\chapter{Proof of the regular realization}\label{s:regularity}\label{s:sigma-regular-proof}

In this appendix, we prove \zcref{p:sigma-regular}, the description \eqref{e:sigma-domain-p} of the domains, and \zcref{l:sigma-chains}, which are stated in \zcref{s:sigma-partition-sec}. We use the notation of \zcref{s:sigma}, and the proofs use only the material of \zcref{s:sigma} that precedes these statements. The operator $\mathsf G$ is a second-order elliptic system on the bundle of superfunctions over $\mathcal M_k^0$, with a potential that grows like $v$. We realize it by Kato's form method, and we follow the standard arguments for elliptic operators on manifolds of bounded geometry \cite{shubin1992spectral}. Two features need care. The principal part of $\mathsf C_k$ is not a multiple of the identity on the odd coefficients, and the potential is not real. Both problems are solved by the choice of a small weight $\omega$ in the coefficient norms, since in the norm $| \cdot |_{2; \omega}$ the nilpotent parts of the principal symbol and of the potential become small.

Throughout the proof, constants may depend on $k$, $X$, $\eps$, $R_{\mathrm F}$, $p$, and $\omega$, and they are bounded for $(X, \eps)$ in compact subsets of $(0, \infty)^2$. We write $\| \cdot \| = \| \cdot \|_0$ and $\langle \cdot, \cdot \rangle = \langle \cdot, \cdot \rangle_0$.

\section{Local structure}\label{s:sigma-local}

By \zcref{l:sigma-casimir}\ref{i:sigma-casimir-local} and the homogeneity of \zcref{s:sigma-weighted}, in every translated chart the generator has the form
\begin{equation}\label{e:sigma-generator-local}
\mathsf G = - X \Delta_{\mathcal M_k^0} + X \sum_{\mu, \nu} n^{\mu \nu} \partial_\mu \partial_\nu + \sum_\mu \tilde b^\mu \partial_\mu + \tilde c + \pi \Str(Q \Gamma) ,
\end{equation}
where the coefficients $n^{\mu \nu}$, $\tilde b^\mu$, and $\tilde c$ are linear maps of $\Grass{4k^2}$ with smooth coefficients, the same in all translated charts. Each $n^{\mu \nu}$ raises the degree by at least two. In the norm $| \cdot |_{2; \omega}$, a derivative in $\chi_a$ costs a factor $\omega^{-1}$ and a factor $\chi_a$ gains a factor $\omega$, by \zcref{l:grass-norm}. Therefore
\begin{equation}\label{e:sigma-n-small}
\|n^{\mu \nu}\|_{| \cdot |_{2; \omega} \to | \cdot |_{2; \omega}} \le C \omega^2, \qquad \|\tilde b^\mu\| + \|\tilde c\| \le C_\omega ,
\end{equation}
where $C$ is uniform in $\omega \in (0, 1]$, and $C_\omega$ may depend on $\omega$. For the potential, \eqref{e:sigma-potential}, \zcref{l:sigma-multipliers}, and the bound $|\Tr(Q^{\mathrm{FF}} M)| \le 2k \|M\|_{\mathrm{op}}$ for the body imply
\begin{equation}\label{e:sigma-potential-bounds}
\begin{gathered}
\Re \big( \pi \Str(Q \Gamma) \big)_\emptyset \ge 2 \pi \eps (v - 1 - k) - C R_{\mathrm F}, \\
\big| \pi \Str(Q \Gamma) - (\pi \Str(Q \Gamma))_\emptyset \big|_{1; \omega} \le C \omega^2 (\eps v + R_{\mathrm F}) .
\end{gathered}
\end{equation}
Here we used that the soul of $Q^{\mathrm{FF}}$ has degree at least two, since odd powers of $\Theta$ have vanishing diagonal blocks, and that the imaginary part of the body comes only from the bounded fermionic term.

\section{The form}\label{s:sigma-form}

For compactly supported smooth superfunctions $f$ and $f'$, we set $\mathfrak t[f, f'] = \langle \mathsf G f, f' \rangle$ and $\|f\|_V^2 = \|\nabla f\|^2 + \|v^{1/2} f\|^2 + \|f\|^2$. We claim that there exist $c > 0$ and $C$ such that
\begin{equation}\label{e:sigma-coercive}
\Re \mathfrak t[f, f] + C \|f\|^2 \ge c \|f\|_V^2, \qquad |\mathfrak t[f, f']| \le C \|f\|_V \|f'\|_V ,
\end{equation}
for all compactly supported smooth $f$ and $f'$, provided $\omega$ is small enough.

To prove \eqref{e:sigma-coercive}, we fix a partition of unity $1 = \sum_\ell \phi_\ell^2$ on $\mathcal M_k^0$ subordinate to a cover by the domains of translated charts, with uniformly bounded multiplicity, such that each $\phi_\ell$ is the transport of one of finitely many fixed functions by the action of $G_0$. Such a cover exists since $\mathcal M_k^0$ is homogeneous. In each chart, we integrate the second-order terms by parts once. The term $-X \Delta_{\mathcal M_k^0}$ gives $X \|\nabla f\|^2$, up to terms bounded by $C \|\nabla f\| \|f\|$ that come from the derivatives of the $\phi_\ell$ and from the change of the odd frame between charts. Using \eqref{e:sigma-n-small}, we bound the terms with $n^{\mu \nu}$ by $C \omega^2 \|\nabla f\|^2 + C_\omega \|\nabla f\| \|f\|$, and the first-order and zeroth-order terms by $C_\omega \|\nabla f\| \|f\| + C_\omega \|f\|^2$.

For the potential, \eqref{e:sigma-potential-bounds} implies that
\[
\Re \langle \pi \Str(Q \Gamma) f, f \rangle \ge 2 \pi \eps \|v^{1/2} f\|^2 - C \omega^2 (\eps + R_{\mathrm F}) \|v^{1/2} f\|^2 - C \|f\|^2 ,
\]
where we used \zcref{l:grass-norm}\ref{i:grass-norm-sub} for the soul. We now choose $\omega$ so small that $C \omega^2 \le X/2$ in the bound for the principal part and $C \omega^2 (\eps + R_{\mathrm F}) \le \pi \eps$ in the bound for the potential. Absorbing the products $\|\nabla f\| \|f\|$ by Young's inequality then yields the first inequality in \eqref{e:sigma-coercive}. These two conditions are the only requirements on $\omega$ in the proof, and they hold uniformly for $(X, \eps)$ in compact subsets of $(0, \infty)^2$. The second inequality follows from the same integration by parts and the bound $|\pi \Str(Q \Gamma)|_{1; \omega} \le C v$.

By \eqref{e:sigma-coercive}, the form $\mathfrak t + C$ is sectorial and closable, and the domain of its closure is the completion of the compactly supported smooth superfunctions in $\| \cdot \|_V$. This completion is the space $V_0$ of part \ref{i:sigma-regular-kato}, since a superfunction with $\|f\|_V < \infty$ is the limit in $\| \cdot \|_V$ of $\psi_R f$, with the cutoffs $\psi_R$ of \eqref{e:sigma-cutoff}, followed by mollification in the charts. By Kato's first representation theorem \cite[Theorem~VI.2.1]{kato1995perturbation}, there exists an m-sectorial operator $\mathsf G_{(0)} + C$ such that $f \in D(\mathsf G_{(0)})$ if and only if $f \in V_0$ and there exists $g \in L^2_0$ with $\bar{\mathfrak t}[f, f'] = \langle g, f' \rangle$ for all $f' \in V_0$, and then $\mathsf G_{(0)} f = g$. Since the compactly supported smooth superfunctions are dense in $V_0$, it suffices to test with these $f'$, and an integration by parts yields the characterization
\begin{equation}\label{e:sigma-domain}
D(\mathsf G_{(0)}) = \{ f \in V_0 : \mathsf G f \in L^2_0 \} ,
\end{equation}
where $\mathsf G f$ is taken in the sense of distributions, and $\mathsf G_{(0)} f = \mathsf G f$. In particular, $\mathsf G_{(0)}$ extends $\mathsf G$ on compactly supported smooth superfunctions. Since $\mathsf G_{(0)} + C$ is m-sectorial, the operator $-\mathsf G_{(0)}$ generates a holomorphic semigroup with $\|e^{-t \mathsf G_{(0)}}\| \le e^{C t}$ \cite{kato1995perturbation}.

\section{The core}\label{s:sigma-core}

We now show that the compactly supported smooth superfunctions are a core for $\mathsf G_{(0)}$, and we first record the local regularity of the elements of $D(\mathsf G_{(0)})$. Let $f \in D(\mathsf G_{(0)})$. By \eqref{e:sigma-generator-local}, the principal part of $\mathsf G$ in a chart is triangular with respect to the degree in the $\chi_a$, with diagonal part $-X \Delta_{\mathcal M_k^0}$, since each $n^{\mu \nu}$ raises the degree. Since $f \in V_0$ and $\mathsf G f \in L^2_0$, the component $f_j$ of $f$ of degree $j$ satisfies $-X \Delta_{\mathcal M_k^0} f_j = g_j$ in each chart, where $g_j$ is square integrable once the components of degree less than $j$ have square integrable second derivatives. By induction on $j$ and interior regularity for scalar elliptic equations \cite{gilbarg2001elliptic}, $f$ has locally square integrable second derivatives.

We claim that $\psi_R f \to f$ in the graph norm of $\mathsf G_{(0)}$. We have $\psi_R f \in V_0$ and
\[
\mathsf G(\psi_R f) = \psi_R \mathsf G f + [\mathsf G, \psi_R] f .
\]
The potential commutes with $\psi_R$. The commutator is therefore a first-order operator whose coefficients are derivatives of $\psi_R$ of order one and two, multiplied by bounded coefficients of $\mathsf G$, and
\[
\|[\mathsf G, \psi_R] f\| \le C \big( \|\nabla \psi_R\|_\infty + \|\nabla^2 \psi_R\|_\infty \big) \big( \|\nabla f\| + \|f\| \big) \le C R^{-1} \|f\|_V \longrightarrow 0 .
\]
Moreover $\psi_R f \in D(\mathsf G_{(0)})$ by \eqref{e:sigma-domain}, and $\psi_R f \to f$ and $\psi_R \mathsf G f \to \mathsf G_{(0)} f$ in $L^2_0$, which proves the claim. Finally, a compactly supported element of $D(\mathsf G_{(0)})$ is the limit in the graph norm of its mollifications in finitely many charts, since its second derivatives are square integrable. We have proved part \ref{i:sigma-regular-kato} for $p = 0$.

\section{Weights}\label{s:sigma-weights-proof}

Let $p \in \R$. Multiplication by $v^p$ is a unitary map from $L^2_p$ onto $L^2_0$ that preserves the compactly supported smooth superfunctions. We have
\[
v^p \mathsf G v^{-p} = \mathsf G + \mathcal R_p ,
\]
where $\mathcal R_p$ is a first-order operator whose coefficients are polynomials in the derivatives of $\log v$ of order one and two, multiplied by coefficients of the differential part of $\mathsf G$. In particular, conjugating the first-order term contributes the bounded zeroth-order term $-p \sum_\mu \tilde b^\mu \partial_\mu \log v$. These coefficients are bounded, by \zcref{l:sigma-body}\ref{i:sigma-body-derivatives} and \eqref{e:sigma-n-small}, and \eqref{e:sigma-coercive} and the arguments of \zcref{s:sigma-form,s:sigma-core} hold for $\mathsf G + \mathcal R_p$ on $L^2_0$. Transporting back by multiplication by $v^{-p}$, we obtain part \ref{i:sigma-regular-kato} for all $p$, with $V_p = v^{-p} V_0$, which is the space displayed there, and the description \eqref{e:sigma-domain-p} of the domain of $\mathsf G_{(p)}$.

We now prove the consistency in part \ref{i:sigma-regular-weights}. Let $q \ge p$. Then $V_q \subset V_p$, since $v^p f = v^{p - q} (v^q f)$, and multiplication by $v^{p - q} \le 1$ preserves $V_0$ by \zcref{l:sigma-body}\ref{i:sigma-body-derivatives}. Let $\lambda$ be larger than the shifts of $\mathsf G_{(p)}$ and $\mathsf G_{(q)}$, let $f \in L^2_q$, and let $g = (\mathsf G_{(q)} + \lambda)^{-1} f$. Then $g \in V_q \subset V_p$, and $\mathsf G g = f - \lambda g \in L^2_p$ in the sense of distributions. Using \eqref{e:sigma-domain-p}, we find that $g \in D(\mathsf G_{(p)})$ and $(\mathsf G_{(p)} + \lambda) g = f$. The resolvents of $\mathsf G_{(p)}$ and $\mathsf G_{(q)}$ agree on $L^2_q$, and so do the semigroups, by the exponential formula $e^{-t \mathsf G} = \lim_{n \to \infty} (\Id + t \mathsf G/n)^{-n}$ \cite{kato1995perturbation}.

Next, we prove the smoothing estimate. Let $f \in L^2_0$ and $g(t) = e^{-t \mathsf G_{(0)}} f$. Since the semigroup is holomorphic, $\|\mathsf G_{(0)} g(t)\| \le C t^{-1} \|f\|$ for $0 < t \le T$, and \eqref{e:sigma-coercive} implies that
\[
c \|v^{1/2} g(t)\|^2 \le \Re \langle \mathsf G_{(0)} g(t), g(t) \rangle + C \|g(t)\|^2 \le C (t^{-1} + 1) \|f\|^2 .
\]
This is the case $p = 0$, $m = 1$ of the estimate, and the same argument for $\mathsf G + \mathcal R_p$ yields the case $m = 1$ for all $p$. For general $m$, we write $e^{-t \mathsf G} = (e^{-t \mathsf G/m})^m$, which is legitimate by consistency, and we apply the case $m = 1$ with the weights $p, p + \frac12, \dots, p + \frac{m - 1}{2}$.

It remains to show that $e^{-t \mathsf G} f$ is rapidly decreasing for $t > 0$ and $f \in L^2_p$. Recall the domains $D(\mathsf G^j_{(p)})$ of the powers of $\mathsf G$ from \zcref{s:sigma-partition-sec}. By the smoothing estimate, consistency, and holomorphy, $g = e^{-t \mathsf G} f$ belongs to $D(\mathsf G_{(p')}^n)$ for all $p'$ and all $n$. We use interior elliptic estimates in the translated charts. Let $x \in \mathcal M_k^0$, and let $B_1 \subset B_2$ denote the images of the balls of radii one and two centered at $\Lambda$ in the translated chart centered at $x$. We claim that for every integer $j \ge 1$ and every $p \in \R$, there exists $C$, independent of $x$, such that
\begin{equation}\label{e:sigma-chains-interior}
\sum_{i = 0}^{2j} \|\nabla^i g\|_{L^2(B_1)} \le C \, v(x)^j \sum_{l = 0}^j \|\mathsf G^l g\|_{L^2(B_2)} , \qquad g \in D(\mathsf G^j_{(p)}) ,
\end{equation}
where the norms are those of the coefficients in the chart.

To prove \eqref{e:sigma-chains-interior}, we first note that the local form of the coercivity estimate \eqref{e:sigma-coercive}, with a cutoff function, bounds the first derivatives of $g$ on a ball between $B_1$ and $B_2$. The part of degree $s$ of the equation $\mathsf G g = h$ is then an equation $-X \Delta_{\mathcal M_k^0} g_s = h_s - (\dots)$, in which the omitted terms contain second derivatives only of the parts of $g$ of degree at most $s - 2$, since each $n^{\mu \nu}$ in \eqref{e:sigma-generator-local} raises the degree by at least two, and first derivatives of all parts. By induction on $s$ and interior regularity for $-X \Delta_{\mathcal M_k^0}$ \cite{gilbarg2001elliptic}, we obtain the case $j = 1$, and the general case follows by differentiating the equation. The constant is uniform in $x$, since the coefficients of $\mathsf G - \pi \Str(Q \Gamma)$ are the same in all translated charts, the potential is a multiplier of degree one by \zcref{l:sigma-multipliers}, and $v$ varies by at most a bounded factor on $B_2$, since $\log v$ is Lipschitz.

Multiplying \eqref{e:sigma-chains-interior} for $g = e^{-t \mathsf G} f$ by $v(x)^{p'}$, summing the squares over a cover of bounded multiplicity, and using the Sobolev embedding theorem, we find that $g$ is smooth and that $\|\nabla^j g\|_{p'} < \infty$ for all $j$ and $p'$. This completes the proof of part \ref{i:sigma-regular-weights}.

\section{Symmetry, the split pairing, and small damping}\label{s:sigma-symmetry-proof}

We begin with part \ref{i:sigma-regular-transpose}. Recall from \eqref{e:sigma-integral} that $\mathcal B_k(f, f') = \int_{\mathcal M_k^0} \int \rho f f' \dd \chi \dd \mathrm{vol}$. The inner Berezin integral is the top coefficient of $\rho f f'$, and
\[
\Big| \int \rho f f' \dd \chi \Big| \le \omega^{-4k^2} |\rho f f'|_{1; \omega} \le C |f|_{2; \omega} |f'|_{2; \omega} ,
\]
where we used \zcref{l:grass-norm}\ref{i:grass-norm-sub} and the Cauchy--Schwarz inequality over the $2^{4k^2}$ monomials. Writing $1 = v^p v^{-p}$ and applying the Cauchy--Schwarz inequality on $\mathcal M_k^0$, we obtain $|\mathcal B_k(f, f')| \le C \|f\|_p \|f'\|_{-p}$.

By \zcref{l:sigma-casimir}\ref{i:sigma-casimir-symmetric}, we have $\mathcal B_k(\mathsf G f, f') = \mathcal B_k(f, \mathsf G f')$ for compactly supported smooth $f$ and $f'$, since the potential acts by multiplication by an even superfunction. These superfunctions are cores for $\mathsf G_{(p)}$ and $\mathsf G_{(-p)}$. Since $\mathcal B_k$ is bounded on $L^2_p \times L^2_{-p}$, the identity extends to $f \in D(\mathsf G_{(p)})$ and $f' \in D(\mathsf G_{(-p)})$. For such $f$ and $f'$, the function $s \mapsto \mathcal B_k(e^{-s \mathsf G} f, e^{-(t - s) \mathsf G} f')$ on $[0, t]$ is differentiable, with derivative
\[
- \mathcal B_k(\mathsf G e^{-s \mathsf G} f, e^{-(t - s) \mathsf G} f') + \mathcal B_k(e^{-s \mathsf G} f, \mathsf G e^{-(t - s) \mathsf G} f') = 0 .
\]
Comparing $s = 0$ and $s = t$, and using that the domains are dense, we obtain part \ref{i:sigma-regular-transpose}.

We now prove part \ref{i:sigma-regular-split}. By the layer cake formula and \zcref{l:sigma-body}\ref{i:sigma-body-volume}, we have $\|1\|_{-q}^2 = \int v^{-2q} \dd \mathrm{vol} < \infty$ for $2q > q_1$, and part \ref{i:sigma-regular-weights} shows that $e^{-t \mathsf G} 1$ is rapidly decreasing for $t > 0$. Applying part \ref{i:sigma-regular-transpose} with $p = -q$, $f = 1$, $t = a$, and $f' = e^{-(T - a) \mathsf G} 1 \in L^2_q$, and using consistency, we find
\[
\mathcal B_k(e^{-a \mathsf G} 1, e^{-(T - a) \mathsf G} 1) = \mathcal B_k(1, e^{-T \mathsf G} 1) = \int e^{-T \mathsf G} 1 .
\]

Finally, we prove part \ref{i:sigma-regular-uniform}, in which the main point is that the soul of the damping is controlled by its body, uniformly as $z \downarrow 0$. Let $0 \le z \le z_0$. Using \zcref{l:sigma-multipliers}\ref{i:sigma-mult-d} and \eqref{e:sigma-dk-body}, we find
\[
|d_k - (d_k)_\emptyset|_{1; \omega} \le C \omega^2 v \le C \omega^2 \big( (d_k)_\emptyset + 1 + k \big) .
\]
Choose $\omega_0 > 0$ small enough for the principal-part estimate of \zcref{s:sigma-form} and for $C \omega_0^2 \le \frac12$, and fix $\omega \in (0, \omega_0]$ for the rest of the argument. Then half of the nonnegative body $z (d_k)_\emptyset$ absorbs the soul of $z d_k$, up to the constant $(1 + k) z_0$, and the argument of \zcref{s:sigma-form} gives, for compactly supported smooth $f$,
\[
\Re \langle \mathsf G_{X, z} f, f \rangle + C_\omega \|f\|^2 \ge c \|\nabla f\|^2 + \frac z2 \langle (d_k)_\emptyset f, f \rangle \ge c \|\nabla f\|^2 ,
\]
\[
|\Im \langle \mathsf G_{X, z} f, f \rangle| \le C_\omega \big( \|\nabla f\|^2 + \|f\|^2 \big) + \frac z2 \langle (d_k)_\emptyset f, f \rangle ,
\]
with $c$ and $C_\omega$ uniform in $z$. The dependence on $\omega$ comes from the first-order and zeroth-order coefficient bounds $C_\omega$ in \eqref{e:sigma-n-small}; Young's inequality retains this dependence in the form shift. The same holds with the weight $v^{2p}$, by the argument of \zcref{s:sigma-weights-proof}. For $z = 0$, these bounds show, as in \zcref{s:sigma-form,s:sigma-core,s:sigma-weights-proof}, that $X \mathsf C_k$ is closable in $L^2_p$, and that its closure has the form domain $H^1_p$ and the domain $\{ f \in H^1_p : \mathsf C_k f \in L^2_p \}$; the potential plays no role in these arguments. The numerical range of $\mathsf G_{X, z} + C_\omega$ lies in a fixed sector for all $z \in [0, z_0]$, and for $0 < s \le T$ and $j \ge 0$,
\[
\|e^{-s \mathsf G_{X, z}}\|_{L^2_p \to L^2_p} \le e^{C_\omega s}, \qquad \|\mathsf G_{X, z}^j e^{-s \mathsf G_{X, z}}\|_{L^2_p \to L^2_p} \le C_{j,\omega} s^{-j} ,
\]
uniformly in $z$, with $\omega$ fixed; the first bound holds for all $s \ge 0$. For $X$ in a compact subset of $(0, \infty)$, the threshold $\omega_0$ and, at fixed admissible $\omega$, these constants can be chosen uniformly in $X$.

The bound on the coefficients of the superfunctions $e^{-s_n \mathsf G_{X, z}} m_n \cdots e^{-s_1 \mathsf G_{X, z}} m_1 1$ in \ref{i:sigma-regular-uniform} is part \ref{i:sigma-chains-compact} of \zcref{l:sigma-chains}, whose proof in \zcref{s:sigma-chains} below uses only the parts of \zcref{p:sigma-regular} proved so far. This completes the proof of \zcref{p:sigma-regular}.

\section{Semigroup products at all times}\label{s:sigma-chains}

We now prove \zcref{l:sigma-chains}, which gives bounds on the semigroup products \eqref{e:sigma-chain} that are uniform in the times.

\begin{proof}[Proof of \zcref{l:sigma-chains}]
We write $\mathsf G = \mathsf G_{X, z}$, and we fix $\omega$ so small that \zcref{p:sigma-regular} holds for the finitely many weights used below, uniformly in $z \in [0, z_0]$. We use three facts.

\emph{(A) Interior estimates.} Recall the interior estimate \eqref{e:sigma-chains-interior} from the proof of \zcref{p:sigma-regular}\ref{i:sigma-regular-weights}, for the balls $B_1 \subset B_2$ of the translated chart centered at $x$. Its proof uses only that $g \in D(\mathsf G^j_{(p)})$, and it applies to $\mathsf G = \mathsf G_{X, z}$ for $z \in [0, z_0]$, with the realization of \zcref{p:sigma-regular}\ref{i:sigma-regular-uniform} if $z = 0$. The constant $C$ in \eqref{e:sigma-chains-interior} is then uniform in $z \le z_0$, since $z d_k$ is $z$ times a multiplier of degree one.

Multiplying \eqref{e:sigma-chains-interior} by $v(x)^{p - j}$, squaring, and summing over the centers of a cover of bounded multiplicity, as in \zcref{s:sigma-form}, we obtain
\begin{equation}\label{e:sigma-chains-elliptic}
\sum_{i = 0}^{2j} \|\nabla^i g\|_{p - j} \le C \sum_{l = 0}^j \|\mathsf G^l g\|_p , \qquad g \in D(\mathsf G^j_{(p)}) .
\end{equation}

\emph{(B) Multipliers.} Let $m$ be a multiplier of degree $s$, let $j \ge 1$, and let $u$ be a smooth superfunction with $\|\nabla^i u\|_p \le C$ for $i \le 2j + 1$. In a translated chart, $\mathsf G^l (m u)$ is a sum of terms $a_\beta \partial^\beta u$ with $|\beta| \le 2l$, whose coefficients are products of derivatives of $m$, of the coefficients of $\mathsf C_k$, and of $z d_k$. The coefficients of $\mathsf C_k$ are bounded with all their derivatives in the translated charts, and $m$ and $z d_k$ are multipliers of degrees $s$ and one, so that
\[
|a_\beta|_{1; 1} \le C v^{s + l}, \qquad |\beta| \le 2l ,
\]
with all derivatives, uniformly in $z \le z_0$. Let $p' = p - s - j - 1$. Then $\mathsf G^l (m u) \in V_{p'}$ for $l < j$, with $H^1_{p'}$ in place of $V_{p'}$ if $z = 0$, and $\mathsf G^j (m u) \in L^2_{p'}$, with norms bounded by a constant. We apply \eqref{e:sigma-domain-p}, or its analogue for $z = 0$ in the proof of \zcref{p:sigma-regular}\ref{i:sigma-regular-uniform}, in turn to $\mathsf G^{j - 1}(m u), \mathsf G^{j - 2}(m u), \dots, m u$, and we obtain
\[
m u \in D(\mathsf G^j_{(p')}), \qquad \sum_{l = 0}^j \|\mathsf G^l (m u)\|_{p'} \le C .
\]

\emph{(C) The semigroup.} For $u \in D(\mathsf G^j_{(p)})$ and $s \ge 0$, we have $e^{-s \mathsf G} u \in D(\mathsf G^j_{(p)})$ and $\mathsf G^l e^{-s \mathsf G} u = e^{-s \mathsf G} \mathsf G^l u$ for $l \le j$, and the first bound of \zcref{p:sigma-regular}\ref{i:sigma-regular-uniform} implies that
\[
\| \mathsf G^l e^{-s \mathsf G} u \|_p \le e^{C s} \|\mathsf G^l u\|_p , \qquad l \le j, \ s \in [0, T] .
\]

We now prove \ref{i:sigma-chains-weighted} by induction over the construction of the semigroup product. Let $q > q_1/2$, so that $1 \in L^2_{-q}$ by \zcref{p:sigma-regular}\ref{i:sigma-regular-split}. The derivatives of the constant $1$ vanish in the translated charts, and (B) with $m = 1$ and $u = 1$ shows that $1 \in D(\mathsf G^J_{(p_J)})$ for every $J$, with $p_J = -q - J - 1$. If $u$ satisfies the bounds of \ref{i:sigma-chains-weighted} for some $J \ge 2$, then (A) bounds $\|\nabla^i u\|_{p - J}$ for $i \le 2J$, and (B) shows that $m u$ satisfies them with $J - 1$ in place of $J$ and a smaller weight. By (C), the bounds are preserved by a semigroup factor. Starting with $J = j + n + 1$, we obtain $f \in D(\mathsf G^{j + 1}_{(p)})$ for some $p$, with bounds uniform in $z$ and in the times, and (A) then implies \eqref{e:sigma-chains-weighted}, after lowering $p$. For \ref{i:sigma-chains-compact}, let $K$ be compact. On a compact neighborhood of $K$, the weight $v$ is bounded above and below, and \ref{i:sigma-chains-weighted} with $j$ large and the Sobolev embedding theorem in the translated charts bound the coefficients of $f$ and their derivatives of order at most $j$ on $K$.

For \ref{i:sigma-chains-time}, fix $z$ and the times. By \ref{i:sigma-chains-weighted}, for every $j$ there exists $p$ with $f \in D(\mathsf G^{j + 1}_{(p)})$. The map $t \mapsto e^{-t \mathsf G} f$ is then continuously differentiable on $[0, \infty)$ in the graph norm of $\mathsf G_{(p)}^j$, with derivative $- \mathsf G e^{-t \mathsf G} f = - e^{-t \mathsf G} \mathsf G f$. Iterating, we find that $t \mapsto e^{-t \mathsf G} f$ is $l$ times continuously differentiable in the graph norm of $\mathsf G_{(p)}^{j - l}$, for $l \le j$, with
\[
\partial_t^l e^{-t \mathsf G} f = (-1)^l e^{-t \mathsf G} \mathsf G^l f .
\] By \eqref{e:sigma-chains-elliptic} and the Sobolev embedding theorem, the coefficients of $\partial_t^l e^{-t \mathsf G} f$ have continuous derivatives in the body variables up to an order that tends to infinity with $j - l$, jointly continuous in $t \in [0, \infty)$. Since $j$ is arbitrary, the coefficients are smooth up to $t = 0$, and $\partial_t u = - \mathsf G u$ holds classically. The bound on $\|\nabla^i e^{-t \mathsf G} f\|_p$ follows from (C) and \eqref{e:sigma-chains-elliptic}.

For \ref{i:sigma-chains-rapid}, let $f$ be rapidly decreasing. Then (B) with $m = 1$ and $u = f$ shows that $f \in D(\mathsf G^j_{(p)})$ for all $j$ and $p$, and the proof of \ref{i:sigma-chains-time} applies with every $p$.

Finally, \ref{i:sigma-chains-scalar} follows from the proofs above and those of \zcref{p:sigma-regular}, in which there are then no odd coordinates, the terms $n^{\mu \nu}$ are absent, and the potential $z (d_k)_\emptyset$ has no soul. In this case the condition on $\omega$ disappears, and the form $f \mapsto X \|\nabla f\|^2 + z \langle (d_k)_\emptyset f, f \rangle$ of $A_{X, z}$ on $L^2_0$ is real and nonnegative. For $z>0$, its form domain is $V_0$, and the positive-damping realization and smoothing proofs apply. At $z=0$, its form domain is $H^1_0$ and the corresponding weighted realization has form domain $H^1_p$. The small-damping estimates and the semigroup-product arguments (A)--(C) apply throughout $0\le z\le z_0$, using this $H^1_p$ domain at zero, as in (B). They give the asserted weighted, compact, time-regularity, and rapid-data bounds also at $z=0$.
\end{proof}