\xi)\). This is again Borel. To verify this without choosing or
parametrizing the roots, for each fixed \(w,R_\ell\) the function

\[
 \xi\longmapsto\min_{0\leq t\leq2\pi}|p(\xi+R_\ell e^{it}w)|
\]

is continuous: uniform continuity on the product of a compact
neighborhood in \(\xi\) and the compact circle proves continuity of the
minimum. Divide by the positive continuous function \(\widetilde p\),
test the closed inequality against \(\kappa_{m,n}\), and use the finite
tie rule. This proves all the measurability needed for the subsequent
integrals.

Write

\[
 \zeta(\xi,t)=\rho(\xi)e^{it}\theta(\xi),
 \qquad |\zeta_j(\xi,t)|\leq2\ (1\leq j\leq n).                \tag{CS19}
\]

There is no root of the original denominator on any selected circle, and
(CS9) is uniform in \(\xi\) and \(t\).

\ReaderArmAliases{\ReaderOriginalAnchorStart{exact-complex-shift-comparison-and-the-fundamental-distribution}\ReaderOriginalAnchorEnd}
\subsubsection{14.4. Exact complex-shift comparison and the fundamental
distribution}\label{exact-complex-shift-comparison-and-the-fundamental-distribution}

If \(q\in V_p\), its degree is at most \(m\). Its full Taylor formula
and Cauchy--Schwarz give, for the shifts (CS19),

\[
 |q(\xi+\zeta)|
 \leq\sum_{|\alpha|\leq m}
           |\partial^\alpha q(\xi)|\frac{2^{|\alpha|}}{\alpha!}
 \leq J_{m,n}\widetilde q(\xi),
 \qquad
 J_{m,n}:=\left(\sum_{|\alpha|\leq m}
                         \frac{2^{2|\alpha|}}{(\alpha!)^2}\right)^{1/2}.
                                                                    \tag{CS20}
\]

In particular, with

\[
 a_q(\xi,t)=\frac{q(\xi+\zeta(\xi,t))}{p(\xi+\zeta(\xi,t))},
 \qquad A_q=J_{m,n}\kappa_{m,n}^{-1}\|q\|_p,
\]

we have

\[
 |a_q(\xi,t)|\leq A_q.                                       \tag{CS21}
\]

No comparison at complex zeros is asserted: the selected circles and
their proved denominator bound are essential.

Define \(F\in\mathcal D'(\mathbb R^n)\) by

\[
 \langle F,\varphi\rangle
 =(2\pi)^{-n}\int_{\mathbb R^n}\frac1{2\pi}\int_0^{2\pi}
       \frac{\widehat\varphi(-\xi-\zeta(\xi,t))}
            {p(\xi+\zeta(\xi,t))}\,dt\,d\xi,
 \qquad \varphi\in C_c^\infty(\mathbb R^n).                    \tag{CS22}
\]

We prove this is a distribution rather than merely writing an
oscillatory expression. Fix a compact set \(K\subset[-R,R]^n\), an
integer \(N\) with \(2N>n\), and the seminorm

\[
 s_{2N,K}(\varphi)=\max_{|\beta|\leq2N}\sup_{y\in K}
                                      |\partial^\beta\varphi(y)|.
\]

The function \(\widehat\varphi(-\xi-\zeta)\) is the Fourier transform at
\(-\xi\) of \(e^{iy\cdot\zeta}\varphi(y)\). Integration by parts with
\((1-\Delta_y)^N\) gives

\[
 |\widehat\varphi(-\xi-\zeta)|
       \leq C_{N,K,R}\,s_{2N,K}(\varphi)(1+|\xi|^2)^{-N},       \tag{CS23}
\]

uniformly in (CS19), where one explicit finite constant is

\[
 C_{N,K,R}=|K|e^{2nR}
   \sum_{j=0}^N\binom Nj\sum_{|\gamma|=j}\frac{j!}{\gamma!}
          \sum_{\delta\leq2\gamma}\binom{2\gamma}{\delta}2^{|\delta|}.
                                                                    \tag{CS24}
\]

Indeed \((1-\Delta)^N=\sum_{j=0}^N\binom Nj(-1)^j
\sum_{|\gamma|=j}(j!/\gamma!)\partial^{2\gamma}\); the last sum in
(CS24) is exactly the product-rule bound when derivatives of the
exponential have absolute value at most \(2^{|\delta|}e^{2nR}\). Test
functions supported in \(K\) have no boundary term in this integration
by parts.

Since (CS5) and (CS9) give \(|p(\xi+\zeta)|\geq\kappa_{m,n}d_p\), (CS22)
is absolutely convergent and

\[
 |\langle F,\varphi\rangle|
 \leq(2\pi)^{-n}(\kappa_{m,n}d_p)^{-1}C_{N,K,R}I_{N,n}
                                     s_{2N,K}(\varphi),
 \quad I_{N,n}=\int_{\mathbb R^n}(1+|\xi|^2)^{-N}\,d\xi<\infty.
                                                                    \tag{CS25}
\]

Integrability follows from polar coordinates: the integrand has radial
behavior \(r^{n-1}\) at zero and \(r^{n-1-2N}\) at infinity. These
seminorm bounds on every compact support set prove \(F\in\mathcal D'\).
They also retain the \((2\pi)^{-n}\) and circle measure factors in its
definition.

For a test function, entire Fourier transformation satisfies

\[
 \widehat{p(-D)\varphi}(\eta)=p(-\eta)\widehat\varphi(\eta).
\]

This follows by integration by parts, including the factor
\(D_j=-i\partial_j\), for each monomial. Therefore the denominator
cancels exactly in the pairing of \(p(D)F\):

\[
 \langle p(D)F,\varphi\rangle
 =(2\pi)^{-n}\int_{\mathbb R^n}\frac1{2\pi}\int_0^{2\pi}
             \widehat\varphi(-\xi-\rho(\xi)e^{it}\theta(\xi))\,dt\,d\xi.
                                                                    \tag{CS26}
\]

For each fixed \(\xi\), the integrand is the restriction to the circle
of the entire one-variable function
\(z\mapsto\widehat\varphi(-\xi-z\theta(\xi))\). Its circle average
equals its value at zero. This can be proved directly by its uniformly
convergent power series on the circle: the integral of \(e^{ikt}\) is
zero for every positive integer \(k\), and is one for \(k=0\) after
dividing by \(2\pi\). The averages therefore equal
\(\widehat\varphi(-\xi)\). The absolute bound (CS23) justifies
integration. Fourier inversion at zero yields

\[
 \langle p(D)F,\varphi\rangle
       =(2\pi)^{-n}\int\widehat\varphi(-\xi)\,d\xi
       =\varphi(0),\qquad p(D)F=\delta_0.                     \tag{CS27}
\]

This proves the fundamental-solution assertion itself, with the original
full \(p\).

\ReaderArmAliases{\ReaderOriginalAnchorStart{convolution-both-frozen-inverse-identities-and-all-weaker-estimates}\ReaderOriginalAnchorEnd}
\subsubsection{14.5. Convolution, both frozen inverse identities, and
all weaker
estimates}\label{convolution-both-frozen-inverse-identities-and-all-weaker-estimates}

For \(f\in C_c^\infty(\mathbb R^n)\), convolution in the original
coordinates is

\[
 (F*f)(x)=\langle F, f(x-\cdot)\rangle
 =(2\pi)^{-n}\int_{\mathbb R^n}\frac1{2\pi}\int_0^{2\pi}
     e^{ix\cdot(\xi+\zeta(\xi,t))}
     \frac{\widehat f(\xi+\zeta(\xi,t))}
          {p(\xi+\zeta(\xi,t))}\,dt\,d\xi.                    \tag{CS28}
\]

The first equality is the definition for a test-function second factor.
The second follows by substituting \(y=x-z\) in the Fourier transform of
\(z\mapsto f(x-z)\) evaluated at \(-\xi-\zeta\); the exponential is
exactly the one displayed. The uniform shifted Fourier decay proved in
(CS23), on any compact \(x\) set, justifies (CS28) and every fixed
number of its \(x\)-derivatives. Consequently

\[
 (q(D)(F*f))(x)
 =(2\pi)^{-n}\int\frac1{2\pi}\int_0^{2\pi}
     e^{ix\cdot(\xi+\zeta)}a_q(\xi,t)\widehat f(\xi+\zeta)
                                                \,dt\,d\xi.   \tag{CS29}
\]

For general derivatives the numerator has polynomial growth; arbitrarily
large \(N\) in (CS23) supplies integrability. Formula (CS29) uses a
bounded quotient when \(q\in V_p\).

The differential transfers in convolution have no missing sign or
conjugation. For one coordinate,

\[
 D_{x_j}\langle F,f(x-\cdot)\rangle
  =\langle F,(D_jf)(x-\cdot)\rangle
  =\langle D_jF,f(x-\cdot)\rangle.
\]

The second equality follows because \(D_{z_j}f(x-z)=-D_{x_j}f(x-z)\) and
\(\langle D_jF,\psi\rangle=-\langle F,D_j\psi\rangle\). Iterating for
each original monomial proves

\[
 p(D)(F*f)=(p(D)F)*f=F*(p(D)f).                               \tag{CS30}
\]

Together with (CS27), this gives the global frozen identities

\[
 p(D)(F*f)=f,\qquad F*(p(D)u)=u
                       \quad(f,u\in C_c^\infty(\mathbb R^n)). \tag{CS31}
\]

We next prove the \(L^2\) bound, including the convergence step that an
integral formula alone would not supply. Fix \(x_0\in\mathbb R^n\),
\(R>0\), and a bounded open \(U\) contained in

\[
 Q_{x_0,R}=\{x:|x_j-x_{0j}|<R\text{ for every }j\}.             \tag{CS32}
\]

Let \(Z_U:L^2(U)\to L^2(\mathbb R^n)\) be extension by zero and \(R_U\)
be restriction. Their norms are respectively one and at most one. Write
\(X^\alpha(x)=(x-x_0)^\alpha\), retaining the original \(x_0\). For
every pair of multiindices define the bounded measurable symbol

\[
 b_{\alpha\beta,q}(\xi)=\frac1{2\pi}\int_0^{2\pi}
          a_q(\xi,t)\zeta(\xi,t)^{\alpha+\beta}\,dt.
 \qquad |b_{\alpha\beta,q}(\xi)|
                  \leq A_q2^{|\alpha|+|\beta|}.                \tag{CS33}
\]

For a bounded measurable symbol \(b\), define \(b(D)\) on
\(L^2(\mathbb R^n)\) by Fourier multiplication. Plancherel, with (CS4),
proves \(\|b(D)\|\leq\operatorname*{ess\,sup}|b|\). Consider

\[
 T_{q,U}=
  \sum_{\alpha,\beta\in\mathbb N^n}
       \frac{i^{|\alpha|}(-i)^{|\beta|}}{\alpha!\beta!}
       M_{X^\alpha}R_U b_{\alpha\beta,q}(D)
                                      Z_U M_{X^\beta}.        \tag{CS34}
\]

Every multiplication in (CS34) takes place on \(U\). Its individual
operator norm is bounded by

\[
 A_q\frac{(2R)^{|\alpha|+|\beta|}}{\alpha!\beta!}.
\]

The double series of these bounds sums exactly to \(A_qe^{4nR}\). Thus
(CS34) converges absolutely in operator norm and

\[
 \|T_{q,U}\|_{L^2(U)\to L^2(U)}
 \leq K_{m,n,R}\|q\|_p,
 \qquad K_{m,n,R}=J_{m,n}\kappa_{m,n}^{-1}e^{4nR}.             \tag{CS35}
\]

The identification with (CS29) needs no regularity of the circle
selector in \(\xi\). Put a cutoff
\(\chi_N(\xi)=\mathbf1_{\{|\xi|\leq N\}}\) in the frequency integral
(CS29). For finite \(N\), all integrations in \(\xi,t,y\) are over
finite measure or compact sets, so Fubini is valid. Expand the exact
exponential in its kernel as

\[
 e^{i(x-y)\cdot\zeta}
 =e^{i(x-x_0)\cdot\zeta}e^{-i(y-x_0)\cdot\zeta}
 =\sum_{\alpha,\beta}
    \frac{i^{|\alpha|}(-i)^{|\beta|}}{\alpha!\beta!}
    X^\alpha(x)X^\beta(y)\zeta^{\alpha+\beta}.                 \tag{CS36}
\]

This series converges uniformly and absolutely on
\(U\times U\times[0,2\pi]\times\mathbb R^n\), by (CS19) and (CS32).
Termwise integration for finite \(N\) gives exactly (CS34) with each
\(b\) replaced by \(\chi_N b\). For each fixed pair \(\alpha,\beta\),
these multiplier operators converge strongly to
\(b_{\alpha\beta,q}(D)\), since \(\chi_N\to1\), the symbols are
uniformly bounded, and dominated convergence applies to
\(|\widehat g|^2\). The same summable operator-norm majorant works for
every \(N\). A finite-term/tail argument therefore makes the cutoff
series converge strongly to (CS34).

For \(f\in C_c^\infty(U)\), its zero extension is a test function, and
(CS23) applied to \(f\) proves that the cutoff version of (CS29)
converges uniformly for \(x\) in \(U\) to (CS29): the integrand is
bounded by a constant times \((1+|\xi|^2)^{-N_0}\), with \(2N_0>n\),
uniformly in \(x,t\). The cutoff series and integral consequently have
the same \(L^2(U)\) limit. We have proved

\[
 T_{q,U}f=(q(D)(F*Z_Uf))|_U
                         \quad(f\in C_c^\infty(U)),           \tag{CS37}
\]

including (CS35). This avoids an unjustified interchange of an
infinite-frequency integral and the exponential series.

Here the test-function density on the original open set \(U\) follows
from the proved global approximation without any boundary regularity.
Put
\(K_j=\{x\in U:|x|\leq j,\ \operatorname{dist}(x,\mathbb R^n\setminus U)\geq1/j\}\).
These compact sets increase to \(U\). Thus \(1_{K_j}f\to f\) in
\(L^2(U)\) by dominated convergence. Their zero extensions have compact
support. Fix the compact smooth approximate identity \(r\), of integral
one, from the linked integration lesson, and a finite \(R_r>0\) with
\(\operatorname{supp}r\subset\{|x|\leq R_r\}\). Choose
\(0<\varepsilon_j<(2jR_r)^{-1}\) so that
\(r_{\varepsilon_j}*Z_U(1_{K_j}f)\) differs from \(Z_U(1_{K_j}f)\) by at
most \(1/j\) in \(L^2(\mathbb R^n)\); the proved approximate-identity
limit permits this choice. The convolution is smooth by differentiation
under its compact-support integral and has compact support within
distance \(1/(2j)\) of \(K_j\), hence inside \(U\). Restricting it gives
the required \(C_c^\infty(U)\) approximation. All original support
margins and scale factors are retained.

Now set \(S_U=T_{1,U}:L^2(U)\to L^2(U)\). Density gives the unique
bounded extension from the test-function convolution solver. If
\(f_j\in C_c^\infty(U)\) tends to \(f\) in \(L^2(U)\), then
\(S_Uf_j\to S_Uf\) and \(T_{q,U}f_j\to T_{q,U}f\) in \(L^2(U)\).
Distributional differentiation is continuous, and \(L^2\) convergence
implies distributional convergence by Cauchy--Schwarz against each
compactly supported test function. From (CS37), it follows that

\[
 q(D)S_Uf=T_{q,U}f\in L^2(U),\qquad
 \|q(D)S_Uf\|_2\leq K_{m,n,R}\|q\|_p\|f\|_2.                 \tag{CS38}
\]

In particular \(p(D)S_Uf=f\), because this identity holds for \(f_j\) by
(CS31) and passes to distributions and then to \(L^2\). Also, for
\(u\in C_c^\infty(U)\), the zero extension of \(p(D)u\) is precisely
\(p(D)\) applied to the zero extension of \(u\), with no boundary
contribution since \(u\) vanishes near \(\partial U\). Hence (CS31)
proves

\[
 p(D)S_U=I\text{ on }L^2(U),\qquad
 S_Up(D)u=u\text{ for }u\in C_c^\infty(U).                   \tag{CS39}
\]

The same construction can also be interpreted directly as
\((F*Z_Uf)|_U\) for every \(f\in L^2(U)\). To see this, \(Z_Uf\) is a
compactly supported distribution of order zero. Define its convolution
against a test function \(\psi\) by pairing \(Z_Uf\) with the smooth
function \(y\mapsto\langle F,\psi(y+\cdot)\rangle\), with a cutoff equal
to one on \(\overline U\). This pairing is continuous in \(f\) for the
\(L^2(U)\) norm because that smooth function is bounded on
\(\overline U\). Applying this to \(f_j\) gives convergence to
\(F*Z_Uf\) in distributions, so its restriction equals \(S_Uf\) by the
already proved \(L^2\) convergence. The distributional inverse
identities extend by the same continuity. No unspecified global
temperedness of \(F\) is needed.

\ReaderArmAliases{\ReaderOriginalAnchorStart{the-graph-domain-and-continuous-coefficients}\ReaderOriginalAnchorEnd}
\subsubsection{14.6. The graph domain and continuous
coefficients}\label{the-graph-domain-and-continuous-coefficients}

For a basis \(q_1,\ldots,q_s\) of \(V_p\), define

\[
 \mathcal H_p(U)=\{u\in L^2(U):q_j(D)u\in L^2(U)
                              \ (1\leq j\leq s)\},
 \qquad
 \|u\|_{\mathcal H_p(U)}^2
        =\|u\|_2^2+\sum_{j=1}^s\|q_j(D)u\|_2^2.              \tag{CS40}
\]

This is a Hilbert space: a Cauchy sequence has \(L^2\) limits of \(u\)
and all displayed derivatives, and the distributional limit verifies the
derivative identities. The graph is thus a closed subspace of the finite
product of \(L^2\) spaces. Each \(q\in V_p\) has a finite basis
expansion, so every \(q(D)u\) belongs to \(L^2(U)\), and the space is
independent of the chosen basis. This is the precise common derivative
domain. Equation (CS38) shows \(S_U:L^2(U)\to\mathcal H_p(U)\) is
bounded, with graph-norm bound

\[
 \|S_U\|_{L^2\to\mathcal H_p}
 \leq K_{m,n,R}\left(\|1\|_p^2+\sum_j\|q_j\|_p^2\right)^{1/2}.
                                                                    \tag{CS41}
\]

Fix \(R_0>0\) small enough that the closed cube about \(x_0\) of radius
\(R_0\) lies in \(\Omega\); a smaller ball would also suffice. Set
\(K_0=K_{m,n,R_0}\). By the continuity and vanishing of the original
\(c_\nu\), choose a bounded open neighborhood \(U\) with
\(\overline U\subset\Omega\), \(U\subset Q_{x_0,R_0}\), and

\[
 b:=K_0\sum_{\nu=1}^r\|p_\nu\|_p\sup_{x\in U}|c_\nu(x)|<1.
                                                                    \tag{CS42}
\]

For example choose it so \(b\leq1/2\); finitely many continuous
functions vanishing at \(x_0\) make this possible. Terms with
\(p_\nu=0\) contribute zero. Multiplication
\(M_{c_\nu}:L^2(U)\to L^2(U)\) has norm at most \(\sup_U|c_\nu|\), even
though \(c_\nu\) need not be differentiable.

The original operator in (CS3) now has the well-defined graph-domain
realization

\[
 P_U:\mathcal H_p(U)\to L^2(U),\qquad
 P_Uu=p(D)u+\sum_{\nu=1}^r M_{c_\nu}p_\nu(D)u.               \tag{CS43}
\]

This construction never multiplies an arbitrary distribution by a
continuous function. All products in (CS43) are products of an \(L^2\)
function and a bounded continuous function.

Define

\[
 B_U=\sum_{\nu=1}^r M_{c_\nu}p_\nu(D)S_U:L^2(U)\to L^2(U).
                                                                    \tag{CS44}
\]

Equations (CS38) and (CS42) give \(\|B_U\|\leq b<1\). The geometric
series

\[
 T_U^{-1}=\sum_{k=0}^{\infty}(-B_U)^k,
 \qquad T_U=I+B_U,
 \qquad \|T_U^{-1}\|\leq(1-b)^{-1}                            \tag{CS45}
\]

converges in operator norm. Its finite sums multiplied on either side by
\(I+B_U\) give \(I+(-1)^NB_U^{N+1}\); the last term tends to zero. Thus
(CS45) is an actual two-sided bounded inverse.

Finally put

\[
 E=S_UT_U^{-1}:L^2(U)\longrightarrow\mathcal H_p(U)
                                      \longrightarrow L^2(U). \tag{CS46}
\]

The exact bounds, for every original weaker polynomial, are

\[
 \|q(D)E\|_{L^2(U)\to L^2(U)}
      \leq\frac{K_0\|q\|_p}{1-b}\quad(q\in V_p),
 \qquad
 \|E\|_{L^2(U)\to L^2(U)}
      \leq\frac{K_0\|1\|_p}{1-b}.                            \tag{CS47}
\]

For \(f\in L^2(U)\), (CS39) and (CS44) show

\[
 P_UEf
 =\left(p(D)S_U+\sum_{\nu=1}^rM_{c_\nu}p_\nu(D)S_U\right)T_U^{-1}f
 =(I+B_U)T_U^{-1}f=f.                                        \tag{CS48}
\]

For \(u\in C_c^\infty(U)\), each \(p_\nu(D)u\) is smooth with compact
support, and \(Pu\in L^2(U)\). Applying the second identity in (CS39),
including its distributional derivatives, gives

\[
 T_Up(D)u
  =p(D)u+\sum_{\nu=1}^r c_\nu p_\nu(D)S_Up(D)u
  =p(D)u+\sum_{\nu=1}^r c_\nu p_\nu(D)u=Pu.                  \tag{CS49}
\]

Consequently

\[
 EPu=S_UT_U^{-1}T_Up(D)u=S_Up(D)u=u.                          \tag{CS50}
\]

Equations (CS47), (CS48), and (CS50) are all three parts of the
requested assertion. No coefficient derivative, unproved antecedent,
missing fundamental-solution theorem, or conditional solvability
hypothesis has entered the proof.

\ReaderArmAliases{\ReaderOriginalAnchorStart{why-the-one-sided-hypothesis-gives-the-local-reverse-comparison}\ReaderOriginalAnchorEnd}
\subsubsection{14.7. Why the one-sided hypothesis gives the local
reverse
comparison}\label{why-the-one-sided-hypothesis-gives-the-local-reverse-comparison}

The hypothesis used to construct (CS7) is weaker in its stated
direction: each original frozen polynomial \(p_x\) need only belong to
\(V_p\), where \(p=p_{x_0}\ne0\). The continuity of every original
monomial coefficient then proves the missing reverse comparison locally.
Keep the exact basis and coefficients in (CS7)--(CS8), and put \[
 \varepsilon(x)=\sum_{\nu=1}^r |c_\nu(x)|\|p_\nu\|_p.
 \tag{CS51}
\] This is continuous and vanishes at \(x_0\). The full derivative
arrays, with every multiindex included, obey the ordinary Euclidean
triangle and reverse triangle inequalities. Therefore, at every original
real frequency, \[
 \begin{split}
 \widetilde{p_x-p}(\xi)&\leq\varepsilon(x)\widetilde p(\xi),\\
 (1-\varepsilon(x))\widetilde p(\xi)
 &\leq\widetilde p_x(\xi)
 \leq(1+\varepsilon(x))\widetilde p(\xi).
 \end{split}\tag{CS52}
\] The first inequality follows from the exact finite derivative sum in
(CS8) and the defining norms (CS2); the second follows by adding the
original derivative array of \(p\). No principal part replaces either
polynomial. Given any fixed \(0<\delta<1\), shrink the original
neighborhood so that \(\varepsilon(x)\leq\delta\) there. Positivity
(CS5) now makes each \(p_x\) nonzero. For any two points in that same
neighborhood, retain both comparison factors: \[
 \widetilde p_x(\xi)
 \leq\frac{1+\delta}{1-\delta}\widetilde p_y(\xi),
 \qquad
 \widetilde p_y(\xi)
 \leq\frac{1+\delta}{1-\delta}\widetilde p_x(\xi).
 \tag{CS53}
\] Indeed the upper bound in (CS52) at the first point and the lower
bound at the second give the first inequality, and exchanging the points
gives the second. Thus the one-sided condition and continuous full
coefficients imply actual pairwise constant strength on a smaller
neighborhood. Conversely pairwise constant strength makes \(p_x\in V_p\)
by taking one point to be \(x_0\). These are exact maps between the two
local hypotheses, rather than an assertion that the original families or
polynomial formulas have been replaced.

The same neighborhood may be shrunk further to satisfy (CS42). All
estimates (CS47) and both inverse identities (CS48), (CS50) then hold
for the unchanged original family. This corollary explains the full
relation between the original constant-strength assertion and the
finite-dimensional construction used to prove it.

\ReaderArmAliases{\ReaderOriginalAnchorStart{the-selected-circles-for-an-original-polynomial}\ReaderOriginalAnchorEnd}
\subsubsection{14.8. The selected circles for an original
polynomial}\label{the-selected-circles-for-an-original-polynomial}

This worked example illustrates the complete construction in Section14,
equations (CS1)--(CS50). Frequency coordinates \(\xi=(\xi_1,\xi_2)\) and
spatial coordinates \(x=(x_1,x_2)\) retain their original meanings. The
frequency sample below is \(\xi_*=(1,0)\); the spatial frozen point in
Section14.8.5 is \(x_0=(0,0)\). These are different points in different
spaces.

\ReaderArmAliases{\ReaderOriginalAnchorStart{the-original-polynomial-and-every-strength-contribution}\ReaderOriginalAnchorEnd}
\paragraph{14.8.1. The original polynomial and every strength
contribution}\label{the-original-polynomial-and-every-strength-contribution}

Use

\[
 p(\xi_1,\xi_2)=\xi_1^2+i\xi_2,
 \qquad n=2,\quad m=2,\quad D_j=-i\partial_{x_j}.                \tag{CE1}
\]

Its degree-two homogeneous part is \(p_2(\xi)=\xi_1^2\). It vanishes at
the nonzero real vector \((0,1)\), so this differential operator is
nonelliptic in the ordinary total-order sense. Its full real zero set is
exactly \(\{(0,0)\}\): the real and imaginary parts of \(p(\xi)=0\)
respectively give \(\xi_1^2=0\) and \(\xi_2=0\). Neither the principal
part nor the real zero set replaces the full polynomial in the
construction.

The complete derivative array is

\[
 p=\xi_1^2+i\xi_2,\quad
 \partial_{\xi_1}p=2\xi_1,\quad
 \partial_{\xi_2}p=i,\quad
 \partial_{\xi_1}^2p=2,
\]

with \(\partial_{\xi_1}\partial_{\xi_2}p=0\), \(\partial_{\xi_2}^2p=0\),
and all derivatives of total order above two equal to zero. Thus the
original full strength is

\[
 \widetilde p(\xi)^2
 =|\xi_1^2+i\xi_2|^2+|2\xi_1|^2+|i|^2+|2|^2
 =\xi_1^4+\xi_2^2+4\xi_1^2+1+4.                              \tag{CE2}
\]

In particular \(\widetilde p(0,0)=\sqrt5\),
\(\widetilde p(1,0)=\sqrt{10}\), whereas \(|p(1,0)|=1\). On the
displayed real square \([-2,2]^2\), the minimum of \(\widetilde p\) is
\(\sqrt5\), attained at zero, and its maximum is \(\sqrt{41}\), attained
at the four corners. Indeed every summand in (CE2) is nonnegative and
increases with \(|\xi_1|\) or \(|\xi_2|\). This proves the colorbar
endpoint labels in the figure. The dashed line \(\xi_1=0\) depicts zeros
of \(p_2\); the black diamond depicts the single real zero of the full
\(p\).

\ReaderArmAliases{\ReaderOriginalAnchorStart{the-actual-finite-grid-and-the-global-construction-constant}\ReaderOriginalAnchorEnd}
\paragraph{14.8.2. The actual finite grid and the global construction
constant}\label{the-actual-finite-grid-and-the-global-construction-constant}

For \(m=n=2\), the grid of CS10 is exactly

\[
 W=\{0,\tfrac12,1\}^2.                                       \tag{CE3}
\]

Order it lexicographically, first by \(w_1\) and then by \(w_2\). At the
original frequency sample \(\xi_*=(1,0)\),

\[
 |p(\xi_*+w)|^2=(1+w_1)^4+w_2^2.                            \tag{CE4}
\]

All nine exact values, also printed beside the grid points in the
figure, are:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lrrr@{}}
\toprule\noalign{}
\(w_2\backslash w_1\) & \(0\) & \(1/2\) & \(1\) \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(0\) & \(1\) & \(81/16\) & \(16\) \\
\(1/2\) & \(5/4\) & \(85/16\) & \(65/4\) \\
\(1\) & \(2\) & \(97/16\) & \(17\) \\
\end{longtable}
}

Because \((1+w_1)^4\) and \(w_2^2\) both increase strictly over these
nonnegative coordinates, the unique maximizer is \(w=(1,1)\). The
first-maximizer rule of CS13 therefore selects

\[
 \theta(\xi_*)=(1,1),\qquad |p(\xi_*+\theta)|=\sqrt{17}.        \tag{CE5}
\]

The construction's global constant uses the entire interpolation grid,
not merely this one sample. The one-variable evaluation matrix and its
inverse are

\[
 V=\begin{pmatrix}1&0&0\\1&1/2&1/4\\1&1&1\end{pmatrix},
 \qquad
 V^{-1}=\begin{pmatrix}1&0&0\\-3&4&-1\\2&-4&2\end{pmatrix}.
                                                                    \tag{CE6}
\]

Their product is the identity by direct multiplication. With the product
grid and product monomial ordering, CS10's full \(9\times9\) matrix is
\(A=V\otimes V\). Put

\[
 C=(V^{-1})^*V^{-1}
  =\begin{pmatrix}14&-20&7\\-20&32&-12\\7&-12&5\end{pmatrix},
 \qquad \lambda_*:=\text{the largest eigenvalue of }C.         \tag{CE7}
\]

The matrix \(C\) is positive definite because \(V^{-1}\) is invertible.
The eigenvalues of \(C\otimes C\) are all products of eigenvalues of
\(C\): if \(Cu_j=\lambda_j u_j\) for an orthonormal eigenbasis, then
\((C\otimes C)(u_j\otimes u_k)=\lambda_j\lambda_k(u_j\otimes u_k)\), and
these nine vectors form an orthonormal basis. Consequently

\[
 L=\|A^{-1}\|_{2\to2}=\lambda_*,\qquad
 H=2!\,L\sqrt9=6\lambda_*.                                  \tag{CE8}
\]

Expanding \(\det(\lambda I-C)\) gives

\[
 f(\lambda)=\lambda^3-51\lambda^2+85\lambda-16.                \tag{CE9}
\]

This is an exact algebraic specification of \(\lambda_*\). For
orientation, \(49<\lambda_*<50\): \(f(49)=-653\), \(f(50)=1734\), and
\(f'(\lambda)=3\lambda^2-102\lambda+85\) is positive and increasing for
\(\lambda\geq49\), so the root in that interval is the only root above
49 and hence the largest eigenvalue. More precisely, substitution of the
rational endpoints gives

\[
 49.28181370<\lambda_*<49.28181372,
\]

because \(f(49.28181370)=-25343869120302647/10^{21}<0\) and\linebreak[4]
\(f(49.28181372)=336610322876968607/15625000000000000000000>0\), with
the same strict monotonicity. Decimal positions in the drawings are
computed from the exact matrices and formulas; the statements use those
exact objects.

Substitution into CS18 retains every original factor:

\[
 \kappa_{2,2}
 =\frac{(1/(24\cdot2))^2}{(6\lambda_*)(2+1)}
 =\frac1{41472\lambda_*}.                                   \tag{CE10}
\]

This is the global constant used uniformly for every real \(\xi\), not
the sharp constant for the present sample. Also, CS20 gives

\[
 J_{2,2}^2=1+(4+4)+(4+16+4)=33,
 \qquad J_{2,2}=\sqrt{33}.                                   \tag{CE11}
\]

The six terms in the second expression are precisely the indices
\((0,0),(1,0),(0,1),(2,0),(1,1),(0,2)\); no derivative weight is
suppressed.

\ReaderArmAliases{\ReaderOriginalAnchorStart{first-admissible-radius-every-original-root-and-the-image-circle}\ReaderOriginalAnchorEnd}
\paragraph{14.8.3. First admissible radius, every original root, and the
image
circle}\label{first-admissible-radius-every-original-root-and-the-image-circle}

The exact one-variable polynomial selected at \(\xi_*\) is

\[
 r(z)=p(\xi_*+z\theta)=(1+z)^2+iz
      =z^2+(2+i)z+1.                                        \tag{CE12}
\]

The candidate radii, in their original order from CS15, are \(1,3/2,2\).
On the first circle \(z=e^{it}\), use \(z^2+1=2z\cos t\) to obtain the
exact comparison, retaining the original factor \(z\):

\[
 r(e^{it})=e^{it}(2+2\cos t+i),
 \qquad
 |r(e^{it})|^2=(2+2\cos t)^2+1.                              \tag{CE13}
\]

Thus

\[
 \min_t|r(e^{it})|=1\quad(t=\pi),\qquad
 \max_t|r(e^{it})|=\sqrt{17}\quad(t=0\bmod2\pi).              \tag{CE14}
\]

Since \(\lambda_*\geq C_{22}=32\) by the Rayleigh quotient and
\(\sqrt{10}<4\),

\[
 \kappa_{2,2}\widetilde p(\xi_*)
   =\frac{\sqrt{10}}{41472\lambda_*}
   \leq\frac{\sqrt{10}}{41472\cdot32}<1.                     \tag{CE15}
\]

The first circle already satisfies the exact admissibility test of CS9.
Hence the actual first-admissible-radius rule selects \(\rho(\xi_*)=1\).
The sample-specific comparison is
\(\min|r|/\widetilde p(\xi_*)=1/\sqrt{10}\); it must not be substituted
for the proved global \(\kappa_{2,2}\).

For explicit root avoidance, let

\[
 u=\sqrt{\frac{\sqrt{17}-1}{2}},\qquad
 v=\sqrt{\frac{\sqrt{17}+1}{2}},\qquad
 s=\frac{1+\sqrt{17}}2=v^2.                                  \tag{CE16}
\]

Both square roots are positive, \(u^2-v^2=-1\), and \(uv=2\), so
\((u+iv)^2=-1+4i\). The quadratic formula applied to the original
coefficients in (CE12) gives exactly

\[
 \lambda_+=\frac{u-2+i(v-1)}2,
 \qquad
 \lambda_-=-\frac{u+2+i(v+1)}2.                              \tag{CE17}
\]

Their product is 1 and their sum is \(-2-i\), which also directly
verifies \(r(z)=(z-\lambda_+)(z-\lambda_-)\). Their moduli are

\[
 r_{\rm in}:=|\lambda_+|=\frac{s-\sqrt s}{2},\qquad
 r_{\rm out}:=|\lambda_-|=\frac{s+\sqrt s}{2}.                 \tag{CE18}
\]

Here is an explicit verification of (CE18), without replacing the
original complex roots. From (CE17),
\(|\lambda_+|^2=(\sqrt{17}+5-4u-2v)/4\) and
\(|\lambda_-|^2=(\sqrt{17}+5+4u+2v)/4\). The identity \(2u=v(s-1)\)
follows by squaring both positive sides:
\(4u^2=2(\sqrt{17}-1)=s(s-1)^2\). Therefore \(4u+2v=2sv\). Also
\(s^2=(9+\sqrt{17})/2\), so \(s^2+s=5+\sqrt{17}\). Substitution gives
\(|\lambda_\pm|^2=(s\mp\sqrt s)^2/4\). Since \(s>1\), both stated moduli
are positive, proving (CE18).

As \(5/2<s<3\), we have \(3/2<\sqrt s<2\), and hence
\(0<r_{\rm in}<3/4<1\) and \(r_{\rm out}>2>1\). Each original root is
therefore separated from the circle. The exact shortest distances, drawn
radially in the root-plane panel, are

\[
 d_{\rm in}=1-r_{\rm in},\qquad
 d_{\rm out}=r_{\rm out}-1.                                 \tag{CE19}
\]

For any root \(\lambda\neq0\), the triangle inequality proves
\(|e^{it}-\lambda|\geq||\lambda|-1|\); equality occurs at
\(e^{it}=\lambda/|\lambda|\). This proves the labelled radial-distance
segments. Their product gives the additional sample lower bound

\[
 |r(e^{it})|\geq d_{\rm in}d_{\rm out}
       =r_{\rm in}+r_{\rm out}-2
       =s-2=\frac{\sqrt{17}-3}{2}.                           \tag{CE20}
\]

The equality in the middle uses \(r_{\rm in}r_{\rm out}=1\). This
product bound is weaker than the exact minimum 1 in (CE14); the closest
point to each of the two roots need not occur at the same \(t\). The
first-radius proof uses the exact minimum, and the global proof uses
(CE10). These three quantities have distinct proved roles.

The right lower panel shows the actual image under the original \(p\) of
the complex line circle \((\xi_1,\xi_2)=(1+e^{it},e^{it})\), rather than
an unrelated circle in the \(p\)-plane. Its marked values are

\[
\begin{array}{c|c|c}
t & z & p(1+z,z)\\\hline
0 & 1 & 4+i\\
\pi/2 & i & -1+2i\\
\pi & -1 & -i\\
3\pi/2 & -i & 1-2i
\end{array}                                                   \tag{CE21}
\]

All values follow by substitution in (CE12), with the factor \(i\xi_2\)
retained. The dashed unit circle in the image plane marks the exact
lower bound \(|r|\geq1\); the image touches it only at \(-i\) and avoids
zero. The arrows follow increasing \(t\), the original positive
orientation of the selected circle.

\ReaderArmAliases{\ReaderOriginalAnchorStart{what-the-fundamental-solution-picture-actually-constructs}\ReaderOriginalAnchorEnd}
\paragraph{14.8.4. What the fundamental-solution picture actually
constructs}\label{what-the-fundamental-solution-picture-actually-constructs}

The selected circle at \(\xi_*\) is one member of the measurable family
used in CS22. It does not by itself define the fundamental solution. The
complete original formula is

\[
 \langle F,\varphi\rangle
 =(2\pi)^{-2}\int_{\mathbb R^2}\frac1{2\pi}\int_0^{2\pi}
 \frac{\widehat\varphi(-\xi-\rho(\xi)e^{it}\theta(\xi))}
 {p(\xi+\rho(\xi)e^{it}\theta(\xi))}\,dt\,d\xi.               \tag{CE22}
\]

The finite grid and radii are selected separately at every real \(\xi\).
CS23--CS27 prove convergence and \(p(D)F=\delta_0\). For every weaker
polynomial \(q\), the exact Taylor comparison gives

\[
 \left|\frac{q(\xi+\rho e^{it}\theta)}
              {p(\xi+\rho e^{it}\theta)}\right|
       \leq\sqrt{33}\,(41472\lambda_*)\|q\|_p.               \tag{CE23}
\]

The bounded complex shift is expanded in both original spatial variables
\(x-x_0\) and \(y-x_0\), as in CS36. It produces the bounded multiplier
series, rather than requiring derivatives of the measurable selector. On
a square of coordinate radius \(R\), CS35's exact bound is

\[
 \|q(D)S_U\|_{L^2(U)\to L^2(U)}
 \leq K_R\|q\|_p,\qquad
 K_R=41472\lambda_*\sqrt{33}\,e^{8R},
 \quad S_Uf=(F*Z_Uf)|_U.                                    \tag{CE24}
\]

Here \(Z_U\) is extension by zero; the graph-domain derivative meaning
and density passage are CS38--CS40. The exponential is the full sum of
both multiplication-series bounds with \(n=2\); it is not omitted or
absorbed into another definition. Both frozen identities are
\(p(D)S_Uf=f\) on \(L^2(U)\) and \(S_Up(D)u=u\) for
\(u\in C_c^\infty(U)\).

\ReaderArmAliases{\ReaderOriginalAnchorStart{a-fully-specified-continuous-coefficient-example-and-the-solver-maps}\ReaderOriginalAnchorEnd}
\paragraph{14.8.5. A fully specified continuous-coefficient example and
the solver
maps}\label{a-fully-specified-continuous-coefficient-example-and-the-solver-maps}

Set \(x_0=(0,0)\), \(R=1/4\), and \(U=(-1/4,1/4)^2\). Use the actual
weaker polynomial \(q_0(\xi)=\xi_1\), whose full strength is
\(\widetilde q_0^2=\xi_1^2+1\). Its exact strength norm is

\[
 a:=\|q_0\|_p=\sqrt{\frac{\sqrt2-1}{2}}.                      \tag{CE25}
\]

To prove this constant, let \(t=\xi_1^2\geq0\) and \(h=\xi_2^2\geq0\).
The squared norm is the supremum of

\[
 \frac{t+1}{t^2+4t+5+h}.
\]

For fixed \(t\), its maximum over \(h\geq0\) is at \(h=0\). The
derivative in \(t\) has numerator \(-t^2-2t+1\), positive on
\([0,\sqrt2-1)\) and negative thereafter. At \(t=\sqrt2-1\), the ratio
equals \((\sqrt2-1)/2\); the endpoints give no larger value. This proves
the exact supremum with no discarded strength term.

Define all remaining labelled constants by

\[
 K:=K_{1/4}=41472\lambda_*\sqrt{33}\,e^2,
 \qquad \varepsilon=\frac{2}{Ka},\qquad c(x)=\varepsilon x_1.   \tag{CE26}
\]

Thus \(c\) is continuous, \(c(x_0)=0\), and
\(\sup_U|c|=\varepsilon/4=1/(2Ka)\). The actual spatial operator and
full frozen family are

\[
 P(x,D)=D_1^2+iD_2+\varepsilon x_1D_1,
 \qquad p_x(\xi)=\xi_1^2+i\xi_2+\varepsilon x_1\xi_1,          \tag{CE27}
\]

with the full strength

\[
 \widetilde p_x(\xi)^2
 =(\xi_1^2+\varepsilon x_1\xi_1)^2+\xi_2^2
     +(2\xi_1+\varepsilon x_1)^2+1+4.                       \tag{CE28}
\]

Here \(c\) is real; every derivative term in (CE28) is displayed. The
triangle and reverse triangle inequalities on the full derivative
arrays, together with (CE25), prove

\[
 \left(1-\frac1{2K}\right)\widetilde p(\xi)
 \leq\widetilde p_x(\xi)
 \leq\left(1+\frac1{2K}\right)\widetilde p(\xi)
                 \quad(x\in U,\ \xi\in\mathbb R^2).          \tag{CE29}
\]

The lower constant is positive because \(K\geq41472\cdot32>1/2\). Thus
this is an actual constant-strength family, not merely a proposed
perturbation.

In particular, for any two original spatial points \(x,y\in U\), (CE29)
gives the explicit pairwise bound

\[
 \widetilde p_x(\xi)
 \leq\frac{1+1/(2K)}{1-1/(2K)}\widetilde p_y(\xi)
                  \quad(\xi\in\mathbb R^2),                  \tag{CE29a}
\]

and interchanging \(x,y\) gives the reverse comparison. This is also the
concrete instance of the proved comparison corollary (CS51)--(CS53) in
Section14.7; no extra condition is needed.

The maps in the second figure are exactly

\[
 B=M_cD_1S_U:L^2(U)\to L^2(U),\qquad
 \|B\|\leq\left(\frac1{2Ka}\right)(Ka)=\frac12,
 \qquad T=I+B,\qquad \|T^{-1}\|\leq2,                       \tag{CE30}
\]

where \(T^{-1}=\sum_{k\geq0}(-B)^k\) in operator norm. Define

\[
 E=S_UT^{-1}:L^2(U)\to\mathcal H_p(U).                         \tag{CE31}
\]

The graph space \(\mathcal H_p(U)\) is CS40's space of \(L^2\) functions
whose derivatives for every polynomial in \(V_p\) lie in \(L^2\),
equivalently for any finite basis. Continuous \(c\) multiplies only the
already controlled function \(D_1Ef\). Consequently the actual output
satisfies

\[
 PEf=(I+B)T^{-1}f=f\quad(f\in L^2(U)).                        \tag{CE32}
\]

For \(u\in C_c^\infty(U)\), the exact frozen left identity gives
\(T p(D)u=p(D)u+cD_1S_Up(D)u=Pu\), and therefore

\[
 EPu=S_UT^{-1}T p(D)u=S_Up(D)u=u.                            \tag{CE33}
\]

All weaker maps in the diagram have the bound

\[
 \|q(D)E\|\leq2K\|q\|_p\quad(q\in V_p),\qquad
 \|E\|\leq\frac{2K}{\sqrt5}.                                \tag{CE34}
\]

The second bound uses the exact \(\|1\|_p=1/\sqrt5\), because (CE2) has
minimum \(\sqrt5\). Equations (CE30)--(CE34) prove the diagram's
domains, arrows, constants, right identity, and compactly supported left
identity. No global inverse, boundary support property, or uniqueness
claim is depicted.

\ReaderArmAliases{\ReaderOriginalAnchorStart{reading-and-reproducing-the-figures}\ReaderOriginalAnchorEnd}
\paragraph{14.8.6. Reading and reproducing the
figures}\label{reading-and-reproducing-the-figures}

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={Full strength, the finite grid, both original complex roots and the image of the selected circle}]{../figures/constant_strength_circle_example.png}}
\caption{Full strength, the finite grid, both original complex roots and
the image of the selected circle}
\end{figure}

The four panels show the original full \(p\), its strength, all nine
grid values, the actual selected circle and its image. Equations
(CE1)--(CE21) prove every displayed coordinate, root, orientation and
bound; (CE22)--(CE24) give the complete distributional interpretation.
The colors follow increasing circle parameter. Numerical sampling draws
the exact formulas and does not supply their proofs.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={The original continuous-coefficient perturbation and the proved local inverse maps}]{../figures/constant_strength_solver_maps.png}}
\caption{The original continuous-coefficient perturbation and the proved
local inverse maps}
\end{figure}

The second figure shows the proved \(S_U,T^{-1},E,P,q(D)\) maps, with
their original domains and constants. Equations (CE25)--(CE34) prove the
coefficient, strength, graph-domain realization and both inverse
identities. The specified local square and the nonzero imaginary
coefficient remain part of the calculation.

\href{../figures/render_constant_strength_example.py}{Reproducible
figure source} retains the original polynomial and every strength
contribution. The exact formulas, finite selectors, matrix and roots
above determine its coordinates. The two vector versions are
\href{../figures/constant_strength_circle_example.svg}{the circle
figure} and \href{../figures/constant_strength_solver_maps.svg}{the
solver maps}.

The new Section14 text, these two figures and their reproducible source
are available under CC0-1.0.

\ReaderArmAliases{\ReaderOriginalAnchorStart{reading-routes-and-scope-of-the-imports}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{reading-routes-and-scope-of-the-imports}\label{reading-routes-and-scope-of-the-imports}\endgroup\ReaderOriginalAnchorStart{references}\ReaderOriginalAnchorEnd}
\subsection{References}\label{references}

In particular, the finite-part construction, the scaled Hölder proof and
the explicit annular cover expose steps needed for the stated
generality. Section 14 also proves the broader constant-strength
comparison with its exact common derivative domain.

Tao's
\href{https://www.math.ucla.edu/~tao/247a.1.06f/notes4.pdf}{\emph{Lecture
Notes 4 for 247A}}, Theorem 4.4 and Corollary 2.10, is the specified
multiplier reading. His
\href{https://www.math.ucla.edu/~tao/247a.1.06f/notes2.pdf}{\emph{Lecture
Notes 2 for 247A}}, Proposition 6.1 and Corollary 6.3, is the specified
fractional-integration reading. Their normalization uses \(2\pi\) in the
Fourier phase; the change of frequency variable gives our convention
without changing the multiplier or exponent conditions. The cited
multiplier and fractional-integration statements are used with the
hypotheses stated in Section 1.

In the multiplier proof on page 20 of Notes 4, read the two intermediate
gradient formulas with the frequency and scale factors restored: \[
\nabla K_j=(2\pi i\xi m_j)^\vee,\qquad
|\nabla K_j(x)|\lesssim 2^{j(d+1)}
       \min\bigl(1,(2^j|x|)^{-(d+2)}\bigr).
\] Here \(d\) is that reading's dimension. Its subsequent expression
with the factor \(|x|^{-d-1}\) already has the correct scaling. These
printing corrections preserve the theorem and its \(d+2\)-derivative
hypothesis.

For a complementary proof technique, John K. Hunter's
\href{https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf}{\emph{Notes
on Partial Differential Equations}, revised 18 June 2014}, Theorem 2.28,
printed pages 40--43, estimates the Newtonian potential in Hölder spaces
by splitting spatial integrals. Its scope is the Laplacian; (L24)--(L26)
handle arbitrary homogeneous elliptic order. Hunter's Lemma 8.11,
printed pages 228--229, treats a commutator for first-order systems with
\(C^1\) coefficients in \(L^2\). Our proof in (L38)--(L39) gives the
scalar Lipschitz, arbitrary finite \(p\) version used here. The
comparisons do not substitute the smoother assumptions of those examples
for the hypotheses of this lesson.

The earlier external multiplier and fractional-integration routes remain
identifiable in the citations. Section 12 supplies an owned
fractional-integration proof with its original kernel, finite endpoint,
full constants and exact elliptic application. Section 13 supplies the
full multiplier proof, including the complete Fourier maps and every
top-order elliptic receiving constant. Section 14 supplies its complete
constant-strength construction, using only the earlier programme proofs
linked there.

\end{document}