Looping Directions and Integrals of Eigenfunctions over Submanifolds

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.,

$$\begin{aligned} -\Delta _g e_\lambda = \lambda ^2 e_\lambda \quad \text {and} \quad \left\| e_\lambda \right\| _{L^2(M)} = 1. \end{aligned}$$

Let $\varSigma$ be a d-dimensional submanifold and $\mathrm{d}\mu$ a smooth, compactly supported measure on $\varSigma .$ It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992 in far greater generality) that

$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^\frac{n-d-1}{2}\right) . \end{aligned}$$

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

$$\begin{aligned} {{\mathcal {L}}}_{\varSigma } = \{ (x,\,\xi ) \in \mathrm{SN}^*\varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*\varSigma \text { for some } t > 0 \} \end{aligned}$$

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

1 Introduction

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

$$\begin{aligned} -\varDelta _g e_\lambda = \lambda ^2 e_\lambda \quad \text {and} \quad \left\| e_\lambda \right\| _{L^2(M)} = 1. \end{aligned}$$

In [7], Sogge and Zelditch investigate which manifolds have a sequence of eigenfunctions $e_\lambda$ with $\lambda \rightarrow \infty$ which saturate the standard sup-norm bound

$$\begin{aligned} \left\| e_\lambda \right\| _{L^\infty (M)} = O\left( \lambda ^\frac{n-1}{2}\right) . \end{aligned}$$

They show this bound necessarily improves to $o\left( \lambda ^\frac{n-1}{2}\right)$ if at each x,  the set of looping directions through x, 

$$\begin{aligned} {{\mathcal {L}}}_x = \left\{ \xi \in S_x^* M : \varPhi _t(x,\,\xi ) \in S_x^*M \text { for some } t > 0 \right\} \end{aligned}$$

has measure zero¹ as a subset of $S^*_xM$ for each $x \in M.$ Here, $\varPhi _t$ denotes the geodesic flow on the unit cosphere bundle $S^*M$ after time t. The hypotheses were later weakened by Sogge et al. [8], where they showed

$$\begin{aligned} \left\| e_\lambda \right\| _{L^\infty (M)} = o\left( \lambda ^\frac{n-1}{2}\right) \end{aligned}$$

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

We are interested in extending the result in [7] to integrals of eigenfunctions over submanifolds. Let $\varSigma$ be a submanifold of dimension d with $d < n$ and a measure $\mathrm{d}\mu (x) = h(x)\mathrm{d}\sigma (x)$ where $\mathrm{d}\sigma$ is the surface measure on $\varSigma$ and h is a smooth function supported on a compact subset of $\varSigma .$ In his 1992 paper [12], Zelditch proves, among other things, a Kuznecov asymptotic formula

$$\begin{aligned} \sum _{\lambda _j \le \lambda } \left| \int _\varSigma e_j \, \mathrm{d}\mu \right| ^2 \sim \lambda ^{n-d} + O\left( \lambda ^{n-d-1}\right) , \end{aligned}$$

(1.1)

where $e_j$ for $j = 0,\,1,\,2,\ldots$ form a Hilbert basis of eigenfunctions on M with corresponding eigenvalues $\lambda _j.$ From (1.1) follows the standard bound

$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^{\frac{n-d-1}{2}}\right) \end{aligned}$$

(1.2)

which is sharp² in general. However, it should be noted that (1.1) implies that generic eigenfunctions satisfy much better bounds. Indeed for any function $R(\lambda ) \rightarrow +\infty ,$ an extraction argument shows there exists a density one sequence of eigenfunctions satisfying

$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^{-\frac{d}{2}} R(\lambda )\right) . \end{aligned}$$

Though (1.2) is already well known and has been proven in stronger terms, we state it here as a theorem. We do this for two reasons. First, it provides a baseline with which to compare our main result. Second, we end up providing a direct proof in the form of Proposition 2.1, which we will need for our main argument anyway.

Theorem 1.1

Let $\varSigma$ be a d-dimensional submanifold with $0 \le d < n,$ and $\mathrm{d}\mu (x) = h(x) \mathrm{d}\sigma (x)$ where h is a smooth, real-valued function supported on a compact neighborhood in $\varSigma .$ Then,

$$\begin{aligned} \sum _{\lambda _j \in [\lambda ,\, \lambda +1]} \left| \int _\varSigma e_j \, \mathrm{d}\mu \right| ^2 = O\left( \lambda ^{n-d-1}\right) . \end{aligned}$$

(1.2) follows.

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

$$\begin{aligned} {{\mathcal {L}}}_\varSigma = \{ (x,\,\xi ) \in \mathrm{SN}^* \varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*\varSigma \text { for some } t > 0 \}. \end{aligned}$$

A covector in ${{\mathcal {L}}}_\varSigma$ is the initial data of a geodesic which departs $\varSigma$ conormally and eventually arrives again at $\varSigma$ conormally. Our main result is as follows.

Theorem 1.2

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

$$\begin{aligned} \sum _{\lambda _j \in [\lambda ,\, \lambda + \delta ]} \left| \int _\varSigma e_\lambda \, \mathrm{d}\mu \right| ^2 \le C\delta \lambda ^{n-d-1} + C_\delta \lambda ^{n-d-2}, \end{aligned}$$

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

Sogge and Zelditch’s result [7] implies the theorem for $d = 0.$ We adapt their strategy to provide the proof for the remaining cases $d \ge 1.$ The following theorem is an immediate corollary of Theorem 1.2 and shows (1.2) cannot be saturated whenever ${{\mathcal {L}}}_\varSigma$ has measure zero in $\mathrm{SN}^*\varSigma .$

Theorem 1.3

Assume the hypotheses of Theorem 1.2. Then,

$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = o\left( \lambda ^\frac{n-d-1}{2}\right) . \end{aligned}$$

We illustrate Theorem 1.3 in three settings: the torus, the sphere, and surfaces with negative sectional curvature.

The torus Let ${\mathbb {T}}^n = {\mathbb {R}}^n / 2\pi {\mathbb {Z}}^n$ denote the flat, n-dimensional torus. Let $\varSigma$ be a small patch of a sphere centered at 0 in ${\mathbb {T}}^n.$ Since all geodesics passing through $\varSigma$ in the conormal direction intersect the origin, the set of looping directions ${{\mathcal {L}}}_\varSigma$ is countable by the countability of ${\mathbb {Z}}^n$ and hence has measure 0. The conclusion of Theorem 1.3 is easily verified by the result [3, Proposition 3.5], which provides an essentially optimal bound on the integrals of eigenfunctions over hypersurfaces in the torus.

Proposition 1.4

([3]) Suppose $\varSigma$ has nonvanishing Gaussian curvature in the torus. Then,

$$\begin{aligned} \left| \int _\varSigma e_\lambda \mathrm{d}\mu (x)\right| = O\left( \lambda ^{-1/2 + \varepsilon }\right) \end{aligned}$$

for all $\varepsilon > 0,$ where we may set $\varepsilon = 0$ if $n \ge 5.$

Applying this result to spheres yields a much better bound than suggested by Theorem 1.3.

On the other hand if $\varSigma$ is a closed hyperplane in ${\mathbb {T}}^n,$ neither the hypotheses nor the conclusion of Theorem 1.3 are satisfied. All geodesics departing $\varSigma$ conormally arrive again conormally after some fixed, uniform time. At the same time, one can construct a sequence of exponentials which are all identically 1 along $\varSigma .$

The sphere Let $S^n$ denote the n-dimensional sphere equipped with the standard metric. Every geodesic in $S^n$ is periodic, so ${{\mathcal {L}}}_\varSigma = \mathrm{SN}^*\varSigma$ for every submanifold $\varSigma .$ Hence, no submanifold of $S^n$ satisfies the hypotheses of Theorem 1.3. As we will find, no submanifold of $S^n$ enjoys the little-o improvement of Theorem 1.3 either.

The functional ideas here are Zelditch’s–Kuznecov formula (1.1) and the fact that all eigenvalues are of the form

$$\begin{aligned} \lambda = \sqrt{k(k + n - 1)} \quad \text {for some } k = 0,\,1,\,2,\ldots \end{aligned}$$

(see [5, Sect. 3.4] or [2, Theorem 3.1]). Let $e_j$ for $j = 0,\,1,\,2,\ldots$ denote some Hilbert basis of eigenfunctions on M with corresponding eigenvalues $\lambda _j.$ For each distinct eigenvalue $\lambda ,$ we construct an eigenfunction $e_\lambda$ by

$$\begin{aligned} e_\lambda = \frac{\displaystyle \sum \nolimits _{\lambda _j = \lambda } \left( \int _\varSigma \overline{e_j} \, \mathrm{d}\sigma \right) e_j }{\displaystyle \left( \sum \nolimits _{\lambda _j = \lambda } \left| \int _{\varSigma } \overline{e_j} \, \mathrm{d}\sigma \right| ^2 \right) ^{1/2}}. \end{aligned}$$

Note $\Vert e_\lambda \Vert _{L^2(M)} = 1$ and

$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\sigma = \left( \sum _{\lambda _j = \lambda } \left| \int _{\varSigma } e_j \, \mathrm{d}\sigma \right| ^2 \right) ^{1/2}. \end{aligned}$$

(1.3)

By Zelditch’s–Kuznecov formula (1.1), we have

$$\begin{aligned} \sum _{\lambda _j \in [\lambda , \,\lambda + C]} \left| \int _\varSigma e_j \, \mathrm{d}\sigma \right| ^2 \ge \lambda ^{n-d-1} \end{aligned}$$

for some large enough constant C. Moreover there are no more than $C + 1$ distinct eigenvalues in the interval $[\lambda ,\, \lambda + C]$ for each $\lambda .$ Hence in every interval of length C,  there exists $\lambda$ such that the right-hand side of (1.3) is bounded below by $\lambda ^{\frac{n-d-1}{2}}/\sqrt{C+1},$ i.e., the bound in (1.2) is saturated.

Negatively curved surfaces We can use recent results to verify Theorem 1.3 for some examples where M is a surface ($n = 2$) with negative sectional curvature. Chen and Sogge [1] proved that if $\varSigma$ is a geodesic in such a manifold M, 

$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = o(1). \end{aligned}$$

They consider a lift ${\tilde{\varSigma }}$ of $\varSigma$ to the universal cover of M. Using the Gauss–Bonnet theorem, they show for each deck transformation $\alpha ,$ there is at most one geodesic which intersects both ${\tilde{\varSigma }}$ and $\alpha ({\tilde{\varSigma }})$ perpendicularly. Since there are only countably many deck transformations, ${{\mathcal {L}}}_\varSigma$ is at most a countable subset of $\mathrm{SN}^*\varSigma$ and so satisfies the hypotheses of Theorem  1.3. Since [1], Sogge et al. [9] have obtained an explicit decay of $O(1/\sqrt{\log \lambda })$ while also allowing the sectional curvature of M to vanish of finite order. Recently the author [10, 11] obtained Chen and Sogge’s o(1) bound, and more recently the explicit bound $O(1/\sqrt{\log \lambda }),$ if M has nonpositive sectional curvature and the geodesic curvature of $\varSigma$ avoids that of circles of infinite radius. These curves similarly have countable ${{\mathcal {L}}}_\varSigma$ provided they are sufficiently short.

2 Microlocal Tools

The hypotheses on the looping directions in Theorem 1.2 ensure that the wavefront sets of $\mu$ and $\mathrm{e}^{it\sqrt{-\varDelta _{g}}} \mu$ have minimal intersection for any given t away from 0. We can then use pseudodifferential operators to break the measure $\mu$ into two parts, the first which has small essential support and the second whose wavefront set is disjoint from that of $\mathrm{e}^{it\sqrt{-\varDelta _{g}}}\mu .$ The following propositions will allow us to handle these cases, respectively. The first proposition generalizes both [5, Lemma 5.2.2] and the standard Theorem 1.1 and is our main technical result.

Before we proceed, we lay out some Fermi local coordinates which we will return to repeatedly. Fix $p \in \varSigma ,$ and consider local coordinates $x = (x_1,\ldots , x_n) = (x',\,{\bar{x}})$ centered about p, where $x'$ denotes the first d coordinates and ${\bar{x}}$ the remaining $n-d$ coordinates. We let $(x',\,0)$ parametrize $\varSigma$ on a neighborhood of p in such a way that $\mathrm{d}x'$ agrees with the surface measure on $\varSigma .$ Let g denote the metric tensor with respect to our local coordinates. We require

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

where I here is the $(n-d) \times (n-d)$ identity matrix. This is ensured after inductively picking smooth sections $v_j(x')$ of $\mathrm{SN}\varSigma$ for $j = d+1,\ldots , n$ with $\langle v_i,\, v_j \rangle = \delta _{ij},$ and then using

$$\begin{aligned} \left( x_1,\ldots , x_n\right) \mapsto \exp \left( x_{d+1} v_{d+1}(x') + \cdots + x_{n} v_{n}(x')\right) \end{aligned}$$

(2.1)

as our coordinate map. In these coordinates we write $\mathrm{d}\mu (x) = h(x') \ \mathrm{d}x'$ where h is a smooth, compactly supported function on ${\mathbb {R}}^d.$

Proposition 2.1

Let $b(x,\,\xi )$ be smooth for $\xi \ne 0$ and homogeneous of degree 0 in the $\xi$ variable. We define $b \in \varPsi _\mathrm{{cl}}^0(M)$ by

$$\begin{aligned} b(x,\,D)f(x) = \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n} \mathrm{e}^{i\langle x - y , \xi \rangle } b(x,\,\xi ) \, \mathrm{d}y \, \mathrm{d}\xi \end{aligned}$$

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

$$\begin{aligned}&\sum _{\lambda _j \in [\lambda ,\,\lambda +1]} \left| \int _\varSigma b e_j \, \mathrm{d}\mu \right| ^2\\&\quad \le C \left( \int _{{\mathbb {R}}^d} \int _{S^{n-d-1}} |b(x',\,\omega )|^2 h(x')^2 \, \mathrm{d}\omega \, \mathrm{d}x' \right) \lambda ^{n-d-1} + C_{b}\lambda ^{n-d-2}, \end{aligned}$$

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

Note Theorem 1.1 follows by setting $b \equiv 1.$ The proof is based largely on that of [5, Lemma 5.2.2]. We will come to a point in our argument where it seems like we may have to perform a stationary phase argument involving an eight-by-eight Hessian matrix. Instead, we appeal to [4, Theorem 7.7.6] to break this process into two steps involving two four-by-four Hessian matrices.

Proof

For simplicity, we assume without loss of generality that $\mathrm{d}\mu$ is a real measure. Let $\chi$ be a nonnegative Schwartz-class function on ${\mathbb {R}}$ with $\chi (0) = 1$ and ${\hat{\chi }}$ supported on a small neighborhood of 0.³ It suffices to show

$$\begin{aligned}&\sum _j \chi \left( \lambda _j - \lambda \right) \left| \int _\varSigma b e_j \, \mathrm{d}\mu \right| ^2\\&\quad \sim \left( \int _{{\mathbb {R}}^d} \int _{S^{n-d-1}} |b(y',\,\omega )|^2 h(y')^2 \, \mathrm{d}\omega \, \mathrm{d}y' \right) \lambda ^{n-d-1} + O_{b}\left( \lambda ^{n-d-2}\right) . \end{aligned}$$

We may by a partition of unity assume that $b(x,\,D)$ has small x-support. The left-hand side is

$$\begin{aligned}&=\sum _{j} \int _\varSigma \int _\varSigma \chi \left( \lambda _j - \lambda \right) b(x,\,D) e_j(x) \overline{b(y,\,D) e_j(y)} \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \nonumber \\&= \frac{1}{2\pi } \sum _j \int _\varSigma \int _\varSigma \int _{-\infty }^\infty {\hat{\chi }}(t) \mathrm{e}^{it(\lambda _j - \lambda )} b(x,\,D) e_j(x) \overline{b(y,\,D) e_j(y)} \, \mathrm{d}t \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \nonumber \\&= \frac{1}{2\pi } \int _\varSigma \int _\varSigma \int _{-\infty }^\infty {\hat{\chi }}(t) \mathrm{e}^{-it \lambda } b e^{it \sqrt{-\varDelta _{g}}} b^*(x,\,y) \, \mathrm{d}t \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y), \end{aligned}$$

(2.2)

where here $b\mathrm{e}^{it\sqrt{-\varDelta _{g}}}b^*(x,\,y)$ is the kernel

$$\begin{aligned} \sum _j \mathrm{e}^{it\lambda _j} b(x,\,D)e_j(x) \overline{b(y,\,D) e_j(y)} \end{aligned}$$

of the half-wave operator $\mathrm{e}^{it\sqrt{-\varDelta _{g}}}$ conjugated by b. Set $\beta \in C_0^\infty ({\mathbb {R}})$ with small support and where $\beta \equiv 1$ near 0. Then,

$$\begin{aligned}&\int _{{\mathbb {R}}^d} b(x',\,D)f(x')h(x') \, \mathrm{d}x'\nonumber \\&\quad = \frac{\lambda ^n}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \mathrm{e}^{i \lambda \langle x' - w, \eta \rangle } b(x',\,\eta ) f(w) h(x') \, \mathrm{d}x' \, \mathrm{d}w \, \mathrm{d}\eta \nonumber \\&\quad = \frac{\lambda ^n}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \mathrm{e}^{i \lambda \langle x' - w, \eta \rangle } \beta (\log |\eta |) b(x',\,\eta ) f(w) h(x') \, \mathrm{d}x' \, \mathrm{d}w \, \mathrm{d}\eta \nonumber \\&\qquad + O\left( \lambda ^{-N}\right) , \end{aligned}$$

(2.3)

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 $x'.$ Additionally,

$$\begin{aligned} b^*(z,\,D)d\mu (z)&= \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \mathrm{e}^{i\langle z - y', \zeta \rangle } b(y',\,\zeta ) h(y') \, \mathrm{d}y' \, \mathrm{d}\zeta \nonumber \\&= \frac{\lambda ^n}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \mathrm{e}^{i \lambda \langle z - y', \zeta \rangle } b(y',\,\zeta ) h(y') \, \mathrm{d}y' \, \mathrm{d}\zeta \nonumber \\&= \frac{\lambda ^n}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \mathrm{e}^{i \lambda \langle z - y', \zeta \rangle } \beta (\log |\zeta |) b(y',\,\zeta ) h(y') \, \mathrm{d}y' \, \mathrm{d}\zeta + O\left( \lambda ^{-N}\right) \nonumber \\&= \frac{\lambda ^n}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \mathrm{e}^{i \lambda \langle z - y', \zeta \rangle } \beta (\log |\zeta |) \beta (|z - y'|) b(y',\,\zeta ) h(y') \, \mathrm{d}y' \, \mathrm{d}\zeta \nonumber \\&\quad + O\left( \lambda ^{-N}\right) , \end{aligned}$$

(2.4)

where the second and third lines are obtained similarly as before and the fourth line is obtained after multiplying by $\beta (\log |z - y'|)$ and integrating the remainder by parts in $\zeta .$

We will use Hörmander’s parametrix of the half-wave operator $\mathrm{e}^{it\sqrt{-\varDelta _{g}}}$ on M to treat the integral in t. In local coordinates (2.1), we have

$$\begin{aligned} \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(x,\,y) = \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} \mathrm{e}^{i(\varphi (x,y,\xi ) + tp(y,\xi ))} q(t,\,x,\,y,\,\xi ) \, \mathrm{d}\xi \end{aligned}$$

modulo a smooth kernel where

$$\begin{aligned} p(y,\,\xi ) = \sqrt{\sum _{j,k} g^{ij}(y) \xi _i \xi _j}, \end{aligned}$$

where q is a symbol in $\xi$ satisfying bounds

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

for all multiindices $\alpha$ and $\beta ,$ and where $\varphi$ is homogeneous of degree 1 in $\xi$ and smooth for $\xi \ne 0$ and satisfies

$$\begin{aligned} \varphi (x,\,y,\,\xi ) = \langle x - y ,\,\xi \rangle + O\left( |x - y|^2 |\xi |\right) . \end{aligned}$$

(2.5)

This parametrix is valid only when |t| and $|x - y|$ are small. In fact, we may take q to be supported on an arbitrarily small neighborhood of the diagonal $x = y$ of our choosing provided we only consider times t on a correspondingly small neighborhood of 0. (See [6, Chap. 4] for a treatment of Hörmander’s parametrix.)

Using Hörmander’s parametrix,

$$\begin{aligned}&\frac{1}{2\pi } \int _{-\infty }^\infty {\hat{\chi }}(t) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(w,\,z) \, \mathrm{d}t \nonumber \\&\quad = \frac{1}{(2\pi )^{n+1}} \int _{-\infty }^\infty \int _{{\mathbb {R}}^n} \mathrm{e}^{i(\varphi (w,z,\xi ) + tp(z,\xi ) - t\lambda )} {\hat{\chi }}(t) q(t,\,w,\,z,\,\xi ) \, \mathrm{d}\xi \, \mathrm{d}t \nonumber \\&\quad = \frac{\lambda ^n}{(2\pi )^{n+1}} \int _{-\infty }^\infty \int _{{\mathbb {R}}^n} \mathrm{e}^{i\lambda (\varphi (w,z,\xi ) + t(p(z,\xi ) - 1))} {\hat{\chi }}(t) q(t,\,w,\,z,\,\lambda \xi ) \, \mathrm{d}\xi \, \mathrm{d}t \nonumber \\&\quad = \frac{\lambda ^n}{(2\pi )^{n+1}} \int _{-\infty }^\infty \int _{{\mathbb {R}}^n} \mathrm{e}^{i\lambda (\varphi (w,z,\xi ) + t(p(z,\xi ) - 1))} \beta (\log p(z,\,\xi )) {\hat{\chi }}(t) q(t,\,w,\,z,\,\lambda \xi ) \, \mathrm{d}\xi \nonumber \\&\qquad + O\left( \lambda ^{-N}\right) . \end{aligned}$$

(2.6)

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 (2.3), (2.4), and (2.6), we write (2.2) as

$$\begin{aligned}&\lambda ^{3n} \int \cdots \int \mathrm{e}^{i\varPhi (t,x',y',w,z,\eta ,\zeta ,\xi )} a(\lambda ;\,t,\,x',\,y',\,w,\,z,\,\eta ,\,\zeta ,\,\xi ) \nonumber \\&\quad \mathrm{d}x' \, \mathrm{d}y' \, \mathrm{d}w \, \mathrm{d}z \, \mathrm{d}\eta \, \mathrm{d}\zeta \, \mathrm{d}\xi + O\left( \lambda ^{-N}\right) \end{aligned}$$

(2.7)

with amplitude

$$\begin{aligned}&a(\lambda ;\,t,\,x',\,y',\,w,\,z,\,\eta ,\,\zeta ,\,\xi ) = \frac{1}{(2\pi )^{3n+1}} {\hat{\chi }}(t) q(t,\,w,\,z,\,\lambda \xi ) \beta (\log p(z,\,\xi ))\beta (\log |\eta |) \\&\quad \beta (\log |\zeta |)\beta (|z - y'|) b(x',\,\eta ) b(y',\,\zeta ) h(x') h(y') \end{aligned}$$

and phase

$$\begin{aligned} \varPhi (t,\,x',\,y',\,w,\,z,\,\eta ,\,\zeta ,\,\xi ) = \langle x' - w, \,\eta \rangle + \varphi (w,\,z,\,\xi ) + t(p(z,\,\xi ) - 1) + \langle z - y',\, \zeta \rangle . \end{aligned}$$

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

$$\begin{aligned} \nabla _w \varPhi&= - \eta + \nabla _w \varphi (w,\,z,\,\xi ), \\ \nabla _z \varPhi&= \nabla _z \varphi (w,\,z,\,\xi ) + t \nabla _z p(z,\,\xi ) + \zeta , \\ \nabla _\eta \varPhi&= x' - w, \\ \nabla _\zeta \varPhi&= z - y' \end{aligned}$$

which all simultaneously vanish if and only if

$$\begin{aligned} (w,\,z,\,\eta ,\,\zeta ) = \left( x',\,y',\, \nabla _x \varphi (x',\,y',\,\xi ),\, -\nabla _y \varphi (x',\,y',\,\xi ) - t\nabla _y p(y',\,\xi )\right) . \end{aligned}$$

(2.8)

At such a critical point we have the Hessian matrix

$$\begin{aligned} \nabla _{w,z,\eta ,\zeta }^2 \varPhi = \left[ \begin{array}{cccc} * &{}\quad * &{}\quad -I &{}\quad 0 \\ * &{}\quad * &{}\quad 0 &{}\quad I \\ -I &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad I &{}\quad 0 &{}\quad 0 \end{array} \right] , \end{aligned}$$

which has determinant $-1.$ By [4, Theorem 7.7.6], (2.7) is equal to a complex constant times

$$\begin{aligned}&\lambda ^n \int _{-\infty }^\infty \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \int _{{\mathbb {R}}^d} \mathrm{e}^{i\lambda \varPhi (t,\,x',\,y',\,\xi )} a(\lambda ; \,t,\, x',\, y',\, \xi ) \, \mathrm{d}x' \, \mathrm{d}y' \, \mathrm{d}\xi ' \, \mathrm{d}t \nonumber \\&\quad + \lambda ^{n-1} \int _{-\infty }^\infty \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^d} \int _{{\mathbb {R}}^d} \mathrm{e}^{i\lambda \varPhi (t,x',y',\xi )} R_N(\lambda ;\, t, \,x',\, y',\, \xi ) \, \mathrm{d}x' \, \mathrm{d}y' \, \mathrm{d}\xi ' \, \mathrm{d}t\nonumber \\&\quad + O\left( \lambda ^{-N}\right) \end{aligned}$$

(2.9)

where we have phase

$$\begin{aligned} \varPhi (t,\,x',\,y',\,\xi ) = \varphi (x',\,y',\,\xi ) + t(p(y',\,\xi ) - 1), \end{aligned}$$

amplitude

$$\begin{aligned}&a(\lambda ;\, t,\, x',\, y',\, \xi ) = a(\lambda ;\,t,\,x',\,y',\,w,\,z,\,\eta ,\,\zeta ,\,\xi ) \end{aligned}$$

with $w,\,z,\,\eta ,$ and $\zeta$ subject to the constraints (2.8), where $R_N$ is a compactly supported smooth function in $t,\,x',\,y',$ and $\xi ,$ whose derivatives are bounded uniformly with respect to $\lambda ,$ and where N can be taken to be as large as desired.

Write $\xi = (\xi ',\, {\bar{\xi }})$ and write ${\bar{\xi }} = r \omega$ in polar coordinates with $r \ge 0$ and $\omega \in S^{n-d-1}.$ The first integral in (2.9) is then written

$$\begin{aligned}&\lambda ^n \int _{-\infty }^\infty \int _{{\mathbb {R}}^d} \int _{{\mathbb {R}}^d} \int _{{\mathbb {R}}^d} \int _{S^{n-d-1}} \int _0^\infty \mathrm{e}^{i \lambda \varPhi (t,x',y',\xi )} a(\lambda ;\, t,\, x',\, y',\, \xi )\\&\quad r^{n-d-1} \,\mathrm{d}r \, \mathrm{d}\omega \, \mathrm{d}\xi ' \, \mathrm{d}x' \, \mathrm{d}y' \, \mathrm{d}t. \end{aligned}$$

We will fix $y'$ and $\omega$ and use the method of stationary phase in the remaining variables t,  $x',$$\xi ',$ and r (a total of $2d + 2$ dimensions). We assert that, for fixed $y'$ and $\omega ,$ there is a nondegenerate stationary point at $(t,\,x',\,\xi ',\,r) = (0,\,y',\,0,\,1).$$\varPhi = 0$ at such a stationary point, and after perhaps shrinking the support of a we apply [4, Theorem 7.7.6] again to write the first integral in (2.9) as constant times

$$\begin{aligned} \lambda ^{n-d-1} \int _{{\mathbb {R}}^d} \int _{S^{n-d-1}} a(\lambda ;\, 0,\, y',\, y',\, \omega ) \, \mathrm{d}y' \, \mathrm{d}\omega + O\left( \lambda ^{n-d-2}\right) . \end{aligned}$$

The proposition will follow after noting $a(\lambda ;\, 0,\, y',\, y',\, \omega ) = |b(y',\,\omega )|^2 h(y')^2$ and applying the same stationary phase argument to the second integral in (2.9).

We have

$$\begin{aligned} \partial _t \varPhi&= p(y',\,\xi ) - 1, \\ \nabla _{x'} \varPhi&= \nabla _{x'} \varphi (x',\,y',\,\xi ) , \\ \nabla _{\xi '} \varPhi&= \nabla _{\xi '} \varphi (x',\,y',\,\xi ) + t \nabla _{\xi '}p(y',\,\xi ), \\ \partial _r \varPhi&= \partial _r \varphi (x',\,y',\,\xi ) + t \partial _r p(y',\,\xi ). \end{aligned}$$

Note for fixed $y'$ and $\omega ,$$(t,\,x',\,\xi ',\,r) = (0,\,y',\,0,\,1)$ is a critical point of $\varPhi .$ Now we compute the second derivatives at this point. We immediately see that $\partial _t^2 \varPhi ,$$\partial _t \nabla _{x'}\varPhi ,$$\nabla _{\xi '}^2 \varPhi ,$$\partial _r \nabla _{\xi '} \varPhi ,$ and $\partial _r^2 \varPhi$ all vanish. Moreover, $\partial _r \partial _r \varPhi = 1$ since $p(y',\,\xi ) = r$, where $\xi ' = 0.$ By our coordinates (2.1) and the fact that $[g^{ij}]_{i,j \le d}$ is necessarily positive definite,

$$\begin{aligned} p(y',\,\xi ) = \sqrt{\sum _{j,k} g^{jk} \xi _j \xi _k} = \sqrt{r^2 + \sum _{j,k \le d} g^{jk} \xi _j' \xi _k' } \ge r = p(y',\,r\omega ). \end{aligned}$$

Hence, $\partial _t \nabla _{\xi '} \varPhi = \nabla _{\xi '} p(y',\,\xi ) = 0.$ Since $\varphi$ is homogeneous of degree 1 in $\xi ,$ at $\xi ' = 0$ and $t = 0,$

$$\begin{aligned} \nabla _{x'}\partial _r \varPhi = \nabla _{x'} \partial _r \varphi (x',\,y',\,\xi ) = \nabla _{x'} \varphi (x',\,y',\,\omega ) = 0 \end{aligned}$$

since $\varphi (x',\,y',\,\omega ) = O(|x' - y'|^2)$ by (2.5) and the fact that $\langle x' - y', \,\omega \rangle = 0.$ Finally by (2.5),

$$\begin{aligned} \nabla _{\xi '} \varphi (x',\,y',\,\xi ' + \omega ) = x' + O\left( |x' - y'|^2\right) \end{aligned}$$

whence at the critical point

$$\begin{aligned} \nabla _{x'} \nabla _{\xi '} \varPhi = I, \end{aligned}$$

the $d \times d$ identity matrix. In summary, the Hessian matrix of $\varPhi$ at the critical point $(t,\,x',\,\xi ',\,r) = (0,\,y',\,0,\,1)$ is

$$\begin{aligned} \nabla _{t,x',\xi ',r}^2 \varPhi = \left[ \begin{array}{cccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad * &{}\quad I &{}\quad 0 \\ 0 &{}\quad I &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{array} \right] \end{aligned}$$

which has full rank. This concludes the proof of Proposition  2.1. $\square$

The second proposition, below, allows us to deal with the partition of $\mu$ whose wavefront set is disjoint from that of $\mathrm{e}^{it\sqrt{-\varDelta _{g}}}\mu$ for $t > 0.$

Proposition 2.2

Let u and v be distributions on M for which

$$\begin{aligned} {\text {WF}}(u) \cap {\text {WF}}(v) = \emptyset . \end{aligned}$$

Then

$$\begin{aligned} t \mapsto \int _M \mathrm{e}^{it\sqrt{-\varDelta _{g}}} u(x) \overline{v(x)} \, \mathrm{d}x \end{aligned}$$

is a smooth function of t on some neighborhood of 0.

Proof

Using a partition of unity, we write

$$\begin{aligned} I = \sum _j A_j \end{aligned}$$

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

$$\begin{aligned} \int _{M} \mathrm{e}^{it \sqrt{-\varDelta _{g}}} u(x) \overline{v(x)} \, \mathrm{d}x = \sum _{j,k} \int _{M} A_j\mathrm{e}^{it\sqrt{-\varDelta _{g}}}u(x) \overline{A_k v(x)} \, \mathrm{d}x. \end{aligned}$$

We are done if for each i and j, 

$$\begin{aligned} \int _M A_j \mathrm{e}^{it\sqrt{-\varDelta _{g}}} u(x) \overline{A_k v(x)} \, \mathrm{d}x \quad \text {is smooth for } |t| \ll 1. \end{aligned}$$

(2.10)

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

$$\begin{aligned} \int _{M} u(x) \overline{\mathrm{e}^{it\sqrt{-\varDelta _{g}}} A_j^* A_k v(x)} \, \mathrm{d}x \end{aligned}$$

is smooth is t. Assume the essential support of $A_j$ is small enough so that for each j there exists a small conic neighborhood $\varGamma _j$ which fully contains the essential support of $A_k$ if it intersects the essential support of $A_j.$ We in turn take $\varGamma _j$ small enough so that for each j,  $\overline{ \varGamma _j}$ either does not intersect ${\text {WF}}(u)$ or does not intersect ${\text {WF}}(v).$ In the latter case, $A_k v$ is smooth and we have (2.10) as before. In the former case,

$$\begin{aligned} \overline{\varGamma _j} \cap {\text {WF}}\left( \mathrm{e}^{it\sqrt{-\varDelta _{g}}}u\right) = \emptyset \quad \text {for } |t| \ll 1 \end{aligned}$$

since both sets above are closed and the geodesic flow is continuous. Then $A_j \mathrm{e}^{it\sqrt{-\varDelta _{g}}} u(x)$ is smooth as a function of t and x,  and we have (2.10). $\square$

3 Proof of Theorem 1.2

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 $\mathrm{d}\mu$ 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 $\mathrm{d}\mu$ as needed, finitely many times.

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

$$\begin{aligned} \sum _j \chi \left( T\left( \lambda _j - \lambda \right) \right) \left| \int _\varSigma e_\lambda \, \mathrm{d}\mu \right| ^2 \le CT^{-1} \lambda ^{n-d-1} + C_T \lambda ^{n-d-2} \end{aligned}$$

for $T > 1.$ Similar to the reduction in the proof of Proposition  2.1, the left-hand side is equal to

$$\begin{aligned}&\sum _j \int _\varSigma \int _\varSigma \chi \left( T\left( \lambda _j - \lambda \right) \right) e_j(x) \overline{e_j(y)} \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y)\\&\quad = \frac{1}{2\pi } \sum _j \int _{-\infty }^\infty \int _\varSigma \int _\varSigma {\hat{\chi }}(t) \mathrm{e}^{itT(\lambda _j - \lambda )} e_j(x) \overline{e_j(y)} \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \, \mathrm{d}t\\&\quad = \frac{1}{2\pi T} \sum _j \int _{-\infty }^\infty \int _\varSigma \int _\varSigma {\hat{\chi }}(t/T) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\lambda _j} e_j(x) \overline{e_j(y)} \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \, \mathrm{d}t\\&\quad = \frac{1}{2\pi T} \int _{-\infty }^\infty \int _\varSigma \int _\varSigma {\hat{\chi }}(t/T) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(x,\,y) \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \, \mathrm{d}t. \end{aligned}$$

Hence, it suffices to show

$$\begin{aligned}&\left| \int _{-\infty }^\infty \int _\varSigma \int _\varSigma {\hat{\chi }}(t/T) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(x,\,y) \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \, \mathrm{d}t \right| \nonumber \\&\quad \le C \lambda ^{n-d-1} + C_T \lambda ^{n-d-2}. \end{aligned}$$

(3.1)

Let $\beta \in C_0^\infty ({\mathbb {R}})$ be supported on a small interval about 0 with $\beta \equiv 1$ near 0. We cut the integral in (3.1) into $\beta (t)$ and $1 - \beta (t)$ parts. Since $\beta (t){\hat{\chi }}(t/T)$ and its derivatives are all bounded independently of $T \ge 1,$

$$\begin{aligned} \left| \int _{-\infty }^\infty \int _\varSigma \int _\varSigma \beta (t) {\hat{\chi }}(t/T) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(x,\,y) \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \, \mathrm{d}t \right| \le C \lambda ^{n-d-1} \end{aligned}$$

by the arguments in Proposition 2.1. Hence, it suffices to show

$$\begin{aligned}&\left| \int _{-\infty }^\infty \int _\varSigma \int _\varSigma (1 - \beta (t)) {\hat{\chi }}(t/T) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(x,\,y) \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \, \mathrm{d}t \right| \nonumber \\&\quad \le C \lambda ^{n-d-1} + C_T \lambda ^{n-d-2}. \end{aligned}$$

(3.2)

Here we shrink the support of $\mu$ so that $\beta (d_g(x,\,y)) = 1$ for $x,\,y\in {\text {supp}}\,\mu .$ We now state and prove a useful decomposition based off of those in [7, 8], and [5, Chap. 5]. We let ${{\mathcal {L}}}_\varSigma ({\text {supp}}\,\mu ,\,T)$ denote the subset of ${{\mathcal {L}}}_\varSigma$ relevant to the support of $\mu$ and the timespan $[1,\,T],$ specifically

$$\begin{aligned}&{{\mathcal {L}}}_\varSigma ({\text {supp}}\,\mu ,\,T) = \left\{ (x,\,\xi ) \in \mathrm{SN}^*\varSigma : \varPhi _t(x,\,\xi ) = (y,\,\eta ) \in \mathrm{SN}^*\varSigma \right. \\&\quad \text {for some } t \in [1,\,T]\quad \text {and where } x,\,y \in {\text {supp}}\,\mu \}. \end{aligned}$$

Lemma 3.1

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

• (1) $b(x,\,D) + B(x,\,D) = I$ on ${\text {supp}}\,\mu .$

• (2) Using coordinates (2.1),

  $$\begin{aligned} \int _{{\mathbb {R}}^d} \int _{S^{n-d-1}} |b(x',\,\omega )|^2 \, \mathrm{d}\omega \, \mathrm{d}x' < \varepsilon , \end{aligned}$$

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

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

Proof

As shorthand, we write

$$\begin{aligned} \mathrm{SN}^*_{{\text {supp}}\,\mu } \varSigma = \{ (x,\,\xi ) \in \mathrm{SN}^* \varSigma : x \in {\text {supp}}\,\mu \}. \end{aligned}$$

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

$$\begin{aligned} \left\{ (t,\,x,\,\xi ) \in [1,\,T] \times \mathrm{SN}^*_{{\text {supp}}\,\mu } \varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*_{{\text {supp}}\,\mu } \varSigma \right\} \end{aligned}$$

(3.3)

onto $\mathrm{SN}^*_{{\text {supp}}\,\mu }\varSigma ,$ and since $[1,\,T]$ is compact it suffices to show that (3.3) is closed. However, (3.3) is the intersection of $[1,\,T] \times \mathrm{SN}^*_{{\text {supp}}\,\mu }\varSigma$ with the preimage of $\mathrm{SN}^*_{{\text {supp}}\,\mu } \varSigma$ under the continuous map

$$\begin{aligned} (t,\,x,\,\xi ) \mapsto \varPhi _t(x,\,\xi ). \end{aligned}$$

Since $\mathrm{SN}^*_{{\text {supp}}\,\mu }\varSigma$ is closed, (3.3) is closed.

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

$$\begin{aligned} \int _{{\mathbb {R}}^d}\int _{S^{n-d-1}} |{\tilde{b}}(x',\,\omega )|^2 \, \mathrm{d}\omega \, \mathrm{d}x' < \varepsilon . \end{aligned}$$

(3.4)

We set $\psi \in C_0^\infty (\varSigma )$ to be a cutoff function supported on a neighborhood of ${\text {supp}}\,\mu$ in M with $\psi \equiv 1$ on ${\text {supp}}\,\mu .$ We use the coordinates in (2.1) and define symbols

$$\begin{aligned} b(x,\,\xi ) = \psi (x) {\tilde{b}}(x,\,\xi /|\xi |) \end{aligned}$$

and

$$\begin{aligned} B(x,\,\xi ) = \psi (x) (1 - {\tilde{b}}(x',\,\xi /|\xi |)), \end{aligned}$$

along with their associated operators

$$\begin{aligned} b(x,\,D)f(x) = \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n} \mathrm{e}^{i\langle x - y, \xi \rangle } b(x,\,\xi ) f(y) \, \mathrm{d}y \, \mathrm{d}\xi \end{aligned}$$

and

$$\begin{aligned} B(x,\,D)f(x) = \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n} \mathrm{e}^{i\langle x - y, \xi \rangle } B(x,\,\xi ) f(y) \, \mathrm{d}y \, \mathrm{d}\xi . \end{aligned}$$

By construction,

$$\begin{aligned} B(x,\,D) + b(x,\,D) = \psi (x), \end{aligned}$$

whose restriction to ${\text {supp}}\,\mu$ is 1,  yielding (1). (2) Follows from the definition of $b(x,\,D)$ and (3.4). We have (3) since the support of $1 - {\tilde{b}}(x,\,\xi )$ contains no elements of ${{\mathcal {L}}}_\varSigma ({\text {supp}}\,\mu , \,T).$$\square$

Returning to the proof of Theorem 1.2, let $X_{T}$ denote the function with

$$\begin{aligned} {\hat{X}}_{T}(t) = (1 - \beta (t)) {\hat{\chi }}(t/T), \end{aligned}$$

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

$$\begin{aligned} X_{T,\lambda }(x,\,y) = \frac{1}{2\pi } \int _{-\infty }^\infty {\hat{X}}_T(t) \mathrm{e}^{-it\lambda } \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(x,\,y) \, \mathrm{d}t. \end{aligned}$$

We use part (1) of Lemma 3.1 to write the integral in (3.2) as

$$\begin{aligned} \int _\varSigma \int _\varSigma X_{T,\lambda }(x,\,y) \, \mathrm{d}\mu (y) \, \mathrm{d}\mu (x)&= \int _\varSigma \int _\varSigma B X_{T,\lambda } B^* (x,\,y) \, \mathrm{d}\mu (y) \, \mathrm{d}\mu (x) \\&\quad + \int _\varSigma \int _\varSigma B X_{T,\lambda } b^* (x,\,y) \, \mathrm{d}\mu (y) \, \mathrm{d}\mu (x) \\&\quad + \int _\varSigma \int _\varSigma b X_{T,\lambda } B^* (x,\,y) \, \mathrm{d}\mu (y) \, \mathrm{d}\mu (x) \\&\quad + \int _\varSigma \int _\varSigma b X_{T,\lambda } b^* (x,\,y) \, \mathrm{d}\mu (y) \, \mathrm{d}\mu (x). \end{aligned}$$

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

$$\begin{aligned}&\int _\varSigma \int _\varSigma B X_{T,\lambda } B^* (x,\,y) \, \mathrm{d}\mu (y) \, \mathrm{d}\mu (x)\nonumber \\&\quad = \int _M \int _M X_{T,\lambda }(x,\,y) B^*\mu (y) \overline{B^*\mu (x)} \, \mathrm{d}x \, \mathrm{d}y \nonumber \\&\quad = \frac{1}{2\pi }\int _{-\infty }^\infty {\hat{X}}_{T}(t) \mathrm{e}^{-it\lambda } \int _M \mathrm{e}^{it\sqrt{-\varDelta _{g}}}(B^* \mu )(x) \overline{B^* \mu (x)} \, \mathrm{d}x \, \mathrm{d}t. \end{aligned}$$

(3.5)

Once we show

$$\begin{aligned} {\text {WF}}\left( \mathrm{e}^{it\sqrt{-\varDelta _{g}}}B^*\mu \right) \cap {\text {WF}}(B^*\mu ) = \emptyset \quad \text {for all } t \in {\text {supp}}\,{\hat{X}}_T, \end{aligned}$$

(3.6)

the integral over M will be smooth in t by Proposition 2.2. Integration by parts in t then gives the desired bound of $O_T(\lambda ^{-N}).$ By the calculus of wavefront sets and pseudodifferential operators,

$$\begin{aligned} {\text {WF}}(B^* \mu ) \subset {\text {esssupp}} B \cap N^*_{{\text {supp}}\,\mu } \varSigma . \end{aligned}$$

To prove (3.6), suppose $(x,\,\xi )$ is a unit covector in ${\text {WF}}(B^*\mu ).$ By part (3) of Lemma 3.1, $\varPhi _t(x,\,\xi )$ is not in $\mathrm{SN}^*_{{\text {supp}}\,\mu }\varSigma$ for any $1 \le |t| \le T.$ By propagation of singularities,

$$\begin{aligned} {\text {WF}}\left( \mathrm{e}^{it\sqrt{-\varDelta _{g}}} B^*\mu \right) = \varPhi _t {\text {WF}}(B^*\mu ), \end{aligned}$$

hence

$$\begin{aligned} {\text {WF}}\left( \mathrm{e}^{it\sqrt{-\varDelta _{g}}} B^*\mu \right) \cap {\text {WF}}(B^* \mu ) = \emptyset \quad \text {for } 1 \le |t| \le T. \end{aligned}$$

(3.7)

Since the support of $\mu$ has been made small, if there is $(x,\,\xi ) \in \mathrm{SN}_{{\text {supp}}\,\mu }^*\varSigma$ and some $t > 0$ in the support of $(1 - \beta (t)) {\hat{\chi }}(t/T)$ for which $\varPhi _t(x,\,\xi ) \in \mathrm{SN}_{{\text {supp}}\,\mu }^* \Sigma ,$ then $t \ge 1$ since the diameter of ${\text {supp}}\,\mu$ is small and the injectivity radius of M is at least 1. We now have (3.6), from which follows (3.5) as promised.

What remains is to bound

$$\begin{aligned} \left| \int _\varSigma \int _\varSigma b X_{T,\lambda }b^*(x,\,y) \, \mathrm{d}\mu (x) \, \mathrm{d}\mu (y) \right| \le \lambda ^{n-d-1} + C_{T,b} \lambda ^{n-d-2}. \end{aligned}$$

(3.8)

We have

$$\begin{aligned} b X_{T,\lambda }b^*(x,\,y) = \sum _j X_{T}\left( \lambda _j - \lambda \right) b e_j(x) \overline{b e_j(y)}, \end{aligned}$$

and so we write the integral in (3.8) as

$$\begin{aligned} \sum _j X_T\left( \lambda _j - \lambda \right) \left| \int _\varSigma b(x,\,D)e_j(x) \, \mathrm{d}\mu (x) \right| ^2. \end{aligned}$$

(3.9)

Note $X_T$ satisfies bounds

$$\begin{aligned} \left| X_T(\tau )\right| \le C_{T,N}(1 + |\tau |)^{-N} \quad \text {for } N = 1,\,2,\ldots \end{aligned}$$

(3.10)

We dominate $|X_T|$ by a step function

$$\begin{aligned} \sum _{k \in {\mathbb {Z}}} a_{T,k} \chi _{[k,k+1)} \end{aligned}$$

satisfying similar bounds as $|X_T|$ with coefficients

$$\begin{aligned} a_{T,k} = \sup _{[k,k+1]} \left| X_T\right| . \end{aligned}$$

Now,

$$\begin{aligned}&\left| \sum _j X_T\left( \lambda _j - \lambda \right) \left| \int _\varSigma b(x,\,D)e_j(x) \, \mathrm{d}\mu (x) \right| ^2 \right| \nonumber \\&\quad \le \sum _{k \in {\mathbb {Z}}} a_{T,k} \sum _{\lambda _j - \lambda \in [k,k+1)} \left| \int _\varSigma b(x,\,D)e_j(x) \, \mathrm{d}\mu (x) \right| ^2. \end{aligned}$$

(3.11)

Using Proposition 2.1 and part (2) of Lemma  3.1, we write

$$\begin{aligned}&\sum _{\lambda _j - \lambda \in [k,k+1)} \left| \int _\varSigma b(x,\,D) e_j(x) \, \mathrm{d}\mu (x) \right| ^2\\&\quad \le \varepsilon (|\lambda + k| + 1)^{n-d-1} + C_b (|\lambda + k| + 1)^{n-d-2}. \end{aligned}$$

Hence, (3.11) is bounded by

$$\begin{aligned}&\le C_T \sum _{k \in {\mathbb {Z}}} a_{T,k} \left( \varepsilon (|\lambda + k| + 1)^{n - d - 1} + C_b (|\lambda + k| + 1)^{n - d - 2} \right) \\&\le \varepsilon C_T \lambda ^{n-d-1} + C_{T,b} \lambda ^{n-d-2} \quad \text {for } \lambda \ge 1 \end{aligned}$$

by the bounds (3.10). Taking $\varepsilon$ in part (2) of Lemma 3.1 small enough so that $\varepsilon C_T \le 1$ yields (3.8). This concludes the proof of Theorem 1.2.