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\begin{document}

\section{From local energy to global divergence
equations}\label{from-local-energy-to-global-divergence-equations}

A divergence equation can be meaningful before its solution has even one
square-integrable derivative. Its positive energy then supplies that
derivative, and local elliptic regularity supplies the next one. Global
existence has a different difficulty: data may grow arbitrarily towards
infinity or the boundary of an open set. We solve that difficulty by
assigning a sufficiently large weight to the adjoint residual, with
controlled changes on each previously treated compact region.

The argument permits complex coefficients and uses only local Lipschitz
bounds. On a manifold it requires a density to specify divergence. We
make that convention explicit and separate the existence theorem on
noncompact components from the exact obstruction on compact components.
No boundary condition or global integrability of the solution is
imposed.

\subsection{1. The equation and the tools used
below}\label{the-equation-and-the-tools-used-below}

Let \(X\) be a Hausdorff, second-countable \(C^2\) manifold without
boundary of dimension \(n\geq1\). An arbitrary open subset of
\(\mathbb R^n\) is included. Fix a positive \(C^1\) density \(d\mu\) and
a complex symmetric contravariant tensor \(A\), with locally Lipschitz
coordinate entries. Symmetric means transpose-symmetric,
\(a^{jk}=a^{kj}\), and does not mean Hermitian. We require

\[
 \operatorname{Re}\sum_{j,k}a^{jk}(x)\xi_k\xi_j>0
 \quad(0\ne\xi\in\mathbb R^n).                                  \tag{D1}
\]

Continuity and compactness make the lower bound uniform on each compact
coordinate neighborhood. Since the real and imaginary parts of \(A\) are
real symmetric matrices, (D1) also gives

\[
 \operatorname{Re}\sum_{j,k}a^{jk}z_k\overline{z_j}
       \geq \lambda\sum_j|z_j|^2\quad(z\in\mathbb C^n)            \tag{D2}
\]

on that neighborhood. Indeed the imaginary symmetric part contributes a
purely imaginary number, and the real part acts separately on the real
and imaginary parts of \(z\).

In a chart write \(d\mu=\rho(x)\,dx\). Our operator and its Hilbert
adjoint on compactly supported tests are

\[
 \begin{split}
 Lu&=-\rho^{-1}\partial_j(\rho a^{jk}\partial_k u),\\
 L^*v&=-\rho^{-1}\partial_j(\rho\overline{a^{jk}}\partial_k v).
 \end{split}                                                     \tag{D3}
\]

Repeated indices in this lesson are summed from \(1\) to \(n\). The
pairing \((u,v)_\mu=\int u\overline v\,d\mu\) is linear in its first
entry. For Euclidean Lebesgue density, (D3) is \(D_j(a^{jk}D_k u)\),
with \(D_j=-i\partial_j\), as in \emph{Local inverses and
distance-weighted elliptic estimates}.

For \(u\in L^2_{\mathrm{loc}}\), the equation \(Lu=f\) means

\[
                  (u,L^*\phi)_\mu=(f,\phi)_\mu
                    \quad(\phi\in C_c^2(X)).                     \tag{D4}
\]

This pairing is defined: \(L^*\phi\) is a bounded, compactly supported
measurable function. In each chart (D4) agrees with distributional
differentiation. More explicitly, for a Lipschitz multiplier \(b\),
define

\[
                  b\partial_k u=\partial_k(bu)- (\partial_k b)u.
                                                                    \tag{D5}
\]

The right side belongs locally to \(H^{-1}\). Multiplication by
\(\rho^{-1}\) and changes of charts in (D3) can be interpreted through
(D4); no product of an arbitrary second-order distribution by a merely
\(C^1\) function is being postulated. The equivalent chart equation is
\(-\partial_j(\rho a^{jk}\partial_k u)=\rho f\).

There is no need to choose an unmentioned smooth structure. Compactly
supported \(C^2\) tests suffice for an operator of order two acting on
\(L^2_{\mathrm{loc}}\). In a coordinate chart, approximation in \(H^2\)
shows that an equation against smooth tests also holds against these
\(C^2\) tests. Finite partitions of unity make the equivalence global.
The coordinate changes are \(C^2\); their first derivatives and inverse
first derivatives are bounded on compact subcharts, and their second
derivatives are bounded there. The ordinary chain rule, first on smooth
functions and then by Sobolev approximation, therefore preserves local
\(H^1\) and \(H^2\). The transformation of \(A\) uses first derivatives
of the coordinate map, so its new entries remain locally Lipschitz. The
density has the usual absolute-Jacobian transformation. These facts make
(D3) and (D4) intrinsic, including on a nonorientable manifold.

The following dependencies specify the base of the proofs.

\begin{itemize}
\tightlist
\item
  \textbf{Weak Lipschitz products.} Section 7 of
  \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates} gives the weak Lipschitz product
  (D5) and the Friedrichs commutator bound, including convergence, on
  \(L^2\).
\item
  \textbf{Weak elliptic regularity.} Section 8 of
  \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates} proves that for a second-order
  elliptic principal operator with locally Lipschitz complex
  coefficients, \(u\in H^1_{\mathrm{loc}}\) and
  \(P_2u\in L^2_{\mathrm{loc}}\) imply \(u\in H^2_{\mathrm{loc}}\), with
  the local estimates proved there. Its quantitative estimate comes from
  Section 6 of \href{local-elliptic-coefficients.pdf}{Local inverses and
  distance-weighted elliptic estimates}.
\item
  \textbf{Support continuation.} Sections 7--8 of
  \href{support-continuation.pdf}{Curved weights and the directions in
  which support can end} prove continuation from a nonempty open subset
  of a connected domain for an \(H^1_{\mathrm{loc}}\) solution
  satisfying \(|P_2u|\leq C(|u|+|Du|)\), when every normal is
  admissible. Positive real part gives this admissibility also in
  dimension two: the path \(\operatorname{Re}p+it\operatorname{Im}p\),
  \(0\leq t\leq1\), remains elliptic and preserves the number of roots
  in each half-plane. No reality assumption at a point is needed in this
  positive-real-part case.
\item
  \textbf{Sobolev and compactness facts.} The local \(L^2\) and
  weak-derivative calculus, smooth approximation in integer Sobolev
  spaces, boundedness of convolution on \(H^{-1}(\mathbb R^n)\), weak
  subsequential compactness of bounded Hilbert-space sequences, and
  compactness of bounded \(H^1\) sequences in \(L^2\), or bounded
  \(H^2\) sequences in \(H^1\), when their supports lie in one fixed
  compact interior region. The compactness statements transfer through a
  finite collection of charts. A weakly differentiable function with
  zero gradient on a connected open set is almost everywhere constant.
\item
  \textbf{Manifold geometry.} Locally finite \(C^2\) partitions of unity
  and compactly supported \(C^2\) cutoffs on second-countable \(C^2\)
  manifolds; a positive \(C^1\) density and a \(C^1\) auxiliary
  Riemannian metric exist. Only their local coordinate bounds are used.
  A manifold is locally path connected and locally compact; its
  connected components are open, and each connected component is path
  connected. The compact-hull construction needed below is proved in
  this lesson.
\item
  \textbf{Hilbert-space tools.} The complex Hahn--Banach theorem for a
  bounded linear functional on any linear subspace of a normed space,
  and the Riesz representation theorem for bounded linear functionals on
  a Hilbert space. The extension in the global proof is generally from
  an infinite-dimensional subspace; a finite-dimensional extension
  contract would not suffice. We also use completeness, the
  Cauchy--Schwarz inequality and the convergent geometric series for an
  operator of norm less than one.
\end{itemize}

The Hilbert-space tools have complete proofs in this course. Section 5
of \href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments} proves norm-preserving complex
Hahn--Banach extension from any linear subspace, including one that is
neither closed nor finite dimensional. Section 2.3 of
\href{lower-bounded-spectral-calculus.pdf}{Spectral measures with the
original operator domain retained} proves Hilbert representation with
the linear-first convention; Section 6.5 proves the operator geometric
series and both inverse products. These proofs supply exactly the
functional extension, represented vector and inverse used below.

The argument uses these stated entry results and the named prior
lessons. No global solvability theorem is assumed.

\subsection{2. Energy recovers the first
derivative}\label{energy-recovers-the-first-derivative}

\textbf{Local gain.} If \(u\in L^2_{\mathrm{loc}}(X)\) and
\(Lu\in H^{-1}_{\mathrm{loc}}(X)\), then \(u\in H^1_{\mathrm{loc}}(X)\).

We first work in a Euclidean chart and multiply the equation by
\(\rho\). Set \(b^{jk}=\rho a^{jk}\) and
\(Q=-\partial_j(b^{jk}\partial_k)\). The entries of \(b\) are Lipschitz
and their real part is uniformly positive on any fixed compact subchart.
A \(C^1\) multiplier acts boundedly on local \(H^{-1}\): by duality its
action is the product on \(H^1\), where the first-order product rule
gives the bound. Thus \(Qu\in H^{-1}_{\mathrm{loc}}\).

Choose a smooth cutoff \(\chi\) supported in the chart. Distributional
product rules, using (D5), give

\[
 Q(\chi u)=\chi Qu
 -\partial_j\bigl(b^{jk}(\partial_k\chi)u\bigr)
 -(\partial_j\chi)b^{jk}\partial_k u.                           \tag{D6}
\]

All terms on the right belong to \(H^{-1}\) after zero extension. For
the last term put \(c^k=(\partial_j\chi)b^{jk}\) and write
\(c^k\partial_k u=\partial_k(c^ku)-(\partial_kc^k)u\). Both \(c^ku\) and
\((\partial_kc^k)u\) are \(L^2\) with fixed compact support. The same
observation justifies the product rule itself by local smooth
approximation of \(u\) in \(L^2\).

It therefore suffices to treat \(v\in L^2(\mathbb R^n)\) supported in a
compact set \(K_0\) inside the chart, with
\(Qv\in H^{-1}(\mathbb R^n)\). Extend the coefficients by a cutoff on a
slightly larger chart; they become globally Lipschitz and agree with the
original positive matrix on a fixed compact neighborhood \(K\) of
\(K_0\). Positivity of the extension outside \(K\) is irrelevant.

Let \(J_\epsilon v=v*\eta_\epsilon\), where \(\eta\in C_c^\infty\) has
integral one. For small \(\epsilon\), \(v_\epsilon=J_\epsilon v\) is
smooth with support in \(K\). Define the vector commutators

\[
 r_{\epsilon,j}=b^{jk}\partial_kv_\epsilon
                  -J_\epsilon(b^{jk}\partial_kv).
 \qquad
 \sup_{0<\epsilon<\epsilon_0}\|r_\epsilon\|_2\leq C\|v\|_2.     \tag{D7}
\]

The bound is exactly Section 7 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates}, applied to each entry and
derivative; replacing \(D_k\) by \(\partial_k\) only multiplies by a
scalar of modulus one. Hence

\[
 Qv_\epsilon=J_\epsilon(Qv)-\partial_jr_{\epsilon,j},
 \qquad
 \|Qv_\epsilon\|_{H^{-1}}
       \leq C\bigl(\|Qv\|_{H^{-1}}+\|v\|_2\bigr).               \tag{D8}
\]

In particular no derivative of a Lipschitz coefficient of order two has
appeared. Choose a fixed box containing \(K\). The elementary
compact-support Poincaré estimate \(\|w\|_2\leq C_K\|\nabla w\|_2\)
follows by integrating \(w\) along lines parallel to one coordinate axis
from a face of that box, where its zero extension vanishes, and applying
Cauchy--Schwarz. Integration by parts and (D2) now give

\[
 \begin{split}
 \|v_\epsilon\|_{H^1}^2
 &\leq C\|\nabla v_\epsilon\|_2^2
 \leq C\operatorname{Re}(Qv_\epsilon,v_\epsilon)_{dx}\\
 &\leq C\|Qv_\epsilon\|_{H^{-1}}\|v_\epsilon\|_{H^1}.
 \end{split}                                                     \tag{D9}
\]

If the last \(H^1\) norm is zero there is nothing to divide by;
otherwise cancellation gives a uniform \(H^1\) bound. A sequence
\(\epsilon\downarrow0\) has a weak \(H^1\) limit. Its strong \(L^2\)
limit is \(v\), so uniqueness of distributional limits identifies the
weak limit with \(v\). This proves \(v\in H^1\), and (D6) proves the
local assertion. The argument also yields, for nested compact subcharts,

\[
 \|u\|_{H^1(K_0)}\leq C\bigl(\|Lu\|_{H^{-1}(V)}
                                      +\|u\|_{L^2(V)}\bigr),    \tag{D10}
\]

where \(K_0\Subset V\) and the local negative norm is taken after a
fixed cutoff equal to one near \(K_0\). A finite chart cover supplies
the same statement on \(X\). \(\square\)

\subsection{3. The second derivative and the local
estimate}\label{the-second-derivative-and-the-local-estimate}

\textbf{Regularity theorem.} Under (D1)--(D3), if
\(u,f\in L^2_{\mathrm{loc}}(X)\) satisfy (D4), then
\(u\in H^2_{\mathrm{loc}}(X)\).

By Section 2, \(u\in H^1_{\mathrm{loc}}\). In a chart the weak product
rule gives

\[
 -a^{jk}\partial_j\partial_k u
   =f+\bigl(\partial_j a^{jk}
                 +a^{jk}\partial_j\log\rho\bigr)\partial_k u.   \tag{D11}
\]

The parenthesized coefficients are locally bounded. Its right side is
now \(L^2_{\mathrm{loc}}\), and the left side is the weak nondivergence
principal operator of Section 8 of
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates}, with \(m=p=2\). The real part of
its principal symbol is positive in the sense (D1), so it is elliptic.
That result gives \(H^2_{\mathrm{loc}}\), with no differentiability of
the first-order coefficients beyond boundedness. Its quantitative local
estimate yields

\[
 \|u\|_{H^2(K)}\leq C_{K,V}
       \bigl(\|Lu\|_{L^2(V)}+\|u\|_{H^1(V)}\bigr),
                         \qquad K\Subset V\Subset X,             \tag{D12}
\]

using finitely many nested coordinate neighborhoods when necessary. Here
and below Sobolev norms on relatively compact regions may be defined
using any fixed finite chart partition; the choices give equivalent
norms. Applied after (D10) on a slightly larger region, (D12) also gives
the familiar interior estimate with \(\|u\|_{L^2}\) on that larger
region. The proof has established finiteness of \(H^1\) before using
(D11), and finiteness of \(H^2\) before applying any estimate to its
highest derivatives. \(\square\)

There is a useful support-preserving approximation consequence. If
\(u\in H^2_{\mathrm{loc}}(X)\) has compact support \(K\) inside an open
set \(V\), then there are \(u_\nu\in C_c^2(V)\), supported in one fixed
compact subset of \(V\), with

\[
 u_\nu\longrightarrow u\text{ in }H^2,
 \qquad Lu_\nu\longrightarrow Lu\text{ in }L^2.                  \tag{D13}
\]

To construct them, choose a finite \(C^2\) partition near \(K\),
subordinate to coordinate charts compactly contained in \(V\). Each
partitioned function has compact support in its chart. Convolve its zero
extension there and take the sum after returning to \(X\). The margin
between the supports and chart boundaries keeps all approximants in
\(V\). Chartwise mollification converges in \(H^2\), and multiplication
by a \(C^2\) cutoff is bounded on \(H^2\). Finally (D3) expanded as in
(D11) defines a bounded map from \(H^2\) to \(L^2\) on that fixed
compact region. This proves the second convergence. The same statement
holds with \(L^*\).

\subsection{4. Fixed-support coercivity and
continuation}\label{fixed-support-coercivity-and-continuation}

For the next three sections assume that \textbf{every connected
component of \(X\) is noncompact}. No uniform ellipticity constant on
all of \(X\) is required.

For a fixed compact \(K\subset X\), choose a relatively compact
neighborhood \(V\) of \(K\). Every \(v\in H^1(X)\) supported in \(K\)
satisfies

\[
                  \|v\|_{H^1}\leq C_K\|dv\|_{L^2}.               \tag{D14}
\]

Here \(|dv|\) is measured by an auxiliary \(C^1\) metric and the norm
can be computed within \(V\). For a proof, failure would give supported
\(v_\nu\) with \(\|v_\nu\|_2=1\) and \(\|dv_\nu\|_2\to0\). This sequence
is bounded in \(H^1\). Compactness on a finite chart cover gives a
strongly \(L^2\) convergent subsequence with limit \(v\), of norm one,
supported in \(K\), and with weak gradient zero. On every connected
component it is constant. A component has points outside \(K\), since it
is noncompact; near such a point the supported function is zero. Thus
every constant is zero, a contradiction. A compact set meets only
finitely many components because those components form an open cover of
it. This also justifies the finite chart argument if \(X\) is
disconnected.

The energy identity extends by \(H^1\) approximation from tests to
supported \(H^1\) functions:

\[
 \operatorname{Re}(L^*v,v)_\mu
       =\operatorname{Re}\int\overline{a^{jk}}\partial_kv
                          \overline{\partial_jv}\,d\mu
       \geq c_K\|dv\|_2^2.                                     \tag{D15}
\]

When the first entry is in \(H^{-1}\), the pairing in (D15) denotes its
dual action. Combining (D14) and (D15), and defining the negative norm
by duality with \(H_0^1(V)\), gives

\[
 \|v\|_{H^1}\leq C_K\|L^*v\|_{H^{-1}(V)}
                   \leq C_K'\|L^*v\|_2.                        \tag{D16}
\]

The last inequality is used only when \(L^*v\in L^2\). For compactly
supported \(C^2\) tests, (D12) and (D16) imply

\[
                   \|v\|_{H^2}\leq C_K''\|L^*v\|_2,
                         \qquad\operatorname{supp}v\subset K.   \tag{D17}
\]

The constants depend on a compact neighborhood of \(K\), its coefficient
bounds, and the fixed norm conventions. Formula (D17) is precisely the
compactness bound needed in the global construction.

The second ingredient is \textbf{support propagation}. If
\(w\in H^1_{\mathrm{loc}}(U)\), \(U\subset X\) is connected and open,
and \(L^*w=0\), then vanishing on a nonempty open subset of \(U\)
implies \(w=0\) throughout \(U\). Indeed (D11), conjugating the
principal coefficients, gives \(|P_2w|\leq C|Dw|\) on every compact
subchart. Its principal form has positive real part. Section 8 of
\href{support-continuation.pdf}{Curved weights and the directions in
which support can end} verifies every normal using the
positive-real-part path, and Section 7 of
\href{support-continuation.pdf}{Curved weights and the directions in
which support can end} gives continuation. These are local statements,
so coordinate neighborhoods propagate the conclusion along any path in
\(U\). This checks the complex coefficients and the density term in the
exact use of unique continuation.

\subsection{5. Filling the bounded complementary
components}\label{filling-the-bounded-complementary-components}

For a compact set \(K\subset X\), define its filled hull by adjoining
every connected component of \(X\setminus K\) whose closure in \(X\) is
compact:

\[
 \mathcal H(K)=K\ \cup\!
    \bigcup_{\substack{C\text{ component of }X\setminus K\\
                              \overline C\text{ compact}}} C.    \tag{D18}
\]

We retain the assumption that \(X\) has no compact connected components.
Then \(\mathcal H(K)\) is compact, and \(X\setminus\mathcal H(K)\) has
no relatively compact connected component. We prove compactness rather
than impose it as a further geometric condition.

Choose a relatively compact open neighborhood \(U\) of \(K\), with
\(K\subset U\). Such a neighborhood is a finite union of precompact
chart neighborhoods. Its boundary is compact and disjoint from \(K\).
Cover \(\partial U\) by finitely many connected open neighborhoods
\(W_1,\ldots,W_r\) disjoint from \(K\). Each \(W_i\) lies in one
component of \(X\setminus K\), so at most \(r\) components of that
complement meet \(\partial U\).

If a component \(C\) is relatively compact, the ambient connected
component \(Y\) of \(X\) containing it must meet \(K\). Otherwise
\(C=Y\), contradicting the noncompactness of \(Y\). If also
\(C\not\subset U\), choose \(x\in C\setminus U\) and join \(x\) to a
point of \(K\cap Y\) by a path in \(Y\). Up to its first meeting with
\(K\), the path remains in \(C\). It must cross \(\partial U\) before
meeting \(K\), since \(K\subset U\). Thus \(C\) is among the finitely
many components just found. Consequently every component added in (D18)
is contained either in \(U\) or in one of finitely many relatively
compact components. Their closures together with \(\overline U\) form a
compact set containing \(\mathcal H(K)\).

The hull is closed: its complement is the union of the other components
of \(X\setminus K\), each open by local connectedness. Therefore the
hull is compact. Those remaining components are exactly the components
of its complement and none is relatively compact, by their selection.
Notice that \(K\) need not have a smooth boundary, finitely many
boundary components, or even nonempty interior.

We obtain a sequence of compact sets

\[
 K_j\subset\operatorname{int}K_{j+1},\qquad
 X=\bigcup_{j\geq1}K_j,\qquad
 X\setminus K_j\text{ has no relatively compact component}.    \tag{D19}
\]

Indeed choose a countable cover by precompact coordinate neighborhoods.
Given \(K_j\), enclose it and the closures of the first \(j+1\) members
of the cover in a precompact open set \(U_{j+1}\), and set
\(K_{j+1}=\mathcal H(\overline U_{j+1})\). Start in the same way with
the first neighborhood. The hull contains \(\overline U_{j+1}\), so
\(K_j\) lies in its interior. The countable cover proves exhaustion.
This construction uses no regular-value theorem and preserves the
\(C^2\) setting. \(\square\)

\subsection{6. Extending an adjoint weight without disturbing the
interior}\label{extending-an-adjoint-weight-without-disturbing-the-interior}

Fix \(f\in L^2_{\mathrm{loc}}(X)\) and an exhaustion (D19). Write
\(\mathcal T(K)\) for the \(C_c^2(X)\) functions supported in \(K\).
Suppose \(j\geq3\) and a continuous positive function \(M\) on \(X\)
satisfies

\[
                    |(f,\phi)_\mu|\leq\|ML^*\phi\|_2
                          \quad(\phi\in\mathcal T(K_j)).         \tag{D20}
\]

For every \(\epsilon>0\) there is a continuous positive \(\widetilde M\)
such that

\[
 \begin{gathered}
 \widetilde M\geq(1+\epsilon)M\text{ on }X,
 \qquad \widetilde M=(1+\epsilon)M\text{ on }K_{j-2},\\
 |(f,\phi)_\mu|\leq\|\widetilde M L^*\phi\|_2
                          \quad(\phi\in\mathcal T(K_{j+1})).
 \end{gathered}                                                   \tag{D21}
\]

\textbf{Proof.} Choose \(\chi\in C^2(X)\), \(0\leq\chi\leq1\), equal to
zero on a neighborhood of \(K_{j-2}\) and equal to one outside
\(\operatorname{int}K_{j-1}\). Set

\[
                      M_N=(1+\epsilon)M+N\chi,\qquad N\geq1.    \tag{D22}
\]

If no \(M_N\) satisfies the last line of (D21), choose a violating
\(\phi_N\in\mathcal T(K_{j+1})\). Its pairing with \(f\) is nonzero;
multiply it by the reciprocal of the conjugate of that pairing to
arrange

\[
 (f,\phi_N)_\mu=1,\qquad \|M_NL^*\phi_N\|_2<1.                 \tag{D23}
\]

The positive minimum of \(M\) on \(K_{j+1}\) gives a uniform bound for
\(\|L^*\phi_N\|_2\). Formula (D17) then bounds \(\phi_N\) in \(H^2\),
all with support in that same compact set. Pass to a subsequence weakly
convergent in \(H^2\) and strongly in \(H^1\), with limit \(\Phi\). The
limit has support in \(K_{j+1}\), and \((f,\Phi)_\mu=1\), since \(f\) is
square integrable there. The bounded coefficient formula for
\(L^*:H^2\to L^2\) on a compact neighborhood gives weak \(L^2\)
convergence of \(L^*\phi_N\) to \(L^*\Phi\). Since \(M\) is bounded on
that neighborhood, weak lower semicontinuity gives

\[
                     \|(1+\epsilon)ML^*\Phi\|_2\leq1.          \tag{D24}
\]

On \(X\setminus K_{j-1}\), (D22) and (D23) give

\[
           \|L^*\phi_N\|_{L^2(X\setminus K_{j-1})}\leq N^{-1}.
                                                                    \tag{D25}
\]

Thus \(L^*\Phi=0\) on \(X\setminus K_{j-1}\). Every component \(C\) of
this open set is not relatively compact. It cannot be contained in the
compact set \(K_{j+1}\); hence \(C\setminus K_{j+1}\) is a nonempty open
set. There \(\Phi=0\). Continuation from Section 4 gives \(\Phi=0\)
throughout \(C\). This establishes

\[
                         \operatorname{supp}\Phi\subset K_{j-1}.
                                                                    \tag{D26}
\]

One must use continuation from an open zero set at this point:
\(\Phi|_C\) need not have compact support relative to \(C\), since its
support may approach \(\partial C\).

Since \(K_{j-1}\subset\operatorname{int}K_j\), approximation (D13) gives
\(\psi_\nu\in\mathcal T(K_j)\) converging to \(\Phi\) in \(H^2\). It
also gives \(ML^*\psi_\nu\to ML^*\Phi\) in \(L^2\), because \(M\) is
bounded on the fixed approximation support. Passing to the limit in
(D20) yields

\[
                 1=|(f,\Phi)_\mu|\leq\|ML^*\Phi\|_2
                                      \leq(1+\epsilon)^{-1},    \tag{D27}
\]

which is impossible. At least one \(M_N\) therefore works; take it as
\(\widetilde M\). \(\square\)

The argument explains the two inner compact sets in (D21). The weight
remains controlled on \(K_{j-2}\); the limiting adjoint solution is
supported in \(K_{j-1}\); the extra margin to \(K_j\) permits
approximation in the previously proved estimate.

\textbf{The weight step when compact components are present.} The
implication (D20)--(D21) holds on any manifold satisfying (D1)--(D3)
with an exhaustion (D19), even if compact connected components are
present. A filled exhaustion does not itself exclude them. Each compact
connected component \(Y\) must lie entirely in \(K_1\): otherwise a
component of \(Y\setminus K_1\) would be a relatively compact component
of \(X\setminus K_1\), contradicting (D19). There are finitely many such
\(Y\), because they all meet the compact set \(K_1\) and the components
of \(X\) are open. Their indicator functions \(1_Y\) belong to
\(C_c^2(X)\). Testing (D20) with \(1_Y\) shows that
\(\int_Y f\,d\mu=0\).

For any test \(\phi\), replace it by

\[
 \Pi\phi=\phi-\sum_{Y\text{ compact component}}
       \left(\frac{1}{\mu(Y)}\int_Y\phi\,d\mu\right)1_Y.
\]

For every \(r\geq1\), this operation preserves \(\mathcal T(K_r)\), the
residual \(L^*\phi\), and the pairing \((f,\phi)_\mu\). Its image has
mean zero on each compact component. The proof of (D14) applies to
supported functions in this image: a limit with zero gradient is
constant on each component, vanishes on each noncompact component by its
support, and vanishes on each compact component by its mean. Therefore
(D16)--(D17) hold for these projected functions. In the contradiction
argument, project each violating test before imposing (D23). The
projection leaves both quantities in (D23) unchanged, and its mean-zero
conditions pass to the weak limit. The remaining steps (D24)--(D27) are
unchanged; in particular, (D19) supplies the same nonempty open zero set
on every complementary component. This proves the weight extension in
its full filled-exhaustion setting. It does not construct an initial
estimate for data of nonzero mean on a compact component.

\subsection{7. Global existence on noncompact
components}\label{global-existence-on-noncompact-components}

\textbf{Solvability theorem.} Assume (D1)--(D3) and that every connected
component of \(X\) is noncompact. For every
\(f\in L^2_{\mathrm{loc}}(X)\) there exist a continuous function
\(M>0\), a function \(g\in L^2(X,d\mu)\) with \(\|g\|_2\leq1\), and
\(u=Mg\in H^2_{\mathrm{loc}}(X)\) satisfying \(Lu=f\).

\textbf{Construction of the weight.} By Cauchy--Schwarz and (D16),

\[
 |(f,\phi)_\mu|\leq\|f\|_{L^2(K_3)}\|\phi\|_2
                  \leq C_3\|f\|_{L^2(K_3)}\|L^*\phi\|_2
                    \quad(\phi\in\mathcal T(K_3)).               \tag{D28}
\]

Thus a positive constant function \(M_3\) larger than the last
coefficient establishes (D20) for \(j=3\). Choose positive numbers
\(\epsilon_j\), \(j\geq3\), with \(\sum_j\epsilon_j<\infty\), for
example \(\epsilon_j=2^{-j}\). Inductively use Section 6 to obtain
\(M_{j+1}\) from \(M_j\). Put

\[
 P_3=1,\qquad P_j=\prod_{k=3}^{j-1}(1+\epsilon_k),
 \qquad N_j=P_j^{-1}M_j.                                        \tag{D29}
\]

The positive numbers \(P_j\) increase to a finite \(P\): use
\(\log(1+t)\leq t\) for \(t\geq0\). Moreover \(N_{j+1}\geq N_j\)
everywhere and \(N_{j+1}=N_j\) on \(K_{j-2}\). On any fixed compact
neighborhood this sequence is therefore eventually stationary. Its limit
\(N\) is continuous and positive, since locally it is exactly one of the
continuous positive functions \(N_j\). Set \(M=PN\). Equivalently,
\(M_j\to M\) locally uniformly: after stationarity, only the scalar
\(P_j\) is changing. In particular \(M\geq M_j\), and every compactly
supported test belongs to \(\mathcal T(K_j)\) for some \(j\). We have
proved

\[
                  |(f,\phi)_\mu|\leq\|ML^*\phi\|_2
                                   \quad(\phi\in C_c^2(X)).    \tag{D30}
\]

\textbf{Dual construction of the solution.} Let \(E\) be the linear
subspace of \(L^2(X,d\mu)\) consisting of the vectors \(ML^*\phi\),
\(\phi\in C_c^2(X)\). Define a complex linear functional on it by

\[
                  F(ML^*\phi)=(\phi,f)_\mu.                    \tag{D31}
\]

If two tests give the same vector in \(E\), their difference has zero
right side in (D30), so (D31) is well-defined. Also
\(|F(h)|\leq\|h\|_2\). Hahn--Banach extends \(F\) to \(L^2\) with norm
at most one. Riesz representation gives \(g\in L^2\), \(\|g\|_2\leq1\),
such that \(F(h)=(h,g)_\mu\). Conjugating (D31) now gives the correct
pairing order:

\[
          (f,\phi)_\mu=(g,ML^*\phi)_\mu=(Mg,L^*\phi)_\mu.       \tag{D32}
\]

Since \(M\) is locally bounded, \(u=Mg\) belongs to
\(L^2_{\mathrm{loc}}\); (D32) is its equation (D4). Section 3 supplies
\(u\in H^2_{\mathrm{loc}}\). The earlier sentence that differentiation
of \(M\) was not justified was too weak: the actual recursion gives a
twice continuously differentiable weight, as the separately identified
strengthening in Section14 proves. Regularity comes from the equation,
even though the weight itself was only continuous. \(\square\)

The theorem gives existence for each datum. It does not assert
uniqueness, a globally bounded inverse on unweighted \(L^2\), a solution
satisfying prescribed boundary values, or a datum-independent choice of
\(M\). These distinctions matter when this theorem is used as a local
ingredient in a boundary or Fredholm problem.

\subsection{8. The exact compact-component
obstruction}\label{the-exact-compact-component-obstruction}

The local regularity theorem applies on compact manifolds too. The
unrestricted global existence assertion does not: on a compact connected
component \(Y\), the test \(\phi=1\) has compact support in \(X\) and
\(L^*1=0\). Thus a necessary condition is

\[
                              \int_Y f\,d\mu=0.                 \tag{D33}
\]

For the pure divergence operator (D3) this is also sufficient. We
include the proof to identify the entire obstruction rather than merely
give a counterexample.

Let \(H\) be the closed subspace of \(H^1(Y)\) with mean zero. On \(H\)
the gradient norm is equivalent to the full \(H^1\) norm. Indeed failure
of Poincaré would give a normalized sequence converging strongly in
\(L^2\) to a constant of mean zero, exactly as in the proof of (D14), a
contradiction. Define

\[
                  B(v,w)=\int_Y a^{jk}\partial_kv
                                      \overline{\partial_jw}\,d\mu.
                                                                    \tag{D34}
\]

Its boundedness follows from the bounded coefficients on \(Y\). Its real
part satisfies \(\operatorname{Re}B(v,v)\geq c\|v\|_H^2\) for some
\(c>0\). Fix any Hilbert norm on \(H\) equivalent to \(H^1\), with
pairing linear first. Riesz representation in the second entry supplies
a bounded operator \(T:H\to H\) with \((Tv,w)_H=B(v,w)\). Let
\(\|T\|\leq C\), where \(C\geq c\). For \(t=c/C^2\),

\[
 \|(I-tT)v\|_H^2
 \leq(1-2tc+t^2C^2)\|v\|_H^2
 =(1-c^2/C^2)\|v\|_H^2.                                       \tag{D35}
\]

Thus \(I-tT\) has norm less than one (possibly zero), and the geometric
series gives an inverse for \(T\). The bounded conjugate-linear
functional \(w\mapsto(f,w)_\mu\) on \(H\) is represented by a vector
\(z\in H\). Solve \(Tv=z\); then \(B(v,w)=(f,w)_\mu\) for every
\(w\in H\). Any \(H^1\) test differs from an element of \(H\) by a
constant. Both sides vanish on constants, the right side by (D33). Hence
\(Lv=f\) on \(Y\), and Section 3 gives \(v\in H^2(Y)\).

If \(Lw=0\) on \(Y\), local regularity and compactness give
\(w\in H^2(Y)\) whenever \(w\in L^2(Y)\). Its energy is zero, so (D2)
implies \(dw=0\); it is constant. The same proof applies to \(L^*\).
Consequently the adjoint obstruction is exactly the constants on each
compact connected component, and the normalized solution there is
unique.

We have proved the following complete form for a possibly disconnected
manifold:

\[
 \begin{split}
 &\exists u\in H^2_{\mathrm{loc}}(X):Lu=f\\
 &\hspace{1em}\Longleftrightarrow\quad
 \int_Y f\,d\mu=0\text{ for every compact connected component }Y.
 \end{split}                                                     \tag{D36}
\]

For sufficiency solve separately on compact components by (D34)--(D35),
and on noncompact components by Section 7. There are at most countably
many components by second countability. Every compact subset meets only
finitely many of them, so their componentwise solutions assemble to an
\(H^2_{\mathrm{loc}}\) function. Necessity was (D33).

For Euclidean open sets no component is compact, so (D36) gives
arbitrary-data solvability without an additional condition. For the
manifold formulation, the noncompact-component convention or the
obstruction in (D36) must be stated. An unqualified assertion of
arbitrary-data solvability on every boundaryless manifold would be
false, even for the Laplacian. This is a mathematical qualification of
that formulation, independent of any unstated convention in a reference.

\subsection{9. Worked models}\label{worked-models}

\textbf{Example 1: a complex Lipschitz coefficient and rapidly growing
data.} On \(\mathbb R\), take

\[
 a(x)=2+|x|+ix,\qquad \rho(x)=e^{\sin x},\qquad
 f(x)=e^{x^2}(1+i\sin x).                                       \tag{D37}
\]

The real part of \(a\) is at least two, but \(a\) is not differentiable
at zero. Define

\[
 F(x)=\int_0^x\rho(s)f(s)\,ds,\qquad
 u(x)=-\int_0^x\frac{F(t)}{\rho(t)a(t)}\,dt.                     \tag{D38}
\]

These integrals are finite on every compact interval. Since \(F\) is
continuously differentiable and \(1/(\rho a)\) is locally Lipschitz,
\(u'=-F/(\rho a)\) is locally Lipschitz and \(u\in H^2_{\mathrm{loc}}\).
Its flux is \(\rho a u'=-F\), so differentiating once gives
\(-\rho^{-1}(\rho a u')'=f\), including across zero. No pointwise value
of \(a'(0)\) is required. Arbitrary constants may be added both to \(F\)
and to \(u\), producing the two local homogeneous degrees of freedom.
The theorem accommodates the growth of \(f\) because it asks for local,
rather than global, square integrability.

\textbf{Example 2: bounded in coordinates need not mean relatively
compact.} Let \(X=\mathbb R^2\setminus\{0\}\) and
\(K=\{x:1\leq|x|\leq2\}\). Its inner complementary component \(0<|x|<1\)
is bounded in \(\mathbb R^2\) but is not relatively compact in \(X\): a
sequence tending to the missing origin has no convergent subsequence in
\(X\). The outer complementary component is also not relatively compact.
Thus \(\mathcal H(K)=K\) in this manifold. The distinction explains why
the support argument is expressed intrinsically, with compact closure in
\(X\), rather than by a Euclidean size test. For any later compact
\(K'\subset X\), the inner component has points outside \(K'\), as the
continuation step requires.

\textbf{Example 3: mixed compact and noncompact components.} Let
\(X=\mathbb R^2\sqcup\mathbb T^2\), use the standard densities, and
choose \(A=(1+i)I\) on both components; both torus coordinates have
period \(2\pi\). Give the plane the datum \(f_1(x)=1\), and the torus
the datum \(f_2(\theta)=\cos\theta_1\). A solution is

\[
 u_1(x)=-\frac{x_1^2}{2(1+i)},\qquad
 u_2(\theta)=\frac{\cos\theta_1}{1+i}.                            \tag{D39}
\]

Each is locally \(H^2\); direct differentiation verifies the equation.
Replacing \(f_2\) by \(1+\cos\theta_1\) makes its integral nonzero and
destroys solvability on that component, while the plane equation remains
solvable. A compact-component obstruction is a condition on that
component's datum, not on the behavior at infinity of other components.

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

\textbf{Problem 1.} On the line let \(a(x)=2+|x|\) and \(u(x)=|x|\).
Compute \(-\partial_x(a\partial_xu)\) as a distribution. Verify that its
local regularity is compatible with Section 2 but does not trigger
Section 3.

\textbf{Solution.} Almost everywhere \(u'=\operatorname{sgn}x\), and
\(au'=2\operatorname{sgn}x+x\). Since the distributional derivative of
\(\operatorname{sgn}x\) is \(2\delta_0\),

\[
                       -\partial_x(au')=-4\delta_0-1.           \tag{D40}
\]

The Dirac mass belongs to \(H^{-1}_{\mathrm{loc}}(\mathbb R)\). For
instance, for a smooth compactly supported \(\psi\), integrate
\((|\psi|^2)'\) from an endpoint beyond its support to zero; this gives
\(|\psi(0)|^2\leq2\|\psi\|_2\|\psi'\|_2\leq\|\psi\|_{H^1}^2\).
Evaluation therefore extends continuously to \(H^1\). Thus (D40) has the
negative regularity needed for the first gain. In fact
\(|x|\in H^1_{\mathrm{loc}}\). But \(u''=2\delta_0\) is not represented
by an \(L^2\) function, so \(u\notin H^2_{\mathrm{loc}}\) near zero. The
datum in (D40) is likewise not \(L^2_{\mathrm{loc}}\). The two stages of
the theorem have different data assumptions, and this example
distinguishes them.

\textbf{Problem 2.} Start with \(L=-\partial_x^2\), Lebesgue density,
and the coordinate change \(y=F(x)=x+x^3\). Determine the transformed
density and tensor coefficient and check the first-order term of the
transformed operator.

\textbf{Solution.} Write \(x=G(y)=F^{-1}(y)\). Since \(F'=1+3x^2>0\),
this is a global smooth coordinate change. The density and coefficient
are

\[
 \rho_y(y)=\frac1{F'(G(y))},\qquad
 a_y(y)=F'(G(y))^2.                                             \tag{D41}
\]

Thus \(\rho_y a_y=F'\), with the right side evaluated at \(G(y)\). The
divergence formula gives

\[
 -\rho_y^{-1}\partial_y(\rho_y a_y\partial_y)
           =-F'^2\partial_y^2-F''\partial_y.                    \tag{D42}
\]

Here \(\partial_y F'(G(y))=F''/F'\). The same result follows from
\(\partial_x=F'\partial_y\), applied twice. If the density were omitted
and one wrote \(-\partial_y(a_y\partial_y)\), the first-order
coefficient would instead be \(-2F''\), giving a different operator.
This is why a divergence formula with coordinate-dependent coefficients
needs its density convention.

\textbf{Problem 3.} For a fixed real \(\kappa\), consider on
\(\mathbb R^2\) the form
\(q(\xi)=(1+i\kappa)\xi_1^2+(2-i\kappa)\xi_2^2\). Check the normal-root
condition for \(N=e_2\), then explain why it holds for every real normal
without choosing roots continuously over all normals.

\textbf{Solution.} For \(\xi=e_1\) the line polynomial is
\((1+i\kappa)+(2-i\kappa)z^2\). Its two roots are opposite, nonzero
numbers. Neither is real, since a real \(z\) would give real part
\(1+2z^2>0\). Hence one lies in each half-plane and the roots are
distinct. For arbitrary independent real \(\xi,N\), use
\(q_t=\operatorname{Re}q+it\operatorname{Im}q\), \(0\leq t\leq1\). The
leading coefficient \(q_t(N)\) is nonzero and the line polynomial has no
real root throughout the path. At \(t=0\) its real coefficients yield a
nonreal conjugate pair. Continuity of the unordered pair of roots
preserves the number in each half-plane, so the same is true at \(t=1\).
When \(\xi\) is parallel to \(N\), the root corresponds to the zero
argument and is excluded from the normal criterion. The path argument is
pointwise in the chosen pair and requires no global root labeling.

\textbf{Problem 4.} Determine solvability of \(-u''=1\) on \(\mathbb R\)
if the desired solution is (a) locally \(H^2\), (b) globally \(L^2\).
Compare with the same equation on the circle. Explain what the global
theorem actually supplies.

\textbf{Solution.} Integrating distributionally twice gives every
solution on the line as \(u(x)=-x^2/2+cx+d\), where \(c,d\in\mathbb C\).
These functions are smooth and therefore locally \(H^2\). None is
globally \(L^2\): its quadratic leading coefficient cannot be canceled
by the affine part, so for sufficiently large \(|x|\),
\(|u(x)|\geq |x|^2/4\). On the circle, integration against the constant
test gives \(0=\int1\), an impossibility. Thus arbitrary-data
local-Sobolev existence on a noncompact component gives neither
unweighted global integrability nor arbitrary-data existence on a
compact component. In the weighted representation \(u=Mg\), one may have
\(g\in L^2\) precisely because the allowed \(M\) grows; boundedness of
\(M\) was never asserted.

\textbf{Problem 5.} In the weight construction, replace
\(\epsilon_j=2^{-j}\) by \(\epsilon_j=1/j\), \(j\geq3\). Show why the
argument no longer produces a finite continuous weight on the previously
treated interior. Does this refute the solvability theorem?

\textbf{Solution.} On \(K_1\) every step \(j\geq3\) satisfies the exact
equality \(M_{j+1}=(1+1/j)M_j\), since \(K_1\subset K_{j-2}\). Hence
there

\[
              M_j=M_3\prod_{k=3}^{j-1}\frac{k+1}{k}
                               =\frac j3M_3.                   \tag{D43}
\]

The initial weight is strictly positive, so this tends to infinity at
every point of \(K_1\). Normalized weights still stabilize there, but
the compensating scalar product does not have a finite limit. The
argument therefore needs a bounded product of interior enlargement
factors, supplied by summability of the positive \(\epsilon_j\). This
failure concerns that choice of iterative construction. Choosing the
summable sequence used in Section 7 proves the theorem and avoids the
divergence.

\textbf{Problem 6.} Explain concretely why one cannot replace the filled
exhaustion by an arbitrary compact exhaustion in the continuation step.
Use the Laplacian on \(\mathbb R^2\) and a compact annulus.

\textbf{Solution.} Take \(K=\{1\leq|x|\leq2\}\) and a smooth radial
function \(\Phi\) equal to one on \(|x|\leq1\) and zero on \(|x|\geq2\),
with its transition strictly between these radii. Then
\(\operatorname{supp}(-\Delta\Phi)\subset K\). Thus \(-\Delta\Phi=0\) on
\(X\setminus K\), and \(\Phi\) has compact support. But \(\Phi\) is
nonzero on the bounded complementary component \(|x|<1\). Continuation
does not force zero there, because there is no nonempty open zero set
within that connected component. The support conclusion
\(\operatorname{supp}\Phi\subset K\) would be false. Filling the inner
disk replaces \(K\) by \(\{|x|\leq2\}\), for which the support
conclusion is correct. The failure occurs in the support inference, not
in local regularity or the energy estimate.

\subsection{11. The manifold geometry used by the
equation}\label{the-manifold-geometry-used-by-the-equation}

We prove all the geometry used in Section 1 on its original Hausdorff,
second-countable \(C^2\) manifold without boundary and of dimension
\(n\geq1\); the notation below retains those original objects. If \(X\)
is empty, the empty partition has sum one at every point vacuously, the
unique empty metric and density have all the stated positivity and
regularity properties, every compact subset is empty and has zero
cutoff, and there are no components. All the assertions therefore hold
in that case. For the remaining construction assume \(X\) is nonempty.
The proof uses the original coordinate compactness and finite scalar
calculus; it does not select a smoother atlas.

\subsubsection{11.1. Countable compact exhaustion from the original
charts}\label{countable-compact-exhaustion-from-the-original-charts}

Every point has a coordinate neighborhood and a smaller coordinate ball
whose closed ball is contained in the coordinate image. Its inverse
image is compact, since the inverse chart is continuous and the closed
Euclidean ball is compact. It is closed in \(X\), since \(X\) is
Hausdorff. Its interior is an open neighborhood of the point. This
proves local compactness and gives an open cover by relatively compact
coordinate balls.

Here is the countable-subcover argument used in the construction. Fix
the given countable topological base. For each base member contained in
some member of the cover, choose one such cover member. This gives at
most countably many chosen members. Any point of the covered space
belongs to a base neighborhood contained in a cover member, so belongs
to one of the chosen members. The chosen family covers \(X\). This
argument applies to every open cover, with no countability assumption on
that cover.

Write the resulting relatively compact coordinate balls as
\(V_1,V_2,\ldots\), allowing a finite list when it is finite. Put
\(K_0=K_{-1}=\varnothing\). Choose \(K_1=\overline V_1\). Given compact
\(K_j\), cover it by finitely many relatively compact coordinate balls
and take the union of their compact closures together with the compact
closures of the first \(j+1\) members of the original cover. Call this
finite union \(K_{j+1}\). It is compact; the open balls covering \(K_j\)
show \(K_j\subset\operatorname{int}K_{j+1}\). Including the first
\(j+1\) original balls makes the interiors exhaust \(X\). In a finite
cover, include every available member once the index exceeds its length;
the same construction works, and may become constant at \(X\) if \(X\)
is compact. Thus \[
 K_j\subset\operatorname{int}K_{j+1},\qquad
 X=\bigcup_{j\geq1}\operatorname{int}K_j .
                                                               \tag{MG1}
\] There is no connectedness requirement. Each compact subset of \(X\)
lies in the interior of some \(K_N\), by a finite subcover of these
increasing interiors.

Define the original compact shells and their open neighborhoods by \[
 A_j=K_j\setminus\operatorname{int}K_{j-1},\qquad
 W_j=\operatorname{int}K_{j+1}\setminus K_{j-2}.
                                                               \tag{MG2}
\] The shell is compact, and \(A_j\subset W_j\): it lies in
\(\operatorname{int}K_{j+1}\), while
\(K_{j-2}\subset\operatorname{int}K_{j-1}\) makes it disjoint from
\(K_{j-2}\). Every point lies in some shell, by choosing the first index
for which it belongs to \(K_j\).

\subsubsection{\texorpdfstring{11.2. Compactly supported \(C^2\)
partition subordinate to any open
cover}{11.2. Compactly supported C\^{}2 partition subordinate to any open cover}}\label{compactly-supported-c2-partition-subordinate-to-any-open-cover}

Let \(\mathcal O\) be the original open cover. For every point of
\(A_j\), choose a coordinate ball with closed outer ball inside \(W_j\),
inside one assigned member \(O\in\mathcal O\), and inside its original
chart. Choose concentric coordinate balls of radii \(r,2r,3r\), with the
closed radius-\(3r\) ball inside this open intersection. The
radius-\(r\) balls cover \(A_j\); compactness selects finitely many,
indexed by \(r'\in\{1,\ldots,N_j\}\). Denote the smaller balls by
\(V_{j,r'}\) and the radius-\(3r\) balls by \(U_{j,r'}\). The chosen
finite families together cover \(X\), and
\(\overline U_{j,r'}\subset W_j\cap O_{j,r'}\). Empty shells need no
balls.

This family is locally finite. If \(x\in\operatorname{int}K_N\), that
same open set is a neighborhood meeting none of the \(U_{j,r'}\) with
\(j\geq N+2\), because their \(W_j\)'s exclude \(K_{j-2}\supset K_N\).
Only finitely many shells remain, and each has finitely many balls. This
proves local finiteness on the original space, rather than assuming a
paracompactness theorem.

To construct the needed coordinate functions, put \[
 \theta(t)=
 \begin{cases}e^{-1/t},&t>0,\\0,&t\leq0,\end{cases}
 \qquad
 s(t)=\frac{\theta(t)}{\theta(t)+\theta(1-t)} .
                                                               \tag{MG3}
\] For \(t>0\), each derivative of \(\theta\) is a finite polynomial in
\(1/t\) times \(e^{-1/t}\), by induction using the product and chain
rules. Every such expression tends to zero as \(t\downarrow0\): for each
positive integer \(k\), the exponential series gives \(e^u\geq u^k/k!\),
and choosing \(k\) larger than the polynomial degree proves the limit.
Thus extension by zero makes \(\theta\) smooth with all derivatives zero
at zero. The denominator in (MG3) is positive for every real \(t\);
\(s\) is smooth, equals zero for \(t\leq0\), equals one for \(t\geq1\),
and lies between zero and one.

In the original chart of a selected ball with center \(a\) and small
radius \(r\), define \[
 \beta_{j,r'}(x)
   =s\!\left(\frac{(2r)^2-|x-a|^2}{(2r)^2-r^2}\right).
                                                               \tag{MG4}
\] It equals one on the closed radius-\(r\) ball, and is zero outside
the radius-\(2r\) ball. Its support is compact inside \(U_{j,r'}\).
Composing with the \(C^2\) chart and extending by zero gives a \(C^2\)
function on \(X\): the support has an open neighborhood inside the
chart, and the function is identically zero near every point outside
that compact support. This retains the manifold's \(C^2\) structure and
introduces no unmentioned smoother atlas.

Let \(I\) denote these countably many indices and put \[
 S(x)=\sum_{i\in I}\beta_i(x),\qquad
 \phi_i(x)=\frac{\beta_i(x)}{S(x)},\qquad
 \sum_{i\in I}\phi_i(x)=1 .
                                                               \tag{MG5}
\] Each sum is a finite sum on a neighborhood of each point by local
finiteness. The inner balls cover \(X\), so \(S(x)\geq1\). The quotient
is therefore \(C^2\). The \(\phi_i\)'s are nonnegative, their compact
supports stay in the original assigned cover members, and their family
is locally finite. Their exact derivatives in any original chart are \[
 \begin{split}
 \partial_a\phi_i
 &=\frac{\partial_a\beta_i}{S}
       -\frac{\beta_i\partial_aS}{S^2},\\
 \partial_a\partial_b\phi_i
 &=\frac{\partial_a\partial_b\beta_i}{S}
 -\frac{(\partial_a\beta_i)(\partial_bS)
        +(\partial_b\beta_i)(\partial_aS)
        +\beta_i\partial_a\partial_bS}{S^2}
 +\frac{2\beta_i(\partial_aS)(\partial_bS)}{S^3}.
 \end{split}                                                   \tag{MG6}
\] All sums and all denominator factors remain present. On a compact
subchart only finitely many supports meet the compact set, so every
derivative through order two has a finite supremum there. This proves
precisely the subordinate locally finite \(C^2\) partition asserted by
the original contract.

\subsubsection{11.3. The original compact cutoff and positive
local-radius
minorant}\label{the-original-compact-cutoff-and-positive-local-radius-minorant}

If a compact \(K\subset X\) lies in an open \(O\), apply the partition
construction to the cover \(\{O,X\setminus K\}\). A locally finite
family meets a compact set in only finitely many members: cover the
compact set by finitely many neighborhoods each meeting finitely many
supports. Take the finite collection \(J\) of supports assigned to \(O\)
that meet \(K\), and define \(\chi=\sum_{i\in J}\phi_i\). Then
\(0\leq\chi\leq1\), \(\chi\in C_c^2(O)\), and its support is contained
in the finite union of the selected original compact supports.

In fact \(\chi=1\) on a neighborhood of \(K\), not just on \(K\). Every
unselected support is disjoint from \(K\): a support assigned to
\(X\setminus K\) is contained there, and every other support disjoint
from \(K\) was not selected. A locally finite union of closed sets is
closed, because near each point that union is a finite closed union. The
union of the unselected supports is therefore closed and misses \(K\).
Its open complement is a neighborhood of \(K\); there every unselected
\(\phi_i\) vanishes, so (MG5) gives \(\chi=1\). If \(K=\varnothing\),
use \(\chi=0\). This proves every required compact cutoff.

There is also a useful proved consequence for the local radii used in
the continuation and kernel constructions. Let \(R:X\to(0,\infty)\) have
a positive lower bound on some neighborhood of every point, as does a
positive lower-semicontinuous function. Choose such neighborhoods
\(O_i\), with constants \(a_i>0\) for which \(R(x)\geq a_i\) on \(O_i\),
and take the proved subordinate partition, assigning each support its
original \(a_i\). Then \[
 r(x)=\sum_i\phi_i(x)\frac{a_i}{2},\qquad
 0<r(x)\leq\frac{R(x)}2<R(x).
                                                               \tag{MG7}
\] The sum is locally finite and \(C^2\), and at least one positive
summand occurs at every point. Each term active at \(x\) has
\(a_i\leq R(x)\), and summing with (MG5) proves the upper bound. No
positive global lower bound or compactness of \(X\) is assumed.

\subsubsection{11.4. Positive metric and the exact density
transition}\label{positive-metric-and-the-exact-density-transition}

Take a locally finite partition as above subordinate to the original
coordinate neighborhoods. In each such neighborhood with coordinates
\(x_i^1,\ldots,x_i^n\), retain the coordinate metric
\(g_i=\sum_{a=1}^n dx_i^a\otimes dx_i^a\). Its coefficients in another
\(C^2\) chart \(y\) are \(C^1\), since \[
 (g_i)_{ab}(y)
   =\sum_{c=1}^n
        \frac{\partial x_i^c}{\partial y^a}
        \frac{\partial x_i^c}{\partial y^b}.
\] The chart derivative is invertible: the chain rule for the original
chart and its \(C^2\) inverse gives both inverse matrix products. Hence
\(g_i(v,v)=\sum_c|dx_i^c(v)|^2>0\) for every nonzero real tangent vector
\(v\) in that neighborhood.

Extend each \(\phi_i g_i\) by zero and define \(g=\sum_i\phi_i g_i\).
The compact support inside its chart makes each extension \(C^1\); local
finiteness makes the sum \(C^1\). At a point and a nonzero tangent
vector, some \(\phi_i\) is positive and its \(g_i(v,v)\) is positive,
while all other terms are nonnegative. Thus \(g\) is a positive \(C^1\)
Riemannian metric on the original manifold. It need not coincide with
any metric already chosen in the divergence equation.

Write its matrix in coordinates \(x\) as \(G_x\), and keep the full
density \[
 d\mu_g=\sqrt{\det G_x(x)}\,|dx^1\cdots dx^n|.
                                                               \tag{MG8}
\] The determinant is a finite polynomial in the matrix entries. It is
positive for a positive real symmetric matrix: the real spectral theorem
gives positive eigenvalues whose product is the determinant. The
ordinary scalar square root is \(C^1\) on the positive axis, so the
coefficient in (MG8) is positive and \(C^1\).

On an overlapping original chart \(y=\kappa(x)\), put \(J=Dx/Dy\). The
tensor chain rule gives \(G_y=J^TG_xJ\), including both matrix factors
in their displayed order. Taking determinants yields \[
 \det G_y=(\det J)^2\det G_x,\qquad
 \sqrt{\det G_y}
   =|\det J|\sqrt{\det G_x}.
                                                               \tag{MG9}
\] This is exactly the density transition factor. It patches the
coefficients in (MG8) to a density, including orientation-reversing
changes of coordinates and nonorientable manifolds. No sign of a
determinant is chosen or discarded.

Every compact subchart has positive finite lower and upper bounds for
this density coefficient, and finite bounds for its first derivatives.
For the metric, the continuous function \(g_x(v,v)\) on the compact
subchart times the original Euclidean unit sphere has a positive minimum
and a finite maximum; its coefficient first derivatives also have finite
suprema. These are precisely the local bounds used by the divergence
chapter. The constructed density proves existence; the original
separately fixed positive density \(\rho(x)\,dx\) in (D3) is retained,
and is not silently replaced by \(\mu_g\). The same compactness argument
gives local bounds for that original positive \(C^1\) coefficient.

\subsubsection{11.5. Components and exact receiving
scope}\label{components-and-exact-receiving-scope}

Coordinate balls are path connected: the inverse chart applied to the
straight segment in the original ball gives a path between its points.
They form a neighborhood base. Thus \(X\) is locally path connected.
Local compactness was proved in Section 11.1.

A path component \(P\) is open: any point in it has a path-connected
coordinate ball, and concatenation of its path with paths in that ball
places the whole ball in \(P\). The other path components are open for
the same reason. A path is connected, because any separation of its
image pulls back to a separation of the interval; the interval's
connectedness follows from completeness and the intermediate-value
theorem. Consequently a path component is connected: it is the union of
the connected images of paths through its fixed point, and a union of
connected sets sharing that point cannot be separated.

The connected component through a point contains its path component. If
it contained a second path component, its intersection with the first
and with the union of the others would be a separation into two nonempty
relatively open sets. This is impossible. Thus each connected component
equals a path component, is open, and is path connected, exactly as the
original contract states.

These proofs supply the geometry entry needed for the finite chart
arguments in (D4)--(D10), the compact cutoff in the energy proof, the
local metric and density bounds, the connected component alternatives
and the exhaustion/partition steps in the global divergence
construction. Section 12 separately proves nonlinear substitution for
the completed coordinate measure and the exact weak Sobolev chain maps
under the original \(C^2\) coordinates, together with their receiving
density and adjoint formulas. The original contract's statement and the
original \(\rho\), \(A\), operator signs and Hilbert pairing are
unchanged.

For a \(C^r\) manifold, integer \(r\geq2\), this same proof gives
\(C^r\) partitions, compact cutoffs and local-radius minorants, and a
positive \(C^{r-1}\) metric and density. Every transition calculation
uses exactly one derivative of the original \(C^r\) chart; every other
step preserves its \(C^r\) regularity. This proved regularity extension
is an identified consequence, with no novelty claim. For a smooth atlas
this same construction is smooth at every order; local finiteness makes
every derivative a finite sum. These are identified regularity
consequences of the complete construction, without a novelty claim.

\subsection{12. The exact coordinate maps in the weak
equation}\label{the-exact-coordinate-maps-in-the-weak-equation}

The geometry in Section 11 supplies the original charts and cutoffs.
This section supplies the analytic maps asserted in Section 1.
Throughout, \(n\geq1\), \(F:U\to V\) is a \(C^2\) diffeomorphism between
the original open subsets of \(\mathbb R^n\), \(G=F^{-1}\), and \[
 P_{ai}(x)=\partial_{x_i}F^a(x),\qquad
 Q_{ia}(y)=\partial_{y_a}G^i(y),\qquad
 J_F(x)=|\det P(x)|,\quad J_G(y)=|\det Q(y)|.
                                                               \tag{CX1}
\] The chain rule for the two actual inverse maps proves both matrix
products and the full determinant identities: \[
 \sum_aQ_{ia}(F(x))P_{aj}(x)=\delta_{ij},\quad
 \sum_iP_{ai}(G(y))Q_{ib}(y)=\delta_{ab},\quad
 J_G(F(x))J_F(x)=1,\quad J_F(G(y))J_G(y)=1.                       \tag{CX2}
\] No orientation is chosen. Empty chart domains give empty maps and
zero integrals; below the domains are nonempty.

\subsubsection{12.1. Substitution for the completed original coordinate
measures}\label{substitution-for-the-completed-original-coordinate-measures}

The completed-measure substitution theorem also has a full proof for C¹
diffeomorphisms in
\href{conormal-transmission.pdf}{Singularities along
a submanifold, Section 16.9}. The argument below is an alternative proof
using contraction on each original cell. The C² coordinate assumption
enters the later second weak derivatives; the measure argument uses only
C¹ regularity of both inverse maps. Its identity then enters the weak
derivative, density, tensor and weak equation calculations in Sections
12.2--12.4.

We first prove the measure identity, rather than assume nonlinear
substitution in a Sobolev approximation argument. An invertible real
matrix \(T\) acts on the original completed coordinate measure by \[
             \lambda_n(TE)=|\det T|\lambda_n(E).                 \tag{CX3}
\] Here is a proof retaining the actual matrix factor. For a coordinate
shear \(x_i\mapsto x_i+c x_j\), with \(i\ne j\), fix the other \(n-1\)
coordinates. Every one-dimensional section is translated, so its length
is unchanged. Integration of these sections, first for nonnegative Borel
indicators, proves preservation of measure. A coordinate permutation
preserves iterated product measure; reflection preserves length;
multiplication of one coordinate by a nonzero scalar \(c\) multiplies
length by \(|c|\). One-dimensional length has these properties directly
from the original interval lengths and their completed-measure
uniqueness. Gaussian elimination expresses every actual invertible \(T\)
as a finite ordered product of these invertible elementary matrices: at
column \(j\), choose a nonzero pivot in the remaining rows (failure
would make the remaining columns linearly dependent), exchange its row
with row \(j\), divide that row by the nonzero pivot, and subtract its
multiples from every other row. After \(n\) steps the actual product
\(E_N\cdots E_1T\) is \(I\). Thus \(T=E_1^{-1}\cdots E_N^{-1}\), in that
order. Applying the preceding section calculations successively
multiplies measure by \(\prod_{r=1}^N|\det E_r^{-1}|=|\det T|\). Every
elementary factor and the determinant of the original \(T\) are present
in this comparison.

A map with Lipschitz constant \(L\) in the coordinate maximum norm sends
null sets to null sets on any cube on which that bound holds. Indeed a
cube of side \(s\) maps into a cube of side \(Ls\), by centering at its
center and bounding every image coordinate by \(Ls/2\). A null subset
has cube covers with total original volume as small as desired; the
image outer measure is at most \(L^n\) times that total. Cubes in these
covers can be taken within a slightly larger fixed cube contained in the
domain: intersect with the interior region and subdivide into
sufficiently small dyadic cubes before taking the cover. Thus the bound
applies to every covering cube. A \(C^1\) map is Lipschitz on a closed
cube compactly inside its domain, by integrating its derivative on the
segments of that cube and using the maximum row-sum matrix norm. Such
cube interiors have a countable subcover of the domain. This proves
local null-set transport for \(F\), and for \(G\), including both
directions. In particular the images of all coordinate cube faces have
measure zero.

Fix a closed cube \(C\Subset U\), and subdivide it into congruent small
cubes \(C_k\), of side \(s\) and centers \(a_k\). Their images meet only
along images of faces, since \(F\) is injective. Put \(T_k=DF(a_k)\).
The inverse matrices \(T_k^{-1}\) have a uniform finite maximum row-sum
bound: the cofactor formula has a nonvanishing determinant on the
compact cube. Uniform continuity of \(DF\) gives numbers
\(\epsilon_s\to0\), such that on each cell \[
 \left\|T_k^{-1}\bigl(DF(x)-T_k\bigr)\right\|_\infty
 \leq\epsilon_s .
\] In the following calculation retain \(F\) and \(T_k\) explicitly.
Define the exact error \[
 h_k(x)=T_k^{-1}\bigl(F(x)-F(a_k)\bigr)-(x-a_k).
\] Then \(h_k(a_k)=0\), its Lipschitz constant on \(C_k\) is at most
\(\epsilon_s\), and \(|h_k(x)|_\infty\leq\epsilon_s s/2\). Hence \[
 F(C_k)\subset
 F(a_k)+T_k[-(1+\epsilon_s)s/2,(1+\epsilon_s)s/2]^n .
\] For the reverse inclusion suppose \(\epsilon_s<1\), and fix
\(|z|_\infty\leq(1-\epsilon_s)s/2\). The map \(x\mapsto a_k+z-h_k(x)\)
sends the closed original cell to itself and contracts distances by
\(\epsilon_s\). Starting at any point, its consecutive differences are
bounded by a geometric sequence. Completeness of the closed cube gives a
limit; continuity gives a fixed point, and the contraction estimate
makes it unique. At that point \(F(x)=F(a_k)+T_kz\). Thus \[
 \begin{split}
 &F(a_k)+T_k[-(1-\epsilon_s)s/2,(1-\epsilon_s)s/2]^n
       \subset F(C_k),\\
 &|\det T_k|(1-\epsilon_s)^n s^n
       \leq\lambda_n(F(C_k))
       \leq|\det T_k|(1+\epsilon_s)^n s^n .
 \end{split}                                                   \tag{CX4}
\] These inequalities use (CX3) with the entire original derivative
matrix. Summing over the cells is valid because their boundary images
are null. The continuous function \(J_F\) has Riemann sums
\(\sum_k|\det T_k|s^n\) tending to its integral on \(C\): upper and
lower step sums differ by at most its uniform oscillation times
\(\lambda_n(C)\). Both factors \((1-\epsilon_s)^n\) and
\((1+\epsilon_s)^n\) tend to one. Therefore \[
              \lambda_n(F(C))=\int_C J_F(x)\,dx.               \tag{CX5}
\]

For completeness this cube calculation determines the full Borel
measure. Define \(\nu(E)=\lambda_n(F(E))\) on Borel subsets of \(U\),
and \(\sigma(E)=\int_E J_F\,dx\). The homeomorphism makes images Borel,
and injectivity makes \(\nu\) countably additive. Both measures are
finite on compactly contained cubes and give zero measure to all dyadic
faces. Every open subset \(O\) of \(U\) is covered, off the countable
union of those faces, by disjoint interiors of dyadic cubes with closure
in \(O\). To see this without a maximal cube at infinity, take integer
levels \(r=0,1,\ldots\). At level \(r\), choose all cells whose closure
lies in \(O\) and whose interior has not already been selected at a
preceding level. Dyadic nesting makes their interiors disjoint. Every
point off the faces eventually lies in such a cell, because its distance
from the complement is positive in some small neighborhood. Equation
(CX5) and countable additivity show \(\nu(O)=\sigma(O)\).

Equality on open sets implies equality on Borel sets here as follows.
Both measures are regular on a relatively compact open \(W\Subset U\).
For \(\nu\), transport compact inner approximation and open outer
approximation for the original coordinate measure through the
homeomorphism \(F:W\to F(W)\). The image \(F(W)\) is bounded after
replacing \(W\) by a relatively compact cube interior, so these
approximations concern finite measure. For \(\sigma\), \(J_F\) is
bounded above on the closure of that cube: the error of a
coordinate-measure open or compact approximation is bounded by that
supremum times the original measure error. Restricting to a cube
interior gives regularity there for both measures. For a Borel subset
\(E\) of that interior, the outer approximating relative open sets have
equal measures for \(\nu\) and \(\sigma\); taking their infima gives
equality on \(E\). Cover \(U\) by countably many relatively compact cube
interiors, disjointizing that cover by removing the preceding members.
The resulting Borel pieces remain subsets of individual interiors, so
countable additivity gives equality for all Borel \(E\subset U\).

The previously proved null-set transport extends this equality to the
completed coordinate measures: a completed-measurable set is a Borel set
modified within a Borel null set, and its image has the corresponding
Borel image and null modification. The inverse map gives the converse.
Approximating nonnegative measurable functions by simple functions, and
then using monotone convergence, proves the exact original substitution
law \[
 \int_V H(y)\,dy=\int_U H(F(x))J_F(x)\,dx                       \tag{CX6}
\] for every nonnegative measurable \(H\), allowing \(+\infty\).
Applying it to \(|H|\) gives equivalence of absolute integrability; the
real and imaginary positive/negative parts then prove (CX6) for complex
integrable \(H\). The analogous identity for \(G\) retains \(J_G\). Null
sets in either chart are null in the other, so all compositions below
are well-defined on actual almost-everywhere equivalence classes.

\subsubsection{12.2. The full first and second weak derivative
maps}\label{the-full-first-and-second-weak-derivative-maps}

Let \(u\in H^2_{\mathrm{loc}}(V)\), with the original complex-valued
weak derivatives, and \(v=u\circ F\). For classical \(C^2\) functions
the ordinary chain rule gives every term in \[
 \begin{split}
 \partial_{x_i}v
  &=\sum_a P_{ai}(\partial_{y_a}u)\circ F,\\
 \partial_{x_i}\partial_{x_j}v
  &=\sum_{a,b}P_{ai}P_{bj}(\partial_{y_a}\partial_{y_b}u)\circ F
      +\sum_a(\partial_{x_i}P_{aj})(\partial_{y_a}u)\circ F .
 \end{split}                                                   \tag{CX7}
\] The second sum is essential on the original \(C^2\) charts. With the
original \(D_i=-i\partial_i\) the exact version is \[
 \begin{split}
 D_i^xv&=\sum_aP_{ai}(D_a^yu)\circ F,\\
 D_i^xD_j^xv
 &=\sum_{a,b}P_{ai}P_{bj}(D_a^yD_b^yu)\circ F
       -i\sum_a(\partial_{x_i}P_{aj})(D_a^yu)\circ F .
 \end{split}                                                   \tag{CX8}
\] No second derivative of \(u\), mixed term, or factor \(-i\) is
suppressed.

Here are complete local bounds, including the original array
multiplicities. For \(U_0\Subset U\), put \(V_0=F(U_0)\),
\(m_F=\inf_{\overline U_0}J_F>0\), and \[
 \begin{split}
 C_i&=\sum_a\sup_{\overline U_0}|P_{ai}|,\\
 C_{ij}&=\sum_{a,b}\sup_{\overline U_0}|P_{ai}P_{bj}|
                 +\sum_a\sup_{\overline U_0}|\partial_{x_i}P_{aj}|.
 \end{split}
\] Use the full original integer norm
\(\|u\|_{H^k(V_0)}^2=\sum_{|\alpha|\leq k}\|D_y^\alpha u\|_{L^2(V_0)}^2\),
\(k=0,1,2\). Substitution with \(G\) gives
\(\|w\circ F\|_{L^2(U_0)}\leq m_F^{-1/2}\|w\|_{L^2(V_0)}\). Apply this
inequality to every summand in (CX8), and use that each original
component norm is at most the full original array norm. Summing the
squares over \(i\) and over \(i\leq j\), which includes each multiindex
of order two exactly once, proves \[
 \begin{split}
 \|u\circ F\|_{H^1(U_0)}
 &\leq m_F^{-1/2}\left(1+\sum_i C_i^2\right)^{1/2}
                   \|u\|_{H^1(V_0)},\\
 \|u\circ F\|_{H^2(U_0)}
 &\leq m_F^{-1/2}
       \left(1+\sum_i C_i^2+\sum_{i\leq j}C_{ij}^2\right)^{1/2}
                   \|u\|_{H^2(V_0)}.                           
 \end{split} \tag{CX9}
\] Within each \(C_{ij}\), the entire ordered sum over \(a,b\) remains
present, including both occurrences of a mixed derivative when its two
indices differ. The norm on the left retains the original unordered
multiindex count. These are bounds, not a replacement of either array.

To prove the weak assertion, choose a coordinate cutoff
\(\eta\in C_c^\infty(V)\) equal to one on a neighborhood of
\(\overline V_0\). The elementary Euclidean cutoff follows from the flat
scalar function in (MG3), or from a finite sum of coordinate ball
cutoffs. The full product rule is \[
 D^\alpha(\eta u)=
   \sum_{\beta\leq\alpha}
       \binom{\alpha}{\beta}(D^\beta\eta)D^{\alpha-\beta}u,
       \qquad |\alpha|\leq2 .
\] It holds weakly by the proved integration/product rules of Section 15
in \href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments}. The zero extension of \(\eta u\)
belongs to \(H^2(\mathbb R^n)\): it vanishes on an open neighborhood of
the chart boundary, so integration against a global test introduces no
boundary term. Original mollifiers give
\(u_\epsilon=(\eta u)*\eta_\epsilon\to\eta u\) in the full
\(H^2(\mathbb R^n)\) norm, by that lesson's support-preserving
derivative and translation estimates. Each \(u_\epsilon\) is smooth; its
pullback is \(C^2\). Apply (CX9) to the differences on \(U_0\). These
pullbacks and both displayed derivative arrays converge in \(L^2(U_0)\).
Integration by parts against any smooth compactly supported test passes
to the limit, proving (CX7)--(CX8) for \(u\). The same proof with only
first derivatives gives the \(H^1\) assertion; order zero is exactly
(CX6). The inverse map \(G\) has the identical proof with \(P,J_F\)
replaced by the actual \(Q,J_G\). Both compositions are the identity on
the actual local equivalence classes, by (CX2) and null-set transport.
Thus the original \(H^1_{\mathrm{loc}}\) and \(H^2_{\mathrm{loc}}\)
chart spaces and their supports agree through the exact maps.

There is an exact negative-order map as well. For a local scalar
distribution \(R_x\in H^{-1}_{\mathrm{loc}}(U)\), define its scalar
coordinate representative by \[
       \langle R_y,\psi\rangle
       =\langle R_x,J_F(\psi\circ F)\rangle,\qquad
                         \psi\in C_c^\infty(V).                 \tag{CX10}
\] The right-hand test is compactly supported \(C^1\), and is admissible
through the \(H^1\) extension of \(R_x\). Multiplication by any \(C^1\)
coefficient \(b\) on the compact support is bounded on \(H^1\): the full
rule \(D_i(bw)=bD_iw-i(\partial_i b)w\) gives the bound with squared
coefficient \[
 \|b\|_\infty^2+
      \sum_i\bigl(\|b\|_\infty+\|\partial_i b\|_\infty\bigr)^2 .
\] Together with (CX9) for \(H^1\), this bounds (CX10) on each compact
subchart and proves \(R_y\in H^{-1}_{\mathrm{loc}}(V)\). The test map
has compact support and is the \(H^1\) limit of smooth compact tests by
the same cutoff and mollifier construction; hence it lies in the actual
\(H_0^1\) test domain. For an \(L^2_{\mathrm{loc}}\) representative
\(R_x=f_x\), (CX6) shows that \(R_y=f_x\circ G\), with precisely the
factor \(J_F\) in its action. Applying the inverse test map introduces
\(J_G\circ F\) and \(J_F\); their full product in (CX2) is one, so the
two distribution maps are inverses. A density-valued distribution
instead transforms by
\(\langle F_*R,\psi\rangle=\langle R,\psi\circ F\rangle\); this second
map also is bounded on \(H^{-1}_{\mathrm{loc}}\), but has a different
test formula. The scalar and density-valued maps have both been
constructed rather than identified by dropping a determinant.

\subsubsection{12.3. The original density and contravariant tensor in
the
equation}\label{the-original-density-and-contravariant-tensor-in-the-equation}

Keep the density and coefficients fixed in (D1)--(D3), rather than
substitute the auxiliary metric density. Their exact coordinate
transforms are \[
 \begin{split}
 \rho_y(y)&=\rho_x(G(y))J_G(y),\\
 a_y^{ab}(y)
  &=\sum_{i,j}P_{ai}(G(y))a_x^{ij}(G(y))P_{bj}(G(y)).
 \end{split}                                                   \tag{CX11}
\] The first is positive \(C^1\), because \(G\) is \(C^2\), the original
\(\rho_x\) is positive \(C^1\), and its nonzero determinant has locally
constant sign. Each tensor entry is locally Lipschitz: \(G\) and
\(P\circ G\) are \(C^1\) and have bounded derivatives on compact
subcharts, and the original entries are locally Lipschitz. To check this
directly, subtract a product at two points as \[
 \begin{split}
 p(y)a(y)q(y)-p(z)a(z)q(z)
  &=[p(y)-p(z)]a(y)q(y)\\
  &\quad+p(z)[a(y)-a(z)]q(y)
             +p(z)a(z)[q(y)-q(z)] .
 \end{split}
\] Every summand has one Lipschitz difference and two bounded factors.
Sum this identity over the original \(i,j\) entries in (CX11); it
retains every factor and proves the assertion.

The original transpose symmetry is preserved by the ordered product
\(P A_xP^T\). For a real nonzero covector \(\zeta\) in the \(y\) chart,
the real vector \(P^T\zeta\) is nonzero by (CX2), and \[
 \operatorname{Re}\sum_{a,b}a_y^{ab}\zeta_b\zeta_a
 =\operatorname{Re}\sum_{i,j}
        a_x^{ij}\left(\sum_bP_{bj}\zeta_b\right)
                   \left(\sum_aP_{ai}\zeta_a\right)>0 .          \tag{CX12}
\] On compact subcharts the minimum of \(|P^T\zeta|^2\) on the original
unit sphere is positive. Thus an original local lower bound
\(\lambda_x\) transfers to \(\lambda_x\) times that actual minimum. The
density factor is still separate.

For \(w,\phi\in C_c^2(V)\), write \(w_x=w\circ F\),
\(\phi_x=\phi\circ F\). The scalar substitution law, full chain rule and
(CX11) give the entire energy equality \[
 \begin{split}
 \int_U\sum_{i,j}a_x^{ij}\partial_{x_j}w_x
                         \overline{\partial_{x_i}\phi_x}\rho_x\,dx
 &=\int_U\sum_{i,j,a,b}
    a_x^{ij}P_{bj}P_{ai}
       (\partial_{y_b}w)\circ F
       \overline{(\partial_{y_a}\phi)\circ F}\rho_x\,dx\\
 &=\int_V\sum_{a,b}a_y^{ab}\partial_{y_b}w
                         \overline{\partial_{y_a}\phi}\rho_y\,dy .
 \end{split}                                                   \tag{CX13}
\] In the last step the factors \(\rho_x\circ G\) and \(J_G\) form the
original transformed density exactly. Neither a tensor derivative term
nor a density factor was discarded.

Integration by parts in the original charts, which is valid for locally
Lipschitz coefficients and compactly supported \(C^2\) tests by the
proved weak product rule, identifies the left side of (CX13) with
\((w_x,L_x^*\phi_x)_{\mu_x}\), and the right side with
\((w,L_y^*\phi)_{\mu_y}\). After substitution this proves \[
                 L_x^*(\phi\circ F)=(L_y^*\phi)\circ F
                     \quad\hbox{almost everywhere}.             \tag{CX14}
\] Indeed both sides are bounded and compactly supported on the relevant
subcharts; testing their difference against all \(w_x\in C_c^2(U)\)
forces that difference to vanish. Such tests include every smooth
compactly supported coordinate test; the positive density may be divided
out using its local lower bound and the \(L^2\) density of those tests.
Every \(C_c^2(U)\) is \(w\circ F\) for \(w=w_x\circ G\in C_c^2(V)\), so
no test class was omitted.

For an arbitrary original \(u_x,f_x\in L^2_{\mathrm{loc}}(U)\), put
\(u_y=u_x\circ G\), \(f_y=f_x\circ G\). Equations (CX6) and (CX14) give
the exact receiving identity \[
 (u_y,L_y^*\phi)_{\mu_y}
  =(u_x,L_x^*(\phi\circ F))_{\mu_x}
  =(f_x,\phi\circ F)_{\mu_x}
  =(f_y,\phi)_{\mu_y}.                                          \tag{CX15}
\] Thus the original weak equation (D4) holds in one chart precisely
when it holds in the other. This proves the original intrinsic operator
and its actual \(L^2\) test domain without multiplying an arbitrary
second-order distribution by a \(C^1\) function. For
\(u\in H^1_{\mathrm{loc}}\), the same equality follows from (CX13) by
the proved approximation and energy bounds; the scalar negative-order
map is exactly (CX10). Orientation-reversing charts retain the absolute
determinants throughout.

\subsubsection{12.4. Gluing the actual measures, supports and finite
chart
norms}\label{gluing-the-actual-measures-supports-and-finite-chart-norms}

The positive \(C^1\) density now defines a completed measure on the
original \(C^2\) manifold. Choose the locally finite coordinate
partition \(\phi_i\) proved in Section 11. In chart \(x_i\) with
coefficient \(\rho_i\), set for a Borel \(E\subset X\) \[
 \mu(E)=\sum_i\int_{x_i(E\cap U_i)}
       \phi_i(x_i^{-1}(z))\rho_i(z)\,dz .                        \tag{CX16}
\] Each summand is a Borel measure, and nonnegative countable sums
preserve countable additivity. In any one chart, apply (CX6) and (CX11)
to the intersections with \(U_i\). Each summand becomes the integral
there of \(\phi_i\rho\), with its full absolute determinant. Summing and
using the locally finite identity \(\sum_i\phi_i=1\) proves that (CX16)
is exactly \(\rho\,dx\) on that chart. This also proves independence of
the chosen partition. Its completion agrees with the local completed
coordinate measures: on relatively compact subcharts \(\rho\) has
positive lower and finite upper bounds, so its null sets are exactly the
original coordinate null sets. The proved bidirectional chart null
transport patches this assertion over the countable cover. Each compact
set has finite measure, because it meets only finitely many partition
supports and each corresponding coordinate integral has bounded
coefficient and compact support. The separately chosen original density
has been used everywhere.

A compactly supported \(H^2\) function may now be partitioned in its
actual charts. For each \(C^2\) partition function the full weak product
formula, for every \(|\alpha|\leq2\), is \[
 D^\alpha(\phi_i u)=
   \sum_{\beta\leq\alpha}\binom{\alpha}{\beta}
                   (D^\beta\phi_i)D^{\alpha-\beta}u.             \tag{CX17}
\] It follows by the first-order product rule and its repeated
application; for \(\alpha=e_a+e_b\) this retains both first-derivative
products, with coefficient two when \(a=b\). All derivatives of
\(\phi_i\) through order two are bounded on the fixed compact support.
The squared original multiindex norm is bounded by the sum over
\(|\alpha|\leq2\) of the squared constants
\(\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\|D^\beta\phi_i\|_\infty\),
times the full original \(H^2\) norm. Zero extension within each
compactly supported chart introduces no boundary term. Mollification of
those extensions gives smooth coordinate approximants; composing with
the actual \(C^2\) inverse charts and summing the finitely many pieces
gives \(C_c^2(X)\) approximants in the full chart \(H^2\) norm. The
supports stay in the chosen compact neighborhood because each original
cutoff support has a positive margin to its chart boundary. The
order-one version gives the corresponding \(H^1\) approximation.

For two fixed finite chart partitions near a compact set, apply (CX17)
to their products on overlaps and (CX9) to each actual transition. If
\(b_{ji}\) is the product of these displayed finite product and
transition bounds for overlap \((j,i)\), and \(r_i\) is the \(H^2\) norm
of original piece \(i\), the new piece \(j\) has norm at most
\(\sum_i b_{ji}r_i\). Cauchy--Schwarz and summing over \(j\) give the
full comparison \[
       \sum_j\left(\sum_i b_{ji}r_i\right)^2
          \leq\left(\sum_{j,i}b_{ji}^2\right)\sum_i r_i^2.        \tag{CX18}
\] Reverse the two partitions and use the actual inverse transitions to
obtain the opposite bound. This proves equivalence of the finite chart
norms used in (D10)--(D17), retaining every overlap, derivative and
determinant constant rather than replacing the original norm. It also
proves the stated support-preserving \(H^2\) part of (D13).

Finally on its fixed compact support the original formula for \(L\)
expands to \[
 Lu=-\sum_{j,k}a^{jk}\partial_j\partial_k u
       -\sum_{j,k}\bigl(\partial_j a^{jk}
                  +a^{jk}\rho^{-1}\partial_j\rho\bigr)\partial_k u .
                                                               \tag{CX19}
\] Every coefficient on the right is locally bounded. The triangle
inequality bounds its \(L^2\) norm by \[
 \sum_{j,k}\|a^{jk}\|_\infty\|\partial_j\partial_k u\|_2
 +\sum_{j,k}
   \bigl(\|\partial_j a^{jk}\|_\infty
       +\|a^{jk}\|_\infty\|\rho^{-1}\partial_j\rho\|_\infty\bigr)
          \|\partial_k u\|_2 .
\] Thus the constructed full \(H^2\) approximation implies convergence
of \(Lu_\nu\) to \(Lu\) in \(L^2\), with all original density
derivatives and both signs present. Conjugating the actual \(a^{jk}\)
gives the same conclusion for \(L^*\). This supplies the exact
coordinate and approximation maps used in (D4), (D10)--(D13) and the
weighted adjoint argument. Section 13 gives separate complete proofs of
the Euclidean compactness and zero-gradient assertions and proves their
actual finite-chart transfer.

\subsection{13. The full Sobolev and compactness proofs used by the
equation}\label{the-full-sobolev-and-compactness-proofs-used-by-the-equation}

We prove every Sobolev and compactness fact stated in Section 1, keeping
the original complex functions, Hilbert product, differential operators,
kernel scales, supports, coordinate maps and chosen density. No
positivity or reality is imposed on the mass-one smoothing kernel.

The complete entry bases are Section 15 of
\href{banach-foundation-bridges.pdf}{Banach estimates, quotient spaces
and compact parameter arguments}, Section 2.3 of
\href{lower-bounded-spectral-calculus.pdf}{Spectral measures with the
original operator domain retained}, Sections 13.7--13.10 of
\href{geometric-microlocal-calculus.pdf}{Detecting regularity without
choosing coordinates}, and Sections 11--12 of this lesson. They provide
the actual completed measures, integration, norm completeness,
derivative approximation, Hilbert representation, distribution
convolution and original coordinate formulas used below.

\subsubsection{13.1. A weak subsequence in the original Hilbert
space}\label{a-weak-subsequence-in-the-original-hilbert-space}

Let \(H\) be any complex Hilbert space, with \((u,v)\) linear in \(u\),
and let the original sequence satisfy \(\|u_j\|\leq M<\infty\). No
separability of \(H\) is required. If every \(u_j\) is zero, the
assertion holds with the original sequence and limit zero.

Construct orthogonal nonzero residuals without changing their lengths.
At stage \(k\), choose the smallest index \(m_k\) whose vector is not in
the span of the preceding residuals, and set \[
 r_k=u_{m_k}
       -\sum_{\ell<k}\frac{(u_{m_k},r_\ell)}{\|r_\ell\|^2}r_\ell .
                                                               \tag{HS1}
\] Before the first stage that span is \(\{0\}\). If no such index
remains, stop; then every original vector belongs to the finite span
constructed. Otherwise the \(m_k\)'s strictly increase, and every
\(u_j\) belongs to some finite span: after \(m_k\geq j\), every earlier
index has either been selected or has an exactly zero residual. For
every \(v\in H\), orthogonality gives the full residual identity \[
 \left\|v-\sum_{k\leq N}
                \frac{(v,r_k)}{\|r_k\|^2}r_k\right\|^2
       +\sum_{k\leq N}\frac{|(v,r_k)|^2}{\|r_k\|^2}
       =\|v\|^2 .                                               \tag{HS2}
\] Indeed expansion of the square cancels both cross pairings with each
selected residual, while pairings between different residuals are zero.
Thus the scalar series is bounded and the vector series is Cauchy by the
same identity applied to its finite tails. Write its limit as \(P v\).
For each residual, \((v-Pv,r_k)=0\), hence \(v-Pv\) is orthogonal to
every original \(u_j\).

For each \(k\), \((u_j,r_k)\) lies in the closed complex disk of radius
\(M\|r_k\|\). It has a convergent subsequence: repeatedly divide a
containing closed square into four closed squares and retain one
containing infinitely many values; choose increasing indices in those
nested squares. Their diameters tend to zero, so real completeness
supplies a single limit, also in the disk. Apply this construction
successively to the countably many residuals. Choosing the \(\ell\)-th
index from the \(\ell\)-th nested subsequence, larger than its
predecessor, gives one subsequence \(u_{j_\ell}\) for which \[
 (u_{j_\ell},r_k)\longrightarrow c_k\quad\hbox{for every }k,
 \qquad
 \sum_{k\leq N}\frac{|c_k|^2}{\|r_k\|^2}\leq M^2.                \tag{HS3}
\] The inequality is the limit of the finite inequality (HS2) for the
original vectors. Completeness therefore defines \[
 z=\sum_k\frac{c_k}{\|r_k\|^2}r_k,\qquad
 \|z\|^2=\sum_k\frac{|c_k|^2}{\|r_k\|^2}\leq M^2.                \tag{HS4}
\] Finite stopping uses the corresponding finite sum. For an arbitrary
original \(v\in H\), its orthogonal residual contributes zero to its
pairings with both \(u_{j_\ell}\) and \(z\). The finite part of \(Pv\)
converges by (HS3); Cauchy--Schwarz bounds the rest by \[
 \left|\left(u_{j_\ell}-z,\,
          \sum_{k>N}\frac{(v,r_k)}{\|r_k\|^2}r_k\right)\right|
 \leq 2M
       \left(\sum_{k>N}\frac{|(v,r_k)|^2}{\|r_k\|^2}\right)^{1/2}
       \longrightarrow0.                                     \tag{HS5}
\] Taking \(N\) large and then \(\ell\) large proves
\((u_{j_\ell},v)\to(z,v)\) for every \(v\). Hilbert representation
identifies these pairings with all continuous linear functionals. This
is a weakly convergent subsequence in the actual ambient Hilbert space.
Every zero residual, length and original vector was accounted for.
\(\square\)

\subsubsection{13.2. The original negative Sobolev dual and
convolution}\label{the-original-negative-sobolev-dual-and-convolution}

Use \[
 (h,\phi)_{H^1}=(h,\phi)_{L^2}
                   +\sum_{i=1}^n(D_i h,D_i\phi)_{L^2},\qquad
 \|h\|_{H^1}^2=\|h\|_2^2+\sum_i\|D_i h\|_2^2.                 \tag{HS6}
\] Completeness and density are LP16--LP19. The anti-dual convention
compatible with the energy equation is a continuous conjugate-linear
action \(R[\phi]\) on \(H^1\), with norm
\(\|R\|_{H^{-1}}=\sup_{\|\phi\|_{H^1}\leq1}|R[\phi]|\). Its linear
distribution pairing is exactly
\(\langle R,\psi\rangle=R[\overline\psi]\). For an \(L^2\) function the
action is \(\int R\overline\phi\). Apply the proved Hilbert
representation to the linear functional
\(\phi\mapsto\overline{R[\phi]}\). There is a unique \(h\in H^1\) with
\[
 R[\phi]=(h,\phi)_{H^1},\qquad
 \|h\|_{H^1}=\|R\|_{H^{-1}},\qquad
 R=h+\sum_iD_i(D_i h)
       =h-\sum_i\partial_i^2h\quad\hbox{distributionally}.       \tag{HS7}
\] To verify the sign, linear distribution differentiation gives
\(\langle D_i f,\psi\rangle=-\langle f,D_i\psi\rangle
=i\int f\partial_i\psi\). With \(\psi=\overline\phi\) this is
\((f,D_i\phi)\). Applying it to \(f=D_i h\) gives precisely the \(i\)-th
positive summand in (HS6). Thus no derivative or conjugation convention
has changed.

For arbitrary \(f_0,\ldots,f_n\in L^2\), the distribution
\(f_0+\sum_iD_i f_i\) extends to this anti-dual, and finite
Cauchy--Schwarz gives \[
 \left\|f_0+\sum_iD_i f_i\right\|_{H^{-1}}
       \leq\left(\|f_0\|_2^2+\sum_i\|f_i\|_2^2\right)^{1/2}.     \tag{HS8}
\] For (HS7) the right side is exactly \(\|h\|_{H^1}\). This proves a
representation, rather than assuming an \(L^p\) dual theorem.

Let \(J_\epsilon f=\eta_\epsilon*f\), with the actual compact smooth,
possibly complex kernel. Its \(L^2\) adjoint has the complete kernel \[
 \eta_\epsilon^*(x)=\overline{\eta_\epsilon(-x)},\qquad
 (J_\epsilon^*\phi)(y)
       =\int\overline{\eta_\epsilon(t)}\,\phi(y+t)\,dt .          \tag{HS9}
\] Absolute Fubini proves the adjoint identity first on compact tests;
Young and density extend it to \(L^2\). Each \(D_i\) commutes with
convolution by LP19. Young on every original component in (HS6)
consequently gives \[
 \|J_\epsilon^*\phi\|_{H^1}
       \leq\|\eta\|_1\|\phi\|_{H^1},\qquad
 (J_\epsilon R)[\phi]=R[J_\epsilon^*\phi],\qquad
 \|J_\epsilon R\|_{H^{-1}}\leq\|\eta\|_1\|R\|_{H^{-1}}.          \tag{HS10}
\] Changing variables \(t=\epsilon s\) retains the factor
\(\epsilon^{-n}\epsilon^n\), giving \(\|\eta_\epsilon\|_1=\|\eta\|_1\),
not an assumed value one. In the linear-test convention the test
operator is \(\psi\mapsto\int\eta_\epsilon(t)\psi(\,\cdot+t)\,dt\).
Indeed its complex conjugate is (HS9) applied to \(\overline\psi\). This
also identifies the construction with the compact-factor distribution
convolution GC1--GC24.

Using the exact representation (HS7), derivative commutation and (HS8)
prove \[
 \begin{split}
 J_\epsilon R-R
  &=(J_\epsilon h-h)
           +\sum_iD_i(J_\epsilon D_i h-D_i h),\\
 \|J_\epsilon R-R\|_{H^{-1}}
  &\leq\left(\|J_\epsilon h-h\|_2^2
              +\sum_i\|J_\epsilon D_i h-D_i h\|_2^2\right)^{1/2}
       \longrightarrow0.                                    
 \end{split} \tag{HS11}
\] The last limit is LP14 for all \(n+1\) original \(L^2\) components.
The sole mass assumption is \(\int\eta=1\).

The smooth output has the exact point formula
\(J_\epsilon R(x)=R[\overline{\eta_\epsilon(x-\cdot)}]\). Translation
and differentiation of this compact test are continuous in \(H^1\), by
its complete derivative array and dominated convergence. The dual bound
therefore permits every \(x\)-derivative. For every ordinary multiindex
\(\alpha\), \[
 |\partial_x^\alpha J_\epsilon R(x)|
 \leq\|R\|_{H^{-1}}
 \left(\epsilon^{-n-2|\alpha|}\|\partial^\alpha\eta\|_2^2
       +\sum_i\epsilon^{-n-2|\alpha|-2}
                     \|D_i\partial^\alpha\eta\|_2^2\right)^{1/2}.
                                                               \tag{HS12}
\] No derivative scale was omitted. Its support is contained in
\(\operatorname{supp}R+\epsilon\operatorname{supp}\eta\): a compact test
supported outside this closed sum has its transformed test supported
away from \(\operatorname{supp}R\), by (HS9) and its exact sign. Hence
the distribution pairing there is zero. The sum is closed because the
second set is compact: any convergent sequence of sums has a subsequence
with convergent kernel components, leaving a limit in the closed first
set. \(\square\)

\subsubsection{13.3. Compactness for the fixed original interior
support}\label{compactness-for-the-fixed-original-interior-support}

Let \(\Omega\subset\mathbb R^n\) be open, \(K\Subset\Omega\) compact,
and let the original \(u_j\in H^1(\Omega)\) have distributional support
in \(K\) and \(\|u_j\|_{H^1(\Omega)}\leq M\). If \(K\) is empty, all
vectors are zero. A cutoff equal to one near \(K\) shows their zero
extensions belong to \(H^1(\mathbb R^n)\) with exactly the same function
and derivative norms: LP18 and its zero-extension test proof show that
the cutoff derivative terms are supported away from the original
function, hence zero. Distribution derivatives also vanish off \(K\).
Thus no boundary assumption on \(\Omega\) was introduced.

For a smooth compact function the segment integral, scalar
Cauchy--Schwarz, integration and translation give \[
 \|\tau_h u-u\|_2
       \leq |h|\left(\sum_i\|\partial_i u\|_2^2\right)^{1/2}
       =|h|\left(\sum_i\|D_i u\|_2^2\right)^{1/2}.              \tag{HS13}
\] Here \(\tau_h u(x)=u(x-h)\); differentiating the segment \(x-th\)
contributes the original minus sign, whose modulus is used only in the
inequality. Specifically its squared integral is at most
\(|h|^2\int_0^1|\nabla u(x-th)|^2dt\). LP19 approximates every \(H^1\)
input in the original full norm, and translations preserve each \(L^2\)
norm, so (HS13) passes to every input.

Put \(m_1(\eta)=\int|t|\,|\eta(t)|dt\). Minkowski, the exact mass one
and (HS13) give \[
 \|J_\epsilon u-u\|_2
   \leq\epsilon m_1(\eta)
               \left(\sum_i\|D_i u\|_2^2\right)^{1/2}
   \leq\epsilon m_1(\eta)M .                                  \tag{HS14}
\] This is uniform in the original sequence, with the actual first
moment.

For fixed \(\epsilon>0\), Cauchy--Schwarz in the convolution integral
gives \[
 \|J_\epsilon u\|_\infty
       \leq\epsilon^{-n/2}\|\eta\|_2M=:B_\epsilon,\qquad
 \|\nabla J_\epsilon u\|_\infty
       \leq\epsilon^{-n/2-1}
             \left(\sum_i\|\partial_i\eta\|_2^2\right)^{1/2}M
       =:L_\epsilon .                                        \tag{HS15}
\] The output support is in \(K+\epsilon\operatorname{supp}\eta\).
Choose one bounded coordinate box \(Q=\prod_i[a_i,b_i]\), with positive
side lengths, containing these supports for \(0<\epsilon\leq\epsilon_0\)
in its interior. Its full volume is \(|Q|=\prod_i(b_i-a_i)\).

There are finite \(L^2\) nets for the outputs at fixed \(\epsilon\).
Subdivide each original side into \(N_i\) equal intervals of length
\(h_i=(b_i-a_i)/N_i\leq h\). On each resulting cell choose its center.
Quantize the real and imaginary parts of the output value there to
multiples of \(\delta>0\), using the finite range
\(\{-\lceil B_\epsilon/\delta\rceil\delta,\ldots,
\lceil B_\epsilon/\delta\rceil\delta\}\). Nearest choices have complex
error at most \(\sqrt2\delta\). The finite family of all cellwise
constant functions using these choices, zero outside \(Q\), has error \[
 \|J_\epsilon u-v_{\epsilon,h,\delta}\|_2
 \leq |Q|^{1/2}
        \left(L_\epsilon\Big(\sum_i h_i^2\Big)^{1/2}
                           +\sqrt2\delta\right)
 \leq |Q|^{1/2}(L_\epsilon\sqrt n\,h+\sqrt2\delta).             \tag{HS16}
\] Cell faces are null by LM1--LM6. All cell volumes and both real
components are included.

For any \(r>0\), choose \(\epsilon\) with the first bound (HS14) less
than \(r/2\), then \(h,\delta\) with (HS16) less than \(r/2\). This is a
finite radius-\(r\) net for the original sequence in \(L^2\). Repeated
finite covers at radii \(2^{-k}\), retaining an infinite subset inside
one ball at each step and choosing increasing original indices, give a
Cauchy subsequence: any two sufficiently late terms have distance at
most \(2^{1-k}\). LP10 supplies its \(L^2\) limit. That limit is zero
outside \(K\), since the \(L^2\) norm there is bounded by its distance
from each original term. The proof establishes the fixed-support compact
inclusion into \(L^2\) without a boundary regularity or equicontinuity
theorem being imported. If a limit in \(H^1\) is needed, (HS1)--(HS5)
give a further weak \(H^1\) subsequence; its \(L^2\) pairings identify
it with this same strong limit. \(\square\)

\subsubsection{13.4. Every integer Euclidean order and its full
multiplicities}\label{every-integer-euclidean-order-and-its-full-multiplicities}

The original order-two consequence can be proved component by component.
For \(u_j\in H^2\) with the same compact support, each \(D_i u_j\) is a
bounded \(H^1\) sequence. The exact ordered array satisfies \[
 \sum_i\|D_i u_j\|_{H^1}^2
   =\sum_i\|D_i u_j\|_2^2
      +\sum_i\|D_i^2u_j\|_2^2
      +2\sum_{i<k}\|D_iD_k u_j\|_2^2
   \leq2\|u_j\|_{H^2}^2 .                                    \tag{HS17}
\] The factor two records both ordered mixed-derivative occurrences; the
original \(H^2\) norm keeps each multiindex once. Apply the preceding
compactness proof to \(u_j\) and all \(n\) first derivatives, selecting
finitely many nested subsequences. Their \(L^2\) limits
\(u,g_1,\ldots,g_n\) satisfy \(D_i u=g_i\) by LP16 with its exact sign.
Thus the original subsequence converges strongly in the full \(H^1\)
norm. A weak \(H^2\) subsequence, if required, identifies the same limit
by (HS1)--(HS5).

More generally, for each fixed integer \(m\geq1\) and fixed interior
support, bounded subsets of the original \(H^m\) are relatively compact
in the original \(H^{m-1}\). Here is the complete array count: \[
 \begin{split}
 \sum_{|\alpha|\leq m-1}\|D^\alpha u\|_{H^1}^2
 &=\sum_{|\beta|\leq m-1}\|D^\beta u\|_2^2
   +\sum_{1\leq|\beta|\leq m}
       k(\beta)\|D^\beta u\|_2^2,\\
 k(\beta)&=\#\{i:\beta_i>0\}\leq\min(n,|\beta|),\\
 \sum_{|\alpha|\leq m-1}\|D^\alpha u\|_{H^1}^2
 &\leq(1+\min(n,m))\|u\|_{H^m}^2 .                            
 \end{split} \tag{HS18}
\] Indeed each ordered pair \((\alpha,i)\) yields \(\beta=\alpha+e_i\),
with exactly one such pair for every positive coordinate of \(\beta\).
The first sum supplies its separate order-\((m-1)\) components,
including \(\beta=0\); the second retains all multiplicities. There are
exactly \(\binom{n+m-1}{m-1}\) original \(\alpha\)'s. Apply Section 13.3
to each, select this finite number of subsequences, and use LP16 to
identify every limit with \(D^\alpha u\). Their finite squared sum is
strong \(H^{m-1}\) convergence. This is a proved Euclidean
strengthening; the original \(C^2\) manifold supports only the order-one
and order-two coordinate transfers established in CX7--CX9, and is not
asserted to support arbitrary higher orders. \(\square\)

\subsubsection{13.5. The actual finite chart
transfer}\label{the-actual-finite-chart-transfer}

Keep the original \(X,d\mu=\rho_i\,dx_i\), compact \(K\subset X\), and a
finite \(C^2\) partition \(\phi_1,\ldots,\phi_N\) equal in sum to one
near \(K\), each supported in a compact subchart. MG1--MG9 construct
these cutoffs; CX17 gives every product term. The chart expressions
\((\phi_i u_j)\circ x_i^{-1}\), extended by zero, have one fixed compact
support in their original chart and a bounded \(H^r\) norm for \(r=1\)
or \(2\). Apply the proved Euclidean inclusion to each of these \(N\)
original pieces and retain finitely many nested subsequences.

For \(r=1\), the resulting coordinate pieces converge in \(L^2(dx_i)\).
Original density coefficients have finite upper bounds \(C_{\rho,i}\) on
these supports, and for any difference \(w\) of two original terms, \[
 \|w\|_{L^2(d\mu)}^2
   =\left\|\sum_i\phi_i w\right\|_{L^2(d\mu)}^2
   \leq N\sum_i C_{\rho,i}
               \|(\phi_i w)\circ x_i^{-1}\|_{L^2(dx_i)}^2 .
                                                               \tag{HS19}
\] The original \(\rho_i\)'s remain present. This gives a global
\(L^2(d\mu)\) Cauchy subsequence. Measure completeness is LP10 on the
actual completed measure CX16. For \(r=2\), the coordinate pieces
converge in their full \(H^1\) norms. In every other fixed finite chart
partition, CX17 and the order-one actual transition CX8 bound each piece
by the sum of its overlaps, with the actual product and Jacobian
constants \(b_{ji}\). The entire estimate is \[
 \sum_j\left(\sum_i b_{ji}
       \|(\phi_i w)\circ x_i^{-1}\|_{H^1}\right)^2
 \leq \left(\sum_{j,i}b_{ji}^2\right)
         \sum_i\|(\phi_i w)\circ x_i^{-1}\|_{H^1}^2 .             \tag{HS20}
\] It is precisely the order-one instance of CX18, rather than an
identification of different arrays. The coordinate limits agree on
overlaps: their zeroth-order limits are restrictions of the global
\(L^2\) limit from (HS19), and their weak derivatives are those of that
same function by LP16 and CX8. They therefore define a supported \(H^1\)
limit, with strong convergence in every displayed finite chart norm. The
reverse transition proves equivalence for the original selected norm as
well. Thus both original compactness statements transfer to the actual
\(C^2\) manifold with its chosen density. \(\square\)

\subsubsection{13.6. Zero weak gradient at its full function
scope}\label{zero-weak-gradient-at-its-full-function-scope}

Let \(\Omega\subset\mathbb R^n\) be a connected nonempty open set and
\(u\in L^1_{\mathrm{loc}}(\Omega)\) have all distributional first
derivatives zero. This includes every weakly differentiable function
used in the lesson.

Choose any ball \(B(x,2r)\Subset\Omega\), and a compact smooth \(\chi\)
equal to one near \(\overline{B(x,3r/2)}\). Put \(v=\chi u\), extended
by zero. Its distribution derivative is
\(\partial_i v=(\partial_i\chi)u+\chi\partial_i u=(\partial_i\chi)u\).
For \(\epsilon\max_{t\in\operatorname{supp}\eta}|t|<r/2\), convolution
at any point of \(B(x,r)\) encounters only the region where \(\chi=1\).
Thus every derivative of \(J_\epsilon v\) there is zero. Its smooth
segment formula on this convex ball makes \(J_\epsilon v=c_\epsilon\)
there.

LP14 at \(p=1\) proves \(J_\epsilon v\to u\) in \(L^1(B(x,r))\). Its
constants are Cauchy, because \[
 |B(x,r)|\,|c_\epsilon-c_{\epsilon'}|
       =\|c_\epsilon-c_{\epsilon'}\|_{L^1(B(x,r))}
       \leq\|J_\epsilon v-u\|_1+
                         \|J_{\epsilon'}v-u\|_1 .              \tag{HS21}
\] The original ball measure is finite and positive: it is contained in
one bounded coordinate box and contains another box with positive full
product. Hence \(c_\epsilon\to c\) and \(u=c\) almost everywhere on that
ball.

Cover \(\Omega\) by a countable collection of such smaller balls;
rational centers and radii give this collection with doubled closures
contained in \(\Omega\). If two of the balls intersect, their
intersection is nonempty and open, contains a box of positive measure,
and is subject to both almost-everywhere constant statements. Their
constants agree. Declare two balls related if they can be joined by a
finite chain of intersections. The union of each class is open; unions
of distinct classes are disjoint. More than one class would disconnect
\(\Omega\), so there is one class and one constant. Countability permits
taking the union of all local exceptional null sets without losing the
global almost-everywhere assertion. On an empty domain the unique empty
equivalence class is already constant; on a disconnected domain the
proof applies separately to its open components, which are countable.

On the original manifold, \(du=0\) means zero weak coordinate gradient.
The preceding coordinate conclusion first shows that each such function
is locally constant, hence belongs to local \(H^1\). Formula (CX8) and
invertibility of its actual first derivative matrices then preserve the
zero-gradient assertion through the original coordinate changes.
Positive local density bounds give precisely the same null sets as
coordinate measure. A countable chart-ball cover and the same
intersection argument show that \(u\) is constant almost everywhere on
each connected component of \(X\); Section 11.5 proves the component
topology required; MG8 supplies the exact coordinate density formula.
This covers the original \(H^1_{\mathrm{loc}}\) use in (D14) at the
stronger \(L^1_{\mathrm{loc}}\) function scope. \(\square\)

\subsubsection{13.7. The elementary energy receiver and exact completion
scope}\label{the-elementary-energy-receiver-and-exact-completion-scope}

The compact-support estimate used in (D9) also has a direct exact
coefficient. Choose the original coordinate box \(Q=\prod_i(a_i,b_i)\)
containing the compact support in its interior, and write
\(\ell_1=b_1-a_1\). For a smooth supported \(w\), \[
 w(x_1,x')=\int_{a_1}^{x_1}\partial_1w(t,x')\,dt,\qquad
 |w(x_1,x')|^2\leq(x_1-a_1)
                   \int_{a_1}^{b_1}|\partial_1 w(t,x')|^2dt .
\] Integration over the original box, with all other coordinates
unchanged, gives \[
 \|w\|_2^2\leq\frac{\ell_1^2}{2}\|\partial_1w\|_2^2,\qquad
 \|w\|_{H^1}^2
     \leq\left(1+\frac{\ell_1^2}{2}\right)
                                  \sum_i\|D_iw\|_2^2 .         \tag{HS22}
\] The support-preserving approximation LP19 extends this to every
original supported \(H^1\) function. For the original
\(Q=-\partial_j(b^{jk}\partial_k)\), with (D2) lower bound \(\lambda>0\)
on that support neighborhood, integration by parts with the full complex
matrix gives \[
 \|w\|_{H^1}^2
 \leq \frac{1+\ell_1^2/2}{\lambda}
                  \operatorname{Re}(Qw,w)
 \leq \frac{1+\ell_1^2/2}{\lambda}
                    \|Qw\|_{H^{-1}}\|w\|_{H^1}.               \tag{HS23}
\] The zero norm case requires no division; otherwise this proves the
precise bound in (D9). HS10 supplies the actual convolution estimate for
(D8); HS1--HS5 supply the weak \(H^1\) subsequence in (D9); HS19--HS21
supply finite-chart compactness and the constant-gradient step in (D14)
and (D17). Each receiving map acts on the original objects, supports,
density and operators.

Together with LP1--LP19 and CX1--CX19, HS1--HS23 give every assertion in
the original DEP-DVS-SOBOLEV entry. The whole Sobolev and compactness
entry in Section 1 is therefore proved, with the stated full derivative
arrays and original support and density conventions. Other named entry
results retain their own separate proofs and hypotheses.

\subsection{14. Visible corrections and the full accretive
extension}\label{visible-corrections-and-the-full-accretive-extension}

The original symmetric operator and its proofs above remain
identifiable. The following editorial supplement proves its exact
nonsymmetric extension under a complex energy hypothesis, strengthens
the regularity of the constructed weight, and compares the
original-author regularity source without silently rewriting it.

\subsubsection{1. The nonsymmetric tensor, its coordinate map and its
actual
adjoint}\label{the-nonsymmetric-tensor-its-coordinate-map-and-its-actual-adjoint}

Keep the lesson's Hausdorff, second-countable, boundaryless \(C^2\)
manifold \(X\), of dimension \(n\geq1\), and its positive \(C^1\)
density \(d\mu=\rho\,dx\). Let \(\widetilde A=(\widetilde a^{jk})\) be a
contravariant tensor with complex locally Lipschitz entries. Impose no
transpose or Hermitian symmetry. The required hypothesis, on each
compact coordinate neighborhood \(K\), is \[
 \begin{gathered}
 \operatorname{Re}\sum_{j,k=1}^n\widetilde a^{jk}(x)z_k\overline{z_j} \geq\lambda_K\sum_{j=1}^n|z_j|^2, \quad x\in K,\quad z\in\mathbb C^n,\quad\lambda_K>0.
 \end{gathered}\tag{DA1}
\] This is an assumption about all complex covectors. The original
real-covector condition alone does not imply it without the original
symmetry. For example, the complex matrix with diagonal entries one and
off-diagonal entries \(2i,-2i\) has real quadratic form
\(\xi_1^2+\xi_2^2\), whereas at \(z=(1,i)\) its sesquilinear quadratic
form equals \(-2\). Thus that weaker hypothesis cannot be imported into
this extension.

Define both full operators by \[
 \begin{gathered}
 \widetilde L u=-\rho^{-1}\sum_{j,k}\partial_j(\rho\widetilde a^{jk}\partial_k u),\\
  \widetilde L^*v=-\rho^{-1}\sum_{j,k}\partial_j(\rho\overline{\widetilde a^{kj}}\partial_k v).
 \end{gathered}\tag{DA2}
\] For compactly supported \(C^2\) functions, integration by parts gives
\[
 (\widetilde L u,v)_\mu
 =\int\sum_{j,k}\widetilde a^{jk}\partial_k u\,\overline{\partial_j v}\,\rho\,dx
 =(u,\widetilde L^*v)_\mu.\tag{DA3}
\] Indeed, conjugating the coefficient inside the second pairing and
then interchanging its two indices gives exactly the middle integral.
This proves the transpose in DA2. If
\(q_{\widetilde A}(z)=\sum_{j,k}\widetilde a^{jk}z_k\overline{z_j}\),
the same interchange proves \[
 \begin{gathered}
 q_{\widetilde A^*}(z)=\overline{q_{\widetilde A}(z)},\\
 \widetilde a^{*,jk}=\overline{\widetilde a^{kj}},
 \qquad\operatorname{Re}q_{\widetilde A^*}(z)\geq\lambda_K|z|^2.
 \end{gathered}\tag{DA4}
\]

For a change of real coordinates \(y=\Phi(x)\), put
\(P_{rj}=\partial_j\Phi_r\) and \(J=|\det D\Phi|\). The exact maps are
\[
 \begin{gathered}
 \widetilde A_y=P\widetilde A_xP^{\mathsf T},\\
 \rho_y=\rho_x/J,\\
 d_xu=P^{\mathsf T}d_yu,\\
 (\widetilde A_y)^*=P\widetilde A_x^*P^{\mathsf T}.
 \end{gathered}\tag{DA5}
\] The last equality uses that \(P\) is real and reverses the order
under adjunction before transposing back. The energy integral in DA3 is
unchanged: substituting DA5 gives the same two ordered derivatives and
coefficient, and \(\rho_y\,dy=\rho_x\,dx\). On a compact subchart the
least singular value of \(P^{\mathsf T}\) is positive, so DA1 transfers
with lower bound \(\lambda_K\inf_K\sigma_{\min}(P)^2\). The entries
remain locally Lipschitz because \(P\) is \(C^1\) with bounded first
derivatives on compact subcharts; the transformed density is \(C^1\).
Sections 11--13 of the lesson prove the weak coordinate and Sobolev
transfers used here, including the absolute Jacobian on a nonorientable
manifold. Thus DA2 is intrinsic. For \(u\in L^2_{\mathrm{loc}}\), its
equation is precisely \((u,\widetilde L^*\phi)_\mu=(f,\phi)_\mu\) for
\(\phi\in C_c^2(X)\); no product of a general second-order distribution
by a \(C^1\) multiplier is introduced.

\subsubsection{2. Finite regularity and support continuation for the
full
tensor}\label{finite-regularity-and-support-continuation-for-the-full-tensor}

Set \(\widetilde b^{jk}=\rho\widetilde a^{jk}\) and
\(\widetilde Q=\rho\widetilde L=-\sum_{j,k}\partial_j(\widetilde b^{jk}\partial_k)\).
These coefficients are locally Lipschitz. Their complex energy lower
bound is \(\inf_K\rho\,\lambda_K\). The weak product is still
\(b\partial_k u=\partial_k(bu)-(\partial_kb)u\). For a smooth compact
cutoff \(\chi\), its exact consequence is \[
 \widetilde Q(\chi u)=\chi\widetilde Q u
 -\sum_{j,k}\partial_j(\widetilde b^{jk}(\partial_k\chi)u)
 -\sum_{j,k}(\partial_j\chi)\widetilde b^{jk}\partial_k u.\tag{DA6}
\] Each term belongs to \(H^{-1}\) when \(u\in L^2\) locally and
\(\widetilde Q u\in H^{-1}\) locally: for the last one apply the weak
product again with \(c^k=\sum_j(\partial_j\chi)\widetilde b^{jk}\). A
\(C^1\) factor such as \(\rho\) acts boundedly on \(H^{-1}\) by the dual
\(H^1\) product rule. Every coefficient and cutoff derivative in DA6 is
retained, with its original ordered indices.

For \(v=\chi u\), zero extend and cut off the coefficients beyond a
larger compact subchart. Let \(J_\varepsilon\) be convolution with a
compactly supported smooth approximate identity of integral one. The
entrywise commutator proved in Local inverses and distance-weighted
elliptic estimates, L37--L39, gives \[
 \begin{gathered}
 r_{\varepsilon,j}=\sum_k\{\widetilde b^{jk}\partial_k J_\varepsilon v
                    -J_\varepsilon(\widetilde b^{jk}\partial_kv)\},\\
 \|r_\varepsilon\|_2\leq C\|v\|_2,\\
 \widetilde QJ_\varepsilon v=J_\varepsilon\widetilde Qv-\sum_j\partial_jr_{\varepsilon,j}.
 \end{gathered}\tag{DA7}
\] The commutator concerns one entry at a time and requires no symmetry.
The smooth function \(v_\varepsilon=J_\varepsilon v\) has support in one
fixed compact set on which DA1 holds. Compact-support Poincare,
integration by parts and DA1 give the finite-norm estimate \[
 \|v_\varepsilon\|_{H^1}^2
 \leq C\operatorname{Re}(\widetilde Qv_\varepsilon,v_\varepsilon)_{dx}
 \leq C\{\|\widetilde Qv\|_{H^{-1}}+\|v\|_2\}\|v_\varepsilon\|_{H^1}.
 \tag{DA8}
\] Every norm on the smooth left side is finite. If its last factor is
zero there is no division; otherwise cancellation gives a uniform bound.
A weak \(H^1\) subsequential limit is the strong \(L^2\) limit \(v\).
This proves the first gain and the local D10 estimate with the full
nonsymmetric operator.

For \(\widetilde L u=f\in L^2_{\mathrm{loc}}\), the first gain gives
\(u\in H^1_{\mathrm{loc}}\). The exact next equation is \[
 -\sum_{j,k}\widetilde a^{jk}\partial_j\partial_k u
 =f+\sum_{j,k}\{\partial_j\widetilde a^{jk}
                +\widetilde a^{jk}\partial_j\log\rho\}\partial_k u.
 \tag{DA9}
\] The right side is \(L^2_{\mathrm{loc}}\). The scalar principal symbol
is \(p_x(\xi)=\sum_{j,k}\widetilde a^{jk}(x)\xi_j\xi_k\), whose real
part is at least \(\lambda_K|\xi|^2\) for real \(\xi\). The weak
elliptic proof L40--L43 therefore applies, with order two and exponent
two. That proof applies its finite smooth estimate to differences of
mollified functions. The commutator converges in \(L^2\); their highest
derivatives are Cauchy on an inner region; their distributional limit is
the original derivative. It proves \(H^2_{\mathrm{loc}}\), and then its
estimate D12, without assuming finiteness of the unknown second
derivatives. DA4 proves the same assertions for the actual adjoint in
DA2. Compact-support \(H^2\) approximation and convergence of both full
operators in \(L^2\) follow from the bounded coefficients in DA9,
exactly as in D13.

If \(\widetilde L^*w=0\), DA9 for its actual adjoint makes its principal
operator an \(L^2\) function bounded by a compact-dependent constant
times \(|dw|\). The principal form has positive real part. The path
\(\operatorname{Re}p_x+is\operatorname{Im}p_x\), \(0\leq s\leq1\), has
no nonzero real zero. For any two real independent covectors \(\xi,N\),
the two line roots start in opposite open half-planes, cannot cross the
real axis, and depend continuously as an unordered pair. Hence they stay
separated, including in dimension two. A collinear pair produces only
the excluded zero covector; dimension one has no independent pair. The
full support proof S39--S52 in Curved weights and the directions in
which support can end therefore gives continuation from a nonempty open
zero set along the connected domain. No symmetry of the coefficient
matrix or deletion of its lower-order terms has entered this
verification.

\subsubsection{3. Global solvability and the full compact-component
obstruction}\label{global-solvability-and-the-full-compact-component-obstruction}

Under DA1--DA2, for every \(f\in L^2_{\mathrm{loc}}(X)\), \[
 \exists u\in H^2_{\mathrm{loc}}(X):\widetilde Lu=f
 \quad\Longleftrightarrow\quad
 \int_Y f\,d\mu=0\ \text{for every compact connected component }Y.
 \tag{DA10}
\] Here is the proof with the exact receiving hypotheses. On a manifold
with only noncompact components, D14 is geometric and unchanged. DA4 and
DA3 give \(\operatorname{Re}(\widetilde L^*v,v)_\mu\geq c_K\|dv\|_2^2\)
for supported \(H^1\) functions. It follows by duality that
\(\|v\|_{H^1}\leq C_K\|\widetilde L^*v\|_2\). The finite local
second-derivative estimate just proved gives
\(\|v\|_{H^2}\leq C_K'\|\widetilde L^*v\|_2\) for supported tests. The
filled compact exhaustion D18--D19 is geometric and uses no coefficient.

Use the weight step D20--D27 with \(\widetilde L^*\). Its contradiction
sequence has pairing one and weighted residual less than one; the
supported coercivity just proved bounds it in \(H^2\). On that fixed
support, DA9 defines a bounded map \(\widetilde L^*:H^2\to L^2\), so the
weak residual limit and its weighted lower semicontinuity are valid. The
added cutoff equals one outside the preceding inner compact set.
Consequently the limit solves the adjoint equation on every
complementary component and is zero on that component's nonempty open
part beyond the fixed support. The continuation proved in Section 2
forces zero on the entire component. Compact-support approximation then
allows the preceding weight inequality on the limit, giving the same
contradiction \(1\leq(1+\varepsilon)^{-1}<1\). These are all the
operator-dependent steps of the weight extension, now verified for the
actual nonsymmetric adjoint.

Start with the constant \(M_3\) from the supported coercivity estimate
and carry out the full D28--D30 recursion. Hahn--Banach and linear-first
Riesz representation give \(g\in L^2(X,d\mu)\), \(\|g\|_2\leq1\), and \[
 (f,\phi)_\mu=(g,M\widetilde L^*\phi)_\mu
             =(Mg,\widetilde L^*\phi)_\mu,
 \qquad \phi\in C_c^2(X).\tag{DA11}
\] The locally bounded positive weight makes \(u=Mg\) locally \(L^2\),
and Section 2 gives its \(H^2_{\mathrm{loc}}\) regularity. This proves
arbitrary-data existence on all noncompact components.

On a compact connected component, \(\widetilde L^*1=0\) makes the mean
condition necessary. For sufficiency take the mean-zero Hilbert subspace
of \(H^1(Y)\) and the full form \[
 \widetilde B(v,w)=\int_Y\sum_{j,k}\widetilde a^{jk}\partial_kv\,
                                     \overline{\partial_jw}\,d\mu.
 \tag{DA12}
\] Its boundedness and DA1 give a representing operator \(T\) with
\(\|T\|\leq C\) and \(\operatorname{Re}(Tv,v)\geq c\|v\|^2\),
\(C\geq c>0\). For \(t=c/C^2\), expansion gives
\(\|(I-tT)v\|^2\leq(1-c^2/C^2)\|v\|^2\). The full operator geometric
series in D35 gives both inverse products. The represented mean-zero
datum is therefore attained; constants contribute zero to both sides, so
this is the equation against all tests. Section 2 gives \(H^2(Y)\). An
\(L^2\) solution of either homogeneous operator is \(H^2\); its energy
and DA1 or DA4 force zero gradient, so it is constant. Thus no further
adjoint obstruction occurs. Second countability gives countably many
open components, and a compact set meets finitely many. The
componentwise solutions assemble with local \(H^2\) regularity and prove
DA10 in its full disconnected form.

\subsubsection{4. The constructed weight is twice continuously
differentiable}\label{the-constructed-weight-is-twice-continuously-differentiable}

In the noncompact-component construction, the original weight can be
chosen in \(C^2(X)\). This strengthens the original assertion of
continuity; it does not replace the source operator or require a
smoother atlas. Start with its constant \(M_3>0\). The actual recursion
D22 is \[
 \begin{gathered}
 M_{j+1}=(1+\varepsilon_j)M_j+N_j^{\mathrm{add}}\chi_j,\\
 \chi_j\in C^2(X),\quad\chi_j=0\text{ near }K_{j-2},
 \quad N_j^{\mathrm{add}}\geq1.
 \end{gathered}\tag{DA13}
\] Use the same \(\varepsilon_j>0\) with finite sum and retain every
selected scalar \(N_j^{\mathrm{add}}\). Induction makes every \(M_j\) a
positive \(C^2\) function. With the original scalar products \(P_j\) of
D29, write \(W_j=P_j^{-1}M_j\). Then \(W_{j+1}=W_j\) on \(K_{j-2}\), and
\(P_j\uparrow P<\infty\). Given \(x\in X\), choose \(r\) with
\(x\in\operatorname{int}K_r\). For every \(j\geq r+2\), the last
equality holds on the same open neighborhood \(\operatorname{int}K_r\).
Hence the limit \(W\) there equals one fixed \(C^2\) function
\(W_{r+2}\), with equality of all derivatives through order two.
Therefore \(M=PW\) is positive and \(C^2\). This is local equality, not
an inference from uniform convergence of values to convergence of
derivatives. It proves the strengthening for the original symmetric
operator as well as for DA2.

The represented vector \(g=u/M\) consequently belongs to
\(H^2_{\mathrm{loc}}\cap L^2(X,d\mu)\). On a compact chart, \(M^{-1}\)
is \(C^2\) with bounded derivatives and positive lower denominator. The
complete product derivatives are \[
 \begin{split}
 \partial_jg&=M^{-1}\partial_ju-M^{-2}(\partial_jM)u,\\
 \partial_j\partial_kg
 &=M^{-1}\partial_j\partial_ku
   -M^{-2}\{(\partial_jM)\partial_ku+(\partial_kM)\partial_ju
                         +(\partial_j\partial_kM)u\}
   +2M^{-3}(\partial_jM)(\partial_kM)u.
 \end{split}\tag{DA14}
\] Approximation in \(H^2\) justifies these weak formulas, and every
displayed term is locally \(L^2\). The full conjugated equation is \[
 \begin{split}
 \widetilde L(Mg)
 &=M\widetilde Lg
   -\sum_{j,k}\widetilde a^{jk}
       \{(\partial_jM)\partial_kg+(\partial_kM)\partial_jg\}
   +(\widetilde LM)g,\\
 \widetilde LM
 &=-\rho^{-1}\sum_{j,k}
   \{\rho\widetilde a^{jk}\partial_j\partial_kM
             +\partial_j(\rho\widetilde a^{jk})\partial_kM\}.
 \end{split}\tag{DA15}
\] To verify it, insert \(\partial_k(Mg)=M\partial_kg+g\partial_kM\)
into the original flux, and apply the first-order product rule to each
of the two terms before multiplying by \(-\rho^{-1}\). DA15 retains both
ordered cross terms and the entire density derivative. For the original
transpose-symmetric tensor their sum equals
\(-2\sum_{j,k}a^{jk}(\partial_jM)\partial_kg\), by an explicit index
interchange; that reduction is not imposed on the nonsymmetric tensor.
Neither \(g\in H^2(X)\), bounded derivatives of \(M\) on all of \(X\),
nor a datum-independent weight is asserted.

\subsubsection{5. A full nonsymmetric example retaining the density
drift}\label{a-full-nonsymmetric-example-retaining-the-density-drift}

On \(X=\mathbb R^2\), with coordinates \((x,y)\), take \[
 \rho(x,y)=e^{\sin x},\qquad h(x)=|x|,\qquad
 \widetilde A(x,y)=\begin{pmatrix}1&h(x)\\-h(x)&1\end{pmatrix}.
 \tag{DA16}
\] For every complex \(z\), the cross term
\(h(z_2\overline z_1-z_1\overline z_2)\) is purely imaginary. Thus the
real energy is exactly \(|z_1|^2+|z_2|^2\), even though the tensor is
nonsymmetric for \(x\ne0\). Expand the two original fluxes separately.
The mixed second derivatives cancel, since \(h\) is independent of
\(y\), while \(\rho'/\rho=\cos x\) and \(h'=\operatorname{sgn}x\) almost
everywhere. The resulting full operators are \[
 \begin{split}
 \widetilde L&=-\partial_x^2-\partial_y^2-\cos x\,\partial_x
                  -(\operatorname{sgn}x+|x|\cos x)\partial_y,\\
 \widetilde L^*&=-\partial_x^2-\partial_y^2-\cos x\,\partial_x
                  +(\operatorname{sgn}x+|x|\cos x)\partial_y,\\
 \widetilde L(y)&=-\operatorname{sgn}x-|x|\cos x.
 \end{split}\tag{DA17}
\] The first weak derivative of \(|x|\) is the locally bounded sign
function. No second derivative of that coefficient occurs, so no delta
contribution is hidden at \(x=0\). The value assigned to the sign
function at zero is irrelevant to these almost-everywhere and
distributional equations. This exact solution is \(H^2_{\mathrm{loc}}\)
for the locally bounded datum in DA17. The skew coefficient has zero
scalar quadratic contribution on real covectors, but it produces the
displayed first-order drift. Keeping only that scalar principal
quadratic form would lose part of the original equation.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={Full nonsymmetric tensor and its density-bearing drift}]{../figures/divergence-full-tensor.png}}
\caption{Full nonsymmetric tensor and its density-bearing drift}
\end{figure}

The fixed vectors on the left use the exact original tensor at the
indicated coordinate. The curves on the right show the original density
and the entire drift in DA16--DA17. The sign function has two open
one-sided endpoints at zero; its chosen point value does not affect the
weak equation. This is a coordinate diagram, not a trajectory or a
physical model.

\subsubsection{6. A bounded comparison with the original-author
regularity
source}\label{a-bounded-comparison-with-the-original-author-regularity-source}

The source here is Sergey E. Mikhailov's original arXiv:0906.3875v3
archive, revised November 21, 2012, and its Part II file
\texttt{Mik-JCE6-HeC4input.tex}. The archive contains the updated author
sources of two published articles. The whole Part II prefix through
line729, including the complete proof of its theorem labeled
\texttt{RegTh} at lines464--725, was read for this comparison. The later
Part II body and the main Part I body are not claimed as read. The exact
freely accessible author version is
\href{https://arxiv.org/abs/0906.3875v3}{arXiv:0906.3875v3}.

\textbf{The exact receiving operator and coefficient classes.} The
source's equation \texttt{2.1-H}, lines242--247, is
\(-\sum_{i,j}\partial_i(a_{ij}\partial_j u)+\sum_jb_j\partial_ju+cu=f\).
Use it here with the full operator \(\widetilde Q=\rho\widetilde L\),
coefficient \(a_{ij}=\rho\widetilde a^{ij}\), lower coefficients
\(b_j=c=0\), and right side \(\rho f\). This is equality of the original
equations, not replacement of the density-bearing operator. Its scalar
Petrovsky ellipticity follows from
\(\operatorname{Re}\sum_{i,j}\rho\widetilde a^{ij}\xi_i\xi_j\geq\rho\lambda_K|\xi|^2\).
The source's actual adjoint coefficient
\(\overline{a_{ji}}^{\mathsf T}\), lines253--256, agrees with DA2 after
multiplication by \(\rho\).

In definition \texttt{DC\^{}sigma}, lines261--266, the full coefficient
conditions are \[
 \begin{gathered}
 a\in\overline C_+^{|\sigma|},\\
 b\in\overline C_+^{\max(0,|\sigma-1/2|-1/2)},\\
 c\in\overline C_+^{\max(0,|\sigma|-1)}.
 \end{gathered}\tag{DA18}
\] At an integer exponent the source uses the corresponding bounded
derivative class; at a noninteger exponent its plus class takes the
union of strictly higher exponents. Its theorem \texttt{RegTh} uses the
intersection of coefficient classes at \(\sigma=s_1-1\) and
\(\sigma=s_2+1\) for the direct equation. For \((s_1,s_2)=(0,-1)\) these
are \(\mathcal C_+^{-1}\cap\mathcal C_+^0\); for \((0,0)\) they are
\(\mathcal C_+^{-1}\cap\mathcal C_+^1\). In both cases the leading
coefficient need only satisfy the exponent-one bounded weak-derivative
requirement, and \(\rho\widetilde a^{ij}\) does so locally. All
first-order and zeroth-order conditions are satisfied by their exact
zero values. Multiplication by \(\rho\) is bounded on the relevant local
\(H^{-1}\) and \(L^2\) spaces. The two source conclusions are therefore
precisely the \(L^2\)-to-\(H^1\) and \(L^2\)-to-\(H^2\) receiving
assertions. This check does not apply an exponent-one condition to the
expanded density drift in DA9, which can be only continuous or bounded;
using the unexpanded full flux is essential.

\textbf{Visible Fourier pairing erratum.} The source fixes the transform
with phase \(-2\pi i x\cdot\theta\), with inverse phase
\(+2\pi i x\cdot\theta\), at lines151--152. Its scalar bilinear pairing
at line140 uses an inverse transform in its first factor. The vector
display at line156 instead integrates two forward transforms at the same
frequency. With the stated bilinear pairing its correction is \[
 \int_{\mathbb R^n}u(x)^{\mathsf T}v(x)\,dx
 =\int_{\mathbb R^n}\widehat u_{\mathrm{src}}(-\theta)^{\mathsf T}
                          \widehat v_{\mathrm{src}}(\theta)\,d\theta,
 \quad
 \widehat u_{\mathrm{src}}(\theta)=\int e^{-2\pi i x\cdot\theta}u(x)\,dx.
 \tag{DA19}
\] For Schwartz functions insert the full inverse formula for the second
factor into the left side. Absolute Fubini is justified by
\(\|u\|_{L^1}\|\widehat v_{\mathrm{src}}\|_{L^1}<\infty\); the first
integral is exactly the forward transform of \(u\) at \(-\theta\). For
dual Bessel Sobolev classes, weighted Cauchy--Schwarz bounds the right
side by their two norms, because \(|-\theta|=|\theta|\). Schwartz
density then extends the identity. Thus the frequency reversal is
required; it is not a convention-dependent extra power of \(2\pi\).

For an explicit counterexample take dimension and vector rank one and
\(u=v=e^{-\pi(x-a)^2}\), with real \(a\ne0\). Its transform is
\(e^{-2\pi ia\theta}e^{-\pi\theta^2}\). To verify the Gaussian factor
directly, the integral of \(e^{-\pi x^2}\) equals one by squaring and
polar integration; differentiation of its transform and integration by
parts give \(F'=-2\pi\theta F\), \(F(0)=1\). Translation then gives the
displayed phase. Hence \[
 \begin{gathered}
 \int u(x)v(x)\,dx=2^{-1/2},\\
 \int\widehat u_{\mathrm{src}}(\theta)\widehat v_{\mathrm{src}}(\theta)\,d\theta
       =2^{-1/2}e^{-2\pi a^2}.
 \end{gathered}\tag{DA20}
\] The second identity follows by scaling the same Gaussian transform
and evaluating it at frequency \(2a\). These values differ. The archived
author source remains unchanged; DA19 is a separately identified
conceptual erratum.

The course uses the original transform phase \(-ix\cdot\xi\) and inverse
multiplier \((2\pi)^{-n}\). Its exact comparison with the source
convention is \[
 \begin{gathered}
 \widehat u_{\mathrm{src}}(\theta)=\widehat u_{\mathrm{course}}(2\pi\theta),
 \quad \xi=2\pi\theta,\quad d\xi=(2\pi)^n d\theta,\\
  \|u\|_{H^1,\mathrm{src}}^2=\int(1+|\theta|^2)|\widehat u_{\mathrm{src}}(\theta)|^2d\theta,\\
  \|u\|_{H^1,\mathrm{course}}^2=\int(1+4\pi^2|\theta|^2)|\widehat u_{\mathrm{src}}(\theta)|^2d\theta.
 \end{gathered}\tag{DA21}
\] The latter is the full weak-derivative norm
\(\|u\|_2^2+\sum_j\|\partial_ju\|_2^2\). Consequently
\(\|u\|_{H^1,\mathrm{src}}\leq\|u\|_{H^1,\mathrm{course}}\leq2\pi\|u\|_{H^1,\mathrm{src}}\);
taking dual unit balls gives
\(\|f\|_{H^{-1},\mathrm{course}}\leq\|f\|_{H^{-1},\mathrm{src}}\leq2\pi\|f\|_{H^{-1},\mathrm{course}}\).
The source inner product is linear in its second entry, while the course
pairing is linear in its first: their value for the same two arguments
is related by complex conjugation. Both conventions and every original
transform factor have now been compared explicitly.

\textbf{The finite-norm justification in the source proof.} In step(i)
of \texttt{RegTh}, lines591--619, the displayed estimate contains the
desired \(H^{s_2+2}\) norm of a cutoff of the original unknown. The
argument makes its coefficient positive and concludes that the norm is
finite. The displayed calculation does not first supply a regularization
or an inverse argument proving that this highest norm is finite, so
absorption of that quantity alone leaves a proof gap at that step. This
identifies a missing justification in the read proof, not a
counterexample to its theorem or a claim to have audited all its more
general systems and Holder cases. For the scalar Lipschitz receiving
case actually used here, DA6--DA8 prove the first gain using finite
smooth norms and a weak limit; DA9 and the complete L40--L43 proof
establish the second gain using smooth differences and Cauchy highest
derivatives. These supply the missing finite-norm justification in this
course's exact scope. No broader source theorem is being silently
substituted for the local proof.

\subsection{References}\label{references}

The local commutator and elliptic estimates used here are proved in
\href{local-elliptic-coefficients.pdf}{Local inverses and
distance-weighted elliptic estimates}; the continuation input is in
\href{support-continuation.pdf}{Curved weights and the directions in
which support can end}. The compact-component qualification above is
part of the statement that should be retained whenever the manifold
clause is used.

For a direct regularity comparison,
\href{https://bura.brunel.ac.uk/handle/2438/7239}{Mikhailov, ``Solution
regularity and co-normal derivatives for elliptic systems with
non-smooth coefficients on Lipschitz domains'' (2013)}, Theorem 4.3 of
the author postprint, treats a broader system setting and Sobolev
indices. Its statement includes the scalar endpoints relevant here.
\href{https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf}{Hunter's
\emph{Notes on Partial Differential Equations}, revision 18 June 2014},
Theorem 4.27, gives a detailed difference-quotient route for real
\(C^1\) coefficients and an initially \(H^1\) weak solution. Those
initial conditions differ from our first regularity step.

\href{https://aif.centre-mersenne.org/item/AIF_1956__6__271_0/}{Malgrange,
``Existence et approximation des solutions des équations aux dérivées
partielles et des équations de convolution'' (1956)}, Chapter III,
relates adjoint continuation, compact hulls and global solvability on a
smooth noncompact manifold. Its hull lemma and Theorem III.5 are useful
comparisons for the global mechanism. The present argument supplies its
own compact-hull construction and weighted estimate in the stated
\(C^2\)/Lipschitz setting.

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

Three research routes follow from the distinctions established here.
First, prescribe a growth class for the data and seek a weight with a
quantitative growth bound; the arbitrary locally finite construction
does not supply one. Second, add lower-order terms and determine the
full adjoint kernel on compact components; constants need no longer
describe it. Third, compare interior surjectivity with a chosen boundary
condition, where traces and complementing conditions introduce
additional constraints. Each question needs its own hypotheses and
proof.

\end{document}