Anticipating that it is legitimate to take the $\zeta\rightarrow 0^{+}$ limit
inside the integral one has

$\displaystyle\lim_{\zeta\rightarrow
0^{+}}\int\\!d\mu_{g}(y)\,|(\mathbb{F}_{\zeta}\psi)(y)|^{2}=\int\\!d\mu_{g}(y)\Big{(}\sum_{l\geq
1}\chi_{l}(y,y)\Big{)}^{2}|\psi(y)|^{2}=\norm{\psi}^{2}\,.$ (5.30)

In the penultimate step we used that the $l=0$ term does not contribute to the
partition of unity on the diagonal.

It remains to justify that the limit can be taken inside the integrals. To
this end we initially take $\psi\in C_{c}^{\infty}(M)$. By the reasoning
before (5.25) then a variant of (5) arises with finite sums and integrals over
compact domains. Specifically, the integrand reads

$I(y,y_{1},y_{2})=\sum_{k,m=1}^{K}\chi_{l_{k}}(y,y_{1})\chi_{l_{m}}(y,y_{2})F_{\zeta}^{(l_{k})}(y,y_{1})F_{\zeta}^{(l_{m})}(y,y_{2})\psi(y_{1})\psi(y_{2})^{*}\,.$
(5.31)

Here, $\chi_{l_{k}}$ has support in $V_{l_{k}}\times V_{l_{k}}$ and
$\chi_{l_{m}}$ has support in $V_{l_{m}}\times V_{l_{m}}$, which forces $y$ in
the $(k,l)$-th term to lie in the intersection $V_{l_{k}}\cap V_{l_{m}}$. In
order to take the limit $\zeta\rightarrow 0^{+}$ inside the triple integral
$d\mu(y)d\mu(y_{1})d\mu(y_{2})$ with integrand (5.31) we show that $I$ can be
bounded by an integrable function. As in the proof of Lemma 5.3a we use
$|F_{\zeta}^{(l_{k})}(y,y^{\prime})|\leq
C_{y}^{(l_{k})}\exp\\{-\frac{c_{-}}{4}|y-y^{\prime}|^{2}\\}$, for some
constants $C_{y}^{(l_{k})}$. We can bound the $\chi_{l_{k}}$ by $1$ to obtain

$|I(y,y_{1},y_{2})|\leq
C_{y}\Big{(}e^{-\frac{c_{-}}{4}|y-y_{1}|^{2}}\psi(y_{1})\Big{)}\Big{(}e^{-\frac{c_{-}}{4}|y-y_{2}|^{2}}\psi(y_{2})\Big{)}^{*}\,.$
(5.32)

Since the domain of the $y$ integration is also compact (given by the union of
the $V_{l_{k}}\cap V_{l_{m}}$) the right hand side is integrable with respect
to $d\mu(y)d\mu(y_{1})d\mu(y_{2})$ with integrand (5.31). This justifies the
step leading to (5.30) for all $\psi\in C_{c}^{\infty}(M)$.

Finally, we use the fact that $C_{c}^{\infty}(M)$ is dense in $L^{2}(M)$. That
is, for a given $\psi\in L^{2}(M)$ there exists a sequence
$\psi_{n},n\in\mathbb{N}$ of $C_{c}^{\infty}(M)$ functions such that
$\|\psi_{n}-\psi\|\rightarrow 0$, $n\rightarrow\infty$. We know that
$\mathbb{F}_{\zeta}\psi_{n}\in C_{c}^{\infty}(M)$ for all $n$ and
$\mathbb{F}_{\zeta}\psi\in L^{2}(M)$. Since by the first part
$\mathbb{F}_{\zeta}$ is a bounded operator on $L^{2}(M)$ also
$\|\mathbb{F}_{\zeta}\psi_{n}-\mathbb{F}_{\zeta}\psi\|\rightarrow 0$,
$n\rightarrow\infty$ holds. ∎

Bounding the difference kernel. We now use the global parametrix
$F_{\zeta}(y,y^{\prime})$ to define a global remainder function
$R_{\zeta}(y,y^{\prime}):=({\partial}_{\zeta}-\Delta_{\theta,y})F_{\zeta}(y,y^{\prime})$,
that is jointly smooth in $\zeta,y,y^{\prime}$. Adapting the strategy in [47]
we define in terms of it

$Q_{\zeta}(y,y^{\prime}):=\int_{0}^{\zeta}\\!d\zeta^{\prime}\int\\!d\mu_{g}(z)\,K^{\theta}_{\zeta-\zeta^{\prime}}(y,z)\,R_{\zeta^{\prime}}(z,y^{\prime})\,.$
(5.33)

By ${\partial}_{\zeta}$ differentiation of (5.33) one finds

${\partial}_{\zeta}\big{(}F_{\zeta}-Q_{\zeta}\big{)}(y,y^{\prime})=\Delta_{\theta,y}\big{(}F_{\zeta}-Q_{\zeta}\big{)}(y,y^{\prime})\,,$
(5.34)

using the heat equation for
${\partial}_{\zeta}K_{\zeta-\zeta^{\prime}}^{\theta}$ and the definition of
$R_{\zeta}$. The difference kernel $(F_{\zeta}-Q_{\zeta})(y,y^{\prime})$
therefore satisfies the same heat equation as
$K^{\theta}_{\zeta}(y,y^{\prime})$. Since $Q_{\zeta}$ manifestly vanishes for
$\zeta\rightarrow 0^{+}$ and $F_{\zeta}$ obeys (5.13) by Proposition 5.4, the
difference is also a fundamental solution to the heat equation.

We wish to conclude that $F_{\zeta}-Q_{\zeta}$ therefore coincides with
$K_{\zeta}$. Note, however, that on a noncompact manifold a fundamental
solution even to the standard heat equation is not necessarily unique. In
order to utilize the results from Section 4 we associate an integral operator
to the difference kernel, defining for $\psi\in L^{2}(M)$

$\displaystyle((\mathbb{F}_{\zeta}-\mathbb{Q}_{\zeta})\psi)(y):=\int\\!d\mu_{g}(y^{\prime})(F_{\zeta}-Q_{\zeta})(y,y^{\prime})\psi(y^{\prime})\,,\quad\zeta>0\,.$
(5.35)

As seen above, the image solves the heat equation
$({\partial}_{\zeta}-\Delta_{\theta})(\mathbb{F}_{\zeta}-\mathbb{Q}_{\zeta})\psi=0$
for all $\psi\in L^{2}(M)$. Further,

$\lim_{\zeta\rightarrow
0^{+}}\|(\mathbb{F}_{\zeta}-\mathbb{Q}_{\zeta})\psi\|=\|\psi\|\,,\quad\psi\in
L^{2}(M)\,.$ (5.36)

To see this, we first note that $\mathbb{F}_{\zeta}\psi$ and
$\mathbb{Q}_{\zeta}\psi$ are separately in $L^{2}(M)$. For
$\mathbb{F}_{\zeta}\psi$ this was established before; for
$\mathbb{Q}_{\zeta}\psi$ we use the definition and Cauchy-Schwarz to write

$\int\\!d\mu(y)|(\mathbb{Q}_{\zeta}\psi)(y)|^{2}\leq\Big{(}\int_{0}^{\zeta}\\!d\zeta^{\prime}\,\norm{e^{(\zeta-\zeta^{\prime})\Delta_{\theta}}\psi_{\zeta^{\prime}}}\Big{)}^{2}\,,$
(5.37)

where
$\psi_{\zeta}(z):=\int\\!d\mu_{g}(y^{\prime})R_{\zeta}(z,y^{\prime})\psi(y^{\prime})$.
Clearly, $\mathbb{Q}_{\zeta}\psi\in L^{2}(M)$ if $\psi_{\zeta}\in L^{2}(M)$,
where the latter is easily seen. To proceed, we anticipate the (much stronger
and independently established) Lemma 5.5 and the ensued bounds on $R_{\zeta}$.
Based on this, it can be shown that
$[0,\zeta]\ni\zeta^{\prime}\mapsto\|e^{(\zeta-\zeta^{\prime})\Delta_{\theta}}\psi_{\zeta^{\prime}}\|$
is bounded as $\zeta\rightarrow 0^{+}$. This implies $\lim_{\zeta\rightarrow
0^{+}}\|\mathbb{Q}_{\zeta}\psi\|=0$. The triangle inequality
$\|(\mathbb{F}_{\zeta}-\mathbb{Q}_{\zeta})\psi\|-\|\mathbb{F}_{\zeta}\psi\|\leq\|\mathbb{Q}_{\zeta}\psi\|$
and (5.30) then give the assertion (5.36).

On the other hand, we know that $(\Delta_{\theta},D(\Delta_{\theta}))$
generates a strongly continuous semigroup, $\mathbb{R}_{+}\ni\zeta\mapsto
e^{\zeta\Delta_{\theta}}$, on $L^{2}(M)$. As such the solution of the initial
value problem ${\partial}_{\zeta}\psi(\zeta)-\Delta_{\theta}\psi(\zeta)=0$
with $\psi(0)=\psi\in L^{2}(M)$, is unique; see, Ch. II, Prop. 6.4 of [15].
Hence, the semigroups $\mathbb{F}_{\zeta}-\mathbb{Q}_{\zeta}$ and
$e^{\zeta\Delta_{\theta}}$ coincide. By Theorem 4.2(iii) this implies that
their kernels coincide as well, i.e.

$K_{\zeta}^{\theta}(y,y^{\prime})-F_{\zeta}(y,y^{\prime})=Q_{\zeta}(y,y^{\prime})\,.$
(5.38)

To complete the proof we now seek to bound $Q_{\zeta}$ appropriately. Since
$Q_{\zeta}$ is defined in terms of $R_{\zeta}$ this amounts to rendering
$R_{\zeta}$ ‘small’. So far, mostly the very broad properties for $s_{\theta}$
and $A_{n}^{\theta}$ mentioned after (5.12) entered, and $R_{\zeta}$ will just
be given by the expression (5.10). We now identify $s_{\theta}$ with
$\tilde{\sigma}_{\theta}$ from Proposition 5.1 and make for
$A_{n}^{\theta}(y,y^{\prime})$ an ansatz in powers of $\Delta y$ to an
appropriate order and adjust the coefficients so as to render all but the term
proportional to $(ie^{-i\theta}\zeta)^{N}Z_{N}$ on the right hand side of
(5.10) ‘small’. The appropriate requirement is

$\displaystyle X_{n}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
O\big{(}|\Delta y|^{2N+8}\big{)}\,,\;\;\;\,n\geq 0\,,\quad
Y_{0}=O\big{(}|\Delta y|^{2N+6}\big{)}\,,$ $\displaystyle Y_{n+1}+Z_{n}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle O\big{(}|\Delta
y|^{2N+4}\big{)}\,,\quad n=0,\ldots,N-1\,,$ (5.39)

where $N$ is the expansion order in $\zeta$.

We formulate the instrumental fact as a Lemma but postpone its proof.

######  Lemma 5.5.

For $(y,y^{\prime})\in V\times V$, some open chart neighborhood of the
diagonal in $M\times M$, let
$s_{\theta}(y,y^{\prime})=\tilde{\sigma}_{\theta}(y,y^{\prime})$ be given by
Proposition 5.1 and consider its suitably truncated Taylor expansion. Then, in
a polynomial ansatz for $A_{n}^{\theta}(y,y^{\prime})$, $n=0,\ldots,N$, to a
suitable $N$-dependent order in $\Delta y$, the coefficients can be adjusted
such that the conditions (5) hold, for any given $N\geq(d+1)/2$. The
adjustment only requires the solution of decoupled linear equations. In
particular, the resulting diagonal coefficients $A^{\theta}_{n}(y,y)$ are
scalar differential polynomials in $g^{\theta}_{\mu\nu}$ that are smooth in
$y$.

Based on this Lemma we can bound the failure of $F_{\zeta}$ to furnish a
solution of the heat equation. Recall the global remainder function
$R_{\zeta}(y,y^{\prime})=({\partial}_{\zeta}-\Delta_{\theta,y})F_{\zeta}(y,y^{\prime})$,
which may be expressed in terms of local remainders
$R_{\zeta}^{(l)}(y,y^{\prime})$ through the partition of unity analagously to
(5.23). Both are clearly jointly smooth in $\zeta,y,y^{\prime}$ for all
$\zeta>0$. From (5.10) and Lemma 5.5 one has

$R_{\zeta}^{(l)}(y,y^{\prime})=\frac{e^{-\frac{1}{2\zeta}\tilde{\sigma}_{\theta}(y,y^{\prime})}}{\zeta^{\frac{d+1}{2}+2}}\Big{[}\Delta
y^{2N+4}\,A(y,\zeta)+\zeta^{N+2}B(y,y^{\prime})\Big{]}\,.$ (5.40)

Here the first term arises from the approximate vanishing of the
$n=0,\ldots,N\\!-\\!1$ terms in (5.10), and the second term is the $n=N$
contribution proportional to $Z_{N}$.

Next, it is convenient to reexpress (5.33) in the form

$Q_{\zeta}(y,y^{\prime})=\int_{0}^{\zeta}\\!d\zeta^{\prime}\Big{(}\\!e^{(\zeta-\zeta^{\prime})\Delta_{\theta}}\psi_{\zeta^{\prime},y^{\prime}}\Big{)}(y)\,,$
(5.41)

where $\psi_{\zeta^{\prime},y^{\prime}}(z)=R_{\zeta^{\prime}}(z,y^{\prime})$
is for fixed $\zeta^{\prime},y^{\prime}$ smooth and of compact support in $z$,
and also the dependence on $\zeta^{\prime},y^{\prime}$ is smooth. The
integrand may be given a pointwise bound (in $y,y^{\prime}$) as follows

$\displaystyle\big{|}(e^{(\zeta-\zeta^{\prime})\Delta_{\theta}}\psi_{\zeta^{\prime},y^{\prime}})(y)\big{|}$
$\displaystyle\leq
B_{y}\sum_{j=0}^{\sigma_{0}}\norm{\Delta_{\theta}^{j}e^{(\zeta-\zeta^{\prime})\Delta_{\theta}}\psi_{\zeta^{\prime},y^{\prime}}}=B_{y}\sum_{j=0}^{\sigma_{0}}\norm{e^{(\zeta-\zeta^{\prime})\Delta_{\theta}}\Delta_{\theta}^{j}\psi_{\zeta^{\prime},y^{\prime}}}$
$\displaystyle\leq
B_{y}\sum_{j=0}^{\sigma_{0}}\norm{\Delta_{\theta}^{j}\psi_{\zeta^{\prime},y^{\prime}}}.$
(5.42)

Here, the first inequality follows from the Sobolev embedding theorem along
the lines of the computation in (4), and $\sigma_{0}$ is the smallest integer
greater than $(d+1)/4$. In the second step we commute the operator
$\Delta_{\theta}$ through the semigroup, and in the final inequality we use
the contractivity of the semigroup.

Thus, it remains to bound the
$\|\Delta_{\theta}^{j}\psi_{\zeta^{\prime},y^{\prime}}\|$ in (5). To this end
we make use of the fact that (by the Lebesgue covering dimension) for each
$y^{\prime}$ only $d+2$ local parametrices contribute to the right hand side
of (5). It is therefore sufficient to find a bound for only a single local
parametrix. To obtain $\Delta_{\theta}^{j}\psi_{\zeta^{\prime},y^{\prime}}$ we
need to operate with $\Delta_{\theta}^{j}$ (acting on $y$) on (5.40). The
structure of $\Delta_{\theta}^{j}$ can be inferred from (5.57) below, it
contains between $2j$ and $j$ partial derivatives. Each derivative acting on
the exponential prefactor in (5.40) brings down a power of $\zeta^{-1}$. Each
derivative acting on the term in square brackets produces a term where the
order of $\Delta y$ is reduced and others. After rescaling of the difference
variable $\Delta y=\sqrt{\zeta}z$ this results in a structure of the following
form

$\Delta_{\theta,y}^{j}R_{\zeta}(y,y^{\prime})\big{|}_{\Delta
y\mapsto\sqrt{\zeta}z}=\frac{e^{-\frac{1}{2\zeta}\tilde{\sigma}_{\theta}(y,y-\sqrt{\zeta}z)}}{\zeta^{\frac{d+1}{2}+2}}\zeta^{N+2-2j}P_{\zeta}^{(j)}(y,z)\,,$
(5.43)

where $P_{\zeta}^{(j)}(y,z)$ is a polynomial in $z$ of degree $2N+4$ and also
a polynomial in $\zeta$. This gives

$\displaystyle\norm{\Delta_{\theta}^{j}\psi_{\zeta,y^{\prime}}}^{2}=\int\\!d^{d+1}y|g|^{1/2}(y)\,\big{|}\Delta_{\theta}^{j}R_{\zeta}(y,y^{\prime})\big{|}^{2}$
$\displaystyle=\zeta^{2N-4j-(d+1)}\int\\!d^{d+1}y\,e^{-\frac{1}{\zeta}\tilde{\sigma}(y,y-\Delta
y)}|g|^{1/2}(y)\big{|}P_{\zeta}^{(j)}\big{(}y,\Delta
y/\sqrt{\zeta}\big{)}\big{|}^{2}\,.$ (5.44)

The cutoff functions in the partition of unity entail that this integral is
over a compact domain, for which we may take $|y-y^{\prime}|<\delta$. We can
bound the $y$-dependence in the non-exponential factors by its supremuum in
this ball and write $\tilde{P}^{(j)}(\Delta y/\sqrt{\zeta})$ for the resulting
function of the difference variable. The exponential can be bounded by
$\exp\\{-c_{-}|\Delta y|^{2}/(2\zeta)\\}$, using (5.15). The integrand is then
a function of $\Delta y$ only and one can change integration variables to
$z=\Delta y/\sqrt{\zeta}$. This yields

$\norm{\Delta_{\theta}^{j}\psi_{\zeta,y^{\prime}}}^{2}\leq\zeta^{2N-4j-\frac{d+1}{2}}\int\\!d^{d+1}z\,e^{-\frac{c_{-}}{2}|z|^{2}}\tilde{P}^{(j)}_{\zeta}(z)\,.$
(5.45)

After the rescaling the integral is effectively over the ball
$0<\sqrt{\zeta}|z|<\delta$. For small $\zeta$ this forces $|z|$ to be large
and we can obtain an upper bound by extending the integration domain to
$\mathbb{R}^{d+1}$. This gives a sum of (polynomially modified) Gaussians in
$z$ with a polynomial $\zeta$ dependence. Performing them results in a
polynomial in $\zeta$ (with non-vanishing constant term), which for
$0<\zeta<1$ can be bounded by a constant. The upshot is that there exist a
$C(\delta)>0$ such that

$\norm{\Delta_{\theta}^{j}\psi_{\zeta,y^{\prime}}}\leq
C(\delta)\zeta^{N-2j-\frac{d+1}{4}}\,.$ (5.46)

Finally, with (5) one obtains upon performing the $d\zeta^{\prime}$ integral
in the bound obtained from (5.41)

$|Q_{\zeta}(y,y^{\prime})|\leq C_{y}\zeta^{N+1-2\sigma_{0}-\frac{d+1}{4}}\,.$
(5.47)

Used in (5.38), we arrive at the following result: for each $N\in{\mathbb{N}}$
and $y\in M$, for $\zeta\in(0,1)$ there is a constant $C_{y,N}$ such that

$\displaystyle|K^{\theta}_{\zeta}(y,y)-F_{\zeta}^{N}(y,y)|\leq
C_{y,N}\zeta^{N+1-c}$ (5.48)

for $c=2\sigma_{0}+(d+1)/2>0$. Here, we specialized to the diagonal
$y=y^{\prime}$, and denote the the truncation order of the parametrix by a
superscript $N$.

For an asymptotic expansion proper, one seeks to have the remainder of the
same order as the first term omitted in the parametrix ansatz. This can be
achieved by the following adjustment. Pick an integer $N_{0}<N$ and note

$\displaystyle
F_{\zeta}^{N}(y,y)=F_{\zeta}^{N_{0}}(y,y)+\frac{1}{(4\pi\zeta)^{\frac{d+1}{2}}}\big{\\{}A_{N_{0}+1}^{\theta}(y)\zeta^{N_{0}+1}+\ldots+A_{N}^{\theta}(y)\zeta^{N}\big{\\}}\,.$
(5.49)

Then

$\displaystyle|K^{\theta}_{\zeta}(y,y)-F_{\zeta}^{N_{0}}(y,y)|$
$\displaystyle\leq|K_{\zeta}^{\theta}(y,y)-F_{\zeta}^{N}(y,y)|+|F_{\zeta}^{N}(y,y)-F_{\zeta}^{N_{0}}(y,y)|$
$\displaystyle\leq
C_{y,N}\zeta^{N_{0}+1-(c-(N-N_{0}))}+D_{y}\zeta^{N_{0}+1-\frac{d+1}{2}}\,.$
(5.50)

The second term dominates when $(N-N_{0})+(d+1)/2>c$; with
$c=2\sigma_{0}+(d+1)/2>0$ and $\sigma_{0}$ the smallest integer greater than
$(d+1)/4$, this can clearly be realized for any given $d$ by adjusting
$N-N_{0}$. This establishes:

######  Theorem 5.6.

The diagonal of the kernel $K_{\zeta}^{\theta}$ from Theorem 4.2 admits an
asymptotic expansion of the form

$\displaystyle
K_{\zeta}^{\theta}(y,y)\asymp\frac{(-ie^{i\theta})^{\frac{d-1}{2}}}{(4\pi\zeta)^{\frac{d+1}{2}}}\sum_{n\geq
0}A_{n}^{\theta}(y)(ie^{-i\theta}\zeta)^{n},$ (5.51)

i.e. for all $N\in{\mathbb{N}}$ and $y\in M$

$\displaystyle(4\pi\zeta)^{\frac{d+1}{2}}K_{\zeta}^{\theta}(y,y)-(-ie^{i\theta})^{\frac{d-1}{2}}\sum_{n=0}^{N}A_{n}^{\theta}(y)(ie^{-i\theta}\zeta)^{n}=O(\zeta^{N+1})$
(5.52)

as $\zeta\to 0^{+}$. Here the coefficients $A_{n}^{\theta}(y)$ from (5.24) are
in each chart map related to the standard heat kernel coefficients by a lapse
Wick rotation.

Remarks.

(i) The assertion about the coefficients is a by-product of the proof of Lemma
5.5. Indeed, the proof will show in particular that in each local chart the
$A_{n,0}$ terms in the polynomial ansatz for $A_{n}^{c_{n}}$ below in (5) are
uniquely determined differential polynomials in the metric $g^{\theta}$ and
its inverse. Since the $X_{n},Y_{n},Z_{n}$ coefficients (5) determining them
are $N\mapsto ie^{-i\theta}N$ Wick rotations of their Euclidean signature
counterparts and only algebraic manipulations are invoked, the solutions
$A_{n,0}$ likewise must be the $N\mapsto ie^{-i\theta}N$ Wick rotations of the
standard diagonal heat kernel coefficients. In the special case when for given
$p$ the $(A_{n}^{\theta})^{(l)}(\varphi_{l}(p),\varphi_{l}(p))$ are
$l$-independent, i.e. a single chart suffices, the expression (5.24) reduces
to the expected $A_{n}^{\theta}(y,y)$ in that chart.

(ii) The result remains valid for complex $\zeta$ in the wedge $|{\rm
Arg}(\zeta)|<\tilde{\theta}$, where the left hand side is defined by Theorem
4.2. The right hand side is up to a $ie^{-i\theta}$ prefactor only a function
of $|\zeta|$ and $\theta-{\rm Arg}(\zeta)$, while the asymptoticity refers to
$|\zeta|$. With some additional work one can also trace the validity of the
individual steps in the proof.

Proof of Lemma 5.5. We use power series ansätze of the form

$\displaystyle s^{L}(y^{\prime},y)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-
ie^{i\theta}\sum_{l=2}^{L-1}\frac{1}{l!}s_{\mu_{1}\ldots\mu_{l}}(y^{\prime})\Delta
y^{\mu_{1}}\ldots\Delta y^{\mu_{l}}\,,$ $\displaystyle
A^{c_{n}}_{n}(y^{\prime},y)$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\sum_{k=0}^{c_{n}-1}\frac{1}{k!}A_{n,\nu_{1}\ldots\nu_{k}}(y^{\prime})\Delta
y^{\nu_{1}}\ldots\Delta y^{\nu_{k}}\,,$ (5.53)

where the orders $L$ and $c_{n}$ will be functions of $N$ (the truncation
order in $\zeta$) to be determined. The correction terms in (5) are understood
to be of order $\Delta y^{L}$ and $\Delta y^{c_{n}}$, respectively. The base
point $y^{\prime}$ is chosen so that the partial derivatives
${\partial}_{\mu}={\partial}/{\partial}y^{\mu}$ only act on the $\Delta y$
products. To unclutter the notation we indicate the order in $\Delta y$
directly by the number of indices, writing
$A_{n,\nu_{1}\ldots\nu_{k}}(y^{\prime})$ for
$A^{(k)}_{n,\nu_{1}\ldots\nu_{k}}(y^{\prime})$ and $A_{n,0}(y^{\prime})$ for
$A_{n}^{(0)}(y^{\prime})$.

Starting with the $X_{0}$ condition in (5) it is plain that if the
coefficients $s_{\mu_{1},\ldots,\mu_{l}}$, $l=2,\ldots,L\\!-\\!1$ are adjusted
as in Proposition 5.1 the remainder in the Eikonal equation will be of order
$|\Delta y|^{L}$. For this to be of order $|\Delta y|^{2N+8}$ we take

$L=2N+8\,,$ (5.54)

from here on. Note that this does not constrain $A_{0}^{c_{0}}$. Moreover, the
same $L=2N\\!+\\!8$ Synge function approximant $s^{L}$ will render all
$X_{n}$, $n\geq 1$, also $O(|\Delta y|^{2N+4+d_{0}})$ without constraining the
$A_{n}^{c_{n}}$ ansatz. Since this is one or two orders higher than the
requirement in the other two conditions in (5) the latter reduce to

$Y_{0}=O(|\Delta y|^{2N+6})\,,\quad Y_{n+1}+Z_{n}=O(|\Delta y|^{2N+4})\,,\quad
n=0,\ldots,N-1\,.$ (5.55)

In the following we regard $s^{L}$ with $L=2N+8$ as a known function and seek
to determine the coefficients in the $A_{n}^{c_{n}}$ ansätze such that (5.55)
holds. Irrespective of the details, both conditions will have an expansion in
powers of $\Delta y$. Consistency with $L=2N+8$ requires that the order
$c_{n}-1$ of the polynomials used must not exceed $2N+6$, see (5.59).

For $Y_{0}$ one needs to take into account that $A_{0,0}=1$; so the ansatz for
$A_{0}^{c_{0}}$ only contains $c_{0}-1$ free parameter functions. On the other
hand, an explicit computation (detailed below) shows that the constant term in
$Y_{0}$ vanishes, so that only positive powers of $\Delta y$ occur,
$Y_{0}=\sum_{p=1}^{c_{0}-1}(\Delta y)^{p}Y_{0,p}+O(|\Delta y|^{c_{0}})$. In
the absence of any coincidental cancellations the coefficients
$Y_{0,p},p=1,\ldots,2N+5$, will have to be made to vanish so as to render the
result of order $|\Delta y|^{2N+6}$. Taking $c_{0}=2N+6$ the ansatz for
$A_{0}^{c_{0}}$ contains $2N+5$ free parameter functions for $2N+5$ linear
conditions to be met. A closer inspection shows that only linear, decoupled
relations arise, which fix the coefficients uniquely.

For $n\geq 1$, the $Z_{n-1}$ term in (5.55) is determined by
$A_{n-1}^{c_{n-1}}$ and thus constitutes an inhomogeneity known from the
preceding order. The condition $Y_{n}+Z_{n-1}=O(|\Delta y|^{2N+4})$ thus
should determine the $c_{n}$ parameters in $A_{n}^{c_{n}}$. Assuming
$c_{n}\leq c_{n-1}$ the expansion will produce a series of the form
$\sum_{p=0}^{c_{n}-1}(\Delta y)^{p}(Y_{n,p}-Z_{n-1,p})$, so that the
coefficients $p=0,\ldots,2N+3$ have to vanish in order to meet the condition.
Taking $c_{n}=2N+4$ matches the number of parameters to the number of
conditions. Again, the conditions turn out to be linear and decoupled so that
the parameters are uniquely fixed.

In summary, the choice

$c_{0}=2N+6\,,\quad c_{n}=2N+4\,,\quad n\geq 1\,,$ (5.56)

will uniquely determine the parameters in the ansätze, provided the expansion
of the $Y_{n}$, $n\geq 0$, terms has an appropriate structure.

We prepare

$\nabla_{\theta}^{2}=g_{\theta}^{\mu\nu}(y)\nabla_{\mu}^{\theta}\nabla_{\nu}^{\theta}=g_{\theta}^{\mu\nu}(y){\partial}_{\mu}{\partial}_{\nu}+\gamma_{\theta}^{\mu}(y){\partial}_{\mu}\,,\quad\gamma_{\theta}^{\mu}:=|g|^{-1/2}{\partial}_{\nu}(|g|^{1/2}g_{\theta}^{\nu\mu})\,,$
(5.57)

where again the $\theta$-independent density $|g|^{1/2}$ enters. For
${\mathcal{L}}(s^{L})$ from (5) this gives

$\displaystyle{\mathcal{L}}(s^{L})$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle
g_{\theta}^{\mu\nu}(y)g^{\theta}_{\mu\nu}(y^{\prime})+\sum_{l=1}^{L-3}\frac{1}{l!}\Delta
y^{\mu_{1}}\ldots\Delta
y^{\mu_{l}}\Big{[}g_{\theta}^{\mu\nu}(y)s_{\mu\nu\mu_{1}\ldots\mu_{l}}(y^{\prime})$
$\displaystyle+$
$\displaystyle\gamma_{\theta}^{\mu}(y)s_{\mu\mu_{1}\ldots\mu_{l}}(y^{\prime})+2g_{\theta}^{\mu\nu}(y)s_{\mu\mu_{1}\ldots\mu_{l}}(y^{\prime}){\partial}_{\nu}\Big{]}$
$\displaystyle+$ $\displaystyle\frac{1}{(L\\!-\\!2)!}\Delta
y^{\mu_{1}}\ldots\Delta
y^{\mu_{L-2}}\Big{[}\gamma_{\theta}^{\mu}(y)s_{\mu\mu_{1}\ldots\mu_{L-2}}(y^{\prime})+2g_{\theta}^{\mu\nu}(y)s_{\mu\nu\mu_{1}\ldots\mu_{L-2}}(y^{\prime}){\partial}_{\nu}\Big{]}\,.$

One could now expand the coefficient functions evaluated at
$y^{\prime}=y-\Delta y$ around $y$ to obtain a local first order linear
differential operator. For our purposes it is better to expand the
coefficients at $y=y^{\prime}+\Delta y$ around $y^{\prime}$. This gives an
expression of the form

$\displaystyle{\mathcal{L}}(s^{L})=d\\!+\\!1+\sum_{l=1}^{L-2}\Delta
y^{\mu_{1}}\ldots\Delta
y^{\mu_{l}}\big{[}d_{\mu_{1}\ldots\mu_{l}}(y^{\prime})+e^{\nu}_{\mu_{1}\ldots\mu_{l}}(y^{\prime}){\partial}_{\nu}\big{]}+O(\Delta
y^{L-1})\,,$ (5.59)

where the explicit form of the coefficients is readily inferred from (5). From
here the structure of the $Y_{n}$ terms follows. Collecting terms of the same
order in $\Delta y$ one finds

$\displaystyle\Delta y^{0}:2nA_{n,0}$ $\displaystyle\Delta
y^{1}:[2n\delta_{\mu_{1}}^{\nu}+e^{\nu}_{\mu_{1}}]A_{n,\nu}+d_{\mu_{1}}A_{n,0}\,,$
(5.60) $\displaystyle\Delta
y^{p}:\vartheta(c_{n}\\!-1\\!-\\!p)\bigg{\\{}\Big{[}\frac{2n}{p!}\delta_{\mu_{1}}^{\nu}+\frac{1}{(p\\!-\\!1)!}e^{\nu}_{\mu_{1}}\Big{]}A_{n,\nu\mu_{2}\ldots\mu_{p}}+$
$\displaystyle+\frac{1}{(p\\!-\\!2)!}e_{\mu_{1}\mu_{2}}^{\nu}A_{n,\nu\mu_{3}\ldots\mu_{p}}+\ldots+\frac{1}{1!}e^{\nu}_{\mu_{1}\ldots\mu_{p-1}}A_{n,\nu\mu_{p}}\vartheta(L\\!-\\!p)$
$\displaystyle+\frac{1}{(p\\!-\\!1)!}d_{\mu_{1}}A_{n,\mu_{2}\ldots\mu_{p}}+\frac{1}{(p\\!-\\!2)!}d_{\mu_{1}\mu_{2}}A_{n,\mu_{3}\ldots\mu_{p}}+\ldots+\frac{1}{1!}d_{\mu_{1}\ldots\mu_{p-1}}A_{n,\mu_{p}}\vartheta(L\\!-\\!1\\!-\\!p)\bigg{\\}}$
$\displaystyle+d_{\mu_{1}\ldots\mu_{p}}A_{n,0}\,\vartheta(L\\!-\\!2\\!-\\!p)+e^{\nu}_{\mu_{1}\ldots\mu_{p}}A_{n,\nu}\,\vartheta(L\\!-\\!1\\!-\\!p)\,.$

In the last relation $p\geq 2$ and we use $\vartheta(x)=1,\,x\geq 0$,
$\vartheta(x)=0,\,x<0$, to transcribe the range of the original sums. Each
order in $\Delta y$ will give rise to one condition constraining the
coefficients in $A_{n}^{c_{n}}$. For $n\\!=\\!0$ there are $2N\\!+5$ free
coefficients in $A_{0}^{c_{0}}$, with $c_{0}=2N\\!+6$. The vanishing of the
$p=1,\ldots,p_{\rm max}$, relations will match the number of parameters for
$p_{\rm max}=2N+5$. The overall step function in the $\Delta y^{p}$ condition
will then always be unity. With $L=2N+6$ also the $L$-dependent step function
is just $1$. The same happens for $n\geq 1$. By (5.56) there are $2N+4$
coefficients and the step functions limit the validity of the given expression
to $p\leq p_{\rm max}=2N+3$. Hence, the $2N+4$ coefficients in the ansatz for
$A_{n}^{2N}$ are are subjected to as many equations. Further, for $L=2N+6$ the
$L$ dependent step function is always unity. In summary, with the choices
(5.54), (5.56) the relations (5) have all step functions evaluate to $1$.

The expressions (5) clearly have a triangular structure, the
$A_{n,\nu\mu_{2}\ldots\mu_{p}}$ coefficient with $p$ indices occurs only once
and is contracted with a matrix indicated in square brackets. This matrix is
always non-singular and is in fact a multiple of the unit matrix. Indeed,
returning to (5) one reads off
$e_{\mu_{1}}^{\nu}=2g_{\theta}^{\nu\mu}(y^{\prime})s_{\mu\mu_{1}}(y^{\prime})=2\delta_{\mu_{1}}^{\nu}$.
The other terms in the $\Delta y^{p}$ expression have $A_{n,\\#}$ coefficients
with a smaller number of indices, which suggests an iterative procedure to
impose the conditions (5.55).

For $n=0$ the vanishing of the $\Delta y^{p}$, $p=1,\ldots,2N+5$, expressions
in (5) indeed determines the coefficients
$A_{0,\nu},A_{0,\nu\mu_{2}},\ldots,A_{0,\nu,\mu_{2}\ldots\mu_{2N+1}}$,
iteratively. For $n\geq 1$, we solve the relations $Y_{n}=-Z_{n-1}+O(|\Delta
y|^{2N})$, inductively in $n$. Since $-Z_{n-1}$ is determined by the
$A_{n-1}^{c_{n-1}}$ coefficients, the right hand side is known as a polynomial
in $\Delta y$ from the preceding order. Equating its coefficients to the ones
on (5) again leads to a system triangular in $p$, which determines the
coefficients
$A_{n,\nu},A_{n,\nu\mu_{2}},\ldots,A_{n,\nu\mu_{2}\ldots\mu_{2N-1}}$,
uniquely. ∎

## 6\. Schrödinger evolution group from the $\theta\rightarrow 0^{+}$ limit

The Wick rotated heat semigroup is well-defined in the near Lorentzian regime,
as long as $\theta>0$. In this section, we study the strict Lorentzian limit
$\theta\to 0^{+}$, where the properties change qualitatively. Based on the
form (3.14) of the generator $\Delta_{\theta}$, one might hope that in some
sense a unitary Schrödinger group will arise in the limit. The following
theorem specifies a sense in which this indeed holds.

######  Theorem 6.1.

Let $(M,g_{-})$ be a globally hyperbolic manifold and assume that the operator
${\mathcal{D}}_{-}=-\nabla_{-}^{2}+V$ is essentially self-adjoint on
$C_{c}^{\infty}(M)$, i.e., its closure $\overline{{\mathcal{D}}}_{-}$ is self-
adjoint. Then for all $s\in{\mathbb{R}}_{\geq 0}$ there is the limiting
behavior

$\displaystyle e^{s\Delta_{\theta}}\xrightarrow{\theta\to
0^{+}}e^{-is\overline{{\mathcal{D}}}_{-}}\,,\quad
e^{s\Delta_{\pi-\theta}}\xrightarrow{\theta\to
0^{+}}e^{+is\overline{{\mathcal{D}}}_{-}}\,,$ (6.1)

converging with respect to the ultra-weak (or weak-star) topology on
${\mathfrak{B}}(L^{2}(M))$. Here
$e^{-is\overline{{\mathcal{D}}}_{-}},s\in{\mathbb{R}},$ is the one-parameter
unitary group generated by $\overline{{\mathcal{D}}}_{-}$.

We recall that $({\mathfrak{B}}(L^{2}(M)),\norm{\cdot}_{\rm op})$ is the
Banach space of continuous endomorphisms on $L^{2}(M)$, while the ultra-weak
topology on ${\mathfrak{B}}(L^{2}(M))$ is the weak-star topology arising from
its pre-dual, the Banach space of trace-class operators
$\mathfrak{B}_{1}(L^{2}(M))$ equipped with the trace norm. We shall also need
the product topological space ${\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq
0}}$, where each component ${\mathfrak{B}}(L^{2}(M))$ is endowed with the
ultra-weak topology. The product topology for an uncountable product is
defined as follows. Starting with a family $(X_{s})_{s\in{\mathbb{R}}_{\geq
0}}$ of topological spaces, the product space $\prod_{s\in{\mathbb{R}}_{\geq
0}}X_{s}$ (also denoted $X^{{\mathbb{R}}_{\geq 0}}$ when all $X_{s}\equiv X$,
as is the case here) is the set of all functions $f:{\mathbb{R}}_{\geq
0}\to\bigcup_{s\in{\mathbb{R}}_{\geq 0}}X_{s}$ such that
$\forall\,s\in{\mathbb{R}}_{\geq 0}:f(s)\in X_{s}$. Then, the canonical base
for the product topology consists of sets of the form
$\prod_{s\in{\mathbb{R}}_{\geq 0}}{\mathcal{V}}_{s}$, with each
${\mathcal{V}}_{s}\subseteq X_{s}$ open, and ${\mathcal{V}}_{s}=X_{s}$ for all
but finitely many $s\in{\mathbb{R}}_{\geq 0}$.

Throughout this section, we use the notation

$\displaystyle\forall\,s\geq 0,\,\theta\in(0,\pi):\quad
T(s,\theta):=e^{s\Delta_{\theta}}\,.$ (6.2)

Clearly, for each $\theta\in(0,\pi)$,
$T(\cdot,\theta)\in{\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$.

We begin by preparing the following lemmas.555We are grateful to K. Taira for
discussions related to Lemma 6.3 below.

######  Lemma 6.2.

Let $(M,g_{-})$ be a globally hyperbolic manifold and
${\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$ as above. Regarding
$((0,\pi),\preceq)$ as a directed set with ordering

$\displaystyle\theta_{1}\preceq\theta_{2}\iff\theta_{1}\geq\theta_{2}\,,\quad\forall\,\theta_{1},\,\theta_{2}\in(0,\pi)\,,$
(6.3)

the family $(T(\cdot,\theta))_{\theta\in(0,\pi)}$ is a net in
${\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$. Then:

  1. (i)

There exists a subnet $(T(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$, and
a strongly continuous family of contractive operators
$T(\cdot)\in{\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$ such that
$T(0)=\mbox{11}$, and
$T(\cdot,\theta_{\alpha})\xrightarrow{\alpha\in\mathscr{A}}T(\cdot)$ wrt. the
product topology on ${\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$. In
particular, this implies component-wise convergence,

$\displaystyle\forall\,s\geq 0:\quad
T(s,\theta_{\alpha})\xrightarrow{\alpha\in\mathscr{A}}T(s)$ (6.4)

in the ultra-weak topology on ${\mathfrak{B}}(L^{2}(M))$.

  2. (ii)

Consider for each $u\in C^{\infty}_{c}(M)$ the mapping

$\displaystyle\forall\,s\in{\mathbb{R}}_{\geq
0}:\quad{\mathfrak{u}}(s):=T(s)u\,.$ (6.5)

Then $\mathfrak{u}\in C({\mathbb{R}}_{\geq 0},L^{2}(M))\cap
C^{1}({\mathbb{R}}_{>0},L^{2}(M))$, and is a distributional solution to the
Schrödinger equation for each $w\in C_{c}^{\infty}(M)$, i.e.

$\displaystyle\left\langle
w\,\middle|\,d_{s}{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}+\left\langle
i{\mathcal{D}}_{-}w\,\middle|\,{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}=0\,,\quad
s>0\,,$ (6.6)

with initial condition ${\mathfrak{u}}(0)=u$.

######  Lemma 6.3.

Let $(M,g_{-})$ be a globally hyperbolic manifold and assume that the operator
${\mathcal{D}}_{-}=-\nabla_{-}^{2}+V$ is essentially self-adjoint on
$C_{c}^{\infty}(M)$. Then the distributional solution ${\mathfrak{f}}\in
C({\mathbb{R}}_{\geq 0},L^{2}(M))\cap C^{1}({\mathbb{R}}_{>0},L^{2}(M))$ to
the Schrödinger equation

$\displaystyle
d_{s}{\mathfrak{f}}(s)+i{\mathcal{D}}_{-}^{\ast}{\mathfrak{f}}(s)=0\,,\quad
s>0\,,$ (6.7)

with initial condition ${\mathfrak{f}}(0)=f\in C_{c}^{\infty}(M)$ has a unique
solution, namely ${\mathfrak{f}}(s)=e^{-is\overline{{\mathcal{D}}}_{-}}f$,
where $\overline{{\mathcal{D}}}_{-}$ is the (unique, self-adjoint) closure of
$({\mathcal{D}}_{-},C^{\infty}_{c}(M))$.

The proof of Lemma 6.3 is omitted, referring to Ch.VI Theorem 1.7 of [4].

Proof of Theorem 6.1. Consider from Lemma 6.2 the subnet
$(T(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$ and its limit
$T(\cdot)\in{\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$. For every $u\in
C_{c}^{\infty}(M)$, it follows from (6.5), (6.6) that
${\mathfrak{u}}(s):=T(s)u$ is a solution to (6.7) with initial condition
${\mathfrak{u}}(0)=u$. This solution is unique by Lemma 6.3, and hence
$\forall\,s\geq 0:\,T(s)u=e^{-is\overline{{\mathcal{D}}}_{-}}u$. Thus, the
action of the families $(T(s))_{s\geq 0}$ and
$(e^{-is\overline{{\mathcal{D}}}_{-}})_{s\geq 0}$ coincide on
$C_{c}^{\infty}(M)$. Since $C_{c}^{\infty}(M)$ is dense in $L^{2}(M)$, it
follows immediately that the families are identical, and so

$\displaystyle\forall\,s\geq 0:\quad
T(s,\theta_{\alpha})\xrightarrow{\alpha\in\mathscr{A}}e^{-is\overline{{\mathcal{D}}}_{-}}\,,$
(6.8)

in the ultra-weak topology on ${\mathfrak{B}}(L^{2}(M))$.

Next, consider an arbitrary subnet
$(T(\cdot,\theta_{\beta}))_{\beta\in\mathscr{B}}$ of the net
$(T(\cdot,\theta))_{\theta\in(0,\pi)}$ in
${\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$. By the same arguments as
in the proof of Lemma 6.2 below, it follows that there is a sub-subnet
$(T(\cdot,\theta_{\beta(\kappa)}))_{\kappa\in\mathscr{K}}$ converging to a
$\widetilde{T}(\cdot)\in{\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$ in
the product topology. This limit
$\widetilde{T}(\cdot)\in{\mathfrak{B}}(L^{2}(M))^{{\mathbb{R}}_{\geq 0}}$ has
the same properties as the $T(\cdot)$ from Lemma 6.2; in particular
$\widetilde{{\mathfrak{u}}}(s):=\widetilde{T}(s)u$, for $u\in
C_{c}^{\infty}(M)$, solves the differential equation (6.7) with initial
condition $u$. As above, this implies that $\forall\,s\geq
0:\,\widetilde{T}(s)=e^{-is\overline{{\mathcal{D}}}_{-}}$. Having shown that
every subnet of $(T(\cdot,\theta))_{\theta\in(0,\pi)}$ has a further sub-
subnet that converges to $(e^{-is\overline{{\mathcal{D}}}_{-}})_{s\geq 0}$, it
follows666The proof of this statement for nets is essentially the same as the
proof of the corresponding result for sequences. Indeed, if the original net
did not converge, then there would be an open neighborhood $V$ of the (sub-
subnet) limit point and a subnet that lives in the complement of $V$. Then,
this subnet cannot have a further sub-subnet that converges to the limit
point, a contradiction. that the original net converges to the same, i.e.
$T(\cdot,\theta)\xrightarrow{\theta\to
0^{+}}e^{-is\overline{{\mathcal{D}}}_{-}}$ in the product topology. This
entails component-wise convergence, i.e.,

$\displaystyle\forall\,s\in{\mathbb{R}}_{\geq 0}:\quad
T(s,\theta)=e^{s\Delta_{\theta}}\xrightarrow{\theta\to
0^{+}}e^{-is\overline{{\mathcal{D}}}_{-}}\,,$ (6.9)

in the ultra-weak topology of ${\mathfrak{B}}(L^{2}(M))$, completing the
proof. ∎

It remains to prove Lemma 6.2.

Proof of Lemma 6.2.

(i) Since for all $s\in{\mathbb{R}}_{\geq
0},\,\theta\in(0,\pi):\,\norm{T(s,\theta)}_{\rm op}\leq 1$, it follows that
$(T(\cdot,\theta))_{\theta\in(0,\pi)}$ is a net in the product topological
space ${\mathcal{B}}^{{\mathbb{R}}_{\geq 0}}$, with ${\mathcal{B}}$ the norm-
closed unit ball of ${\mathfrak{B}}(L^{2}(M))$ equipped with the (relative)
ultra-weak topology inherited from ${\mathfrak{B}}(L^{2}(M))$. Since
${\mathfrak{B}}(L^{2}(M))$ is the dual of the space of trace-class operators
${\mathfrak{B}}_{1}(L^{2}(M))$, ${\mathcal{B}}$ is compact in the ultra-weak
topology by the Banach-Alaoglu theorem, and hence
${\mathcal{B}}^{{\mathbb{R}}_{\geq 0}}$ is compact in the product topology
(Tychonoff’s theorem). Compactness then entails that there is a subnet
$(T(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$ converging to
$T(\cdot)\in{\mathcal{B}}^{{\mathbb{R}}_{\geq 0}}$ in the product topology,
which in particular implies the component-wise convergence

$\displaystyle\forall\,s\geq 0:\quad
T(s,\theta_{\alpha})\xrightarrow{\alpha\in\mathscr{A}}T(s)\,,$ (6.10)

in the ultra-weak topology on ${\mathcal{B}}$, from which (6.4) follows
immediately.

(ii) To proceed, fix arbitrary $S>0$ and $u\in C_{c}^{\infty}(M)$, and
consider the family
$({\mathfrak{u}}(s,\theta))_{s\in[0,S]\atop\theta\in(0,\pi)}$ in $L^{2}(M)$,
defined by

$\displaystyle{\mathfrak{u}}(s,\theta):=T(s,\theta)u\,.$ (6.11)

It is clear that for any fixed $\theta\in(0,\pi)$,
${\mathfrak{u}}(\cdot,\theta)\in C([0,S],L^{2}(M))\cap
C^{\infty}((0,S),L^{2}(M))$, and in particular for any $k\in{\mathbb{N}}$ we
have

$\displaystyle
d^{k}_{s}({\mathfrak{u}}(s,\theta))=(\Delta_{\theta})^{k}T(s,\theta)u=T(s,\theta)(\Delta_{\theta})^{k}u\,,$
(6.12)

leading to the bound

$\displaystyle\norm{d^{k}_{s}({\mathfrak{u}}(s,\theta))}_{L^{2}(M)}=\norm{T(s,\theta)(\Delta_{\theta})^{k}u}_{L^{2}(M)}\leq\norm{(\Delta_{\theta})^{k}u}_{L^{2}(M)}\leq
c_{k}(u)\,.$ (6.13)

Here $c_{k}(u)>0$ is a constant depending on $k\in{\mathbb{N}}$ and $u\in
C_{c}^{\infty}(M)$, but independent of $\theta\in(0,\pi)$, and the final bound
in (6.13) follows since $\Delta_{\theta}$ acts classically on $u\in
C_{c}^{\infty}(M)$.

It then follows that for each $\theta\in(0,\pi)$,
${\mathfrak{u}}(\cdot,\theta)\in H^{2}([0,S],L^{2}(M))$, the Bochner-space of
$L^{2}(M)$ valued Sobolev functions on $[0,S]$, which is in particular a
Hilbert space.777 We refer to [50, 26] for textbook accounts of Bochner
integration, and to [16] for a discussion of the associated Sobolev spaces.
Moreover, the result (6.13) implies that there is a constant $C_{2}(u)>0$ such
that

$\displaystyle\sup_{\theta\in(0,\pi)}\norm{{\mathfrak{u}}(\cdot,\theta)}_{H^{2}([0,S],L^{2}(M))}\leq
C_{2}(u)\,,$ (6.14)

where the $H^{2}$-norm is defined by

$\displaystyle\norm{{\mathfrak{h}}}_{H^{2}([0,S],L^{2}(M))}^{2}:=\sum_{j=0}^{2}\int_{0}^{S}\\!\\!ds\,\norm{d_{s}^{j}{\mathfrak{h}}(s)}_{L^{2}(M)}^{2}\,.$
(6.15)

Returning now to the subnet
$(T(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$ from earlier in the proof,
it follows from (6.14) that
$({\mathfrak{u}}(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$ is a net
contained in the $H^{2}$-norm-closed ball of radius $C_{2}(u)$. Since
$H^{2}([0,S],L^{2}(M))$ is a Hilbert space, this ball is weakly compact, and
hence $({\mathfrak{u}}(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$ has a
subnet $({\mathfrak{u}}(\cdot,\theta_{\alpha(\beta)}))_{\beta\in\mathscr{B}}$
converging weakly to ${\mathfrak{u}}(\cdot)\in H^{2}([0,S],L^{2}(M))$. By
Sobolev embedding (Theorem 2 of Chapter 5.9.2 of [16]),
${\mathfrak{u}}(\cdot)$ has a $C([0,S],L^{2}(M))\cap C^{1}((0,S),L^{2}(M))$
representative, which we continue to denote by ${\mathfrak{u}}(\cdot)$. A
further consequence of Sobolev embedding is that for any $w\in L^{2}(M)$ and
$s\in[0,S]$, the mapping

$\displaystyle H^{2}([0,S],L^{2}(M))\ni{\mathfrak{h}}\mapsto\left\langle
w\,\middle|\,{\mathfrak{h}}(s)\right\rangle_{L^{2}(M)}\in{\mathbb{C}}$ (6.16)

is an element of the dual space $H^{2}([0,S],L^{2}(M))^{\ast}$. Then, the weak
convergence of the subnet
$({\mathfrak{u}}(\cdot,\theta_{\alpha(\beta)}))_{\beta\in\mathscr{B}}$ above
entails that for each $s\in[0,S]$ and $w\in L^{2}(M)$,

$\displaystyle\left\langle
w\,\middle|\,{\mathfrak{u}}(s,\theta_{\alpha(\beta)})\right\rangle_{L^{2}(M)}=\left\langle
w\,\middle|\,T(s,\theta_{\alpha(\beta)})u\right\rangle_{L^{2}(M)}\xrightarrow{\beta\in\mathscr{B}}\left\langle
w\,\middle|\,{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}\,.$ (6.17)

However, the ultra-weak convergence (6.10) of the subnet
$(T(\cdot,\theta_{\alpha}))_{\alpha\in\mathscr{A}}$ when restricted to the
above sub-subnet $(T(\cdot,\theta_{\alpha(\beta)}))_{\beta\in\mathscr{B}}$
implies

$\displaystyle\left\langle
w\,\middle|\,T(s,\theta_{\alpha(\beta)})u\right\rangle_{L^{2}(M)}\xrightarrow{\beta\in\mathscr{B}}\left\langle
w\,\middle|\,T(s)u\right\rangle_{L^{2}(M)}\,.$ (6.18)

Since $w\in L^{2}(M)$ is arbitrary, comparing (6.17) and (6.18) yields

$\displaystyle\forall\,s\in[0,S]:\quad{\mathfrak{u}}(s)=T(s)u\,,$ (6.19)

and hence $T(\cdot)u\in C([0,S],L^{2}(M))\cap C^{1}((0,S),L^{2}(M))$. Finally,
it follows from the density of $C_{c}^{\infty}(M)\subseteq L^{2}(M)$ (and
since $S>0$ is arbitrary) that the family of operators
$(T(s))_{s\in{\mathbb{R}}_{\geq 0}}$ is strongly continuous.

To complete the proof of Lemma 6.2, it only remains to be shown that
${\mathfrak{u}}(s)=T(s)u$ solves the distributional Schrödinger equation
(6.6). To that end, we return to the weakly converging subnet
$({\mathfrak{u}}(\cdot,\theta_{\alpha(\beta)}))_{\beta\in\mathscr{B}}$ in
$H^{2}([0,S],L^{2}(M))$. Fixing an arbitrary $w\in C_{c}^{\infty}(M)$, clearly

$\displaystyle\left\langle
w\,\middle|\,\big{(}{\partial}_{s}-\Delta_{\theta_{\alpha(\beta)}}\big{)}{\mathfrak{u}}(s,\theta_{\alpha(\beta)})\right\rangle_{L^{2}(M)}=0\,,\quad
s\in(0,S)\,.$ (6.20)

On the other-hand, by Sobolev embedding and weak convergence in
$H^{2}([0,S],L^{2}(M))$,

$\displaystyle\left\langle
w\,\middle|\,{\partial}_{s}{\mathfrak{u}}(s,\theta_{\alpha(\beta)})\right\rangle_{L^{2}(M)}=\left\langle
w\,\middle|\,d_{s}{\mathfrak{u}}(s,\theta_{\alpha(\beta)})\right\rangle_{L^{2}(M)}\xrightarrow{\beta\in\mathscr{B}}\left\langle
w\,\middle|\,d_{s}{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}\,,$ (6.21)
$\displaystyle\left\langle
w\,\middle|\,\Delta_{\theta_{\alpha(\beta)}}{\mathfrak{u}}(s,\theta_{\alpha(\beta)})\right\rangle_{L^{2}(M)}=\left\langle\Delta_{\pi-\theta_{\alpha(\beta)}}w\,\middle|\,{\mathfrak{u}}(s,\theta_{\alpha(\beta)})\right\rangle_{L^{2}(M)}\xrightarrow{\beta\in\mathscr{B}}\left\langle
i{\mathcal{D}}_{-}w\,\middle|\,{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}\,,$

and hence

$\displaystyle\left\langle
w\,\middle|\,d_{s}{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}+\left\langle
i{\mathcal{D}}_{-}w\,\middle|\,{\mathfrak{u}}(s)\right\rangle_{L^{2}(M)}=0\,,\quad
s\in(0,S)\,,$ (6.22)

with initial condition ${\mathfrak{u}}(0)=u$. Since $S>0$ is arbitrary, it
follows that ${\mathfrak{u}}(s)=T(s)u$ indeed solves the distributional
Schrödinger equation (6.6) with initial condition ${\mathfrak{u}}(0)=u$. ∎

## 7\. Conclusions

A Wick rotation in the lapse, rather than in time, has been introduced for
real manifolds admitting a co-dimension one foliation. The definition refers
to a fiducial foliation, but covariance under foliation changing
diffeomorphisms is ensured. Further, the interpolating complex metrics are
admissible in the sense of [29] in a foliation independent sense. In this
setting a Wick rotated heat semi-group was constructed and studied in some
detail. The main results obtained are summarized in the introductory
tabulation. Here we mention some further directions.

The most immediate extension is to non-scalar Laplacians, in particular of
Lichnerowicz type. For Euclidean signature the associated heat semigroups are
widely used to investigate the quantum theory of gauge fields and gravity,
often in combination with the non-perturbative Functional Renormalization
Group [39, 40, 42]. We see no principle obstruction to such a generalization,
which would allow one to explore the near Lorentzian regime of such
computations in an apples-to-apples comparison. The widely used de Sitter
background may serve as a test case.

In practice, such computations utilize mainly the coefficients appearing in a
small semigroup time asymptotic expansion of the kernel’s diagonal, as
developed here. Nevertheless, the existence of an asymptotic expansion for the
off-diagonal Wick rotated heat kernel would be of considerable interest.
Although the existence of an off-diagonal expansion can presumably be
established along the lines of [28, 22, 32] without reference to the Synge
function, the formulas for the coefficients are normally expressed in terms of
it. It would therefore be desirable to establish existence results for the
lapse-Wick rotated Synge function under weak assumptions on the metric. A
related issue left unexplored here are generalizations of the known exact heat
kernel bounds [10, 23], usually formulated in terms of the Synge function.

The generalized Laplace transform of the Wick rotated heat kernel’s off-
diagonal expansion relates to a Wick rotated Hadamard parametrix, and the
difference to the exact induced Wick rotated Green’s function may be expected
to code ‘state dependent’ aspects. Here the strict Lorentzian limit warrants
further investigation, particularly the relation to boundary values of the
Lorentzian Hessian’s resolvent and associated Feynman propagator [13, 14]. One
of the relevant questions then being whether or not the $\theta\rightarrow
0^{+}$ limit of the Wick-rotated Hessian’s resolvent singles out a Green’s
function, and if this is associated to a positive quantum state [2, 8].

Acknowledgments. We would like to thank N. Viet Dang, J. Derezinski, I.
Khavkine, A. Strohmaier, R. Verch, A. Vasy, and M. Wrochna for their interest
and helpful discussions at various stages of this work. Additionally, we are
grateful to K. Taira and C. J. Lennard for suggestions on the proof strategy
in Section 6. R.B. thanks the Erwin Schrödinger Institute in Vienna for
hospitality during the ‘Spectral Theory and Mathematical Relativity’ thematic
program, and the Institute Henri Poincaré for hospitality during the ‘Quantum
and classical fields interacting with geometry’ thematic program, where parts
of this work were completed and many stimulating discussions took place. The
authors acknowledge support of the Institute Henri Poincaré (UAR 839 CNRS-
Sorbonne Université), and LabEx CARMIN (ANR-10-LABX-59-01).

##  Appendix A Foliation geometry

In this appendix we set our notation and collect a few basic notions of
foliation geometry in relation to foliation changing diffeomorphisms, as
needed in the main text. Throughout M is a $1\\!+\\!d$ dimensional topological
manifold (locally Euclidean and Haussdorff) that is: smooth, connected,
orientable, $2^{\rm nd}$ countable, and without boundary. We allow it to be
noncompact.

Equivalent foliations. No metric structure is assumed in this part. A co-
dimension-one foliation of $M$ is a collection
$\\{\Sigma_{\alpha}\\}_{\alpha\in A}$ of connected disjoint subsets of $M$
such that: (i) $M=\cup_{\alpha\in A}\Sigma_{\alpha}$, and (ii) every point in
$M$ has a neighborhood $U$ and a system of local coordinates
$y=(y^{0},y^{1},\ldots,y^{d}):U\rightarrow\mathbb{R}^{1+d}$, such that for
each leaf $\Sigma_{\alpha}$, if $\Sigma_{\alpha}\cap U\neq\emptyset$, then its
local coordinate image is a $y^{0}=\text{const.}$ slice of the chart range.
Such a (non-unique) coordinate system is said to be adapted to the foliation.
Criteria for a manifold to admit such a structure can be found in [44] and the
references therein. Here we assume that $M$ admits a co-dimension-one
foliation given by the level sets of a smooth submersion $T:M\to{\mathbb{R}}$
(in particular $dT\neq 0$ everywhere).888In metric geometry $T$ corresponds to
a temporal function and the associated foliations are vorticity-free, see
Section A.2. The foliation can then be parameterized as
$\\{\Sigma_{t}\\}_{t\in I}$, $I\subseteq{\mathbb{R}}$ is the range of $T$, and
$\Sigma_{t}:=T^{-1}(\\{t\\})$; by slight abuse of notation we sometimes denote
such a foliation as $I\ni t\mapsto\Sigma_{t}$. Every leaf is a $d$-dimensional
embedded hypersurface, and we further assume that all leaves $\Sigma_{t}$
arise from embeddings of a single $d$-dimensional manifold $\Sigma$. It
follows readily from the implicit function theorem and the non-vanishing of
the differential $dT$ that each $p\in M$ has a chart neighborhood $U$ such
that in local coordinates $\Sigma_{t}\cap U$ (if non-empty) consists of the
points $(t,y^{1},\ldots,y^{d})$ in the chart range. Such adapted coordinates
are not unique. If $y$ and $y^{\prime}$ are two such coordinate systems
defined on an open set $U\subset M$, then both are related by a diffeomorphism
of the form
${y^{\prime}}^{0}=\chi^{0}(y^{0}),{y^{\prime}}^{a}=\chi^{a}(y)=\chi^{a}(t,x)$,
$a=1,\ldots,d$. By the implicit function theorem we also view $x^{a}(y)$ to be
locally known and such that
$\tilde{y}^{\alpha}=y^{\alpha}(t(\tilde{y}),x(\tilde{y}))$, for all
$\tilde{y}^{\alpha}$. Here and below we often write
$y^{\alpha},\alpha=0,1,\ldots,d$, for $y=(y^{0},y^{a})$.

Two foliations ${I}\ni t\mapsto\Sigma_{t}$, and ${I^{\prime}}\ni
t^{\prime}\mapsto\Sigma^{\prime}_{t^{\prime}}$, defined on $M$ are called
equivalent if there is a diffeomorphism sending the leaves of one into the
leaves of the other. For simplicity we consider only smooth, orientation
preserving diffeomorphisms $\chi:M\rightarrow M$ in the component of the
identity, that reduce to the identity outside a compact set. They form a group
with respect to composition. Sequences of diffeomorphisms and the concomitant
topological considerations will not enter. For short, we just write ${\rm
Diff}(M)$ for the resulting group of diffeomorphisms.

In local charts, we identify points with their coordinates, and write
alternatively $\chi(y)$ and $y^{\prime}$ for the image point of $y\in U$. The
differential $T_{y}\chi$ maps the tangent space at $y$ into the one at
$y^{\prime}$ and is written as
${\partial}{y^{\prime}}^{\alpha}/{\partial}y^{\gamma}$. Similarly, for the
inverse $\chi^{-1}:U^{\prime}\rightarrow U$, the image of $y^{\prime}\in
U^{\prime}$ is written alternatively as $\chi^{-1}(y^{\prime})$ and $y$. For
the differentials one has $T_{y^{\prime}}\chi^{-1}=[T_{y}\chi]^{-1}$. In the
$1\\!+\\!d$ decomposition we write $\chi^{0}$, $\chi^{a}$ and
$(\chi^{-1})^{0}$, $(\chi^{-1})^{a}$ for the projections of $\chi$ and
$\chi^{-1}$ onto an adapted coordinate basis, and whenever unambiguous we
abbreviate those as $t^{\prime}$, ${x^{\prime}}^{a}$ and $t$, $x^{a}$,
respectively. In this notation a generic $\chi\in{\rm Diff}(M)$ changes both
the leaves of the foliation and the coordinatization of the hypersurfaces:

$t\mapsto\Sigma_{t}\;\;\mbox{is mapped
into}\;\;t^{\prime}\mapsto\Sigma^{\prime}_{t^{\prime}}\quad\mbox{by}\;\;t^{\prime}=\chi^{0}(t,x),\,{x^{\prime}}^{a}=\chi^{a}(t,x)\,.$
(A.1)

By the above definition two such foliations are equivalent. However, the
adapted coordinates of one are not adapted to the other. This is to be
contrasted with the subgroup ${\rm Diff}(\\{\Sigma\\})\subset{\rm Diff}(M)$ of
foliation preserving diffeomorphisms

$\chi\in{\rm
Diff}(\\{\Sigma\\})\;\;\;\mbox{iff}\;\;\;t^{\prime}=\chi^{0}(t)\,,\;\;{x^{\prime}}^{a}=\chi^{a}(t,x)\,,$
(A.2)

with $\chi^{0}(t_{i})=t_{i}$, $\chi^{0}(t_{f})=t_{f}$. As noted before, this
is the maximal subgroup that maps adapted coordinates of a given foliation
into each other; merely the labeling of the leaves and their coordinization
changes. The Jacobian matrix in the $1+d$ decomposition is then upper
triangular. We reserve the notation ${\rm Diff}(\Sigma)$ for the subgroup of
$t$ independent diffeomorphisms ${x^{\prime}}^{a}=\chi^{a}(x)$ of $\Sigma$.

Block decomposition of $1\\!+\\!d$ diffeomorphisms. The diffeomorphisms in
$1\\!+\\!d$ form of course still form a group under concatenation.
Concatenating $(t^{\prime},{x^{\prime}}^{a})=(\chi^{0}(t,x),\chi^{a}(t,x))$
with
$(t^{\prime\prime},{x^{\prime\prime}}^{a})=({\chi^{\prime}}^{0}(t^{\prime},x^{\prime}),{\chi^{\prime}}^{a}(t^{\prime},x^{\prime}))$
gives
$(t^{\prime\prime},{x^{\prime\prime}}^{a})=((\chi^{\prime}\circ\chi)^{0}(t,x)$,
$(\chi^{\prime}\circ\chi)^{a}(t,x))$, where
$(\chi^{\prime}\circ\chi)^{0}(t,x)={\chi^{\prime}}^{0}(\chi^{0}(t,x),\chi^{a}(t,x))$
and
$(\chi^{\prime}\circ\chi)^{a}(t,x)={\chi^{\prime}}^{a}(\chi^{0}(t,x),\chi^{a}(t,x))$.
The defining relations for the inverse $\chi^{-1}$ of $\chi$ therefore are
$(\chi^{-1})^{0}(\chi^{0}(t,x),\chi^{b}(t,x))=t$,
$(\chi^{-1})^{a}(\chi^{0}(t,x),\chi^{b}(t,x))=x^{a}$. In general, the temporal
or spatial component of $\chi^{-1}$ also depends on the spatial or temporal
component of $\chi$. An exception are diffeomorphisms trivial in one
component, $(t,x^{a})\mapsto(\chi^{0}(t,x),x^{a})$ or
$(t,x^{a})\mapsto(t,\chi^{a}(t,x)))$, where the inverses depend only
parametrically on $x^{a}$ or $t$, respectively.

Next, consider the composition of the differentials. Written in $1\\!+\\!d$
block form one has

$\frac{{\partial}y^{\gamma}}{{\partial}{y^{\prime}}^{\alpha}}=\left(\begin{array}[]{cc}\dfrac{{\partial}t}{{\partial}t^{\prime}}&\dfrac{{\partial}x^{c}}{{\partial}t^{\prime}}\\\\[8.53581pt]
\dfrac{{\partial}t}{{\partial}{x^{\prime}}^{a}}&\dfrac{{\partial}x^{c}}{{\partial}{x^{\prime}}^{a}}\end{array}\right)\,,\makebox[28.45274pt]{
}\frac{{\partial}{y^{\prime}}^{\alpha}}{{\partial}y^{\gamma}}=\left(\begin{array}[]{cc}\dfrac{{\partial}t^{\prime}}{{\partial}t}&\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}t}\\\\[8.53581pt]
\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}&\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{c}}\end{array}\right)\,.$
(A.3)

The chain rule
$({\partial}{y^{\prime}}^{\gamma}/{\partial}y^{\beta})({\partial}{y^{\prime\prime}}^{\alpha}/{\partial}{y^{\prime}}^{\gamma})=({\partial}{y^{\prime\prime}}^{\alpha}/{\partial}y^{\beta})$
decomposes into blocks according to

$\displaystyle\frac{{\partial}t^{\prime\prime}}{{\partial}t}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{{\partial}t^{\prime\prime}}{{\partial}t^{\prime}}\frac{{\partial}t^{\prime}}{{\partial}t}+\frac{{\partial}t^{\prime\prime}}{{\partial}{x^{\prime}}^{c}}\frac{{\partial}{x^{\prime}}^{c}}{{\partial}t}\,,$
$\displaystyle\frac{{\partial}{x^{\prime\prime}}^{a}}{{\partial}t}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{{\partial}{x^{\prime\prime}}^{a}}{{\partial}t^{\prime}}\frac{{\partial}t^{\prime}}{{\partial}t}+\frac{{\partial}{x^{\prime\prime}}^{a}}{{\partial}{x^{\prime}}^{c}}\frac{{\partial}{x^{\prime}}^{c}}{{\partial}t}\,,$
$\displaystyle\frac{{\partial}t^{\prime\prime}}{{\partial}x^{b}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{{\partial}t^{\prime\prime}}{{\partial}t^{\prime}}\frac{{\partial}t^{\prime}}{{\partial}x^{b}}+\frac{{\partial}t^{\prime\prime}}{{\partial}{x^{\prime}}^{c}}\frac{{\partial}{x^{\prime}}^{c}}{{\partial}x^{b}}\,,$
$\displaystyle\frac{{\partial}{x^{\prime\prime}}^{a}}{{\partial}x^{b}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{{\partial}{x^{\prime\prime}}^{a}}{{\partial}t^{\prime}}\frac{{\partial}t^{\prime}}{{\partial}x^{b}}+\frac{{\partial}{x^{\prime\prime}}^{a}}{{\partial}{x^{\prime}}^{c}}\frac{{\partial}{x^{\prime}}^{c}}{{\partial}x^{b}}\,.$
(A.4)

As a consequence the familiar inversion formula for the full Jacobian matrices
(A.3) does not project to the blocks. Systematically one would want to express
the components of ${\partial}y^{\alpha}/{\partial}{y^{\prime}}^{\beta}$ in
terms of the components of
${\partial}{y^{\prime}}^{\alpha}/{\partial}y^{\beta}$. To do so we specialize
(A) to coinciding initial and final variables and swap the role of the primed
and the unprimed fields. Combining the resulting equations pairwise gives

$\displaystyle\frac{{\partial}x^{c}}{{\partial}{x^{\prime}}^{b}}\bigg{(}\frac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{c}}-\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-1}\frac{{\partial}{x^{\prime}}^{a}}{{\partial}t}\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\bigg{)}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\delta_{b}^{a}\,,$
$\displaystyle\frac{{\partial}x^{c}}{{\partial}t^{\prime}}\bigg{(}\frac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{c}}-\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-1}\,\frac{{\partial}{x^{\prime}}^{a}}{{\partial}t}\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\bigg{)}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-1}\,\frac{{\partial}{x^{\prime}}^{a}}{{\partial}t}\,.$
(A.5)

The inverse of the matrix in brackets can be expressed in terms of the matrix
inverse of ${\partial}{x^{\prime}}^{a}/{\partial}x^{b}$ via the formula for
rank one perturbations (Sherman-Morrison). Writing $Y_{b}^{a}$ for the result
the desired inversion formulas read

$\displaystyle\frac{{\partial}x^{a}}{{\partial}{x^{\prime}}^{b}}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle Y_{b}^{a}\,,$
$\displaystyle\frac{{\partial}x^{a}}{{\partial}t^{\prime}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-1}\,\frac{{\partial}{x^{\prime}}^{b}}{{\partial}t}\,Y^{a}_{b}\,,$
$\displaystyle\frac{{\partial}t}{{\partial}t^{\prime}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-1}+\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-2}\,\frac{{\partial}{x^{\prime}}^{d}}{{\partial}t}\,Y^{c}_{d}\,\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\,,$
$\displaystyle\frac{{\partial}t}{{\partial}{x^{\prime}}^{a}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\Big{(}\frac{{\partial}t^{\prime}}{{\partial}t}\Big{)}^{-1}Y^{c}_{a}\,\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\,.$
(A.6)

In general all components mix under inversion. Upper or lower block diagonal
Jacobian matrices remain so, as required. Only for direct product
diffeomorphism $t^{\prime}=\chi^{0}(t),\,{x^{\prime}}^{a}=\chi^{a}(x)$ does
(A) reduce to the simple variants
${\partial}x^{a}/{\partial}{x^{\prime}}^{b}=[({\partial}{x^{\prime}}/{\partial}x)^{-1}]_{b}^{a}$,
${\partial}t/{\partial}t^{\prime}=({\partial}t^{\prime}/{\partial}t)^{-1}$,
directly entailed by the implicit function theorem.

In summary, the differentials $T_{y}\chi$ and
$T_{y^{\prime}}\chi^{-1}=[T_{y}\chi]^{-1}$ of a generic diffeomorphisms
$\chi\in{\rm Diff}(M)$, admit a block decomposition whose composition and
inverse is governed by the relations (A) and (A). The advantage of this crude
decomposition is that no metric structure is required.

Metric geometry of the foliations. We now assume that the manifold $M$ is in
addition equipped with both a Lorentzian and a Riemannian metric $g$, which we
take to be smooth and similar to $(\epsilon_{g},+,\ldots,+)$,
$\epsilon_{g}=\mp 1$. For Lorentzian signature global hyperbolicity of $(M,g)$
is the instrumental condition. It ensures the existence of smooth time
functions, the attainability of the $N^{a}=0$ gauge, and that the leaves of
the foliation are Cauchy surfaces [5]. Systematic expositions of the
Lorentzian $1+d$ projection formalism in metric geometry can be found in many
textbooks, see e.g. [21]. A temporal function in this context is a smooth
function $T:M\rightarrow\mathbb{R}$ with a timelike past pointing gradient
$dT$, interpreted as a one-form
$dT=({\partial}T/{\partial}y^{\alpha})dy^{\alpha}$. Importantly, any globally
hyperbolic spacetime admits a temporal function such that any level surface
$\Sigma_{t}=\\{y\in M\,|\,T(y)=t\\}$ is a Cauchy surface [5]. For Riemannian
metrics we shall continue to use the term ‘temporal function’ for a smooth
function $T$ with a nowhere vanishing gradient. To fix the notation and to
highlight the dependence on the signature parameter $\epsilon_{g}\in\\{\mp
1\\}$, we display the main relations of the (Arnowitt-Deser-Misner) ADM
formalism.

For a fixed temporal function $T$ and the associated foliation $I\ni
t\mapsto\Sigma_{t}$, one may identify $T$ with $t$ and write
${\partial}_{\alpha}t$ for the components of $dT$. In terms of them we set

$g^{\alpha\beta}{\partial}_{\alpha}t{\partial}_{\beta}t=:\epsilon_{g}N^{-2}\,,\quad
m^{\alpha}:=\epsilon_{g}N^{2}g^{\alpha\beta}{\partial}_{\beta}t\,.$ (A.7)

The first equation defines the lapse $N$, the second defines a vector
conjugate to the temporal gradient, $m^{\alpha}{\partial}_{\alpha}t=1$. Note
that $N$ is scalar and $m^{\alpha}$ a vector as long as $T$ is held fixed.
Further $m^{\alpha}{\partial}_{\alpha}$ has unit coefficient along
${\partial}_{t}$ and

$m^{\alpha}{\partial}_{\alpha}={\partial}_{t}-N^{a}{\partial}_{a}\,,$ (A.8)

defines the shift $N^{a}$. In terms of $m^{\alpha},{\partial}_{\alpha}t$
projectors tangential and transversal to the leaves of the foliation are
defined by

$\Sigma_{\alpha}^{\;\;\beta}:=\delta_{\alpha}^{\beta}-{\partial}_{\alpha}t\,m^{\beta}\,,\quad
T_{\alpha}^{\;\;\beta}:={\partial}_{\alpha}t\,m^{\beta}\,.$ (A.9)

We write ${\rm
g}_{\alpha\beta}:=\Sigma_{\alpha}^{\;\delta}\Sigma_{\beta}^{\;\gamma}g_{\delta\gamma}$
for the induced metric on $\Sigma_{t}$. Since
$m^{\alpha}\Sigma_{\alpha}^{\;\beta}=0$, the natural derivative transversal to
the leaves of the foliation is $e_{0}:={\cal L}_{m}={\partial}_{t}-{\cal
L}_{\vec{N}}$, where ${\cal L}_{\vec{N}}$ is the $d$-dimensional Lie
derivative in the direction of $N^{a}$. When acting on scalars we write
$e_{0}=e_{0}^{\alpha}{\partial}_{\alpha}$, so that
$e_{0}^{\alpha}=m^{\alpha}$. The tangential derivatives acting on scalars are

$e_{\alpha}^{a}\frac{{\partial}}{{\partial}x^{a}}=\Sigma_{\alpha}^{\;\beta}{\partial}_{\beta}={\partial}_{\alpha}-{\partial}_{\alpha}t\,e_{0}\,,\quad\frac{{\partial}}{{\partial}x^{a}}=e_{a}^{\alpha}{\partial}_{\alpha}\,.$
(A.10)

which defines the coefficient matrices $e_{a}^{\alpha}$ and $e_{\alpha}^{a}$.
They are such that

$\displaystyle
e_{a}^{\alpha}\,e^{b}_{\alpha}=\delta_{a}^{b}\,,\makebox[28.45274pt]{
}\;\;\Sigma_{\alpha}^{\;\;\beta}=e^{a}_{\alpha}\,e_{a}^{\beta}\,,$
$\displaystyle e_{a}^{\alpha}:=g^{\alpha\beta}{\rm
g}_{ab}\,e^{b}_{\beta}\,,\quad
g_{\alpha\beta}\,e^{\alpha}_{a}\,m^{\beta}=0=g^{\alpha\beta}e_{\alpha}^{a}\,{\partial}_{\beta}t\,,$
(A.11)

which express the orthogonality and completeness of the component fields. By
(A.7), (A.9), (LABEL:1d4) the metric and its inverse take the block diagonal
form

$\displaystyle g_{\alpha\beta}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\epsilon_{g}N^{2}{\partial}_{\alpha}t{\partial}_{\beta}t+{\rm
g}_{ab}e^{a}_{\alpha}e^{b}_{\beta}\,,$ $\displaystyle g^{\alpha\beta}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\epsilon_{g}N^{-2}m^{\alpha}m^{\beta}+{\rm
g}^{ab}e_{a}^{\alpha}e_{b}^{\beta}\,,$ (A.12)

where ${\rm g}^{ac}{\rm g}_{cb}=\delta^{a}_{b}$. Further $\det
g=\epsilon_{g}N^{2}\det{\rm g}$. For a fixed temporal function in addition to
$N,N^{a}$ also ${\rm g}_{ab}$ is a scalar.

The description in terms of the embedding relations
$\tilde{y}^{\alpha}=y^{\alpha}(t(\tilde{y}),x(\tilde{y}))$ is now secondary,
but still carries over

$\displaystyle{\partial}_{\alpha}t=\frac{{\partial}t}{{\partial}y^{\alpha}}\,,\makebox[28.45274pt]{
}m^{\alpha}=\frac{{\partial}y^{\alpha}}{{\partial}t}-N^{a}e_{a}^{\alpha}\,,$
(A.13a) $\displaystyle
e_{a}^{\alpha}=\frac{{\partial}y^{\alpha}}{{\partial}x^{a}}\,,\makebox[28.45274pt]{
}e_{\alpha}^{a}=\frac{{\partial}x^{a}}{{\partial}y^{\alpha}}+N^{a}{\partial}_{\alpha}t\,,$
(A.13b)

where $t(y)$ is the given temporal function and $x^{a}(y)$ is defined by the
implicit function theorem. The left pair of relations holds by definition.
Further ${\partial}y^{\alpha}/{\partial}t-m^{\alpha}$ is orthogonal to
${\partial}_{\alpha}t$ and thus tangent to $\Sigma_{t}$. As such it can be
written in the form $N^{a}e_{a}^{\alpha}$, which gives the second relation in
(A.13a). The orthogonality (LABEL:1d4) then provides the second relation in
(A.13b). The 1-forms $e^{a}=dx^{a}+N^{a}dt$ span the cotangent space of
$\Sigma$, while ${\partial}_{a}=e_{a}^{\alpha}{\partial}_{\alpha}$ span the
tangent space. The full coordinate 1-forms and associated differentials are
given by

$\displaystyle dy^{\alpha}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
m^{\alpha}dt+e^{\alpha}_{a}e^{a}\,,\,\makebox[28.45274pt]{
}\;e^{a}=dx^{a}+N^{a}dt\,,$
$\displaystyle\frac{{\partial}}{{\partial}y^{\alpha}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle{\partial}_{\alpha}t\,e_{0}+e_{\alpha}^{a}{\partial}_{a}\,,\makebox[28.45274pt]{
}e_{0}={\partial}_{0}-N^{a}{\partial}_{a}\,.$ (A.14)

The one forms $(Ndt,e^{a})$ and dual vector fields
$(N^{-1}e_{0},{\partial}_{a})$ form a moving frame which we refer to as the
foliation frame. As long as the coordinate functions
$t:U\rightarrow\mathbb{R}$ and $x^{a}:U\rightarrow\mathbb{R}^{d}$ are kept
fixed the description is independent of the choice of embedding coordinates
$y^{\alpha}$.

##  Appendix B Distributions and Sobolev spaces

We introduce and summarize the main properties of the function and
distribution spaces used in this paper.

Test functions, distributions and distributional derivatives. As a set, the
test functions on the manifold $M$ is simply
${\mathfrak{D}}(M)=C_{c}^{\infty}(M)$, the compactly supported smooth complex-
valued functions on $M$ (all functions discussed will be complex valued, and
this specification will be henceforth omitted). ${\mathfrak{D}}(M)$ is
equipped with a topology arising from a family of seminorms, rendering it a
locally convex topological vector space; we refer to [3, 19] for further
details. It can be shown that, in order to define distributions, it is
sufficient to specify the mode of convergence in ${\mathfrak{D}}(M)$.

######  Definition B.1.

A sequence $(u_{j})_{j\in{\mathbb{N}}}$ in ${\mathfrak{D}}(M)$ is said to
converge to $u\in{\mathfrak{D}}(M)$ iff the following holds.

  1. (i)

There is a compact set $K\subseteq M$ containing the supports of $u$ and all
the $u_{j}$.

  2. (ii)

There is a covering of $K$ by finitely many charts
$\Omega_{1},\ldots,\Omega_{k}$ such that for each chart in local coordinates,
for all multi-indices $\alpha$,
${\partial}^{\alpha}u_{j}\to{\partial}^{\alpha}u$ uniformly on the chart.

In order to define distributional gradients and Laplacians we introduce the
test vector fields $\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}(M)$ and test
covector fields
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}(M)$. As sets these are respectively
the compactly supported smooth vector and covector fields on $M$, and the mode
of convergence is defined as the straightforward generalization of Definition
B.1.

Next, the distribution space ${\mathfrak{D}}^{\prime}(M)$ comprises the
continuous linear functionals over ${\mathfrak{D}}(M)$, where continuity means
that for any $\varphi\in{\mathfrak{D}}^{\prime}(M)$ and any sequence
$(u_{j})_{j\in{\mathbb{N}}}$ converging to zero in ${\mathfrak{D}}(M)$ we have
$(\varphi,u_{j})\xrightarrow{j}0$ (here $(\varphi,u_{j})$ denotes the action
of the distribution $\varphi$). It is clear that ${\mathfrak{D}}^{\prime}(M)$
is a vector space, and convergence therein is defined as usual:
$\varphi_{j}\xrightarrow{{\mathfrak{D}}^{\prime}}\varphi$ iff
$(\varphi_{j},u)\xrightarrow{j}(\varphi,u)$ for all test functions
$u\in{\mathfrak{D}}(M)$. Similarly, the spaces of distributional vector fields
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}^{\prime}(M)$
and distributional covector fields
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}^{\prime}(M)$ are respectively the
duals of $\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}(M)$ and
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}(M)$, and are linear spaces with
convergence defined analogously to the scalar case.

The above test function and distribution spaces are well-defined on any smooth
manifold, without reference to a metric structure. We now assume that $M$
carries in addition a smooth metric $g$ of Riemannian or Lorentzian signature,
and associated volume form $d\mu_{g}=dy\sqrt{|g|}$. Recall that $f\in
C^{\infty}(M)$ canonically defines a distribution
$(\tilde{f},u):=\int\\!d\mu_{g}fu$. Then the one-form $\nabla f$ defines an
distributional vector field $\widetilde{\nabla
f}\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}^{\prime}(M)$
by an integration by parts, $(\widetilde{\nabla
f},v):=-(\tilde{f},\nabla_{\alpha}v^{\alpha})$ for
$v\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}(M)$, where
$\nabla_{\alpha}v^{\alpha}=|g|^{-1/2}{\partial}_{\alpha}(|g|^{1/2}v^{\alpha})$
is the divergence. Generalizing, the distributional gradient maps
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern
0.5pt}^{\prime}(M)\to\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern
0.5pt}^{\prime}(M)$, where for each $\varphi\in{\mathcal{D}}^{\prime}(M)$, the
distribution $\nabla\varphi$ is defined by

$\displaystyle(\nabla\varphi,v):=-(\varphi,\nabla_{\alpha}v^{\alpha})\,,\quad\forall\,v\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}(M)\,.$ (B.4)

The distributional Laplacian $\nabla^{2}$ is defined analogously: for
$\varphi\in{\mathfrak{D}}^{\prime}(M)$,
$\nabla^{2}\varphi\in{\mathfrak{D}}^{\prime}(M)$ is defined by

$\displaystyle(\nabla^{2}\varphi,u):=(\varphi,\nabla^{2}u)\,,\quad\forall\,u\in{\mathfrak{D}}(M)\,.$
(B.5)

$L^{2}$\- and Sobolev spaces. The $\sigma$-algebra $\mathfrak{M}(M)$ of
measurable sets on the manifold $M$ consists of those $E\subseteq M$ such that
for any chart $U$ on $M$, the image of $E\cap U$ under the chart map is a
Lebesgue measurable subset of ${\mathbb{R}}^{1+d}$; in particular,
$\mathfrak{M}(M)$ contains all Borel sets. Then, measurable functions, vector
fields, and covector fields can be defined on $M$ in the standard way. Next,
the volume form associated to the Lorentzian metric $g_{-}$ induces a
complete, regular measure $\mu_{g}$ on $\mathfrak{M}(M)$; we refer to [23] for
details. Then, as usual, $L^{2}(M)$ consists of (equivalence classes of)
measurable functions $f:M\to{\mathbb{C}}$ for which the $2$-norm
$\norm{f}_{L^{2}(M)}:=(\int\\!d\mu_{g}\,|f|^{2})^{1/2}<\infty$. This is a
Hilbert space with inner product

$\displaystyle\left\langle
f_{1}\,\middle|\,f_{2}\right\rangle_{L^{2}(M)}:=\int\\!d\mu_{g}\,f_{1}^{\ast}f_{2}\,,\quad
f_{1},\,f_{2}\in L^{2}(M)\,.$ (B.6)

To discuss $L^{2}$-spaces of (co)-vector functions, a Riemannian metric
$g_{+}$ is introduced. The space of square-integrable covector fields
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}(M)$ consists of (equivalence classes
of) measurable sections $\omega:M\to T^{\ast}M$ for which
$(g_{+}^{\alpha\beta}\omega_{\alpha}^{\ast}\omega_{\beta})^{1/2}\in L^{2}(M)$.
The definition of the space of square-integrable vector fields
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-2.1pt] L^{2}&\end{array}\kern 0.5pt}(M)$ is analogous,
and moreover the metric $g_{+}$ defines a canonical isometry
$\omega_{\alpha}:=g_{\alpha\beta}^{+}v^{\beta}$ between
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-2.1pt] L^{2}&\end{array}\kern 0.5pt}(M)$ and
$\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}(M)$. Further, as in the scalar case,
these are Hilbert spaces with inner products

$\displaystyle\left\langle\omega\,\middle|\,\eta\right\rangle_{\scriptsize\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern
0.5pt}(M)}:=\int\\!d\mu_{g}\,g_{+}^{\alpha\beta}\omega_{\alpha}^{\ast}\eta_{\beta}\,,\quad\left\langle
v\,\middle|\,w\right\rangle_{\scriptsize\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-4.0pt]
L^{2}&(M)\end{array}}:=\int\\!d\mu_{g}\,g_{\alpha\beta}^{+}v^{\alpha\,\ast}w^{\beta}\,.$
(B.12)

The locally integrable versions of the above spaces consist of measurable
sections that are square-integrable over every open set set $\Omega\Subset M$,
and are denoted e.g. $L^{2}_{\text{loc}}(M)$ in the scalar case; moreover,
clearly $L^{2}(M)\subseteq L^{2}_{\text{loc}}(M)$.

Next, we recall the following characterization of a distribution
$F\in{\mathfrak{D}}^{\prime}(M)$ having an $L^{2}(M)$ realization. In
particular, the right hand side of (B.15) below defines a (possibly infinite)
$L^{2}$-“norm” of a distribution $F\in{\mathfrak{D}}^{\prime}(M)$, without the
$L^{2}$-realization being a-priori well defined.

######  Proposition B.1 (Lemma 2.15 of [23]).

Let $F\in{\mathfrak{D}}^{\prime}(M)$. Then

$\displaystyle\sup_{u\in
C_{c}^{\infty}(M)\atop\norm{u}_{L^{2}}=1}\big{|}(F,u)\big{|}<\infty$ (B.13)

iff there exists a unique $f\in L^{2}(M)$ such that for all $u\in
C_{c}^{\infty}(M)={\mathfrak{D}}(M)$

$\displaystyle(F,u)=\int\\!d\mu_{g}(y)\,f(y)u(y)\,.$ (B.14)

Moreover,

$\displaystyle\norm{f}_{L^{2}}=\sup_{u\in
C_{c}^{\infty}(M)\atop\norm{u}_{L^{2}}=1}\big{|}(F,u)\big{|}\,.$ (B.15)

Proceeding now to Sobolev spaces, every $f\in L^{2}(M)$ defines a distribution
$(\tilde{f},u):=\int\\!\,d\mu_{g}\,fu$, and hence has a well-defined
distributional gradient $\widetilde{\nabla
f}\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern
0.5pt}^{\prime}(M)$,

$\displaystyle(\widetilde{\nabla
f},v)=-\int\\!d\mu_{g}\,f\nabla_{\alpha}v^{\alpha}\,,\quad\forall
v\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}(M)\,.$
(B.19)

The Sobolev space ${\mathcal{W}}^{1}(M)$ consists of those elements $f\in
L^{2}(M)$ for which the distributional gradient $\widetilde{\nabla
f}\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}^{\prime}(M)$
has an $\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}(M)$ realization, denoted by $\nabla
f$ and called the weak derivative of $f\in L^{2}(M)$. Clearly for every test
vector field $v\in\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoonup\\\\[-1.0pt] {\mathfrak{D}}\end{array}\kern 0.5pt}(M)$ one has

$\displaystyle\int\\!d\mu_{g}\,\nabla_{\alpha}f\,v^{\alpha}=-\int\\!d\mu_{g}\,f\nabla_{\alpha}v^{\alpha}\,.$
(B.20)

Clearly, $C_{c}^{\infty}(M)\subseteq{\mathcal{W}}^{1}(M)$, and moreover
${\mathcal{W}}^{1}(M)$ is a Hilbert space with inner product

$\displaystyle\left\langle
f_{1}\,\middle|\,f_{2}\right\rangle_{{\mathcal{W}}^{1}(M)}:=\left\langle
f_{1}\,\middle|\,f_{2}\right\rangle_{L^{2}(M)}+\left\langle\nabla
f_{1}\,\middle|\,\nabla
f_{2}\right\rangle_{\scriptsize\mathord{\begin{array}[t]{@{}c<EMAIL_ADDRESS>\rightharpoondown\end{array}\kern 0.5pt}(M)}\,.$ (B.23)

Finally, the main Sobolev space used in this paper,
${\mathcal{W}}_{0}^{1}(M)$, is defined as the closed subspace of
${\mathcal{W}}^{1}(M)$ obtained by taking the closure of $C_{c}^{\infty}(M)$
in the ${\mathcal{W}}^{1}$-norm. In particular, ${\mathcal{W}}_{0}^{1}(M)$ is
a Hilbert space with inner product (B.23), and embeds densely into $L^{2}(M)$.

##  Appendix C Local regularity results

In this appendix we summarize the salient aspects of the local regularity
theory of the following elliptic operators with complex valued smooth
coefficients, defined on a connected open set
$\Omega\subseteq{\mathbb{R}}^{n}$,

$\displaystyle L(\cdot)$
$\displaystyle={\partial}_{\mu}(a^{\mu\nu}(x){\partial}_{\nu}(\cdot))\,,$
(C.1a) $\displaystyle\bar{L}(\cdot)$
$\displaystyle=b(x){\partial}_{\mu}(a^{\mu\nu}(x){\partial}_{\nu}(\cdot))\,.$
(C.1b)

Here $b\in C^{\infty}(\Omega,{\mathbb{R}})$ and $a\in
C^{\infty}(\Omega,M_{n\times n}({\mathbb{C}}))$ satisfy:

  1. (i)

For all $x\in\Omega:\,b(x)>0$.

  2. (ii)

For each $x\in\Omega:\,a\in M_{n\times n}({\mathbb{C}})$ is symmetric.
Further, there exists a continuous map $\ell:\Omega\to{\mathbb{R}}_{>0}$ such
that

$\displaystyle\real(a^{\mu\nu}(x)\xi_{\mu}^{\ast}\xi_{\nu})\geq\ell(x)\delta^{\mu\nu}\xi_{\mu}^{\ast}\xi_{\nu}\,,\quad\forall\,\xi\in{\mathbb{C}}^{n}\,.$
(C.2)

We shall make copious use of Sobolev embedding:999Here, and throughout this
appendix, all $L^{2}_{\rm loc}(\Omega)$ and Sobolev spaces $W^{l}_{\rm
loc}(\Omega)$ are defined wrt. the Lebesgue measure on $\Omega$. We refer to
[23] for detailed definitions.

Sobolev embedding theorem (Theorem 6.1 of [23]). Let
$\Omega\subseteq{\mathbb{R}}^{n}$ be open and $k,m\in{\mathbb{N}}_{0}$ such
that $k>m+\frac{n}{2}$. Then each $u\in W^{k}_{\rm loc}(\Omega)$ has a
$C^{m}(\Omega)$-representative (also denoted by $u$). Moreover, for any
relatively compact open sets $\Omega_{0},\,\Omega_{1}$ such that
$\Omega_{0}\Subset\Omega_{1}\Subset\Omega$ there exists a constant $C>0$ such
that

$\displaystyle\norm{u}_{C^{m}(\Omega_{0})}\leq
C\norm{u}_{W^{k}(\Omega_{1})}\,,$ (C.3)

and $C$ depends on $\Omega_{0},\Omega_{1},k,m,n$.

The first result of this appendix is that when
$u,\,\bar{L}u,\ldots,\,\bar{L}^{k}u\in L^{2}_{\text{loc}}(\Omega)$ for
sufficiently high $k\in{\mathbb{N}}$, $u$ is classically differentiable. This
is the content of Theorem C.1, and its Corollary C.1.1.

######  Theorem C.1.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be open, and assume that for some
$k\in{\mathbb{N}}$ we have

$\displaystyle u,\,\bar{L}u,\ldots,\,\bar{L}^{k}u\in L^{2}_{\rm
loc}(\Omega)\,,$ (C.4)

where the operator $\bar{L}$ is defined in (C.1b). Then $u\in W^{2k}_{\rm
loc}(\Omega)$, and for any open sets
$\Omega_{0}\Subset\Omega_{1}\Subset\Omega$, there exists $M_{k}>0$ (depending
on $\bar{L},\,\Omega_{0},\,\Omega_{1}$) such that

$\norm{u}_{W^{2k}(\Omega_{0})}\leq
M_{k}\sum_{j=0}^{k}\norm{\bar{L}^{j}u}_{L^{2}(\Omega_{1})}\,.$ (C.5)

Applying the Sobolev embedding theorem then yields

######  Corollary C.1.1.

Given the hypotheses of Theorem C.1, if there exists $m\in{\mathbb{N}}_{0}$
such that $k>m/2+n/4$, then $u\in C^{m}(\Omega)$ (or more precisely $u\in
L^{2}_{\rm loc}(\Omega)$ has a $C^{m}$-representative). Moreover, for open
sets $\Omega_{0}\Subset\Omega_{1}\Subset\Omega$ there is a $B>0$ such that

$\norm{u}_{C^{m}(\Omega_{0})}\leq
B\sum_{j=0}^{k}\norm{\bar{L}^{j}u}_{L^{2}(\Omega_{1})}\,,$ (C.6)

with $\norm{u}_{C^{m}(\Omega_{0})}:=\underset{|\alpha|\leq
m}{\sup}\underset{\Omega_{0}}{\sup}\,|{\partial}^{\alpha}u|$.

The second main result of this appendix concerns analytic maps from the open
ball $B_{S}(0):=\\{\zeta\in{\mathbb{C}}\;|\;|\zeta|<S\\}$ to $L^{2}_{\rm
loc}(\Omega)$ defined by power series of the form

$\displaystyle F(s):=\sum_{j=0}^{\infty}\frac{s^{j}}{j!}f_{j},$ (C.7)

and joint regularity in $(s,x)\in B_{S}(0)\times\Omega$. Indeed, since for
each $s$, $F(s)\in L^{2}_{\rm loc}(\Omega)$ is determined only up to sets of
Lebesgue measure zero, joint regularity is not a well-defined notion for such
maps in general. However, with the additional hypotheses that under the action
of $\bar{L}$, the $F(s),\,f_{j}$ satisfy local integrability conditions of the
form (C.4) for sufficiently large $k\in{\mathbb{N}}$, classical joint
differentiability is enforced.

######  Theorem C.2.

Let there exist a sequence $(f_{j})_{j\in{\mathbb{N}}_{0}}$ in $L^{2}_{\rm
loc}(\Omega)$ and $k\in{\mathbb{N}}$ so that

$\displaystyle\forall\,j\in{\mathbb{N}}_{0}:\,\,f_{j},\,\bar{L}f_{j},\ldots,\,\bar{L}^{k}f_{j}\in
L^{2}_{\rm loc}(\Omega)\,.$ (C.8)

Suppose there exists an analytic function $F:B_{S}(0)\to
L^{2}_{\text{loc}}(\Omega)$, with the following properties for each $\zeta\in
B_{S}(0)$.

  1. (i)

The power series $F(\zeta)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}f_{j}$
converges in $L^{2}_{\text{loc}}(\Omega)$.

  2. (ii)

$F(\zeta),\,\bar{L}F(\zeta),\,\ldots,\,\bar{L}^{k}F(\zeta)\in
L^{2}_{\text{loc}}(\Omega)$.

  3. (iii)

For each $l\in\\{1,\ldots,k\\}$
$\bar{L}^{l}F(\zeta)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}\bar{L}^{l}f_{j}$,
with the sum converging in $L^{2}_{\text{loc}}(\Omega)$.

If there is $m\in{\mathbb{N}}_{0}$ such that $k>m/2+n/4$, then for each
$\zeta\in B_{S}(0)$ there exists a $C^{m}(\Omega)$ representative of
$F(\zeta)$, denoted by $F(\zeta,\cdot)$, and the mapping

$\displaystyle B_{S}(0)\times\Omega\ni(\zeta,x)\mapsto
F(\zeta,x)\in{\mathbb{C}}$ (C.9)

is jointly $C^{m}$.

The technical input required for the proofs of these results are the following
lemmas.

######  Lemma C.3.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be an open set. Then for any
$k\in{\mathbb{N}}_{0},$ if $u\in W^{k}(\Omega)$ with compact support in
$\Omega$,101010$u\in W^{k}(\Omega)$ compactly supported in $\Omega$ means
there exists $K\subseteq\Omega$ such that $K$ is compact and $u=0$ a.e. on
$\Omega\setminus K$. It follows that all weak derivatives up to order $k$ are
compactly supported and are zero a.e. on $\Omega\setminus K$. and $Lu\in
W^{k-1}(\Omega)$, then

$\displaystyle L(u\ast\varphi_{\varepsilon})\xrightarrow{W^{k-1}}Lu\,.$ (C.10)

Here $u\ast\varphi_{\varepsilon}$ is the convolution with the mollifier
$\varphi_{\varepsilon}$.111111Mollifiers are defined by first specifying a
non-negative $\varphi\in C_{c}^{\infty}({\mathbb{R}}^{n})$ supported in the
unit ball $B_{1}(0)\subseteq{\mathbb{R}}^{n}$ and
$\int_{{\mathbb{R}}^{n}}d^{n}x\,\varphi=1$. Then for any $\varepsilon\in(0,1)$
one sets $\varphi_{\varepsilon}(x):=\varepsilon^{-n}\varphi(x/\varepsilon)$,
which is supported in $B_{\varepsilon}(0)$.

Note: since $u\in W^{k}(\Omega)$ is compactly supported in $\Omega$, it may be
extended to an element of $W^{k}({\mathbb{R}}^{n})$ by setting it to zero
outside $\Omega$, thereby rendering the convolution
$u\ast\varphi_{\varepsilon}$ well defined.

######  Lemma C.4.

Let $\Omega,\,\Omega^{\prime}\subseteq{\mathbb{R}}^{n}$ be open and
$\Omega^{\prime}\Subset\Omega$. Then for every
$k\in{\mathbb{N}}:\,\exists\,C_{k}>0$ (depending on $L$ and $\Omega^{\prime}$)
such that

$\displaystyle\norm{u}_{W^{k}(\Omega^{\prime})}\leq
C_{k}\norm{Lu}_{W^{k-2}(\Omega^{\prime})}\,,\quad u\in
C_{c}^{\infty}(\Omega^{\prime})\,.$ (C.11)

These results are (respectively) generalizations to all $k\in{\mathbb{N}}$ of
Lemmas 6.6 and 6.7 of [23], suitably adapted to the complex coefficient
operator (C.1a). The proofs are by induction, with the base case following the
same lines as in [23], and the inductive step straightforward; we omit the
details.

We now proceed to the proof of Theorems C.1 and C.2.

Proof of Theorem C.1. This is by induction on $k\in{\mathbb{N}}$.

Base case $k=1$: Assume that $u,\,\bar{L}u\in L^{2}_{\text{loc}}(\Omega)$,
which implies that $Lu=b^{-1}\bar{L}u\in L^{2}_{\text{loc}}(\Omega)$. Fixing
arbitrary open sets $\Omega_{1}\Subset\Omega_{2}\Subset\Omega$, there is a
further open set $\Omega^{\prime}$ such that
$\Omega_{1}\Subset\Omega^{\prime}\Subset\Omega_{2}$. Next, let $\psi\in
C_{c}^{\infty}(\Omega^{\prime})$ be a cutoff function for $\Omega_{0}$, i.e.
$\psi:\Omega^{\prime}\to[0,1]$ and $\psi|_{\Omega_{0}}\equiv 1$, and define
$v:=\psi u$. Then clearly $v\in L^{2}(\Omega)$, with compact support contained
in $\Omega^{\prime}$.

Next, consider

$\displaystyle Lv=\psi Lu+L\psi
u+2a^{\mu\nu}{\partial}_{\mu}\psi{\partial}_{\nu}u.$ (C.12)

Since both $u,\,Lu\in L^{2}_{\text{loc}}(\Omega)$, the first two terms are in
$L^{2}_{\text{loc}}(\Omega)$ as well. The final term, with a single derivative
acting on $u$ is a priori only in $W^{-1}_{\text{loc}}(\Omega)$. Thus, $Lv\in
W^{-1}_{\text{loc}}(\Omega)$.

Since $v$ is compactly supported in $\Omega^{\prime}$, there is
$\varepsilon_{0}>0$ such that for all $0<\varepsilon<\varepsilon_{0}$ we have
under mollification $v\ast\varphi_{\varepsilon}\in
C_{c}^{\infty}(\Omega^{\prime})$. Thus, Lemma C.4 implies that there is
$C_{1}>0$ such that

$\displaystyle\norm{v\ast\varphi_{\varepsilon}}_{W^{1}(\Omega^{\prime})}\leq
C_{1}\norm{L(v\ast\varphi_{\varepsilon})}_{W^{-1}(\Omega^{\prime})}\,,\quad\forall\,\varepsilon\in(0,\varepsilon_{0})\,.$
(C.13)

Furthermore, $L(v\ast\varphi_{\varepsilon})\xrightarrow{W^{-1}}Lv$ by Lemma
C.3, so

$\displaystyle\limsup_{\varepsilon\to
0^{+}}\norm{v\ast\varphi_{\varepsilon}}_{W^{1}(\Omega^{\prime})}\leq
C_{1}\norm{Lv}_{W^{-1}(\Omega^{\prime})}\,,$ (C.14)

and hence $\liminf_{\varepsilon\to
0^{+}}\norm{v\ast\varphi_{\varepsilon}}_{W^{1}(\Omega^{\prime})}<\infty$. This
implies $v\in W^{1}(\Omega^{\prime})$ with
$\norm{v}_{W^{1}(\Omega^{\prime})}\leq
C_{1}\norm{Lv}_{W^{-1}(\Omega^{\prime})}$ (c.f. [23] Theorem 2.13). Since
$u\equiv v$ on $\Omega_{0}\Subset\Omega^{\prime}$, together with the
expression for $Lv$ in (C.12), it follows that $u\in W^{1}(\Omega_{0})$ and

$\displaystyle\norm{u}_{W^{1}(\Omega_{0})}\leq
C_{1}^{\prime}\big{(}\norm{u}_{L^{2}(\Omega^{\prime})}+\norm{Lu}_{L^{2}(\Omega^{\prime})}\big{)}\,,$
(C.15)

with $C_{1}^{\prime}>0$ independent of $u$. Moreover, since
$\Omega_{1}\Subset\Omega^{\prime}\Subset\Omega$ were arbitrary, $u\in
W^{1}_{\text{loc}}(\Omega)$.

To go from $W^{1}$ to $W^{2}$, note that (the now established) $u\in
W^{1}_{\text{loc}}(\Omega)\implies Lv\in L^{2}_{\text{loc}}(\Omega)$ above.
Then Lemmas C.4 and C.3 in combination yield

$\displaystyle\limsup_{\varepsilon\to
0^{+}}\norm{v\ast\varphi_{\varepsilon}}_{W^{2}(\Omega^{\prime})}\leq
C_{2}\norm{Lv}_{L^{2}(\Omega^{\prime})}\,,$ (C.16)

and hence $v\in W^{2}(\Omega^{\prime})$ with the bound
$\norm{v}_{W^{2}(\Omega^{\prime})}\leq
C_{2}\norm{Lv}_{L^{2}(\Omega^{\prime})}$. Then

$\displaystyle\norm{u}_{W^{2}(\Omega_{0})}$ $\displaystyle\leq
C_{2}\norm{Lv}_{L^{2}(\Omega^{\prime})}\leq
C_{2}^{\prime}\Big{(}\norm{u}_{L^{2}(\Omega^{\prime})}+\norm{Lu}_{L^{2}(\Omega^{\prime})}+\norm{u}_{W^{1}(\Omega^{\prime})}\Big{)}$
$\displaystyle\leq
C_{2}^{\prime\prime}\Big{(}\norm{u}_{L^{2}(\Omega^{\prime})}+\norm{Lu}_{L^{2}(\Omega^{\prime})}\Big{)}$
(C.17)

with the final inequality from the bound (C.15) applied to
$\norm{u}_{W^{1}(\Omega^{\prime})}$. Thus we conclude that $u,\,\bar{L}u\in
L^{2}_{\rm loc}(\Omega)$ implies $u\in W^{2}_{\rm loc}(\Omega)$. Moreover,
since $\bar{L}=bL$, there is $M_{1}>0$ (depending on $\bar{L}$ and the open
sets $\Omega_{0},\,\Omega_{1}$, but independent of $u$) such that

$\displaystyle\norm{u}_{W^{2}(\Omega_{0})}\leq
M_{k}\sum_{j=0}^{1}\norm{\bar{L}^{j}u}_{L^{2}(\Omega_{1})}\,,$ (C.18)

establishing the base case.

The inductive step $k$ to $k+1$: Fixing an arbitrary $k\geq 1$, assume that
the result holds for $k$. Next, assume that

$\displaystyle u,\,\bar{L}u,\ldots,\bar{L}^{k}u,\,\bar{L}^{k+1}u\in
L^{2}_{\text{loc}}(\Omega)\,.$ (C.19)

Then $u,\,\bar{L}u,\ldots,\bar{L}^{k}u\in L^{2}_{\text{loc}}(\Omega)$ and
$\bar{L}u,\ldots,\bar{L}^{k}(\bar{L}u)\in L^{2}_{\text{loc}}(\Omega)$, which
together with the inductive hypothesis implies $u,\,\bar{L}u\in
W^{2k}_{\text{loc}}(\Omega)$ along with the requisite norm bounds (c.f.
(C.5)).

It then follows that, upon fixing an arbitrary multi-index $\alpha$ with
$|\alpha|=2k$, we have ${\partial}^{\alpha}u\in L^{2}_{\text{loc}}(\Omega)$
and using $\bar{L}=bL$,121212Clearly the first term is in $L^{2}_{\rm
loc}(\Omega)$, and since $|\beta_{2}|\leq 2k-1$, $u\in W^{2k}_{\rm
loc}(\Omega)$, the second term is an element of $W^{-1}_{\text{loc}}(\Omega)$.

$\displaystyle
L({\partial}^{\alpha}u)=b^{-1}{\partial}^{\alpha}(\bar{L}u)-\sum_{|\beta_{1}|+|\beta_{2}|=|\alpha|\atop|\beta_{1}|>0}b^{-1}{\partial}^{\beta_{1}}b\,{\partial}^{\beta_{2}}Lu\in
W^{-1}_{\text{loc}}(\Omega)\,.$ (C.20)

Applying the argument in the proof of the base case above to
${\partial}^{\alpha}u$, one concludes that ${\partial}^{\alpha}u\in W^{2}_{\rm
loc}(\Omega)$. Moreover, since $\alpha$ is an arbitrary multi-index with
$|\alpha|=2k$, it follows that $u\in W^{2k+2}_{\text{loc}}(\Omega)$.

Turning now to the bound, having chosen open sets
$\Omega_{0}\Subset\Omega_{1}\Subset\Omega$, choose two further open sets
$\Omega^{\prime},\,\Omega^{\prime\prime}$ satisfying
$\Omega_{0}\Subset\Omega^{\prime}\Subset\Omega^{\prime\prime}\Subset\Omega_{1}$.
The $k=1$ bound (C.18) applied to ${\partial}^{\alpha}u$ on
$\Omega_{0}\Subset\Omega^{\prime}\Subset\Omega$ is

$\displaystyle\norm{{\partial}^{\alpha}u}_{W^{2}(\Omega_{0})}\leq
M_{1}\Big{(}\norm{{\partial}^{\alpha}u}_{L^{2}(\Omega^{\prime})}+\norm{\bar{L}({\partial}^{\alpha}u)}_{L^{2}(\Omega^{\prime})}\Big{)}\,.$
(C.21)

Next, it follows from (C.20) that

$\displaystyle\norm{\bar{L}({\partial}^{\alpha}u)}_{L^{2}(\Omega^{\prime})}$
$\displaystyle\leq
C_{1}^{\prime}\Big{(}\norm{\bar{L}u}_{W^{2k}(\Omega^{\prime})}+\sum_{|\beta|\leq
2k}\norm{{\partial}^{\beta}u}_{W^{1}(\Omega^{\prime})}\Big{)}$
$\displaystyle\leq
C_{1}^{\prime\prime}\Big{(}\norm{u}_{W^{2k}(\Omega^{\prime\prime})}+\norm{\bar{L}u}_{W^{2k}(\Omega^{\prime\prime})}\Big{)}\,,$
(C.22)

where to arrive at the second inequality one applies the $W^{1}$-bound (C.15)
to ${\partial}^{\beta}u$ on
$\Omega^{\prime}\Subset\Omega^{\prime\prime}\Subset\Omega$.

Finally, applying the inductive hypothesis to $u,\,\bar{L}u\in W^{2k}_{\rm
loc}(\Omega)$ on $\Omega^{\prime\prime}\Subset\Omega_{1}\Subset\Omega$ yields

$\displaystyle\norm{u}_{W^{2k}(\Omega^{\prime\prime})}\leq\tilde{C}\sum_{j=0}^{k}\norm{\bar{L}^{j}u}_{L^{2}(\Omega_{1})}\,,\quad\norm{Lu}_{W^{2k}(\Omega^{\prime\prime})}\leq\tilde{C}\sum_{j=0}^{k}\norm{\bar{L}^{j+1}u}_{L^{2}(\Omega_{1})}$
(C.23)

with $\tilde{C}>0$ independent of $u$. This, in combination with (C.21) and
(C) implies the existence of $M_{k+1}>0$ such that

$\norm{u}_{W^{2k+2}(\Omega_{0})}\leq
M_{k+1}\sum_{j=0}^{k+1}\norm{\bar{L}^{j}u}_{L^{2}(\Omega_{1})}\,,$ (C.24)

completing the inductive step, and hence the proof of the theorem. ∎

Proof of Theorem C.2. Assume that there exists $m\in{\mathbb{N}}$ such that
$k>m/2+n/4$. It follows straightforwardly from Corollary C.1.1 that for each
$j\in{\mathbb{N}}_{0}$ and $\zeta\in B_{S}(0)$, $f_{j},\,F(s)$ have
$C^{m}(\Omega)$-representatives, to be denoted $f_{j}(\cdot)$ and
$F(\zeta,\cdot)$ respectively. Moreover, it follows from the convergence of
the series in Theorem C.2 that for each $\zeta\in B_{S}(0)$ the power series

$F(\zeta,\cdot)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}f_{j}(\cdot)$ (C.25)

converges in $C^{m}(\Omega_{0})$ for every open set
$\Omega_{0}\Subset\Omega$.131313Recall that the
$\norm{\cdot}_{C^{m}(\Omega_{0})}$-norm is defined by
$\norm{h}_{C^{m}(\Omega_{0})}:=\sup_{\alpha\leq
m}\sup_{\Omega_{0}}|{\partial}^{\alpha}h|$. The uniformity of this convergence
for each $\zeta\in B_{S}(0)$, together with the basic properties of the
convergence of power series, implies that $F(\cdot,\cdot)$ is jointly $C^{m}$
in $(\zeta,x)$. The detailed argument is by induction.

Base case (jointly $C^{1}$): Fix an arbitrary open set
$\Omega_{0}\Subset\Omega$ and let $\zeta\in B_{S}(0)$ and $\delta\zeta\in
B_{r}(\zeta)\subseteq B_{S}(0)$. Since the $C^{m}(\Omega_{0})$ convergence of
the power series (C.25) holds for any $\zeta\in B_{S}(0)$, the sum
$\sum_{j\geq
0}\frac{|\zeta+\delta\zeta|^{j}}{j!}\norm{f_{j}}_{C^{1}(\Omega_{0})}$ is
uniformly bounded from above for $\delta\zeta\in B_{r}(\zeta)$. Then

$\displaystyle\lim_{\delta\zeta\to
0}F(\zeta+\delta\zeta,\cdot)=\lim_{\delta\zeta\to
0}\sum_{j=0}^{\infty}\frac{(\zeta+\delta\zeta)^{j}}{j!}f_{j}(\cdot)=F(\zeta,\cdot)\,,$
(C.26)

with the limit taken with respect to $\norm{\cdot}_{C^{m}(\Omega_{0})}$, and
the above uniform upper bound allows the limit to be taken inside the sum.
This continuity in $\zeta$, uniformly over $x\in\Omega$, implies joint
continuity in $\zeta\textit{ and }x$.

Having established joint continuity, consider the following series bounding
the difference quotient,

$\displaystyle\sum_{j=1}^{\infty}\frac{1}{j!}\bigg{|}\frac{(\zeta+\delta\zeta)^{j}-\zeta^{j}}{\delta\zeta}\bigg{|}\,\norm{f}_{C^{m}(\Omega_{0})}\leq\sum_{j=1}^{\infty}\frac{r^{j-1}}{(j-1)!}\norm{f}_{C^{m}(\Omega_{0})}<\infty$
(C.27)

with finiteness following from the $C^{m}(\Omega_{0})$ convergence of (C.25)
over $B_{S}(0)$. Analogous to the argument for continuity, this implies that
the difference quotient power series converges to
${\partial}_{\zeta}F(\zeta,\cdot)$ in $C^{m}(\Omega_{0})$ as $\delta\zeta\to
0$. In particular, it yields the power series

$\displaystyle{\partial}_{\zeta}F(\zeta,\cdot)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}f_{j+1}(\cdot)\,,$
(C.28)

converging in $C^{m}(\Omega_{0})$ for $\zeta\in B_{S}(0)$. Applying the above
continuity arguments to this power series implies that $F(\cdot,\cdot)$ is
jointly $C^{1}$, and for indices $i+|\alpha|=1$,

$\displaystyle{\partial}_{\zeta}^{i}{\partial}_{x}^{\alpha}F(\zeta,\cdot)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}{\partial}_{x}^{\alpha}f_{j+1}(\cdot)\,,\quad\zeta\in
B_{S}(0)\,,$ (C.29)

converging in $C^{m-|\alpha|}(\Omega_{0})$.

The inductive step: For $1\leq l<m$, assume that $F(\cdot,\cdot)$ is jointly
$C^{l}$ on $B_{S}(0)\times\Omega$, and that for indices $i+|\alpha|\leq l$ the
series (C.29) converges in $C^{m-|\alpha|}(\Omega_{0})$ for any open set
$\Omega_{0}\Subset\Omega$. To proceed, fix such an open set $\Omega_{0}$. It
is to be shown that for indices $i+|\alpha|=l+1$, the power series expansion
in (C.29) converges in $C^{m-|\alpha|}(\Omega_{0})$. When $i\leq l$, this
holds as a direct consequence of the induction hypothesis. Thus, w.l.o.g.
assume that $i=l+1$, and consider

$\displaystyle{\partial}_{\zeta}^{l}F(\zeta,\cdot)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}f_{j+l}(\cdot)\,.$
(C.30)

By the inductive hypothesis, this converges in $C^{m}(\Omega_{0})$ for each
$\zeta\in B_{S}(0)$. Applying the arguments of the base case above to (C.30)
then implies ${\partial}_{\zeta}^{l}F(\cdot,\cdot)$ is jointly $C^{1}$, and in
particular

${\partial}_{\zeta}^{l+1}F(\zeta,\cdot)=\sum_{j=0}^{\infty}\frac{\zeta^{j}}{j!}f_{j+l+1}(\cdot)\,,\quad\zeta\in
B_{S}(0)\,,$ (C.31)

converging in $C^{m}(\Omega_{0})$ as desired. Thus, $F(\cdot,\cdot)$ is
jointly $C^{l+1}$ in $(\zeta,x)\in B_{S}(0)\times\Omega$, completing the
inductive step, and thereby the proof. ∎

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