To that end we need to understand the geometric nature of the term $b^j \partial_{x^j}$ appearing in $P$, and also to interpret geometrically the first transport equation on $u_0, u_1$. Recall that the operator $P$ is defined in any coordinate system as (see [72, p. 270])

$$Pu = |g|^{-1/2} \partial_{x^j} (|g|^{1/2} g^{jk} \partial_{x^k} u)$$

for all $u \in C^\infty(M)$, where we sum over repeated indices.<fnref n="13" /> Therefore $P = \partial_{x^j} g^{jk} \partial_{x^k} + b^k(x) \partial_{x^k}$ where by [72, p. 270], $b^k(x) = |g(x)|^{-1/2} g^{jk}(x) (\partial_{x^j} |g(x)|^{1/2})$. This leads us to the identity

$$P = \partial_{x^k} g^{kj}(x) \partial_{x^j} + g^{jk}(x) (\partial_{x^j} \log |g(x)|^{1/2}) \partial_{x^k}$$

which holds true in normal coordinates centered at an arbitrary point $x_0$. These formulae are completely analogous to the well-known ones for the Laplace-Beltrami operator on Riemannian manifolds [112, pp. 41-42].

Recall that when we introduced the transport equations to study the parametrix, in Lemma 5.7 there was a function $h$ defined in normal coordinates as $h(x) = b^j(x) \eta_{jk} x^k$. It can be written as

$$h(x) = b^j \eta_{jk} x^k = g^{lj}(x) (\partial_{x^l} \log |g|^{1/2}) \eta_{jk} x^k = x^k \partial_{x^k} \log |g|^{1/2} = \rho \log |g|^{1/2},$$

where $\rho$ is the Euler vector field induced by the pseudo-Riemannian metric. The first transport equation $2\rho u_0 + hu_0 = 0$ now reads [112, (2.4.18), p. 43]

$$2\rho u_0 = -\rho \log |g|^{1/2} u_0, \quad u_0(0) = 1,$$

hence $u_0(x) = |g(0)|^{1/4} |g(x)|^{-1/4}$. The second transport equation is

$$\rho u_1 + u_1 + \frac{h}{2} u_1 = -P u_0.$$

Since both $\rho u_1$ and $h = \rho \log |g|^{1/2}$ vanish at the origin, this implies that

$$u_1(0) = -P u_0(0) = -P(|g(0)|^{1/4} |g(x)|^{-1/4})|_{x=0}.$$

Now in normal coordinates $|g(0)|^{1/4} = 1$ and from the Taylor expansion of the metric in normal coordinates [1, (5.2), p. 82], [9, Proposition 1.28, p. 37],

$$g_{ij}(x) = \eta_{ij} + \frac{1}{3} R_{ikjl} x^k x^l + \mathcal{O}(|x|^3).$$

Therefore [1, p. 84] we get the Taylor expansion of $|g(x)|^{-1/4}$ in normal coordinates:

$$\begin{aligned}
|g(x)|^{-1/4} &= \left||\eta| \exp\left(\text{Tr} \log\left(\delta_{ij} + \eta_i^{i_1} \frac{1}{3} R_{i_1 k j l}(0) x^k x^l + \mathcal{O}(|x|^3)\right)\right)\right|^{-1/4} \\
&= \left(1 + \frac{1}{3} \text{Tr}(\eta_i^{i_1} R_{i_1 k j l}(0) x^k x^l)\right)^{-1/4} + \mathcal{O}(|x|^3) \\
&= 1 + \frac{1}{12} \mathbf{Ric}_{kl}(0) x^k x^l + \mathcal{O}(|x|^3),
\end{aligned}$$

<footnote n="13">¹³Our convention follows Hörmander [72, p. 270].</footnote>
since Tr($\eta_i^{ij} R_{i_1 k j l}(0) x^k x^l$) = $\delta^{ij} \eta_i^{ij} R_{i_1 k j l}(0) x^k x^l = \eta^{ij} R_{i k j l}(0) x^k x^l = -\mathbf{Ric}_{kl}(0) x^k x^l$
where $\mathbf{Ric}_{kl}$ is the Ricci tensor. This implies that

$$ -P|g(x)|^{-1/4} = -\frac{1}{6} g^{kl} \underbrace{\mathbf{Ric}_{kl}(0)}_{=R_g(0)} + \mathcal{O}(|x|), $$

where we recognize $R_g(0) = g^{kl} \mathbf{Ric}_{kl}$ to be the scalar curvature. Finally, $u_1(x, x) = -R_g(x)/6$.

We are done with extracting the scalar curvature from the coefficient $u_1(0)$ of the transport equations. We have thus deduced the following result.

**Proposition 8.5.** As a particular case of Theorem 8.3, if in addition the dimension of M is $n \ge 4$ then

$$ \lim_{\epsilon \to 0^+} \operatorname{res}_{\alpha=n/2-1} (P \pm i\epsilon)^{-\alpha} (x, x) = \pm \frac{i R_g(x)}{6(4\pi)^{n/2}(n/2-2)!}, $$

where $R_g(x)$ is the scalar curvature at $x$.

Put together with Theorem 8.3, this proves our main result stated in the introduction, i.e. Theorem 1.1.

# Appendix A. Propagation estimates

## A.1. Summary

The purpose of this appendix is to supplement the material in Sections 2.1–2.4 with a very brief summary on scattering calculus and propagation estimates.

Propagation estimates in the scattering setting are due to Melrose [87]. The generalization to variable weight orders presented here is due to Vasy [130, 131]; see [50, Sections 2–3] for a concise introduction, cf. [41, Appendix E.4]. The scattering calculus in the model case $\mathbb{R}^n$ was earlier developed among others by Shubin [109] and Parenti [94].

## A.2. Scattering calculus

We use the notation already introduced in Sections 2.1–2.4; recall in particular that $\rho$ is a boundary-defining function and $y$ are local coordinates on $\partial \bar{M}$, extended onto a collar neighborhood of $\partial \bar{M}$. Let $(\rho, y, \varrho, \eta)$ be local coordinates on $\mathring{sc}T^*\bar{M}$ such that $(\varrho, \eta)$ are the dual coordinates of $(\rho, y)$. Recall that we introduced the formal notation $(\xi)^{-1}$ for the boundary-defining function of fiber infinity.

The class of scattering symbols of order $s, \ell \in \mathbb{R}$, denoted by $S_{sc}^{s, \ell}(T^*M)$, is defined away from $\partial \bar{M}$ in the same way as the usual symbol class $S^s(T^*M)$, whereas near the boundary, any $a \in S_{sc}^{s, \ell}(T^*M)$ is a smooth section of $T^*M$ that satisfies the estimate

$$ \forall j, k \in \mathbb{N}, \alpha, \beta \in \mathbb{N}^{n-1}, \quad |(\rho \partial_{\rho})^j \partial_y^{\alpha} \partial_{\varrho}^k \partial_{\eta}^{\beta} a(\rho, y, \varrho, \eta)| \le C_{jk\alpha\beta} \rho^{-\ell} (\xi^{s-k-|\beta|}). \quad (A.1) $$
The model example is as always $\overline{\mathbb{R}^n}$ with standard coordinates $(x, \xi)$ on $T^*\mathbb{R}^n$. In this case, by using spherical coordinates $x = (r, y)$ and setting $\rho = r^{-1}$, $\xi = (\rho, \rho^{-1}\eta)$, $\langle x \rangle = (1 + |x|^2)^{1/2}$ and $\langle \xi \rangle = (1 + |\xi|^2)^{1/2}$, one finds that (A.1) is equivalent to

$$ \forall \alpha, \beta \in \mathbb{N}^n, \quad |\partial_x^\alpha \partial_\xi^\beta a(x, \xi)| \le C_{\alpha\beta} \langle x \rangle^{\ell-|\alpha|} \langle \xi \rangle^{s-|\beta|}. $$

The class $\Psi_{sc}^{s,\ell}(M)$ of scattering pseudo-differential operators is obtained from sc-symbols $a \in S_{sc}^{s,\ell}(T^*M)$ by reduction to quantization of symbols on $T^*\mathbb{R}^n$. This requires choosing a partition of unity $\{\psi_i\}_i$ subordinate to a finite chart covering of $M$ as well as suitable diffeomorphisms close to $\partial \overline{M}$ (one can show that in the end different choices give the same operator modulo an element of $\Psi_{sc}^{s-1,\ell-1}(M)$). One also includes in the definition of $\Psi_{sc}^{s,\ell}(M)$ a class of regularizing operators in the sense that their Schwartz kernels $K_A(x, x')$ are smooth and decrease rapidly (with all derivatives) as the distance between $x \in M$ and $x' \in M$ tends to infinity. We refer the reader to, e.g., [130] and [123, Section 2] for an introduction; cf. [87] for the original, more geometric description of the Schwartz kernels of scattering pseudo-differential operators.

In the sc-calculus, the *principal symbol* of $A \in \Psi_{sc}^{s,\ell}(M)$ is the equivalence class of the symbol of $A$ in $S_{sc}^{s,\ell}(T^*M)/S_{sc}^{s-1,\ell-1}(T^*M)$. It is often useful to consider the more narrow class of *classical pseudo-differential operators* which is obtained from *classical symbols*, i.e. from symbols of the form $a = \langle \xi \rangle^s \rho^{-\ell} \tilde{a}$ with $\tilde{a} \in C^\infty(\mathring{sc}T^*\overline{M})$. In the simplest case of $A \in \Psi_{sc}^{0,0}(M)$ classical, it is possible to identify the principal symbol with the restriction of $a \in C^\infty(\mathring{sc}T^*\overline{M})$ to $\partial\mathring{sc}T^*\overline{M}$. For classical $A \in \Psi_{sc}^{s,\ell}(M)$ of arbitrary order there is also a natural identification of the principal symbol with a function on $\partial\mathring{sc}T^*\overline{M}$; see for instance (2.4) for the explicit formula for the principal symbol $p_z$ of $\square_g - z$.

The *microsupport* WF'$_{sc}(A)$ of $A \in \Psi_{sc}^{s,\ell}(M)$ is the complement of the set of points $q \in \partial\mathring{sc}T^*\overline{M}$ such that the (full) symbol of $A$ coincides in a neighborhood of $A$ with a symbol in $S_{sc}^{-N,-L}(T^*M)$ for all $N, L \in \mathbb{R}$. If $A$ is classical, then its *elliptic set* is the complement Ell$_{sc}(A) = \partial\mathring{sc}T^*\overline{M} \setminus \Sigma_{sc}(A)$ of the *characteristic set* $\Sigma_{sc}(A)$, defined as the closure of the zero set of the principal symbol.

In the context of propagation estimates it is useful to allow for weight orders $\ell$ that vary on $\mathring{sc}T^*\overline{M}$. On top of the obvious modifications of the definitions of $S_{sc}^{s,\ell}(T^*M)$ and $\Psi_{sc}^{s,\ell}(M)$, the cost to pay is that one needs to slightly relax the decay stated in (A.1), and require instead that

$$ |(\rho \partial_{\rho})^j \partial_y^\alpha \partial_\rho^k \partial_\eta^\beta a(\rho, y, \rho, \eta)| \le C_{jka\beta} \rho^{-\ell-\delta(j+k+|\alpha|+|\beta|)} \langle \xi \rangle^{s-k-|\beta|}. \quad (A.2) $$

for some $\delta > 0$. This circumvents logarithmic losses one would otherwise have when differentiating $\ell$. Apart from that, the change of definition (A.2) has however no big practical significance and will be disregarded in the notation entirely.

Now, if $s \ge 0$ and $\ell \in C^\infty(\mathring{sc}T^*\overline{M})$, we define the *weighted Sobolev space of variable weight order* to be

$$ H_{sc}^{s,\ell}(M) = \{u \in L^2(M) \mid Au \in L^2(M)\}, $$
where $A \in \Psi_{sc}^{s,\ell}(M)$ is a classical elliptic operator (i.e., Ell$_{sc}$(A) = $\partial^{\mathrm{sc}}\overline{T}^*\overline{M}$) which can be chosen arbitrarily. One can fix in particular an invertible A, and the norm can then be defined as $\|u\|_{s,\ell} = \|Au\|$ (different choices of A give equivalent norms). This agrees with the definition for $s \in \mathbb{Z}_{\ge 0}$ given in the main part of the paper. For $s \le 0$, $H_{sc}^{s,\ell}(M)$ can be defined as the dual of $H_{sc}^{-s,-\ell}(M)$. Then, for all $s, \ell \in \mathbb{R}$ and any elliptic $A \in \Psi_{sc}^{s,\ell}(M)$,

$$H_{sc}^{s,\ell}(M) = \{u \in \bigcup_{s',\ell'} H_{sc}^{s',\ell'}(M) \mid Au \in L^2(M)\}.$$

### A.3. Propagation estimates

We now consider the setting of the wave or Klein-Gordon operator $P-z$ on non-trapping Lorentzian scattering spaces introduced in Sections 2.1–2.3. We review various microlocal estimates for $P-z$, following [130, 131], with a particular emphasis on the dependence on the complex parameter $z$, which we assume to vary in some set $Z \subset \mathbb{C}$.

Recall that with the notation from Section 2.3, the characteristic set of $P-z$ is $\Sigma_{sc}(P-z) = (\Sigma_0 \cap \partial^{\mathrm{sc}}\overline{T}^*M) \cup (\Sigma_z \cap \partial^{\mathrm{sc}}\overline{T}^*\overline{M})$.

We first state the analogue of Hörmander's propagation of singularities theorem in our setting. The fixed $z$ version is due to Melrose [87]; see [130, Theorem 5.4] and the remarks in [131] for the uniform version below including the Im $z$ term. As in the main part of the text, we write $q \sim q'$ if $q$ and $q'$ are connected by a bicharacteristic in $\Sigma_{sc}(P-z)$, and we denote the closed bicharacteristic segment from $q$ to $q'$ by $\gamma_q \sim q'$. The notation $q > q'$ means that $q \sim q'$ and $q$ comes after $q'$ along the flow.

**Proposition A.1 (Propagation of singularities).** Let $s \in \mathbb{R}$ and let $\ell \in C^{\infty}(\overline{\mathrm{sc}}T^*\overline{M})$ be non-decreasing along the Hamiltonian flow. Let $A_1, A_2, B \in \Psi_{sc}^{0,0}(M)$ be such that WF'$_{sc}$(A$_1$) $\subset$ Ell$_{sc}$(B) and for all $z \in Z$ the following control condition is satisfied:

$$\forall q \in \mathrm{WF}'_{sc}(A_1) \cap \Sigma_{sc}(P-z), \exists q' \in \mathrm{Ell}_{sc}(A_2) : q > q' \text{ and } \gamma_q \sim q' \subset \mathrm{Ell}_{sc}(B). \quad (A.3)$$

Suppose $A_2u \in H_{sc}^{s,\ell}(M)$ and $\{B(P-z)u\}_{z \in Z}$ is bounded in $H_{sc}^{s-1,\ell+1}(M)$. Then for all $u \in H_{sc}^{s,L}(M)$,

$$\|A_1 u\|_{s,\ell} + (\mathrm{Im} z)^{1/2} \|A_1 u\|_{s-1/2,\ell+1/2} \\ \le C (\|A_2 u\|_{s,\ell} + \|B(P-z)u\|_{s-1,\ell+1} + \|u\|_{s,L})$$

uniformly in $z \in Z \cap \{\mathrm{Im} z \ge 0\}$.

The control condition (A.3) means in particular that the knowledge about $u$ being in $H_{sc}^{s,\ell}(M)$ microlocally is propagated forward (from Ell$_{sc}$(A$_2$) to Ell$_{sc}$(A$_1$)), consistently with the sign of Im $z$.

Beside propagation of singularities one can also show a uniform version of the simpler elliptic estimate [130, Corollary 5.5].
Next, recall that in our setting, $L_+$ are the sinks and $L_-$ the sources. Below, $\ell \in C^\infty(\partial^{\mathrm{sc}}T^*M)$ and $\ell_\pm = \ell|_{L_\pm}$ as in the main part of the text.

We now state the radial estimates for $P-z$. The low decay radial estimate can be used to propagate decay properties of $u$ into $L_+$ from a punctured neighborhood $U \setminus U_1$. The higher decay radial estimate serves to gain decay properties in a neighborhood of $L_-$ provided it is already better than the threshold value $-1/2$.

We refer again to [87] and [130, Proposition 5.27] for the fixed $z$ version, and to [131] for the modifications in the proof needed to accomodate the Im $z$ term.

**Proposition A.2 (Low decay radial estimate [131, (5)]).** Let $s \in \mathbb{R}$ and assume that $\ell$ is non-decreasing along the Hamiltonian flow and $\ell_+ < -1/2$. Let $A, B, B_1 \in \Psi_{sc}^{0,0}(M)$ and let $U_1, U$ be open neighborhoods of $L_+$ in $\partial^{\mathrm{sc}}T^*M$ and assume $U_1 \subset \mathrm{Ell}_{sc}(A)$, $\mathrm{WF}'_{sc}(A) \subset \mathrm{Ell}_{sc}(B) \subset U$ and $\mathrm{WF}'_{sc}(B_1) \subset U \setminus U_1$. Assume that for all $z \in Z$ the following control condition is satisfied:

$$ \forall q \in \mathrm{WF}'_{sc}(A) \cap \Sigma_{sc}(P-z) \setminus L_+, \exists q' \in \mathrm{Ell}_{sc}(B_1) : q > q' \text{ and } \gamma_{q \sim q'} \subset \mathrm{Ell}_{sc}(B). $$

Suppose $B_1 u \in H_{sc}^{s,\ell}(M)$ and $\{B(P-z)u\}_{z \in Z}$ is bounded in $H_{sc}^{s-1,\ell+1}(M)$. Then for all $u \in H_{sc}^{s,L}(M)$,

$$ \|Au\|_{s,\ell} + (\mathrm{Im} z)^{1/2} \|Au\|_{s-1/2,\ell+1/2} \le C (\|B_1 u\|_{s,\ell} + \|B(P-z)u\|_{s-1,\ell+1} + \|u\|_{s,L}) \\ \text{uniformly in } z \in Z \cap \{\mathrm{Im} z \ge 0\}. $$

**Proposition A.3 (Higher decay radial estimate [131, (4)])**. Let $s \in \mathbb{R}$ and assume that $\ell$ is non-decreasing along the Hamiltonian flow and $\ell_- > -1/2$. Let $A, B \in \Psi_{sc}^{0,0}(M)$ and let $U$ be a sufficiently small open neighborhood of $L_-$ in $\partial^{\mathrm{sc}}T^*M$. Assume $L_- \subset \mathrm{Ell}_{sc}(A)$ and $\mathrm{WF}'_{sc}(A) \subset \mathrm{Ell}_{sc}(B) \subset U$. Suppose $\{B(P-z)u\}_{z \in Z}$ is bounded in $H_{sc}^{s-1,\ell+1}(M)$. Then for all $s' \in \mathbb{R}$, $\ell' \in ]-1/2, \ell]$ and $u \in H_{sc}^{s,L}(M)$ such that $Bu \in H_{sc}^{s',\ell'}(M)$,

$$ \|Au\|_{s,\ell} + (\mathrm{Im} z)^{1/2} \|Au\|_{s-1/2,\ell+1/2} \le C (\|Bu\|_{s',\ell'} + \|B(P-z)u\|_{s-1,\ell+1} + \|u\|_{s,L}) \\ \text{uniformly in } z \in Z \cap \{\mathrm{Im} z \ge 0\}. $$

## Appendix B. Complex powers via functional calculus

Suppose $P$ is a (possibly unbounded) self-adjoint operator acting in a Hilbert space $\mathcal{H}$.

If $\alpha \in \mathbb{C}$ and $\varepsilon > 0$, or if $\mathrm{Re}\,\alpha < 0$ and $\varepsilon \ge 0$, then the operator $(P - i\varepsilon)^{-\alpha}$ is well-defined by the Borel functional calculus for self-adjoint operators. In the particular case $\mathrm{Re}\,\alpha > 0$ and $\varepsilon > 0$, it satisfies

$$ (P - i\varepsilon)^{-\alpha} = \frac{e^{-i\pi\alpha/2}}{\Gamma(\alpha)} \int_{0}^{\infty} s^{\alpha-1} e^{-\varepsilon s} e^{iPs} ds. $$

in the sense of convergence of the integral in the strong operator topology.
From our point of view it is more useful to express $(P - i\epsilon)^{-\alpha}$ in terms of the resolvent $(P - z)^{-1}$ as a contour integral, the precise form of which we briefly recall. It is actually more instructive to work with $A = iP$ instead of $P$. The reason is that $A$ and $A + \epsilon$ are sectorial operators (of angle $\pi/2$), and therefore their complex powers are special cases of a large and systematically studied functional calculus based on contour integrals; see [60, Section 2] and references therein. By [60, Proposition 7.1.3], the Borel functional calculus definition of $(A + \epsilon)^{-\alpha}$ coincides with the sectorial calculus definition. This has the following immediate consequences.

First, let $\epsilon > 0$. If $\text{Re } \alpha > 0$, then consistency with the sectorial calculus implies the formula

$$ (A + \epsilon)^{-\alpha} = \frac{1}{2\pi i} \int_{\eta_{\delta}} z^{-\alpha} (z - (A + \epsilon))^{-1} dz, \qquad (B.4) $$

where $0 < \delta < \epsilon$ and $\eta_{\delta}$ is an arbitrary contour going from $\text{Im } z \gg 0$ to $\text{Im } z \ll 0$ of the form

$$ \eta_{\delta} = e^{i\theta}] + \infty, \delta] \cup \{\delta e^{i\omega} \mid -\theta < \omega < \theta\} \cup e^{-i\theta}[\delta, +\infty[ $$

for some $\theta \in ]\pi/2, \pi]$. More generally, for any $\alpha \in \mathbb{C}$,

$$ (A + \epsilon)^{-\alpha} = (1 + A)^N \frac{1}{2\pi i} \int_{\eta_{\delta}} \frac{z^{-\alpha}}{(1+z)^N} (z - (A + \epsilon))^{-1} dz, \qquad (B.5) $$

where $N \in \mathbb{N}_{\ge 0}$ is an arbitrary number such that $N > -\text{Re } \alpha$. Furthermore, $\text{Dom } A^N$ is a core for $(A + \epsilon)^{-\alpha}$ [60, Proposition 3.1.1]. Observe that if $\text{Re } \alpha < 1$ then it is not necessary to surround 0 in the integral, and so the contour $\eta_{\delta}$ in (B.5) can be replaced by

$$ \eta_0 = e^{i\theta}] + \infty, 0] \cup e^{-i\theta}[0, +\infty[. $$

For $\text{Re } \alpha < 0$, $(A + \epsilon)^{-\alpha}$ is in general not bounded, it is however still a closed operator, with domain independent of $\epsilon > 0$.

Let now $\epsilon = 0$. If $\text{Re } \alpha < 0$ then again

$$ A^{-\alpha} = (1 + A)^N \frac{1}{2\pi i} \int_{\eta_0} \frac{z^{-\alpha}}{(1+z)^N} (z - A)^{-1} dz, \qquad (B.6) $$

and the domain is $\text{Dom } A^{-\alpha} = \text{Dom } (A + \epsilon)^{-\alpha}$ for $\epsilon > 0$ arbitrary [60, Proposition 3.1.9]. Furthermore,

$$ A^{-\alpha} u = \lim_{\epsilon \to 0^+} (A + \epsilon)^{-\alpha} u, \quad u \in \text{Dom } A^{-\alpha}. $$

In the special situation $0 \notin \text{sp}(A)$, $A^{-\alpha}$ is well-defined for all $\alpha \in \mathbb{C}$, and

$$ A^{-\alpha} = (1 + A)^N \frac{1}{2\pi i} \int_{\eta_{\delta}} \frac{z^{-\alpha}}{(1+z)^N} (z - A)^{-1} dz $$

for all sufficiently small $\delta \ge 0$.

Using back the relation $A = iP$ and changing the integration variable $z \to i(z + \epsilon)$ one finds integrals with $(P - z)^{-1}$ over the contour $\gamma_{\epsilon}$ used in the main part of the text (see Section 2.6).
## Appendix C. The ultrastatic case

### C.1. Resolvent bounds and Feynman wavefront sets for ultrastatic spacetimes

Let $(Y, h)$ be a complete Riemannian manifold of dimension $n-1$ and let

$$M = \mathbb{R} \times Y, \quad g = dt^2 - h \qquad (C.7)$$

be the corresponding *ultrastatic* Lorentzian manifold of dimension $n$.

The wave operator is then $\Box_g = \partial_t^2 - \Delta_h$, where $\Delta_h$ is the Laplace-Beltrami operator on $(Y, h)$ (with the convention $-\Delta_h \ge 0$). As explained in [34], the essential self-adjointness of $\Box_g$ in that case can be shown using Nelson's commutator theorem.

For $s \in \mathbb{R}$ and $p > 1/2$, we recall the definition of weighted Sobolev spaces $L_t^{2,p} H^s(Y)$ on $M$:

$$\|u\|_{L_t^{2,p} H^s(Y)} = \left( \int_{\mathbb{R}} \langle t \rangle^{2p} \|u(t, \cdot)\|_{H^s(Y)}^2 \right)^{1/2},$$

where $H^s(Y) = \langle -\Delta_h \rangle^{-s} L^2(Y)$ is the usual Sobolev space on $Y$.

**Theorem C.1.** Let $(Y, h)$ be a complete Riemannian manifold, let $(M, g)$ be as in (C.7) and let $P = \Box_g$. Let $s \in \mathbb{R}$ and $p > 1/2$. Then for $z \in \{\text{Im } z \ge 0, |\text{Re } z| \ge \varepsilon > 0\}$, $P-z$ admits a Feynman inverse $(P-z)^{-1} : L_t^{2,-p} H_y^s(Y) \to L_t^{2,p} H_y^{s+1}(Y)$ which satisfies a bound of the form

$$\|(P-z)^{-1}u\|_{L_t^{2,-p} H_y^s} \le C \|u\|_{L_t^{2,p} H_y^{s-1}}. \qquad (C.8)$$

Furthermore, if $\text{Re } \alpha > 0$ and $\text{Im } z > 0$ then $(P-z)^{-\alpha} : H^s(M) \to H^s(M)$ is well-defined for all $s \in \mathbb{R}$.

In particular, the above bound (C.8) holds true when $\text{Im } z \to 0^+$, $\text{Re } z \ne 0$, which yields a limiting absorption principle for the Klein-Gordon resolvent $(\Box_g + m^2 - i0)^{-1}$ as also proved in [34].

*Proof of Theorem C.1.* The starting point is the well-known ansatz for $\text{Im } z \ge 0$:

$$((P-z)^{-1}u)(t, \cdot) = -1/2 \int_{\mathbb{R}} \frac{e^{-i|t-s|\sqrt{-\Delta_h-z}}}{\sqrt{-\Delta_h-z}} u(s, \cdot) ds.$$

We first prove a rough bound for low values of $\text{Im } z \ge 0$, $|\text{Re } z| \ge \varepsilon$:

$$\begin{align*}
& \|(P-z)^{-1}u\|_{L_t^{2,-p} H_y^s} \\
&= \frac{1}{4} \int_{\mathbb{R}} \langle t \rangle^{-2p} \left\| \int_{\mathbb{R}} \frac{e^{-i|t-s|\sqrt{-\Delta_h-z}}}{\sqrt{-\Delta_h-z}} u(s, \cdot) ds \right\|_{H_y^s(Y)}^2 dt \\
&\le \frac{1}{4} \int_{\mathbb{R}} \langle t \rangle^{-2p} dt \sup_t \left( \int_{\mathbb{R}} \langle s \rangle^{-p} \langle s \rangle^p \left\| \frac{e^{-i|s-t|\sqrt{-\Delta_h-z}}}{\sqrt{-\Delta_h-z}} u(s, \cdot) \right\|_{H_y^s(Y)}^2 \right) ds.
\end{align*}$$
Then by the Cauchy–Schwarz inequality in s,

$$
\begin{align*}
& \int_{\mathbb{R}} \langle s \rangle^{-p} \langle s \rangle^p \left\| \frac{e^{-i|s-t|\sqrt{-\Delta_h - z}}}{\sqrt{-\Delta_h - z}} u(s, \cdot) \right\|_{H_y^s(Y)} ds \\
&\le \left( \int_{\mathbb{R}} \langle s \rangle^{-2p} ds \right)^{1/2} \sup_t \left( \int_{\mathbb{R}} \langle s \rangle^{2p} \left\| \frac{e^{-i|s-t|\sqrt{-\Delta_h - z}}}{\sqrt{-\Delta_h - z}} u(s, \cdot) \right\|_{H_y^s(Y)}^2 ds \right)^{1/2} \\
&\le C \left( \int_{\mathbb{R}} \langle s \rangle^{2p} \|u(s, \cdot)\|_{H_y^{s-1}(\mathbb{R}^{n-1})}^2 ds \right)^{1/2} = C \|u\|_{L_t^{2,p} H_y^{s-1}}
\end{align*}
$$

using the fact that

$$
\frac{e^{-i|t-s|\sqrt{-\Delta_h - z}}}{\sqrt{-\Delta_h - z}} : H_y^s(Y) \to H_y^{s+1}(Y)
$$

is bounded for all $s \in \mathbb{R}$ uniformly in $\operatorname{Im} z \ge 0$, $\|\operatorname{Re} z\| \ge \varepsilon$. Finally, for small $\operatorname{Im} z$, we
get

$$
\|(P-z)^{-1}u\|_{L_t^{2,-p} H_y^s(\mathbb{R}^{n-1})}^2 \le C^2 \|u\|_{L_t^{2,p} H_y^{s-1}}^2,
$$

which shows that $(P-z)^{-1} : L_t^{2,p} H_y^s(Y) \to L_t^{2,-p} H_y^{s+1}(Y)$ is invertible on the half-plane $\operatorname{Im} z \ge 0$, $|\operatorname{Re} z| \ge \varepsilon$.

Next, we refine the above bounds for large $|z|$ along the contour $\gamma_\epsilon$ defined in Section 2.6 to get decay in $z$. We denote by $E(\lambda)d\lambda$ the projection-valued measure associated to the functional calculus of $-\Delta_h$, which is well-known to be self-adjoint by completeness of $Y$ [24, 114]. For $u \in C_c^\infty(M)$, we define $\hat{u} = \int_{\mathbb{R}} e^{-i\tau t} E_\lambda(u(t, \cdot)) dt$. Then we get

$$
\begin{align*}
\|(P-z)^{-1}u\|_{H^s(M)}^2 &= \int_{\mathbb{R}\times\mathbb{R}_{\ge 0}} \frac{(1+|\tau|^2+\lambda)^s \|\hat{u}(\tau,\lambda)\|_{L^2(Y)}^2}{|-\tau^2+\lambda-z|^2} d\tau d\lambda \\
&\lesssim \frac{1}{|\operatorname{Im}z|^2} \|u\|_{H^s(M)}^2.
\end{align*}
$$

For $\epsilon > 0$, this implies by a contour integration argument as in the proof of Lemma 4.7
that the complex powers $(P - i\epsilon)^{-\alpha}$, $\operatorname{Re}\alpha > 0$, are well-defined and can be represented
as

$$
(P - i\epsilon)^{-\alpha} u = \frac{1}{2\pi} \int_{\mathbb{R} \times \mathbb{R}_{\ge 0}} e^{i\tau t} (-\tau^2 + \lambda - i\epsilon)^{-\alpha} \hat{u}(\tau, \lambda) d\tau d\lambda \quad (C.9)
$$

for all $u \in H^s(M)$.

The bound on the wavefront set of $(P-z)^{-1}$ is an immediate consequence of the
explicit formula for the Feynman inverse and follows the discussion in Section 3.5. ■

C.2. Limiting absorption principle for Feynman powers

In this second part of Appendix C, for the sake of illustration we specialize to ultrastatic
Lorentzian manifolds $\mathbb{R} \times Y$ which are space-compact, i.e. $Y$ is compact Riemannian.
We give an elementary proof of the limiting absorption principle for Feynman powers,
by which we mean that the limit $\lim_{\epsilon \to 0^+} (P - i\epsilon)^{-\alpha} : C_c^\infty(M) \to D'(M)$ exists for $P = \square_g + m^2$ with $m > 0$. This amounts to showing that one can take the $\epsilon \to 0^+$ limit in (C.9), and then to finding suitable weighted Sobolev spaces on which the complex Feynman powers are well-defined.

**Definition C.2** (Weighted anisotropic Sobolev spaces). We define the following product-type weighted Sobolev norms on $M$ depending on three indices, $H^{(s,\ell),p}(M)$, where $s$ is the time regularity, $\ell$ is a weight on the time variable and $p$ is the space regularity:

$$ \|u\|_{H^{(s,\ell),p}(M)} = \left( \int_M |\langle D_t \rangle^s \langle t \rangle^\ell \langle -\Delta_h \rangle^{p/2} u|^2 d\text{vol}_g \right)^{1/2} . $$

An important property of these spaces is that Fourier transform in the time variable exchanges the first two indices, i.e. $u \in H^{(s,\ell),p}(M)$ implies $F_t(u) \in H^{(\ell,s),p}(M)$.

**Lemma C.3.** Let $m > 0$ and $\alpha \in \mathbb{C}$ with $\text{Re} \alpha > 0$. Then $(-\tau^2 + \lambda + m^2 - i0)^{-\alpha}_{\lambda \in \mathbb{R}_{\ge 0}}$ is a family of tempered distributions which satisfies

$$ \|(-\tau^2 + \lambda + m^2 - i0)^{-\alpha}\|_{H^{\ell-\text{Re}\alpha, -s_1}(\mathbb{R})} = \mathcal{O}(\lambda^{-\text{Re}\alpha/2}) \quad (C.10) $$

for all $\ell \in [0, 1/2[$, $s_1 > 1/2$, where $H^{\ell-\text{Re}\alpha, -s_1}(\mathbb{R})$ denotes the weighted Sobolev space $\langle t \rangle^{s_1} H^{\ell-\text{Re}\alpha}(\mathbb{R})$.

*Proof.* We cut the domain into three regions using a smooth partition of unity $1 = \chi_1 + \chi_2 + \chi_3$, where supp $\chi_3 \subset \{\tau \ge \delta > 0\}$, supp $\chi_1 \subset \{\tau \le -\delta < 0\}$ and supp $\chi_2 \subset \{|\tau| \le m/2\}$.

Observe that $\chi_2(-\tau^2 + \lambda + m^2 - i0)^{-\alpha}$ is smooth, compactly supported and uniformly bounded in the space of Schwartz functions of $\lambda \in \mathbb{R}_{\ge 0}$, so this term satisfies (C.10).

Let us examine the term $\chi_1(-\tau^2 + \lambda + m^2 - i0)^{-\alpha}$. On the support of $\chi_3$, $\tau + \sqrt{\lambda + m^2} \ge \delta > 0$ uniformly in $\lambda$, therefore in the factorization

$$ \chi_1(-\tau^2 + \lambda + m^2 - i0)^{-\alpha} = \chi_1(\tau)(\tau + \sqrt{\lambda + m^2})^{-\alpha}(-\tau + \sqrt{\lambda + m^2} - i0)^{-\alpha}, $$

the term $\chi_1(\tau)(\tau + \sqrt{\lambda + m^2})^{-\alpha}$ is $O_{C^\infty(\mathbb{R})}(\lambda^{-\text{Re}\alpha/2})$. By inverse Fourier transform in $\tau$, we get

$$ \mathcal{F}(-\tau + \sqrt{\lambda + m^2} - i0)^{-\alpha} = C e^{it\sqrt{\lambda + m^2}} 1_{\mathbb{R}_{\ge 0}}(t) t^{\alpha-1}, $$

where $C$ is some constant. Thus,

$$ \begin{aligned} |\langle \chi_1(-\tau + \sqrt{\lambda + m^2} - i0)^{-\alpha}, \psi \rangle| &= |C| \left| \int_{\mathbb{R}} e^{it\sqrt{\lambda + m^2}} 1_{\mathbb{R}_{\ge 0}}(t) t^{\alpha-1} \hat{\psi}(t) dt \right| \\ &\le |C| \int_0^\infty |t^{\alpha-1} \hat{\psi}(t)| dt \\ &\le |C| (\|\langle t \rangle^{\text{Re}\alpha-\ell} \hat{\psi} \|_{L^2(\mathbb{R})} + \|\hat{\psi} \|_{H^{s_1}(\mathbb{R})}) \\ &\lesssim \|\hat{\psi} \|_{H^{s_1, \text{Re}\alpha-\ell}(\mathbb{R})} = \|\psi\|_{H^{\text{Re}\alpha-\ell, s_1}(\mathbb{R})} \end{aligned} $$

if $\ell \in [0, 1/2[$, $\ell \le \text{Re} \alpha$ and $s_1 > 1/2$. By duality, $\chi_1(-\tau + \sqrt{\lambda + m^2} - i0)^{-\alpha}$ is bounded in the dual weighted Sobolev space $H^{\ell-\text{Re}\alpha, -s_1}(\mathbb{R})$.

The term with $\chi_3$ is treated in the same way.

<figure><img src="image_4.png" /></figure>
Our objective is to study the regularity of the distribution $\langle u_2, \lim_{\epsilon \to 0^+} (P - i\epsilon)^{-\alpha} u_1 \rangle$
where $P$ is the Klein–Gordon operator. By compactness of $Y$, one has a discrete spectral
resolution of the Laplacian $-\Delta_h : C^\infty(M) \to L^2(M)$; there is an orthonormal basis
$(e_\lambda)_{\lambda \in \text{sp}(-\Delta_h)}$ of $L^2(M)$, where $-\Delta_h e_\lambda = \lambda e_\lambda$. By functional calculus, for any $f \in$
$L^\infty(\mathbb{R}_{\ge 0})$, we have

$$
f(-\Delta_h)u = \sum_{\lambda \in \text{sp}(-\Delta_h)} f(\lambda) \langle u, e_\lambda \rangle e_\lambda,
$$

where $f(-\Delta_h) : L^2(Y) \to L^2(Y)$ acts as a bounded operator.

For all test functions $u \in C_c^\infty(M)$, we define

$$
\hat{u}(\tau, \lambda) = \int_{\mathbb{R}} e^{-i\tau\tau} \langle (u(t, \cdot), e_{\lambda}) \rangle dt \in S(\mathbb{R}).
$$

By Fourier transform we are reduced to studying the pairing

$$
\langle u_2, \lim_{\epsilon \to 0^+} (P - i\epsilon)^{-\alpha} u_1 \rangle = \sum_{\lambda \in \text{sp}(-\Delta_h)} \int_{\mathbb{R}} \frac{\overline{\hat{u}_2}(\tau, \lambda) \hat{u}_1(\tau, \lambda)}{(-\tau^2 + \lambda + m^2 - i0)^\alpha} d\tau
$$

for all test functions $u_1$ and $u_2$. The functions $\hat{u}_1, \hat{u}_2$ are Schwartz in $\tau$, more precisely

$$
\sum_{\lambda \in \text{sp}(-\Delta_h)} \int_{\mathbb{R}} (1 + \lambda + \tau^2)^N \|\langle \partial_{\tau} \rangle^N \hat{u}_i(\tau, \lambda)\|_{L^2(Y)}^2 d\tau < \infty
$$

for all $N$, hence the product $\hat{u}_2(\tau, \lambda)\hat{u}_1(\tau, \lambda)$ is Schwartz in $\tau$ with fast decay in $\lambda$, which implies the distributional pairing $\int_{\mathbb{R}} \frac{\overline{\hat{u}_2(\tau, \lambda)\hat{u}_1(\tau, \lambda)}}{(-\tau^2+\lambda+m^2-i0)^\alpha} d\tau$ is well-defined for all $\lambda \in \sigma(-\Delta_h)$.

Now using the crucial Lemma C.3, we get

$$
\begin{aligned}
\left|\left\langle u_2, \lim_{\epsilon \to 0^+} (P - i\epsilon)^{-\alpha} u_1 \right\rangle\right| &\le \sum_{\lambda \in \text{sp}(-\Delta_h)} \left|\left(\int_{\mathbb{R}} \frac{\overline{\hat{u}_2}(\tau, \lambda) \hat{u}_1(\tau, \lambda)}{(-\tau^2 + \lambda + m^2 - i0)^\alpha} d\tau\right)\right| \\
&\le C \sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^{-\text{Re}\,\alpha/2} \|\overline{\hat{u}_2}(\cdot, \lambda) \hat{u}_1(\cdot, \lambda)\|_{H^{\text{Re}\,\alpha, s_1}(\mathbb{R})}
\end{aligned}
$$

for some $s_1 > 1/2$. Now if $s_2 > 1/2$ and $s_2 \ge \text{Re } \alpha$, the Moser estimates yield

$$
\|uv\|_{H^{s_2}(\mathbb{R})} \lesssim \|v\|_{H^{s_2}(\mathbb{R})} \|v\|_{H^{s_2}(\mathbb{R})},
$$

or in the weighted version,

$$
\|uv\|_{H^{s_2, \cdot 1}(\mathbb{R})} \lesssim \|v\|_{H^{s_2, \cdot \ell_1}(\mathbb{R})} \|v\|_{H^{s_2, \cdot \ell_2}(\mathbb{R})}
$$

where $\ell_1 + \ell_2 = s_1 > 1/2$. This implies that for each $\lambda \in \text{sp}(-\Delta_h)$,

$$
\| \hat{u}_2(\cdot, \lambda) \hat{u}_1(\cdot, \lambda) \|_{H^{\text{Re}\alpha, s_1}(\mathbb{R})} \le C \| \hat{u}_1(\cdot, \lambda) \|_{H^{s_2, \ell_1}(\mathbb{R})} \| \hat{u}_2(\cdot, \lambda) \|_{H^{s_2, \ell_2}(\mathbb{R})}.
$$
Therefore, using the Cauchy–Schwarz inequality we obtain

$$
\begin{aligned}
& \left| \left\langle u_2, \lim_{\epsilon \to 0^+} (P - i\epsilon)^{-\alpha} u_1 \right\rangle \right| \\
& \le \sum_{\lambda \in \text{sp}(-\Delta_h)} C \langle \lambda \rangle^{-\text{Re}\alpha/2} \| \hat{u}_1(\cdot, \lambda) \|_{H_t^{s_2, \ell_1}(\mathbb{R})} \| \hat{u}_2(\cdot, \lambda) \|_{H_t^{s_2, \ell_2}(\mathbb{R})} \\
& \le C \left( \sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^{-2p_1} \| \hat{u}_1(\cdot, \lambda) \|_{H_t^{s_2, \ell_1}(\mathbb{R})}^2 \right)^{1/2} \\
& \quad \times \left( \sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^{-2p_2} \| \hat{u}_2(\cdot, \lambda) \|_{H_t^{s_2, \ell_2}(\mathbb{R})}^2 \right)^{1/2},
\end{aligned}
$$

where $p_1 + p_2 = \text{Re}\alpha/2$.

To estimate the r.h.s. we need the following simple result.

**Lemma C.4.** For all $u \in C_c^\infty(M)$,

$$ \sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^p \|u_\lambda\|_{H_t^{s, \ell}(\mathbb{R})}^2 = \|u\|_{H^{(s, \ell), p}(M)}^2, $$

where $\| \cdot \|_{H^{(s, \ell), p}(M)}$ is the product-type weighted norm from Definition C.2.

*Proof.* By definition of $\| \cdot \|_{H^{(s, \ell), p}(M)}$ and Fubini's theorem,

$$
\begin{aligned}
\sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^p \|u_\lambda\|_{H_t^{s, \ell}(\mathbb{R})}^2 &= \sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^p \int_{\mathbb{R}} |\langle D_t \rangle^s \langle t \rangle^\ell \langle u(t), e_\lambda \rangle|^2 dt \\
&= \int_{\mathbb{R}} \sum_{\lambda \in \text{sp}(-\Delta_h)} \langle \lambda \rangle^p |\langle \langle D_t \rangle^s \langle t \rangle^\ell u(t), e_\lambda \rangle|^2 dt \\
&= \int_{\mathbb{R}} \| (1 - \Delta_h)^{p/2} \langle D_t \rangle^s \langle t \rangle^\ell u \|_{L^2(Y)}^2 dt = \|u\|_{H^{(s, \ell), p}(M)}^2,
\end{aligned}
$$

where we have used functional calculus and the fact that the spectral projection commutes with operators depending only on the $t$ variable. ■

Therefore, we find that

$$ \left| \left\langle u_2, \lim_{\epsilon \to 0^+} (P - i\epsilon)^{-\alpha} u_1 \right\rangle \right| \le C \|u_1\|_{H^{(\ell_1, s_2), p_1}(M)} \|u_2\|_{H^{(\ell_2, s_2), p_2}(M)} $$

for all $s_2 > 1/2$, $s_2 \ge \text{Re}\alpha$, $\ell_1 + \ell_2 > 1/2$ and $p_1 + p_2 = \text{Re}\alpha/2$, which concludes the proof of the limiting absorption principle stated below.

**Theorem C.5.** Let $M = \mathbb{R} \times Y$ be an ultrastatic Lorentzian manifold such that $Y$ is compact, and let $P$ be the Klein–Gordon operator with $m > 0$. The complex Feynman power acts as a continuous map between weighted Sobolev spaces, namely, the weak operator limit

$$ (P - i0)^{-\alpha} : H^{(\ell_1, s_2), p}(M) \to H^{(-\ell_2, -s_2), p - \text{Re}\alpha/2}(M) $$

is well-defined and continuous for all $p \in \mathbb{R}$, $s_2 > 1/2$, $s_2 \ge \text{Re}\alpha$ and $\ell_1 + \ell_2 > 1/2$.
# Appendix D. Various auxiliary proofs

## D.1. A Wick rotation lemma

We state below a lemma used several times in the main part of the text. As in Section 4, $Q$ is the quadratic form $Q(\xi) = -\xi_0^2 + \sum_{i=1}^{n-1} \xi_i^2$ on $\mathbb{R}^n$.

**Lemma D.1.** Let $\alpha \in \mathbb{C}$. When $\theta \to -\pi/2$, the distribution $(e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \to (Q(\xi) - i0)^{-\alpha}$ in $D'_{\Gamma}(\mathbb{R}^n \setminus \{0\})$, $\Gamma = \{(\xi; \tau dQ(\xi)) | Q(\xi) = 0, \tau < 0\}$.

*Proof.* The proof follows closely the proof of [71, Theorem 3.1.15] for the convergence in $D'$. For the control of the wavefront set, the proof follows closely [71, Theorem 8.4.8], which gives a wavefront bound in the sense of quasi-analytic classes. ■

## D.2. Wavefront set of the pull-back $G^*F_\alpha(z, \cdot)$

We compute the wavefront set of the germ of distribution $G^*F_\alpha(z, \cdot)$ as stated in Lemma 5.2. Let us recall that $F_\alpha(z, \cdot)$ is the elementary family of distributions on $\mathbb{R}^n$ introduced in Section 4.1, and the pull-back by the submersion $G$ defined in Section 5.1.2 gives a distribution defined on a neighborhood $U$ of the diagonal $\Delta$ in $M \times M$.

*Proof of Lemma 5.2. Step 1.* An application of the pull-back theorem [71, Theorem 8.2.4] in our situation gives

$$ \mathrm{WF}(G^*F_\alpha) \subset \{(x_1, x_2; k \circ d_{x_1} G, k \circ d_{x_2} G) \mid (G(x_1, x_2), k) \in \mathrm{WF}(F_\alpha)\} \quad (\mathrm{D}.11) $$

We denote by $(x_1, x_2; \eta_1, \eta_2)$ an element of $T^*\mathcal{V} \subset T^*M^2$ and $(h^\mu; k_\mu)$ the coordinates in $T^*\mathbb{R}^n$. The pull-back with indices reads

$$ (x_1, x_2; k \circ d_{x_1} G, k \circ d_{x_2} G) = (x_1, x_2; k_\mu d_{x_1} G^\mu, k_\mu d_{x_2} G^\mu). $$

**Step 2.** We first compute $\mathrm{WF}(G^*F_\alpha)$ outside the set $\Delta = \{x_1 = x_2\}$. The condition $(G(x_1, x_2), k) \in \mathrm{WF}(F_\alpha)$ in (D.11) reads

$$ (G^\mu(x_1, x_2); k_\mu) = (G^\mu(x_1, x_2); \lambda\eta_{\mu\nu}G^\nu(x_1, x_2)), $$

where $\lambda < 0$. We obtain

$$ (x_1, x_2; \lambda k \circ d_{x_1} G, \lambda k \circ d_{x_2} G) = (x_1, x_2; \lambda G^\mu \eta_{\mu\nu} d_{x_1} G^{\nu_2}, \lambda G^\mu \eta_{\mu\nu} d_{x_2} G^{\nu_2}) $$

and also $G^\mu(x_1, x_2)\eta_{\mu\nu}G^\nu(x_1, x_2) = 0$. Now set $\Gamma(x_1, x_2) = G^\mu(x_1, x_2)\eta_{\mu\nu}G^\nu(x_1, x_2)$. The key observation is that $d_{x_1}\Gamma = 2G^\mu\eta_{\mu\nu}d_{x_1}G^\nu$ and $d_{x_2}\Gamma = 2G^\mu\eta_{\mu\nu}d_{x_2}G^\nu$, hence

$$ \mathrm{WF}(G^*F_\alpha) \subset \{(x_1, x_2; \lambda d_{x_1} \Gamma, \lambda d_{x_2} \Gamma) \mid \Gamma(x_1, x_2) = 0, \lambda \in \mathbb{R}_{<0}\}. $$

We first interpret the term

$$ \{(x_1, x_2; \lambda d_{x_1} \Gamma, \lambda d_{x_2} \Gamma) \mid \Gamma(x_1, x_2) = 0, \lambda < 0\} $$
appearing in the last formula as the subset of all elements in $T^*U$ of the conormal bundle of the conoid $\{\Gamma = 0\}$ such that elements of positive energy are propagated in the future and elements of negative energy are propagated in the past: this is exactly the *Feynman condition*. In fact, if we use the metric to lift the indices, $d_{x_1}\Gamma(e_\mu(x_1))\eta^{\mu\nu}e_\nu(x_1)$ and $d_{x_2}\Gamma(e_\mu(x_2))\eta^{\mu\nu}e_\nu(x_2)$ are the Euler vector fields $\nabla_1\Gamma, \nabla_2\Gamma$ defined by Hadamard. The vectors $\nabla_1\Gamma, -\nabla_2\Gamma$ are parallel along the null geodesic connecting $x_1$ and $x_2$ (which is easily checked using normal coordinates centered at $x_1$, proving $(d_{x_1}\Gamma, -d_{x_2}\Gamma)$ are in fact *coparallel* along this null geodesic. Denoting $q_1 = (x_1; \lambda d_{x_1}\Gamma)$ and $q_2 = (x_2; \lambda d_{x_2}\Gamma)$, the relation $\exp_{x_1}(\nabla_1\Gamma) = x_2$ implies that $\nabla_1\Gamma$ points to the future (resp. past) if and only if $q_1 > q_2$ (resp. $q_2 > q_1$), which implies the Feynman condition.

**Step 3 ("diagonal")**. For any function $G$ on $M^2$, we uniquely decompose the total differential into two parts as follows:

$$dG = d_{x_1}G + d_{x_2}G, \quad \text{where} \quad d_{x_1}G|_{\{0\} \times T_{x_2}M} = 0, d_{x_2}G|_{T_{x_1}M \times \{0\}} = 0.$$

Let $i$ be the diagonal inclusion map $i := M \ni x \mapsto (x, x) \in \Delta \subset M$. Then for $x \in M$, $G \circ i(x) = 0$ implies $d_x G \circ i = 0$, which is equivalent to $d_{x_1}G \circ di + d_{x_2}G \circ di = 0$. Since

$$d_{x_2}G^\mu(x, x) = d_{x_2}s_{x_1}^\mu(\exp_{x_1}^{-1}(x_2))|_{x_1=x_2=x} = s_{x_1}^\mu(d_{x_2}\exp_{x_1}^{-1}(x_2))|_{x_1=x_2=x} = s^\mu(x),$$
because $d_{x_2}\exp_{x_1}^{-1}(x_2)|_{x_1=x_2=x} = \text{id}_{T_{x_1}M \to T_{x_2}M} = e_\mu(x)s^\mu(x)$, we see that $d_{x_1}G^\mu(x, x) = -s^\mu(x)$ and

$$\{(x_1, x_2; k \circ d_{x_1}G, k \circ d_{x_2}G) \mid x_1 = x_2\} = \{(x, x; -k_\mu s^\mu(x), k_\mu s^\mu(x)) \mid x \in M\}.$$

This concludes the proof of Lemma 5.2. ■

### D.3. Hölder, scaling and Fourier decay

We now turn our attention to the proof of regularity estimates for $G^*F_\alpha(z, \cdot)$ which are uniform in $z$.

We first recall a position space definition of Hölder $C^s(\mathbb{R}^n)$ functions which coincides with the Fourier definition for non-integer $s > 0$. The equivalence is proved in [122, Proposition 8.1], [72, Proposition 8.6.1]. Let us recall a version adapted to our discussion.

**Lemma D.2.** Let $s \in \mathbb{R}$. Then $u \in C_{\text{loc}}^s(\mathbb{R}^n)$ iff for every test function $\chi \in C_c^\infty(\mathbb{R}^n)$,

$$|\hat{u}\chi(\xi)| \le C(1 + |\xi|)^{-s-n}.$$

As a consequence, we have a continuous injection $C_{\text{loc}}^s(\mathbb{R}^n) \hookrightarrow H_{\text{loc}}^{s+n/2-\varepsilon}(\mathbb{R}^n)$ for all $\varepsilon > 0$.

*Proof.* If $u \in C^s(\mathbb{R}^n)$ with $s > 0$, $k < s < k + 1$ then for any $x$, there exists a polynomial $P$ of degree $k$, which is nothing but the Taylor polynomial of $u$ at $x$, such that for all test
functions $\varphi \in C_c^\infty(\mathbb{R}^n)$ (one could also take $\varphi$ in the Schwartz class)

$$ \left| \int_{\mathbb{R}^n} (u-P)(\lambda(y-x)+x)\varphi(y) \, d^n y \right| \le C \lambda^s \|\varphi\|_{L^\infty}. $$

Now let $u \in C_{\text{loc}}^s(\mathbb{R}^n)$, hence we may multiply $u$ with some cut-off $\chi \in C_c^\infty(\mathbb{R}^n)$ so that $u\chi \in C^s$. In particular, choosing the function $\varphi(x) = e^{i\langle\xi,x\rangle}$ on the r.h.s. yields

$$ \sup_{0<\lambda\le 1} \lambda^{-s} \sup_{1\le|\xi|\le 2} |\langle (u\chi - P)(\lambda\cdot), e^{i\langle\xi,\cdot\rangle} \rangle| \le \|u\chi\|_{C^s} \sup_{1\le|\xi|\le 2} \|e^{i\langle\xi,\cdot\rangle}\|_{L^\infty} = \|u\chi\|_{C^s}. $$

Therefore, using the fact that $\langle (u\chi - P)(\lambda\cdot), e^{i\langle\xi,\cdot\rangle} \rangle = \langle u\chi(\lambda\cdot), e^{i\langle\xi,\cdot\rangle} \rangle$ since the Fourier transform restricted to $|\xi| \ge 1$ does not see the polynomial, and $\langle u\chi(\lambda\cdot), e^{i\langle\xi,\cdot\rangle} \rangle = \lambda^{-n}\widehat{u\chi}(\xi/\lambda)$, we get

$$ \sup_{0<\lambda\le 1} \lambda^{-s-n} \sup_{1\le|\xi|\le 2} |\widehat{u\chi}(\xi/\lambda)| \le \|u\chi\|_{C^s}. $$

Hence for $|\xi| \ge 1$, we get

$$ |\widehat{u\chi}(\xi)| = |\widehat{u\chi}(\xi|\xi|/|\xi|)| \le \|u\chi\|_{C^s} |\xi|^{-s-n}, $$

and finally this means that

$$ |\widehat{u\chi}(\xi)| \le C(1 + |\xi|)^{-s-n}. $$

Conversely, if we have the Fourier decay $|\widehat{u\chi}(\xi)| \le C(1 + |\xi|)^{-r}$ for $r \in \mathbb{R}_{\ge 0}$, then the Littlewood-Paley blocks are bounded by

$$ \begin{aligned} \|\psi(2^{-j}\sqrt{-\Delta})(u\chi)\|_{L^\infty} &= \|\mathcal{F}^{-1}(\psi(2^{-j}|\xi|)\widehat{u\chi}(\xi))\|_{L^\infty} \le \int_{\mathbb{R}^n} |\psi(2^{-j}|\xi|)\widehat{u\chi}(\xi)| \, d^n \xi \\ &\le 2^{jn} \int_{\mathbb{R}^n} |\psi(|\xi|)\widehat{u\chi}(2^{-j}\xi)| \, d^n \xi \le C 2^{jn} \int_{\mathbb{R}^n} \psi(|\xi|)(1 + 2^j|\xi|)^{-r} \, d^n \xi \\ &\le C 2^{j(n-r)} \int_{\mathbb{R}^n} \psi(|\xi|)(2^{-j} + |\xi|)^{-r} \, d^n \xi \le 2^{j(n-r)}. \end{aligned} $$

This means that $u \in C_{\text{loc}}^{r-n}(\mathbb{R}^n)$.

<figure><img src="image_5.png" /></figure>

Let $\alpha \in \mathbb{C}$ with $\text{Re}\,\alpha \ge 0$. We consider the Hölder regularity under pull-back of $G^*F_\alpha \in \mathcal{D}'(\mathcal{U})$ where $\mathcal{U} \subset M \times M$ is the neighborhood of the diagonal and $G: \mathcal{U} \ni (x, y) \mapsto G(x, y) \in \mathbb{R}^n$ is the $C^\infty$ submersive map defined by (5.2).

**Lemma D.3.** Let $k = \lfloor \text{Re}\,\alpha \rfloor + 1$ and $F_\alpha(z, \cdot) \in \mathcal{D}'(\mathbb{R}^n)$ as defined in (4.44). Let $G$ be the $C^\infty$ submersive map defined in (5.2). Then the pull-back $F_\alpha(z, \cdot) = G^*F_\alpha(z, \cdot)$ is in $C_{\text{loc}}^{(2-2a)(\text{Re}\,\alpha+1)-k-n}(\mathcal{U})$ with decay in $z$ of order $\mathcal{O}(|\text{Im}\,z|^{-a(\text{Re}\,\alpha+1)})$ for $a \in [0, 1]$.

*Proof.* Let $U$ be a small geodesically convex open subset in $M$. We choose a test function $\chi \in C_c^\infty(U)$ in such a way that, in the support of $\chi \otimes \chi$, we have a local diffeomorphism
$E: U \times \mathbb{R}^n \ni (x, h) \mapsto (x, \exp_x(h)) \in U \times U$. Then by definition of $G$ and of the exponential map, we have the identity

$$E^*(\chi \otimes \chi^* F_\alpha(z, \cdot))(x; h) = F_\alpha(z, |h|_g)\chi(x)\chi(\exp_x(h)) \in \mathcal{D}'(U \times \mathbb{R}^n).$$

Now observe that $F_\alpha(z, |h|_g) = \mathcal{O}_{cs}((1 + |\mathrm{Im} \, z|)^{-a(\mathrm{Re} \, \alpha + 1)})$ for $a \in [0, 1]$ and $s \le (2 - 2a)(\mathrm{Re} \, \alpha + 1) - k - n$, and that $\chi(x)\chi(\exp_x(h)) \in C_c^\infty(U \times \mathbb{R}^n)$, hence the result follows. ■

**Acknowledgments.** The authors are particularly grateful to Fabien Besnard, Christian Brouder, Jan Dereziński, Peter Hintz, Raphaël Ponge, Kouichi Taira, Gunther Uhlmann and András Vasy for helpful discussions. We thank the MSRI in Berkeley and the Mittag-Leffler Institute in Djursholm for their kind hospitality during thematic programs and workshops in 2019–20.

**Funding.** Support from the grant ANR-16-CE40-0012-01 is gratefully acknowledged. N.V.D. acknowledges the support of the Institut Universitaire de France.

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