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Limiting Absorption Principle and Equivalence of Feynman Propagators on Asymptotically Minkowski Spacetimes

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Abstract

In this paper, we show the limiting absorption principle for the wave operator on asymptotically Minkowski spacetimes. This problem was previously considered by Vasy (J Spectr Theory 10:439–461, 2020). Here, we employ Mourre theory which seems a more transparent tool. Moreover, we also prove that the anti-Feynman propagator defined by Gérard and Wrochna coincides with the outgoing resolvent.

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Acknowledgements

This work is supported by JSPS Research Fellowship for Young Scientists, KAKENHI Grant Number 20J00221. The author would like to thank Christian Gérard for suggesting the problem on the equivalence of the Feynman propagator and to Shu Nakamura for valuable discussions. Michal Wrochna explained the convention of the Feynman propagator in [10, 11], and in physics. The author is also very grateful to anonymous referees for numerous comments which are very helpful to improve the manuscript.

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Correspondence to Kouichi Taira.

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Appendix A: Propagation Estimates

Appendix A: Propagation Estimates

In this appendix, we give a proof for some propagation estimates, that is, the propagation of singularities and the radial estimates although these proofs may be well-known for specialists in geometric scattering theory (see [6, Appendix E.4]). The radial estimates can be regarded as a microlocal alternative of the Mourre theory. However, its proof does not require technical assumptions such as the self-adjointness of the operator and the radial estimates often give an additional information such as regularity of the function. We also mention that the abstract limiting absorption principle is proved by a similar method in [8].

First, we shall explain key dynamical properties of \(H_p\) for the propagation estimates. We set

$$\begin{aligned} \beta _0(x,\xi )=\cos (x,{\tilde{\xi }})=\frac{x\cdot \partial _{\xi }p_0(\xi )}{|x||\partial _{\xi }p_0(\xi )|},\quad \beta (x,\xi )=\frac{x\cdot \partial _{\xi }p(x,\xi )}{|x||\partial _{\xi }p(x,\xi )|}. \end{aligned}$$

A key property of the observable \(\beta _0\) is the following: Setting \(\beta _{\pm }=1\pm \beta _0\), we have

$$\begin{aligned}&|\partial _{\xi }p_0|^{-1}|x|H_{p_0}\beta _{\pm }=\pm (1-\beta _0^2)=\pm \beta _{{\mp }}\beta _{\pm }=\pm 2 \beta _{\pm } \,\, \text {on}\,\, L_{{\mp }}:=\{\beta _{\pm }=0\},\\&|\partial _{\xi }p_0|^{-1}|x|H_{p_0}|x|^{-1}=-\beta _0 |x|^{-1}=\pm |x|^{-1}\quad \text {on}\quad L_{{\mp }}. \end{aligned}$$

Then it turns out that the sets \(L_{{\mp }}=\{\beta _{\pm }=0\}\cap \{|x|=\infty , \xi \ne 0\}\) are attracting/repelling sets along the rescaled Hamiltonian \(|x|H_{p_0}\). The sets \(L_{{\mp }}\cap \{|x|=\infty ,\xi \ne 0\}\) are called the radial source/sink in [6, DEFINITION E.50] (see also [2, Definition 2.3]), which are also called the incoming/outgoing regions in scattering theory.

Similarly to Sect. 3, in this “Appendix”, let P be a self-adjoint operator on the standard \(L^2\)-space \(L^2({\mathbb {R}}^{n+1})\) satisfying

$$\begin{aligned} P=\mathrm {Op}(p)+\mathrm {Op}S^{1,-\mu }. \end{aligned}$$

Since the multiplication operator \(|g|^{\frac{1}{4}}\) preserves all spaces \(H^{k,l}({\mathbb {R}}^{n+1})\) with \(k,l\in {\mathbb {R}}\), all the results hold for P which is defined in (1.1).

1.1 A.1. Preliminary

Now we fix some notation. For symbols ab, we denote \(a\Subset b\) if we have

$$\begin{aligned} \inf _{(x,\xi )\in {\mathrm{supp}\;}a}|b(x,\xi )|>0, \end{aligned}$$

and we denote \(\mathrm {Op}(a)=:A\Subset B:=\mathrm {Op}(b)\) if \(a\Subset b\).

Definition 2

Let \(k,l\in {\mathbb {R}}\).

(i) We call \(a\in S^{k,l}\) (or its quantization \(\mathrm {Op}(a)\)) elliptic in a subset \(\Omega \subset {\mathbb {R}}^{2n+2}\) if there exists \(r\in S^{-\infty ,-\infty }\) such that

$$\begin{aligned} \inf _{(x,\xi )\in \Omega }\langle x \rangle ^{-l}\langle \xi \rangle ^{-k}|a(x,\xi )+r(x,\xi )|>0. \end{aligned}$$

(ii) We call \(a\in S^{k,l}\) (or its quantization \(\mathrm {Op}(a)\)) microlocally negligible outside a subset \(\Omega \subset {\mathbb {R}}^{2n+2}\) if there exists \(r\in S^{-\infty ,-\infty }\) such that

$$\begin{aligned} {\mathrm{supp}\;}(a+r)\subset \Omega . \end{aligned}$$

(iii) We call \(a\in S^{k,l}\) (or its quantization \(\mathrm {Op}(a)\)) microlocally contained in a pair \((\Omega _1,\Omega _2)\) if a is elliptic in \(\Omega _1\) and is microlocally negligible outside \(\Omega _2\).

(iv) Let \(X\subset {\mathcal {S}}'({\mathbb {R}}^{n+1})\) be a function space. For \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\), we say that \(u\in X\) microlocally in a subset \(\Omega \subset {\mathbb {R}}^{2n+2}\) if there exists \(A=\mathrm {Op}(a)\in \mathrm {Op}S^{0,0}\) which is elliptic on \(\Omega \) such that \(Au\in X\).

We set

$$\begin{aligned} \Omega _{\varepsilon ,r,R,in/out}&:= \{(x,\xi )\in {\mathbb {R}}^{2n+2}\mid |x|\ge R,\,\, |\xi |\ge r, \,\, \pm \beta (x,\xi )\le -1+\varepsilon \},\\ \Omega _{r,R}(s)&:= \{(x,\xi )\in {\mathbb {R}}^{2n+2}\mid |x|\ge R,\,\, |\xi |\ge r, \,\, \beta (x,\xi )=s\},\\ \Omega _{r,R}&:= \{(x,\xi )\in {\mathbb {R}}^{2n+2}\mid |x|\ge R,\,\, |\xi |\ge r\},\\&\Omega _{s_1,s_2,r,R,mid}=\cup _{s\in [s_1,s_2]}\Omega _{r,R}(s). \end{aligned}$$

1.2 Estimates for weight functions

For \(k,l\in {\mathbb {R}}\), \(N,\kappa >0\) and \(0\le \delta \le 1\), we set

$$\begin{aligned} \lambda =\lambda _{k,l,\kappa ,N,\delta }=\langle \xi \rangle ^{k-\frac{1}{2}}\langle \delta \xi \rangle ^{-|k|-N-1}\langle x \rangle ^{l+\frac{1}{2}}\langle \delta x \rangle ^{-\kappa }. \end{aligned}$$

Lemma A.1

(i) For \(r>0\), there exists \(C_1>0\) independent of \(0\le \delta \le 1\) such that

$$\begin{aligned} H_p\lambda ^2\le C_1\frac{\langle \xi \rangle }{\langle x \rangle }\lambda ^2\quad \text {for}\quad x\in {\mathbb {R}}^n\quad \text {and}\quad |\xi |\ge r. \end{aligned}$$

(ii) Suppose \(l>-\frac{1}{2}\) and \(2l+1-2\kappa >0\). For \(r>0\), \(\varepsilon \in (0,1)\) and \(R\ge 1\) large enough, there exists \(C_2>0\) independent of \(0\le \delta \le 1\) such that

$$\begin{aligned} H_p\lambda ^2\le -C_2\frac{\langle \xi \rangle }{\langle x \rangle }\lambda ^2\quad \text {for}\quad (x,\xi )\in \Omega _{\varepsilon ,r,R,in}. \end{aligned}$$

(iii) Suppose \(l<-\frac{1}{2}\). For \(r>0\), \(\varepsilon \in (0,1)\) and \(R\ge 1\) large enough, there exists \(C_3>0\) independent of \(0\le \delta \le 1\) such that

$$\begin{aligned} H_p\lambda ^2\le -C_3\frac{\langle \xi \rangle }{\langle x \rangle }\lambda ^2\quad \text {for}\quad (x,\xi )\in \Omega _{\varepsilon ,r,R,out}. \end{aligned}$$

Proof

First, we note \(\partial _{x}p\in S^{2,-1-\mu }\) and \(\partial _{\xi }p\in S^{1,0}\). (i) follows from a simple calculation. A simple calculation gives \(H_p|\xi |^2=O(|x|^{-1-\mu }|\xi |^3)\) and

$$\begin{aligned}&H_p\langle x \rangle ^{2l+1}=(2l+1)|x||\partial _{\xi }p(x,\xi )|\beta (x,\xi )\langle x \rangle ^{2l-1},\\&H_p\langle \delta x \rangle ^{-2\kappa }=-2\kappa \delta ^2|x||\partial _{\xi }p(x,\xi )|\beta (x,\xi )\langle \delta x \rangle ^{-2\kappa -2}. \end{aligned}$$

Moreover,

$$\begin{aligned} \left( \frac{2l+1}{\langle x \rangle ^2}-\frac{\kappa \delta ^2}{\langle \delta x \rangle ^2}\right)&=\frac{(2l+1)(1+\delta ^2|x|^2)-\kappa \delta ^2(1+|x|^2)}{\langle x \rangle ^2\langle \delta x \rangle ^2}\\&{\left\{ \begin{array}{ll} \ge c_1\langle x \rangle ^{-2}\quad \text {if}\quad l>-\frac{1}{2},\,\, 2l+1-2\kappa >0\\ \le -c_2\langle x \rangle ^{-2}\quad \text {if}\quad l<-\frac{1}{2}, \end{array}\right. } \end{aligned}$$

where \(c_1,c_2>0\) are independent of \(0<\delta \le 1\) and \(x\in {\mathbb {R}}^n\). Thus, for \(R>0\) large enough, we have

$$\begin{aligned} H_p\lambda ^2&=\langle \xi \rangle ^{2k-1}\langle \delta \xi \rangle ^{-2|k|-2N-2}H_p\left( \langle x \rangle ^{2l+1}\langle \delta x \rangle ^{-2\kappa }\right) +O\left( \frac{|\xi |}{\langle x \rangle ^{1+\mu }}\lambda ^2\right) \\&=\left( \frac{2l+1}{\langle x \rangle ^2}-\frac{\kappa \delta ^2}{\langle \delta x \rangle ^2}\right) |x||\partial _{\xi }p(x,\xi )|\beta (x,\xi )\lambda ^2 +O\left( \frac{|\xi |}{\langle x \rangle ^{1+\mu }}\lambda ^2\right) \le -C\frac{\langle \xi \rangle }{\langle x \rangle }\lambda ^2 \end{aligned}$$

if \(l>-\frac{1}{2}\), \(2l+1-2\kappa >0\) and \((x,\xi )\in \Omega _{\varepsilon ,r,R,in}\) or if \(l<-\frac{1}{2}\) and \((x,\xi )\in \Omega _{\varepsilon ,r,R,out}\). This proves (ii) and (iii). \(\square \)

1.3 Estimates for cut-off functions

First, we note that for \(0<\varepsilon _0<1\) and \(R\ge 1\) large enough, there exists \(C_4>0\) such that

$$\begin{aligned} H_p\beta \ge C_4|\xi |\langle x \rangle ^{-1}\quad \text {for}\quad |x|\ge R,\,\, \xi \ne 0,\,\, \beta (x,\xi )\in [-1+\varepsilon _0,1-\varepsilon _0] \end{aligned}$$
(A.1)

since \(H_p\beta =|x|^{-1}|\partial _{\xi }p(x,\xi )|(1-\beta ^2)+O(\langle x \rangle ^{-1-\mu }|\xi |)\).

Lemma A.2

(i) Let \(-1<\beta _1<\beta _2<1\). For each \(L>0\), \(\varepsilon >0\) small enough, \(R\ge 1\) large enough, there exist \(a,b_1,b_2\in S^{0,0}\), \(e\in S^{-\infty ,-1/2}\) supported in \(\{r/2\le |\xi |\le \frac{5r}{2}\}\) satisfying the following properties: The symbol a is microlocally contained in \((\Omega _{\beta _1-\varepsilon ,\beta _2+\varepsilon ,2r,2R,mid},\Omega _{\beta _1-2\varepsilon ,\beta _2+2\varepsilon ,r,R,mid})\), the symbols \(b_1,b_2\) are microlocally negligible outside \(\Omega _{\beta _1-2\varepsilon ,\beta _1-\varepsilon ,r,R,mid}\) and \(\Omega _{r,R}\) respectively. Moreover, we have

$$\begin{aligned} H_pa^2\le -L\frac{\langle \xi \rangle }{\langle x \rangle }a^2+\frac{\langle \xi \rangle }{\langle x \rangle }b_1^2+\frac{\langle \xi \rangle }{\langle x \rangle }b_2^2 +\frac{\langle \xi \rangle }{\langle x \rangle }e^2. \end{aligned}$$

In addition, if \(\beta _2<0\), we can take \(b_2=0\).

(ii) For \(0<\varepsilon <\frac{1}{2}\), \(r\ge 1\) and \(R\ge 1\) large enough, there exist \(a\in S^{0,0}\) which is microlocally contained in \((\Omega _{\varepsilon ,2r,2R,in},\Omega _{2\varepsilon ,r,R,in})\) and \(e\in S^{-\infty ,-1/2}\) supported in \(\{r/2\le |\xi |\le \frac{5r}{2}\}\) such that

$$\begin{aligned} H_pa^2\le \frac{\langle \xi \rangle }{\langle x \rangle }e^2. \end{aligned}$$

(iii) For \(0<\varepsilon <\frac{1}{4}\), \(r\ge 1\) and \(R\ge 1\), there exist \(a,b_1,b_2\in S^{0,0}\) and \(e\in S^{-\infty ,-1/2}\) supported in \(\{r/2\le |\xi |\le \frac{5r}{2}\}\) such that the following properties hold: The symbol a is microlocally contained in \((\Omega _{\varepsilon ,2r,2R,out},\Omega _{2\varepsilon ,r,R,out})\), \(b_1\) and \(b_2\) are microlocally negligible outside \(\Omega _{1-2\varepsilon ,1-\varepsilon ,r,R,mid}\) and \(\Omega _{r,R}\) respectively. Moreover, we have

$$\begin{aligned} H_pa^2\le \frac{\langle \xi \rangle }{\langle x \rangle }b_1^2+\frac{\langle \xi \rangle }{\langle x \rangle }b_2^2 +\frac{\langle \xi \rangle }{\langle x \rangle }e^2. \end{aligned}$$

Proof

Take \(\chi \in C^{\infty }({\mathbb {R}}; [0,1])\) such that

$$\begin{aligned} \chi (t)={\left\{ \begin{array}{ll} 1\quad \text {for}\,\, t\le 1,\\ 0\quad \text {for}\,\, t\ge 2, \end{array}\right. }\quad \chi '(t)\le 0. \end{aligned}$$

Moreover, we set \({\bar{\chi }}_R(x)=1-\chi (|x|/R)\) for \(R>0\).

(i) Take \(\varepsilon >0\) small enough and \(\rho _{mid}\in C^{\infty }({\mathbb {R}};[0,1])\) such that

$$\begin{aligned} \rho _{mid}(t)={\left\{ \begin{array}{ll} 1 \quad \text {for}\quad \beta _1-\varepsilon \le t\le \beta _2+\varepsilon \\ 0 \quad \text {for}\quad t\le \beta _1-2\varepsilon \,\,\text {or}\,\, \beta _2+2\varepsilon \le t \end{array}\right. }\,\, \rho _{mid}'(t)\le 0\,\,\text {for}\,\, \beta _2+\varepsilon \le t\le \beta _2+2\varepsilon . \end{aligned}$$

Moreover, take \(R\ge 1\) large enough such that \(H_p\beta \ge C_0|\xi |\langle x \rangle ^{-1}\) for \((x,\xi )\in {\mathrm{supp}\;}\rho _{mid}(\beta )\) with \(|x|\ge R\) and take \(M>0\) such that

$$\begin{aligned} H_pe^{-M\beta }(x,\xi )\le -L\frac{\langle \xi \rangle }{\langle x \rangle } e^{-M\beta }\quad \text {for}\quad (x,\xi )\in \Omega _{\beta _1-2\varepsilon ,\beta _2+2\varepsilon ,r,R,mid}. \end{aligned}$$

Now we set

$$\begin{aligned} a(x,\xi )=e^{-M\beta (x,\xi )}\rho _{mid}(\beta (x,\xi )){\overline{\chi }}_R(x){\overline{\chi }}_r(\xi )\in S^{0,0}. \end{aligned}$$

Then we have \({\mathrm{supp}\;}a\subset \Omega _{\beta _1-2\varepsilon ,\beta _2+2\varepsilon ,r,R,mid}\) and

$$\begin{aligned} H_pa^2\le -L\frac{\langle \xi \rangle }{\langle x \rangle }a^2+e^{-2M\beta }(x,\xi )H_p(\rho _{mid}(\beta )^2{\overline{\chi }}_R^2{\overline{\chi }}_r^2). \end{aligned}$$

Since \(H_p\beta (x,\xi )\ge 0\) on \(\Omega _{\beta _1-2\varepsilon ,\beta _2+2\varepsilon ,r,R,mid}\) and \(\rho _{mid}'(t)\le 0\) for \(\beta _2+\varepsilon \le t\le \beta _2+2\varepsilon \), there exists \(\rho _1\in C^{\infty }({\mathbb {R}};{\mathbb {R}})\) supported in \([\beta _1-3\varepsilon ,\beta _1-\varepsilon /2]\) such that

$$\begin{aligned} H_p\rho _{mid}(\beta )^2\le C\frac{\langle \xi \rangle }{\langle x \rangle }\rho _1(\beta )^2\quad \text {for}\quad (x,\xi )\in \Omega _{r,R}. \end{aligned}$$

Now we construct \(b_1=e^{-M\beta }\rho _1(\beta ){\overline{\chi }}_R{\overline{\chi }}_r\). The symbols \(b_2\) and e are similarly constructed, where we note that \(b_2\) and e come from the term \(H_p({\overline{\chi }}_R)\) and \(H_p({\overline{\chi }}_r)\) respectively. If \(\beta _2<0\) and \(\varepsilon >0\) small enough, then \(H_p({\overline{\chi }}_R)\le 0\) and hence we can take \(b_2=0\).

(ii) Take \(\rho _{in}\in C^{\infty }({\mathbb {R}};[0,1])\) such that \(\rho _{in}(t)=1\) on \(t\le -1+\varepsilon \), \(\rho _{in}(t)=0\) for \(t\ge -1+2\varepsilon \) and \(\rho _{in}'(t)\le 0\). We set

$$\begin{aligned} a(x,\xi )=\rho _{in}(\beta (x,\xi )){\overline{\chi }}_R(x){\overline{\chi }}_r(\xi )\in S^{0,0}. \end{aligned}$$

By virtue of (A.1), we can take \(R\ge 1\) large enough such that \(H_p\beta \ge 0\) if \(-1+\varepsilon \le \beta (x,\xi )\le -1+2\varepsilon \) and \(|x|\ge R\). Then we have

$$\begin{aligned} H_p(\rho _{in}(\beta ))=(H_p\beta )\rho _{in}'(\beta )\le 0,\quad H_p{\overline{\chi }}_R=-R^{-1}\chi '(|x|)|\partial _{\xi }p|\beta \le 0 \end{aligned}$$

for \(|x|\ge R\) and \((x,\xi )\in {\mathrm{supp}\;}\rho _{in}(\beta )\). Then the symbol e can be constructed associated with the term \(a\rho _{in}{\overline{\chi }}_RH_p{\overline{\chi }}_r\) as in (i).

(iii) Take \(\rho _{out}\in C^{\infty }({\mathbb {R}};[0,1])\) such that \(\rho _{out}(t)=1\) on \(t\ge 1-\varepsilon \) and \(\rho _{out}(t)=0\) for \(t\le 1-2\varepsilon \). We set

$$\begin{aligned} a(x,\xi )=\rho _{out}(\beta (x,\xi )){\overline{\chi }}_R(x){\overline{\chi }}_r(\xi )\in S^{0,0}. \end{aligned}$$

Then our claim follows as in (i) and (ii), where we note that the term \(b_1\) comes from \(H_p(\rho _{in}(\beta ))\) and the term \(b_2\) comes from the term \(H_p({\overline{\chi }}_R)\) as in (i). \(\square \)

1.4 A.2. Radial estimates

In this subsection, we do not assume the null non-trapping condition (Assumption A). In the other parts of this “Appendix”, we shall impose Assumption A.

Let \(a,b_1,b_2,e\in S^{0,0}\) be as in Lemma A.2 (i), (ii) and (iii) respectively, where we take \(b_1=b_2=0\) in the case (ii). Set \(A=\mathrm {Op}(a)\), \(B_j=\mathrm {Op}(b_j)\) and \(E=\mathrm {Op}(e)\) and take \(A\in \mathrm {Op}S^{0,0}\) such that

$$\begin{aligned} A, B_j, E\Subset A'. \end{aligned}$$

Correspondingly, we have the following theorem.

Theorem A.3

Let

$$\begin{aligned} z\in \{z\in {\mathbb {C}}\mid {\mathrm{Im}\;}z\ge 0\}:={\mathbb {C}}_{+}. \end{aligned}$$
(A.2)

We consider the estimate

$$\begin{aligned} \Vert Au\Vert _{H^{k,l}}&\le C\Vert A'(P-z)u\Vert _{H^{k-1, l+1}}+C\Vert B_1u\Vert _{H^{k,l}}\nonumber \\&+C\Vert B_2u\Vert _{H^{k,l}}+C\Vert Eu\Vert _{H^{k,l}}+C\Vert u\Vert _{H^{-N,-N}}. \end{aligned}$$
(A.3)

We have the following statements.

\((i)\) \((\)Propagation of singularity) Let \(k,l\in {\mathbb {R}}\), \(N>0\) and \(-1<\beta _1<\beta _2<1\). Suppose that \(u\in H^{-N,-N}\cap H^k_{loc}\) satisfies \(u\in H^{k,l}\) microlocally on \(\{|\xi |\le 3r\}\) and

$$\begin{aligned}&u\in H^{k,l}, \quad \text {microlocally on}\quad \Omega _{r,R}(\beta _2)\\&(P-z)u\in H^{k+1,l-1}\quad \text {microlocally on}\quad \Omega _{\beta _1-2\varepsilon ,\beta _2+2\varepsilon ,r,R,mid} \end{aligned}$$

for \(r>0\), \(\varepsilon >0\) small enough and \(R\ge 1\) large enough. Then we have \(u\in H^{k,l}\) microlocally on \(\Omega _{r',R'}(\beta _1)\) for some \(r',R'>0\).

More precisely, the following statement holds: Then, for \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) such that the right hand side of (A.3) is bounded, the estimate (A.3) hold with a constant \(C>0\) independent of \(z\in {\mathbb {C}}_+\).

\((ii)\) \((\)Radial source estimate) Let \(l>-1/2\), \(N>0\) and \(k\in {\mathbb {R}}\). Suppose that \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) satisfies \(u\in H^{k,l}\) microlocally on \(\{|\xi |\le 3r\}\) and

$$\begin{aligned} u\in H^{-N,l_0}\quad \text {and}\quad (P-z)u\in H^{k+1,l-1}\quad \text {microlocally on}\quad \Omega _{r,R}(-1) \end{aligned}$$

for some \(l_0>-1/2\), \(r>0\) and \(R\ge 1\) large enough. Then we have \(u\in H^{k,l}\) microlocally on \(\Omega _{r',R'}(-1)\) for some \(r',R'>0\).

More precisely, the following statement holds: Set \(B_1=B_2=0\). Consider \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) such that the right hand side of (A.3) is bounded and \(A'u\in H^{-N,l_0}\). Then the estimate (A.3) holds for such u with a constant \(C>0\) independent of \(z\in {\mathbb {C}}_+\).

\((iii)\) \((\)Radial sink estimate) Let \(l<-\frac{1}{2}\), \(N>0\) and \(k\in {\mathbb {R}}\). Suppose that \(u\in H^{-N,-N}\cap H^k_{loc}\) satisfies \(u\in H^{k,l}\) microlocally on \(\{|\xi |\le 3r\}\) and

$$\begin{aligned}&u\in H^{k,l}, \quad \text {microlocally on}\quad \Omega _{r,R}(1-\varepsilon )\\&(P-z)u\in H^{k+1,l-1}\quad \text {microlocally on}\quad \Omega _{1-2\varepsilon ,1,r,R,mid} \end{aligned}$$

for \(r>0\), \(0<\varepsilon <1/2\) and \(R\ge 1\) large enough. Then we have \(u\in H^{k,l}\) microlocally on \(\Omega _{r',R'}(1)\) for some \(r',R'>0\).

More precisely, the following statement holds: Then, for \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) such that the right hand side of (A.3) is bounded, the estimate (A.3) holds for such u with a constant \(C>0\) independent of \(z\in {\mathbb {C}}_+\).

Remark A.4

For the radial source estimate (ii), we do not need the regularity assumption \(u\in H^k_{loc}\), which is different from the propagation of singularity (i) and the radial sink estimate (iii).

Remark A.5

If \(z\ne 0\) and if we take \(r>0\) sufficiently small, then we have \(|p(x,\xi )-z|\ge C>0\) on \(\{|\xi |\le 3r\}\). This implies that under \(z\ne 0\), the assumption \(u\in H^{k,l}\) microlocally on \(\{|\xi |\le 3r\}\) can be removed if we also assume \(A'\) is elliptic on \(\{|\xi |\le 3r\}\).

Remark A.6

For \(z\in {\mathbb {C}}_-=\{w\in {\mathbb {C}}\mid {\mathrm{Im}\;}w\le 0\}\), similar results hold although the direction of the propagation should be reversed.

1.5 Commutator estimates

For the proof of Theorem A.3, the following commutator calculus has an important role: For pseudodifferential operators \(A, \Lambda \), where A is formally self-adjoint and \({\mathrm{Im}\;}z\ge 0\), we have

$$\begin{aligned} {\mathrm{Im}\;}((P-z)u, A\Lambda ^*\Lambda Au)_{L^2}&=-(u,[P,iA\Lambda ^* \Lambda A]u)_{L^2}+{\mathrm{Im}\;}z\Vert \Lambda Au\Vert _{L^2}^2 \end{aligned}$$
(A.4)

for \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\). Moreover, the equation (A.4) with the Cauchy-Schwartz inequality implies that for any small \(\varepsilon _1>0\), there exists \(C>0\) such that

$$\begin{aligned} -(u,[P,iA\Lambda ^*\Lambda A]u)_{L^2}\le C\Vert \Lambda A(P-z)u\Vert _{H^{-\frac{1}{2},\frac{1}{2}}}^2+\varepsilon _1\Vert \Lambda Au\Vert _{H^{\frac{1}{2},-\frac{1}{2}}}^2. \end{aligned}$$
(A.5)

We set

$$\begin{aligned} \Lambda =\Lambda _{k,l,\kappa ,N,\delta }=\mathrm {Op}(\lambda _{k,l,\kappa ,N,\delta }),\quad \,\, \Theta =\langle x \rangle ^{-\frac{1}{2}}\langle D \rangle ^{\frac{1}{2}},\quad \Lambda _{\delta }=\langle \delta x \rangle ^{-\kappa }\langle \delta D \rangle ^{-|k|-N-1}. \end{aligned}$$

Proof of Theorem A.3

We may assume \(N>0\) is sufficiently large. If we take \(R\ge 1\) large enough and \(\varepsilon >0\) small enough, Lemma A.1 and Lemma A.2 yield

$$\begin{aligned} H_p(a^2\lambda ^2)\le (-C_4\frac{\langle \xi \rangle }{\langle x \rangle }a^2+\frac{\langle \xi \rangle }{\langle x \rangle }b_1^2+\frac{\langle \xi \rangle }{\langle x \rangle }b_2^2 +\frac{\langle \xi \rangle }{\langle x \rangle }e^2)\lambda ^2 \end{aligned}$$

with a constant \(C_4>0\), where we take \(L=-2C_0\) in the case (i). The sharp Gårding inequality (Lemma 2.1 (iii)) gives

$$\begin{aligned} C_4A\Lambda _{\delta }\Lambda \Theta ^2\Lambda \Lambda _{\delta } A&\le -[P,iA\Lambda _{\delta }\Lambda ^2\Lambda _{\delta }A] +\sum _{j=1}^2B_j\Lambda _{\delta }\Lambda \Theta ^2\Lambda \Lambda _{\delta } B_j+E\Lambda _{\delta }\Lambda \Theta ^2\Lambda \Lambda _{\delta }E\\&+A_1\Lambda _{\delta }\Lambda \Theta \langle x \rangle ^{-\frac{1}{2}}\langle D \rangle ^{-1}\langle x \rangle ^{-\frac{1}{2}} \Theta \Lambda \Lambda _{\delta }A_1 \end{aligned}$$

up to the term \(\mathrm {Op}S^{-\infty ,-\infty }\) uniformly bounded in \(0\le \delta \le 1\). Here, the last term comes from a remainder term of the sharp Gårding inequality and from \([P-\mathrm {Op}(p),iA\Lambda _{\delta }\Lambda ^2\Lambda _{\delta }A]\). Moreover, \(A_1\in \mathrm {Op}S^{0,0}\) is elliptic in

$$\begin{aligned} {\mathrm{supp}\;}a\cup {\mathrm{supp}\;}b_1\cup {\mathrm{supp}\;}b_2\cup {\mathrm{supp}\;}e. \end{aligned}$$

Now (A.4) and (A.5) imply that for \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\),

$$\begin{aligned} \Vert \Lambda _{\delta } Au\Vert _{H^{k, l}}^2\le C\Vert \Lambda _{\delta } A(P-z)u\Vert _{H^{k-1, l+1}}^2+C\Vert \Lambda _{\delta }B_1u\Vert _{H^{k,l}}^2+C\Vert \Lambda _{\delta }B_2u\Vert _{H^{k,l}}^2\nonumber \\ +C\Vert \Lambda _{\delta }Eu\Vert _{H^{k,l}}^2+C\Vert \Lambda _{\delta }A_1u\Vert _{H^{k-\frac{1}{2},l-\frac{1}{2}}}^2+C\Vert u\Vert _{H^{-N,-N}}^2 \end{aligned}$$

with a constant \(C>0\) independent of \(0\le \delta \le 1\). Now it suffices to prove the lower order term \(\Vert \Lambda _{\delta }A_1u\Vert _{H^{k-\frac{1}{2},l-\frac{1}{2}}}\) and relax the a priori regularity assumption of u.

By a standard bootstrap argument, the term \(\Vert \Lambda _{\delta }A_1u\Vert _{H^{k-\frac{1}{2},l-\frac{1}{2}}}^2\) can be absorbed into the term \(\Vert u\Vert _{H^{-N,-N}}^2\) for the cases (i) and (iii). For the case (ii), using the bootstrap argument, an interpolation argument, and the part (i), the term \(\Vert \Lambda _{\delta }A_1u\Vert _{H^{k-\frac{1}{2},l-\frac{1}{2}}}^2\) can be absorbed into the left hand side and the term \(\Vert u\Vert _{H^{-N,-N}}^2\). For a detail, see [6, Part 2 of proof for Theorem E.52]. Thus, for \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\), we have

$$\begin{aligned} \Vert \Lambda _{\delta } Au\Vert _{H^{k, l}}^2&\le C\Vert \Lambda _{\delta } A(P-z)u\Vert _{H^{k-1, l+1}}^2+C\Vert \Lambda _{\delta }B_1u\Vert _{H^{k,l}}^2+C\Vert \Lambda _{\delta }B_2u\Vert _{H^{k,l}}^2\nonumber \\&+C\Vert \Lambda _{\delta }Eu\Vert _{H^{k,l}}^2+C\Vert u\Vert _{H^{-N,-N}}^2. \end{aligned}$$
(A.6)

Taking \(\delta =0\) and using a standard approximation argument with Lemma 2.2, we have (A.6) for \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) satisfying \(A'u\in H^{k,l}\) and \(A'(P-z)u\in H^{k-1,l+1}\). In fact, take \(a\in C_c^{\infty }({\mathbb {R}}^{2n+2})\) with \(a(x,\xi )=1\) for \(|(x,\xi )|\le 1\) and set \(a_R(x,\xi )=a(x/R,\xi /R)\). Substituting \(\mathrm {Op}(a_R)u\) into (A.6) with \(\delta =0\) and taking \(R\rightarrow \infty \), we obtain (A.6) for such u by Lemma 2.2.

Finally, we consider \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) such that the right hand side of (A.3) is bounded. For the case (ii), we also assume \(A'u\in H^{-N,l_0}\). We recall the notation

$$\begin{aligned} \Lambda _{\delta }=\langle \delta x \rangle ^{-\kappa }\langle \delta D \rangle ^{-|k|-N-1}. \end{aligned}$$

First, we consider the cases (i) and (iii). Take \(\kappa \) and N large enough. Substituting \(\mathrm {Op}(a_R)u\) into (A.6) with \(0<\delta \le 1\), where \(a_R\) is as above, and taking \(R\rightarrow \infty \), we obtain (A.6), which implies (A.3) for such u by taking \(\delta \rightarrow 0\). For the case (ii), we can use this procedure for arbitrary \(N>0\) and for \(\kappa >-l-\frac{1}{2}\) due to Lemma A.1. Thus we need the additional assumption \(A'u\in H^{-N,l_0}\). We omit the detail. For a similar argument, see [6, Exercises E.31, E.35, E.36]. \(\square \)

1.6 A.3. Propagation to the radial source in the past infinity

In order to control the regularity for a bounded region of the x-space, we use the standard propagation of singularity theorem and the null non-trapping condition. To apply it, we need the following dynamical result.

Lemma A.7

Let \((x_0,\xi _0)\in T^*{\mathbb {R}}^{n+1}\) with \(\xi _0\ne 0\) and \(p(x_0,\xi _0)=0\). We denote \(z(t)=z(t,x_0,\xi _0)\), \(\zeta (t)=\zeta (t,x_0,\xi _0)\) and \(\beta (t)=\beta _0(z(t),\zeta (t))\). Then for any \(0<\varepsilon <1\) and \(R\ge 1\), there exists \(T>0\) such that \(|z(-T)|>R\) and

$$\begin{aligned} \beta (-T)<(-1+\varepsilon ). \end{aligned}$$
(A.7)

Remark A.8

The argument below is standard. However, for its justification, we need some estimates for the classical trajectories which are proved in [13, Appendix A].

Proof

Let \(0<\varepsilon <1\) and \(R\ge 1\). Take \(R_0\ge R\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} H_p^2|x|^2\ge C|\xi |^2\quad \text {if}\quad |x|\ge R_0\\ H_p\beta (x,\xi )\ge C\langle x \rangle ^{-1}|\xi |\quad \text {if}\quad |x|\ge R_0\,\, \text {and}\,\, (-1+\varepsilon )\le \beta (x,\xi )\le 0. \end{array}\right. } \end{aligned}$$
(A.8)

By Assumption A, we can choose \(T_0>0\) such that

$$\begin{aligned} |z(-T_0)|\ge 2R_0,\,\, \frac{d}{dt}|z(t)|^2|_{t=-T_0}\le 0. \end{aligned}$$

This inequality and (A.8) imply that \(|z(-t)|\ge R_0\) for \(t\ge T_0\). By the proof of [13, Lemma A.2] and [13, Corollary A.4], we have

$$\begin{aligned} C_1^{-1}\langle t \rangle \le \langle z(t) \rangle \le C_1\langle t \rangle \quad C_2^{-1}\le |\zeta (t)|\le C_2 \end{aligned}$$

for all \(t\in {\mathbb {R}}\) with a constant \(C_1,C_2>0\). Now suppose that (A.7) fails. By the inequality (A.8), we obtain

$$\begin{aligned} \beta (-T_0)=\beta (-t)+\int _{-t}^{-T_0}\beta '(s)ds\ge -1+ CC_1^{-1}C_2^{-1}\int _{-t}^{-T_0}\langle s \rangle ^{-1}ds=\infty \quad \text {as}\quad t\rightarrow \infty \end{aligned}$$

which is a contradiction. \(\square \)

Combining the radial source estimate with the standard propagation of singularities, we have the following corollary which is a generalization of [13, Proposition 3.2].

Corollary A.9

Let \(k\in {\mathbb {R}}\), \(\delta >0\) and \(z\in {\mathbb {C}}_{+}{\setminus } \{0\}\). Suppose that

$$\begin{aligned} u\in H^{k,-\frac{1}{2}+\delta }({\mathbb {R}}^{n+1}) \quad \text {microlocally on}\,\, \Omega _{r,R}(-1) \quad \text {and}\quad (P-z)u\in {\mathcal {S}}({\mathbb {R}}^{n+1}) \end{aligned}$$

with \(R\ge 1\) large enough and \(r>0\) small enough. Then we have \(u\in C^{\infty }({\mathbb {R}}^{n+1})\).

Proof

We shall prove \(u\in C^{\infty }({\mathbb {R}}^{n+1})\) microlocally near \((x_0,\xi _0)\in {\mathbb {R}}^{n+1}{\setminus } \{\xi =0\}\). The last lemma implies that for \(R\ge 1\) large enough, and \(0<\varepsilon <1\), there exists \((x_1,\xi _1)\in T^*{\mathbb {R}}^{n+1}{\setminus } 0\) such that \((x_1,\xi _1)\) lies in the same integral curve of \(H_p\) and

$$\begin{aligned} |x_1|>R,\,\, \beta (x_1,\xi _1)<-1+\varepsilon . \end{aligned}$$

Then it suffices to prove \(u\in C^{\infty }({\mathbb {R}}^{n+1})\) microlocally near \((x_1,\xi _1)\in {\mathbb {R}}^{n+1}{\setminus } \{\xi =0\}\) by the standard propagation of singularities theorem. Moreover, since \(\xi _0\ne 0\) and since p is homogeneous of degree 2, we have \(\xi _1\ne 0\). Moreover, we may assume \(|\xi _1|\) is large enough since the wave front set is invariant under scaling with respect to the \(\xi \)-variable. To use Theorem A.3 (ii), we shall check \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\) microlocally in \(\{|\xi |\le 3r\}\). We note that if \(z\ne 0\), then \(P-z\) is elliptic on \(\{|\xi |\le 4r\}\) with some \(r>0\). Then the standard elliptic parametrix construction yields \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\) microlocally in \(\{|\xi |\le 3r\}\). Now Theorem A.3 (ii) implies that for \(R>>1\) large enough and \(0<\varepsilon <1\), we have \(u\in C^{\infty }({\mathbb {R}}^{n+1})\) microlocally on the incoming region \(\Omega _{\varepsilon ,r,R,in}\). This completes the proof since \((x_1,\xi _1)\in \Omega _{\varepsilon ,r,R,in}\). \(\square \)

Summarizing the above results, we obtain the following corollary.

Corollary A.10

Let \(z\in {\mathbb {C}}_{+}{\setminus } \{0\}\), \(k\in {\mathbb {R}}\) and \(\delta >0\). Suppose \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) satisfies \((P-z)u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\) and \(u\in H^{k,-\frac{1}{2}+\delta }({\mathbb {R}}^{n+1})\) microlocally in the incoming region \(\Omega _{r,R}(-1)\) with \(R\ge 1\) large enough and \(r>0\) small enough. Then we have \(u\in \cap _{k\in {\mathbb {R}}, \delta >0}H^{k,-\frac{1}{2}-\delta }\). Moreover, we have \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\) microlocally away from the outgoing region \(\Omega _{r,R}(1)\) for \(R\ge 1\) large enough and \(r>0\) small enough..

1.7 A.4. Elliptic estimate

The following lemma is proved by a standard parametrix construction.

Lemma A.11

Let \(m_0>0\). If \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) satisfies \((P+m_0^2)u\in H^{k,l}\) with \(k,l\in {\mathbb {R}}\), then we have \(u\in H^{k+2,l}\) microlocally in \(\{(x,\xi )\in {\mathbb {R}}^{2n+2}\mid |x|\ge R,\,\, |p_0(\xi )+m_0^2|\ge \varepsilon |\xi |^2\}\) for \(R\ge 1\) large enough and \(\varepsilon >0\).

1.8 A.5. Absence of resonances

By the proof of [18, Proposition 7], we have the following proposition.

Proposition A.12

Let \(z\in {\mathbb {C}}_{+}{\setminus } \{0\}\). Suppose that a distribution \(u\in {\mathcal {S}}'({\mathbb {R}}^{n+1})\) satisfies \((P-z)u=0\) and \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\) microlocally away from the outgoing region \(\Omega _{r,R}(1)\) for some \(R\ge 1\) large enough and \(r>0\) small enough. Then we have \(u\in {\mathcal {S}}({\mathbb {R}}^{n+1})\).

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Taira, K. Limiting Absorption Principle and Equivalence of Feynman Propagators on Asymptotically Minkowski Spacetimes. Commun. Math. Phys. 388, 625–655 (2021). https://doi.org/10.1007/s00220-021-04196-7

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