Looping directions and integrals of eigenfunctions over submanifolds

TL;DR

This paper improves bounds on eigenfunction integrals over submanifolds by showing that if the set of looping directions has measure zero, then the integrals decay faster than previously known.

Contribution

It establishes a new decay rate for eigenfunction integrals over submanifolds under the condition that looping directions form a measure-zero set.

Findings

Improved decay rate for eigenfunction integrals when looping directions are measure zero.

Conditions under which the integral bounds are sharper than classical estimates.

Connection between geodesic flow properties and eigenfunction behavior.

Abstract

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -\Delta_g e_\lambda = \lambda^2 e_\lambda \qquad \text{ and } \qquad \| e_\lambda \|_{L^2(M)} = 1. \] Let $\Sigma$ be a $d$-dimensional submanifold and $d\mu$ a smooth, compactly supported measure on $\Sigma$. It is well-known (e.g. proved by Zelditch in far greater generality) that \[ \int_\Sigma e_\lambda \, d\mu = O(\lambda^\frac{n-d-1}{2}). \] We show this bound improves to $o(\lambda^\frac{n-d-1}{2})$ provided the set of looping directions, \[ \mathcal{L}_{\Sigma} = \{ (x,\xi) \in SN^*\Sigma : \Phi_t(x,\xi) \in SN^*\Sigma \text{ for some } t > 0 \} \] has measure zero as a subset of $SN^*\Sigma$, where here $\Phi_t$ is the geodesic flow on the cosphere bundle $S^*M$ and $SN^*\Sigma$ is the unit conormal bundle over $\Sigma$.

Equations

$$-{\Delta }_g{e}_{\lambda }={\lambda }^2{e}_{\lambda }\text{ and }\Vert e_{\lambda }{\Vert }_{L^2(M)}=1.$$

$${\int }_{\Sigma }{e}_{\lambda }d\mu =O({\lambda }^{\frac{n-d-1}{2}}).$$

$${\mathcal{L}}_{\Sigma }=\{(x,\xi )\in S{N}^{\ast }\Sigma :{\Phi }_t(x,\xi )\in S{N}^{\ast }\Sigma \text{ for some }t>0\}$$

$$-{\Delta }_g{e}_{\lambda }={\lambda }^2{e}_{\lambda }\text{ and }\Vert e_{\lambda }{\Vert }_{L^2(M)}=1.$$

$$\Vert e_{\lambda }{\Vert }_{L^{\infty }(M)}=O({\lambda }^{\frac{n-1}{2}}).$$

$${\mathcal{L}}_x=\{\xi \in S_x^{\ast }M:{\Phi }_t(x,\xi )\in S_x^{\ast }M\text{ for some }t>0\}$$

$$\Vert e_{\lambda }{\Vert }_{L^{\infty }(M)}=o({\lambda }^{\frac{n-1}{2}})$$

$$\sum _{{\lambda }_j\le \lambda }{\left|{\int }_{\Sigma }{e}_{j}d\mu \right|}^2\sim {\lambda }^{n-d}+O({\lambda }^{n-d-1})$$

$${\int }_{\Sigma }{e}_{\lambda }d\mu =O({\lambda }^{\frac{n-d-1}{2}}).$$

$${\mathcal{L}}_{\Sigma }=\{(x,\xi )\in S{N}^{\ast }\Sigma :{\Phi }_t(x,\xi )\in S{N}^{\ast }\Sigma \text{ for some }t>0\}.$$

$${\int }_{\Sigma }{e}_{\lambda }d\mu =o({\lambda }^{\frac{n-d-1}{2}}).$$

$${\lambda }_j=\sqrt{j(j+n-1)}\text{ for }j=0,1,2,\dots$$

$$\sum _{{\lambda }_j\in [\lambda ,\lambda +1]}{\left|{\int }_{\Sigma }{e}_{j}d\mu \right|}^2\gtrsim {\lambda }^{n-d-1}.$$

$$\left|{\int }_{\Sigma }{e}_{j}d\mu \right|\gtrsim {\lambda }_j^{\frac{n-d-1}{2}}$$

$${\int }_{\Sigma }{e}_{\lambda }d\mu =o(1).$$

$$\sum _{{\lambda }_j\in [\lambda ,\lambda +1]}{\left|{\int }_{\Sigma }{e}_{\lambda }d\mu \right|}^2\le C{\lambda }^{n-d-1}.$$

$$g=\left[\begin{array}[]{cc}*&0\\ 0&I\end{array}\right]\qquad\text{ wherever }\bar{x}=0,$$

$$(x_1,\dots ,x_n)\mapsto \exp (x_{d+1}{v}_{d+1}(x^{\prime })+\cdots +x_n{v}_n(x^{\prime }))$$

$$\sum _j\chi ({\lambda }_j-\lambda ){\left|{\int }_{\Sigma }{e}_{j}d\mu \right|}^2\le C{\lambda }^{n-d-1}.$$

$$\sum _j{\int }_{\Sigma }{\int }_{\Sigma }$$

$$=\frac{1}{2\pi }\sum _j{\int }_{-\infty }^{\infty }{\int }_{\Sigma }{\int }_{\Sigma }\hat{\chi }(t)e^{-it\lambda }{e}^{it{\lambda }_j}{e}_j(x)\overline{e_j(y)}d\mu (x)d\mu (y)dt$$

$$=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\int }_{\Sigma }{\int }_{\Sigma }\hat{\chi }(t)e^{-it\lambda }{e}^{it\sqrt{-{\Delta }_g}}(x,y)d\mu (x)d\mu (y)dt$$

$$e^{it\sqrt{-{\Delta }_g}}(x,y)=\sum _j{e}^{it{\lambda }_j}{e}_j(x)\overline{e_j(y)}.$$

$$=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\hat{\chi }(t)e^{-it\lambda }{e}^{it\sqrt{-{\Delta }_g}}(x^{\prime },y^{\prime })h(x^{\prime })h(y^{\prime })d{x}^{\prime }d{y}^{\prime }dt$$

$$e^{it\sqrt{-{\Delta }_g}}(x,y)=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{e}^{i(\phi (x,y,\xi )+tp(y,\xi ))}q(t,x,y,\xi )d\xi$$

$$p(y,\xi )=\sqrt{\sum _{j,k}{g}^{jk}(y){\xi }_j{\xi }_k}$$

$$|{\partial }_{\xi }^{\alpha }(\phi (x,y,\xi )-⟨x-y,\xi ⟩)|\le C_{\alpha }|x-y{|}^2|\xi {|}^{1-|\alpha |}$$

$$|{\partial }_{\xi }^{\alpha }{\partial }_{t,x,y}^{\beta }q(t,x,y,\xi )|\le C_{\alpha ,\beta }(1+|\xi |{)}^{-|\alpha |},$$

$$=\frac{1}{(2\pi {)}^{n+1}}{\int }_{-\infty }^{\infty }{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^n}{e}^{i(\phi (x^{\prime },y^{\prime },\xi )+tp(y^{\prime },\xi )-t\lambda )}\hat{\chi }(t)q(t,x^{\prime },y^{\prime },\xi )$$

$$d\xi d{x}^{\prime }d{y}^{\prime }dt,$$

$$=\frac{{\lambda }^n}{(2\pi {)}^{n+1}}{\int }_{-\infty }^{\infty }{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^n}{e}^{i\lambda (\phi (x^{\prime },y^{\prime },\xi )+t(p(y^{\prime },\xi )-1))}\hat{\chi }(t)q(t,x^{\prime },y^{\prime },\lambda \xi )$$

$$d\xi d{x}^{\prime }d{y}^{\prime }dt.$$

Full text

[TABLE]

Let $\Sigma$ be a $d$-dimensional submanifold and $d\mu$ a smooth, compactly supported measure on $\Sigma$. It is well-known (e.g. proved by Zelditch in  [15] in far greater generality) that

[TABLE]

We show this bound improves to $o(\lambda^{\frac{n-d-1}{2}})$ provided the set of looping directions,

[TABLE]

has measure zero as a subset of $SN^{*}\Sigma$, where here $\Phi_{t}$ is the geodesic flow on the cosphere bundle $S^{*}M$ and $SN^{*}\Sigma$ is the unit conormal bundle over $\Sigma$.

1. Introduction

In what follows, $(M,g)$ will denote a compact, boundaryless, $n$-dimensional Riemannian manifold. Let $\Delta_{g}$ denote the Laplace-Beltrami operator and $e_{\lambda}$ an $L^{2}$-normalized eigenfunction of $\Delta_{g}$ on $M$, i.e.

[TABLE]

In  [13], Sogge and Zelditch investigate which manifolds have a sequence of eigenfunctions $e_{\lambda}$ with $\lambda\to\infty$ which saturate the bound

[TABLE]

They show that the bound above is necessarily $o(\lambda^{\frac{n-1}{2}})$ if at each each $x$, the set of looping directions through $x$,

[TABLE]

has measure zero111Let $\psi_{j}:U_{j}\subset\mathbb{R}^{n}\to M$ be coordinate charts of a general manifold $M$. We say a set $E\subset M$ has measure zero if the preimage $\psi_{j}^{-1}(E)$ has Lebesgue measure [math] in $\mathbb{R}^{n}$ for each chart $\psi_{j}$. Sets of Lebesgue measure zero are preserved under transition maps, ensuring this definition is intrinsic to the $C^{\infty}$ structure of $M$. as a subset of $S^{*}_{x}M$ for each $x\in M$. Here, $\Phi_{t}$ denotes the geodesic flow on the unit cosphere bundle $S^{*}M$ after time $t$. The hypotheses were later weakened by Sogge, Toth, and Zelditch in  [10], where they showed

[TABLE]

provided the set of recurrent directions at $x$ has measure zero for each $x\in M$.

We are interested in extending the result in  [13] to integrals of eigenfunctions over submanifolds. Let $\Sigma$ be a submanifold of dimension $d$ with $d<n$ and a measure $d\mu(x)=h(x)d\sigma(x)$ where $d\sigma$ is the surface measure on $\Sigma$ and $h$ is a smooth function supported on a compact subset of $\Sigma$. In his 1992 paper  [15], Zelditch proves, among other things, a Weyl law- type bound

[TABLE]

from which follows

[TABLE]

Though (1.2) is already well known, we will give a direct proof which will be illustrative for our main argument.

Theorem 1.1**.**

Let $\Sigma$ be a $d$-dimensional submanifold with $0\leq d<n$, and $d\mu(x)=h(x)d\sigma(x)$ where $h$ is a smooth, real valued function supported on a compact neighborhood in $\Sigma$. Then, (1.2) holds.

We let $SN^{*}\Sigma$ denote the unit conormal bundle over $\Sigma$. We define the set of looping directions through $\Sigma$ by

[TABLE]

Our main result shows the bound (1.2) cannot be saturated whenever the set of looping directions through $\Sigma$ has measure zero.

Theorem 1.2**.**

Assume the hypotheses of Theorem 1.1 and additionally that ${\mathcal{L}}_{\Sigma}$ has measure zero as a subset of $SN^{*}\Sigma$. Then,

[TABLE]

The argument for Theorem 1.2 is modeled after Sogge and Zelditch’s arguments in  [13]. In fact if $d=0$ we obtain the first part of  [13, Theorem 1.2].

We expect the bound (1.2) to be saturated in the case $M=S^{n}$, since ${\mathcal{L}}_{\Sigma}=SN^{*}\Sigma$ always. The spectrum of $-\Delta_{g}$ on $S^{n}$ consists of $\lambda_{j}^{2}$ where

[TABLE]

(see  [9]). For each $\lambda_{j}$ we select an eigenfunction $e_{j}$ maximizing $\left|\int_{\Sigma}e_{j}\,d\mu\right|$. By Zelditch’s Weyl law type bound (1.1), there exists an increasing sequence of $\lambda$ with $\lambda\to\infty$ for which

[TABLE]

Since the gaps $\lambda_{j}-\lambda_{j-1}$ approach a constant width of $1$ as $j\to\infty$, we may pick a subsequence of $\lambda$’s so that only one $\lambda_{j}$ falls in each band $[\lambda,\lambda+1]$. Hence,

[TABLE]

for some subsequence of $\lambda_{j}$.

It is worth remarking that there are some cases where the hypotheses of Theorem 1.2 are naturally fulfilled and we obtain an improvement over (1.2). Chen and Sogge  [2] proved that if $M$ is $2$-dimensional and has negative sectional curvature, and $\Sigma$ is a geodesic in $M$,

[TABLE]

They consider a lift $\tilde{\Sigma}$ of $\Sigma$ to the universal cover of $M$. Using the Gauss-Bonnet theorem, they show for each non-identity deck transformation $\alpha$, there is at most one geodesic which intersects both $\tilde{\Sigma}$ and $\alpha(\tilde{\Sigma})$ perpendicularly. Since there are only countably many deck transformations, ${\mathcal{L}}_{\Sigma}$ is at most a countable subset of $SN^{*}\Sigma$ and so satisfies the hypotheses of Theorem 1.2. This result was extended to a larger class of curves in  [14] which similarly have countable ${\mathcal{L}}_{\Sigma}$.

Acknowledgements. The author would like to thank Yakun Xi for pointing out an error at the end of the proof of Proposition 2.1.

2. Proof of Theorem 1.1

Theorem 1.1 is a consequence of this stronger result.

Proposition 2.1**.**

Given the hypotheses of Theorem 1.1, we have

[TABLE]

We lay out some local coordinates which we will use repeatedly. Fix $p\in\Sigma$, and consider local coordinates $x=(x_{1},\ldots,x_{n})=(x^{\prime},\bar{x})$ centered about $p$, where $x^{\prime}$ denotes the first $d$ coordinates and $\bar{x}$ the remaining $n-d$ coordinates. We let $(x^{\prime},0)$ parametrize $\Sigma$ on a neighborhood of $p$ in such a way that $dx^{\prime}$ agrees with the surface measure on $\Sigma$. Let $g$ denote the metric tensor with respect to our local coordinates. We require

[TABLE]

where $I$ here is the $(n-d)\times(n-d)$ identity matrix. This is ensured after inductively picking smooth sections $v_{j}(x^{\prime})$ of $SN\Sigma$ for $j=d+1,\ldots,n$ with $\langle v_{i},v_{j}\rangle=\delta_{ij}$, and then using

[TABLE]

as our coordinate map.

Now we prove Proposition 2.1. For simplicity, we assume without loss of generality that $d\mu$ is a real measure. We set222This reduction is standard and appears in  [13],  [2], proofs of the sharp Weyl law as presented in  [9] and  [8], and in many other similar problems. $\chi\in C^{\infty}(\mathbb{R})$ with $\chi\geq 0$ and $\hat{\chi}$ supported on a small neighborhood of [math]. It suffices to show

[TABLE]

By Fourier inversion, we write the left hand side as

[TABLE]

where $e^{it\sqrt{-\Delta_{g}}}$ is the half wave operator with kernel

[TABLE]

Using the coordinates $x=(x^{\prime},\bar{x})$ as in (2.1), the last line of (2.2) is written

[TABLE]

where $h$ is a smooth function on $\mathbb{R}^{d}$ such that $d\mu(x)=h(x^{\prime})dx^{\prime}$, and where by abuse of notation $x^{\prime}$ is taken to mean $(x^{\prime},0)$ where appropriate. We now use Hörmander’s parametrix as presented in  [8], i.e

[TABLE]

modulo a smooth kernel, where

[TABLE]

is the principal symbol of $\sqrt{-\Delta_{g}}$ and $\varphi$ is smooth for $|\xi|>0$, homogeneous of degree $1$ in $\xi$, and satisfies

[TABLE]

for multiindices $\alpha\geq 0$ and for $x$ and $y$ sufficiently close. Moreover, $q$ satisfies bounds

[TABLE]

and where for $t\in\operatorname{supp}\hat{\chi}$, $q$ is supported on a small neighborhood of $x=y$. Hence, we write (2.3) as

[TABLE]

and after making a change of coordinates $\xi\mapsto\lambda\xi$ is

[TABLE]

We introduce a function $\beta\in C_{0}^{\infty}(\mathbb{R})$ with $\beta\equiv 1$ near [math] and support contained in a small neighborhood of [math], and cut the integral into $\beta(\log p(y^{\prime},\xi))$ and $1-\beta(\log p(y^{\prime},\xi))$ parts. $|p(y^{\prime},\xi)-1|$ is bounded away from [math] on the support of $1-\beta(\log p(y^{\prime},\xi))$, so integrating by parts in $t$ yields

[TABLE]

for each $N=1,2,\ldots$. What is left to bound is the $\beta(\log p(y^{\prime},\xi))$ part, i.e.

[TABLE]

where we have set the amplitude

[TABLE]

and the phase

[TABLE]

By (2.5) and since $a$ has compact support in $t$, $x^{\prime}$, $y^{\prime}$, and $\xi$, $a$ and all of its derivatives are uniformly bounded in $\lambda$.

We are now in a position to apply stationary phase. Write $\xi=(\xi^{\prime},\bar{\xi})$ and write $\bar{\xi}=r\omega$ in polar coordinates with $r\geq 0$ and $\omega\in S^{n-d-1}$. The integral in (2.6) is then written

[TABLE]

We will fix $y^{\prime}$ and $\omega$ and use the method of stationary phase in the remaining variables $t$, $x^{\prime}$, $\xi^{\prime}$, and $r$ (a total of $2d+2$ dimensions). We assert that, for fixed $y^{\prime}$ and $\omega$, there is a nondegenerate stationary point at $(t,x^{\prime},\xi^{\prime},r)=(0,y^{\prime},0,1)$. $\Phi=0$ at such a stationary point, and after perhaps shrinking the support of $a$ we apply  [8, Corollary 1.1.8] to write the left hand side of (2.6) as

[TABLE]

for some amplitude $a(\lambda;y^{\prime},\omega)$ uniformly bounded with respect to $\lambda$. (2.6) follows.

We have

[TABLE]

Note for fixed $y^{\prime}$ and $\omega$, $(t,x^{\prime},\xi^{\prime},r)=(0,y^{\prime},0,1)$ is a critical point of $\Phi$. Now we compute the second derivatives at this point. We immediately see that $\partial_{t}^{2}\Phi$, $\partial_{t}\nabla_{x^{\prime}}\Phi$, $\nabla_{\xi^{\prime}}^{2}\Phi$, $\partial_{r}\nabla_{\xi^{\prime}}\Phi$, and $\partial_{r}^{2}\Phi$ all vanish. Moreover, $\partial_{r}\partial_{r}\Phi=1$ since $p(y^{\prime},\xi)=r$ where $\xi^{\prime}=0$. By our coordinates (2.1) and the fact that $[g^{ij}]_{i,j\leq d}$ is necessarily positive definite,

[TABLE]

Hence, $\partial_{t}\nabla_{\xi^{\prime}}\Phi=\nabla_{\xi^{\prime}}p(y^{\prime},\xi)=0$. Since $\varphi$ is homogeneous of degree $1$ in $\xi$, at $\xi^{\prime}=0$ and $t=0$,

[TABLE]

since $\varphi(x^{\prime},y^{\prime},\omega)=O(|x^{\prime}-y^{\prime}|^{2})$ by (2.4) and the fact that $\langle x^{\prime}-y^{\prime},\omega\rangle=0$. Finally by (2.4),

[TABLE]

whence at the critical point

[TABLE]

the $d\times d$ identity matrix. In summary, the Hessian matrix of $\Phi$ at the critical point $(t,x^{\prime},\xi^{\prime},r)=(0,y^{\prime},0,1)$ is

[TABLE]

which has full rank.

3. Microlocal tools

The hypotheses on the looping directions in Theorem 1.2 ensure that the wavefront sets of $\mu$ and $e^{it\sqrt{-\Delta_{g}}}\mu$ have minimal intersection for any given $t$. We can then use pseudodifferential operators to break $\mu$ into two parts, the first whose wavefront set is disjoint from that of $e^{it\sqrt{-\Delta_{g}}}\mu$ and the second which contributes a small, controllable term to the bound. The following propositions will allow us to handle these cases, respectively.

Proposition 3.1**.**

Let $u$ and $v$ be distributions on $M$ for which

[TABLE]

Then

[TABLE]

is a smooth function of $t$ on some neighborhood of [math].

Proof.

Using a partition of unity, we write

[TABLE]

modulo a smoothing operator where $A_{j}\in\Psi_{\text{cl}}^{0}(M)$ with essential supports in small conic neighborhoods. We then write, formally,

[TABLE]

We are done if for each $i$ and $j$,

[TABLE]

If the essential supports of $A_{j}$ and $A_{k}$ are disjoint, then $A_{j}^{*}A_{k}$ is a smoothing operator, and so $A_{j}^{*}A_{k}v$ is a smooth function and the contributing term

[TABLE]

is smooth is $t$. Assume the essential support of $A_{j}$ are small enough so that for each $j$ there exists a small conic neighborhood $\Gamma_{j}$ which fully contains the essential support of $A_{k}$ if it intersects the essential support of $A_{j}$. We in turn take $\Gamma_{j}$ small enough so that for each $j$, $\overline{\Gamma_{j}}$ either does not intersect $\operatorname{WF}(u)$ or does not intersect $\operatorname{WF}(v)$. In the latter case, $A_{k}v$ is smooth and we have (3.1) as before. In the former case,

[TABLE]

since both sets above are closed and the geodesic flow is continuous. Then $A_{j}e^{it\sqrt{-\Delta_{g}}}u(x)$ is smooth as a function of $t$ and $x$, and we have (3.1). ∎

The second piece of our argument requires the following generalization of Proposition 2.1, modeled after  [9, Lemma 5.2.2]. In the proof we will come to a point where it seems like we may have to perform a stationary phase argument involving an eight-by-eight Hessian matrix. Instead, we appeal to Proposition 5.1 in the appendix to break the argument into two steps involving two four-by-four Hessian matrices.

Proposition 3.2**.**

Let $b(x,\xi)$ be smooth for $\xi\neq 0$ and homogeneous of degree [math] in the $\xi$ variable. We define $b\in\Psi_{\text{cl}}^{0}(M)$ by

[TABLE]

for $x$, $y$, and $\xi$ expressed locally according to our coordinates (2.1). Then,

[TABLE]

where $C$ is a constant independent of $b$ and $\lambda$ and $C_{b}$ is a constant independent of $\lambda$ but which depends on $b$.

Proof.

We may by a partition of unity assume that $b(x,D)$ has small $x$-support. Let $\chi$ be as in the proof of Proposition 2.1. It suffices to show

[TABLE]

Using the same reduction as in Proposition 2.1, the left hand side is

[TABLE]

Set $\beta\in C_{0}^{\infty}(\mathbb{R})$ with small support and where $\beta\equiv 1$ near [math]. Then,

[TABLE]

where the second line is obtained by a change of variables $\eta\mapsto\lambda\eta$, and the third line is obtained after multiplying in the cutoff $\beta(\log|\eta|)$ and bounding the discrepancy by $O(\lambda^{-N})$ by integrating by parts in $x^{\prime}$. Additionally,

[TABLE]

where the second and third lines are obtained similarly as before and the fourth line is obtained after multiplying by $\beta(\log|z-y^{\prime}|)$ and integrating the remainder by parts in $\zeta$. Using Hörmander’s parametrix,

[TABLE]

Here the third line comes from a change of coordinates $\xi\mapsto\lambda\xi$. The fourth line follows after applying the cutoff $\beta(\log p(z,\zeta))$ and integrating the discrepancy by parts in $t$. Combining (3.3), (3.4), and (3.5), we write (3.2) as

[TABLE]

with amplitude

[TABLE]

and phase

[TABLE]

We pause here to make a couple observations. First, $a$ has compact support in all variables, support which we may adjust to be smaller by controlling the supports of $\hat{\chi}$, $\beta$, $b$, and the support of $q$ near the diagonal. Second, the derivatives of $a$ are bounded independently of $\lambda\geq 1$. We are now in a position to use the method of stationary phase – not in all variables at once, though. First, we fix $t$, $x^{\prime}$, $y^{\prime}$ and $\xi$, and use stationary phase in $w$, $z$, $\eta$, and $\zeta$. We have

[TABLE]

which all simultaneously vanishes if and only if

[TABLE]

At such a critical point we have the Hessian matrix

[TABLE]

which has determinant $-1$. By Proposition 5.1 in the appendix, (3.6) is equal to complex constant times

[TABLE]

where we have phase

[TABLE]

amplitude

[TABLE]

with $w,z,\eta,$ and $\zeta$ subject to the constraints (3.7), and where $R$ is a compactly supported smooth function in $t,x^{\prime},y^{\prime}$, and $\xi$ whose derivatives are bounded uniformly with respect to $\lambda$. Our phase function matches that in the proof of Proposition 2.1, and so we repeat that argument – we write $\bar{\xi}=r\omega$ and fix $y^{\prime}$ and $\omega$. We obtain unique nondegenerate stationary points

[TABLE]

Now,

[TABLE]

Hence, we have

[TABLE]

by Proposition 5.1 as desired. The same argument applied to the remainder term gives

[TABLE]

as desired. ∎

4. Proof of Theorem 1.2

Theorem 1.2 follows from the following stronger statement.

Proposition 4.1**.**

Given the hypotheses of Theorem 1.2, we have

[TABLE]

where $C$ is a constant independent of $\varepsilon$ and $\lambda$, and $C_{\varepsilon}$ is a constant depending on $\varepsilon$ but not $\lambda$.

We make a few convenient assumptions. First, we take the injectivity radius of $M$ to be at least $1$ by scaling the metric $g$. Second, we assume the support of $d\mu$ has diameter less than $1/2$ by a partition of unity. We reserve the right to further scale the metric $g$ and restrict the support of $d\mu$ as needed, finitely many times.

As before, we set $\chi\in C^{\infty}(\mathbb{R})$ with $\chi(0)=1$, $\chi\geq 0$, and $\operatorname{supp}\hat{\chi}\subset[-1,1]$. It suffices to show

[TABLE]

for $T>1$. Similar to the reduction in the proof of Proposition 2.1, we have

[TABLE]

Hence, it suffices to show

[TABLE]

Set $\beta\in C_{0}^{\infty}(\mathbb{R})$ with $\beta(t)\equiv 1$ near [math] and $\beta$. We cut the integral in (4.1) into $\beta(t)$ and $1-\beta(t)$ parts. Since $\beta(t)\hat{\chi}(t/T)$ and its derivatives are all bounded independently of $T\geq 1$,

[TABLE]

by the proof of Proposition 2.1. Hence, it suffices to show

[TABLE]

Here we shrink the support of $\mu$ so that $\beta(d_{g}(x,y))=1$ for $x,y\in\operatorname{supp}\mu$. We now state and prove a useful decomposition based off of those in  [13],  [10], and Chapter 5 of  [9]. We let ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$ denote the subset of ${\mathcal{L}}_{\Sigma}$ relevant to the support of $\mu$ and the timespan $[1,T]$, specifically

[TABLE]

Lemma 4.2**.**

Fix $T>1$ and $\varepsilon>0$. There exist $b,B\in\Psi_{\text{cl}}^{0}(M)$ supported on a neighborhood of $\operatorname{supp}\mu$ with the following properties.

1. (1)

$b(x,D)+B(x,D)=I$* modulo a smoothing operator on $\operatorname{supp}\mu$.* 2. (2)

Using coordinates (2.1),

[TABLE]

where $b(x,\xi)$ is the principal symbol of $b(x,D)$. 3. (3)

The essential support of $B(x,D)$ contains no elements of ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$.

Proof.

As shorthand, we write

[TABLE]

We first argue that ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$ is closed for each $T>1$. However, ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$ is the projection of the set

[TABLE]

onto $SN^{*}_{\operatorname{supp}\mu}\Sigma$, and since $[1,T]$ is compact it suffices to show that (4.3) is closed. However, (4.3) is the intersection of $[1,T]\times SN^{*}_{\operatorname{supp}\mu}\Sigma$ with the preimage of $SN^{*}_{\operatorname{supp}\mu}\Sigma$ under the continuous map

[TABLE]

Since $SN^{*}_{\operatorname{supp}\mu}\Sigma$ is closed, (4.3) is closed.

Since ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$ is closed and has measure zero, there is $\tilde{b}\in C^{\infty}(SN^{*}\Sigma)$ supported on a neighborhood of $SN^{*}_{\operatorname{supp}\mu}\Sigma$ with $0\leq\tilde{b}(x,\xi)\leq 1$, $\tilde{b}(x,\xi)\equiv 1$ on an open neighborhood of ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$, and

[TABLE]

We use the coordinates in (2.1) and define

[TABLE]

hence (2). We set $\psi\in C_{0}^{\infty}(\Sigma)$ to be a cutoff function supported on a neighborhood of $\operatorname{supp}\mu$ with $\psi\equiv 1$ on $\operatorname{supp}\mu$. Defining

[TABLE]

yields (1). We have (3) since the support of $1-\tilde{b}(x,\xi)$ contains no elements of ${\mathcal{L}}_{\Sigma}(\operatorname{supp}\mu,T)$. ∎

Returning to the proof of Proposition 4.1, let $X_{T}$ denote the function with

[TABLE]

and let $X_{T,\lambda}$ denote the operator with kernel

[TABLE]

We use part (1) of Lemma 4.2 to write the integral in (4.2) as

[TABLE]

We claim the first three terms on the right are $O_{T}(\lambda^{-N})$ for $N=1,2,\ldots$. We will only prove this for the first term – the argument is the same for the second term and the bound for the third term follows since $X_{T,\lambda}$ is self-adjoint. Interpreting $\mu$ as a distribution on $M$, we write formally

[TABLE]

Once we show

[TABLE]

the integral over $M$ will be smooth in $t$ by Proposition 3.1. Integration by parts in $t$ then gives the desired bound of $O_{T}(\lambda^{-N})$. To prove (4.5), suppose $(x,\xi)\in\operatorname{WF}(B^{*}\mu)$. By part (3) of Lemma 4.2, $\Phi_{t}(x,\xi)$ is not in $SN^{*}_{\operatorname{supp}\mu}\Sigma$ for any $1\leq|t|\leq T$. By propagation or singularities,

[TABLE]

hence

[TABLE]

Since the support of $\mu$ has been made small, if there is $(x,\xi)\in SN_{\operatorname{supp}\mu}^{*}\Sigma$ and some $t>0$ in the support of $(1-\beta(t))\hat{\chi}(t/T)$ for which $\Phi_{t}(x,\xi)\in SN_{\operatorname{supp}\mu}^{*}\Sigma$, then $t\geq 1$ since the diameter of $\operatorname{supp}\mu$ is small and the injectivity radius of $M$ is at least $1$. We now have (4.5), from which follows (4.4) as promised.

What remains is to bound

[TABLE]

We have

[TABLE]

and so we write the integral in (4.7) as

[TABLE]

By the bounds

[TABLE]

and Proposition 3.2,

[TABLE]

Taking $\varepsilon$ in part (2) of Lemma 4.2 small enough so that $\varepsilon C_{T}\leq 1$ yields (4.7). This concludes the proof of Proposition 4.1.

5. Appendix: Stationary phase tool

The following tool is a combination of Corollary 1.1.8 with the discussion at the end of Section 1.1 in  [8]. Let $\phi(x,y)$ be a smooth phase function on $\mathbb{R}^{m}\times\mathbb{R}^{n}$ with

[TABLE]

and let $a(\lambda;x,y)$ be a smooth amplitude with small, adjustable support satisfying

[TABLE]

for $j=0,1,2,\ldots$ and multiindices $\alpha$ and $\beta$. $\nabla_{y}^{2}\phi\neq 0$ on a neighborhood of [math] by continuity. There exists locally a smooth map $x\mapsto y(x)$ whose graph in $\mathbb{R}^{m}\times\mathbb{R}^{n}$ contains all points in a neighborhood of [math] such that $\nabla_{y}\phi=0$, by the implicit function theorem. Let $p$ and $q$ be integers denoting the number of positive and negative eigenvalues of $\nabla_{y}^{2}\phi$, respectively, counting multiplicity. By continuity, $p$ and $q$ are constant on a neighborhood of [math]. We adjust the support of $a$ to lie in the intersection of these neighborhoods.

Proposition 5.1**.**

Let

[TABLE]

with $\phi$ and $a$ as above. Then,

[TABLE]

for $N=1,2,\ldots$, where $R$ has compact support,

[TABLE]

and the $O(\lambda^{-N})$ term is constant in $x$.

Proof.

We have

[TABLE]

where we have set

[TABLE]

The proof of the Morse-Bott lemma in  [1] lets us construct a smooth map $F:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $y\mapsto F(x,y)$ is a diffeomorphism between neighborhoods of [math] in $\mathbb{R}^{n}$ for each $x$, and for which

[TABLE]

and

[TABLE]

Applying a change of variables in $y$ to (5.1) yields

[TABLE]

where

[TABLE]

The first term evaluates to

[TABLE]

since

[TABLE]

and

[TABLE]

To estimate the second term, we let $\chi$ be a smooth compactly supported cutoff function with $\chi(|y|)=1$ for all $y\in\operatorname{supp}_{y}a$. Then,

[TABLE]

respectively. Since $r(\lambda;x,y)$ vanishes for $y=0$,

[TABLE]

by  [8, Lemma 1.1.6] applied to the $x$-derivatives of $R_{1}$. Finally,

[TABLE]

by integration by parts. ∎