**Proof.** Let

p denote the solution of (1)–(3). We will construct a solution

$\overline{p}$ of the wave equation which is periodic in time with period

$4T$ such that

$\overline{p}=p$ on

${\mathbb{R}}^{3}\times [0,T]$. Once this is done, we obtain

$f=\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)=\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},{4}^{n}T)$ for any

n. Using (

10), we arrive at

It remains to construct the above-mentioned solution

$\overline{p}$ of the wave equation. The idea is to properly reflect the solution

p in the time variable

t through the time moments

$t=T,2T,\cdots ,$ as follows. We first construct

$\overline{p}$ on

$[0,2T]$ by the odd reflection of

p through the moment

$t=T$:

$\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},T)=p(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},T)$ for

$t\in [0,T]$ and

$\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},T)=-p(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}},2T-t)$ for all

$t\in [T,2T]$. Since

$p(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},T)=0$ on

${\mathbb{R}}^{3}$, we obtain that

$\overline{p}$ and

${\overline{p}}_{t}$ are continuous at

$t=T$. Therefore,

p is continuous on

$[0,2T]$ and solves the wave equation on that interval. Next note that

${\overline{p}}_{t}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}},2T)=-{\overline{p}}_{t}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}},0)=0$ on

${\mathbb{R}}^{3}$. By the even reflection through

$t=2T$:

$\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},T)=\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},4T-t)$ for all

$t\in [2T,4T]$, we obtain that

$\overline{p}$ is a solution of the wave equation in

$[0,4T]$. Finally, we extend the solution by periodicity with period

$4T$. Noting that

$\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)=\overline{p}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},4T)$ and

${\overline{p}}_{t}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)={\overline{p}}_{t}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},4T)=0$, we obtain that

$\overline{p}$ and

${\overline{p}}_{t}$ are continuous for all time and

$\overline{p}$ satisfies the wave equation in

${\mathbb{R}}^{3}\times {\mathbb{R}}_{+}$. This finishes our proof. □

**Proof.** Let

p denote the solution of wave Equations (1)–(3) and recall the parametrix formula

$p(\mathbf{y},t)=\frac{1}{{\left(2\pi \right)}^{3}}{\sum}_{\sigma =\pm}{\int}_{{\mathbb{R}}^{3}}{a}_{\sigma}(\mathbf{y},t,\xi ){e}^{i{\varphi}_{\pm}(\mathbf{y},T,\xi )}\widehat{f}\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}d\xi ={\sum}_{\sigma =\pm}{p}_{\sigma}(\mathbf{y},t)$; see [

47]. Here, the phase function

${\varphi}_{\pm}$ solves the eikonal equation

${\partial}_{t}{\varphi}_{\pm}(\mathbf{y},t,\xi )\pm c\left(\mathbf{y}\right)\phantom{\rule{0.166667em}{0ex}}\left|{\nabla}_{\mathbf{y}}{\varphi}_{\pm}(\mathbf{y},t,\xi )\right|=0$ for

$(\mathbf{y},t)\in {\mathbb{R}}^{3}\times {\mathbb{R}}_{+}$ with the initial condition

${\varphi}_{\pm}(\mathbf{x},0,\xi )=\mathbf{x}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\xi $. The amplitude function is a classical symbol

${a}_{\pm}(\mathbf{y},t,\xi )={\sum}_{k=0}^{\infty}{a}_{-k,\pm}(\mathbf{y},t,\xi )$, where

${a}_{-k}$ is homogeneous of order

$-k$ in

$\xi $. Its leading term

${a}_{0,\pm}$ satisfies the transport equation

with the initial condition

${a}_{\pm ,0}(\mathbf{x},0,\xi )=1/2$, see [

41]. Here,

$C(\mathbf{y},\xi ,t)$ only depends on the sound speed

c and the phase function

${\varphi}_{\pm}$. Let us denote by

${\gamma}_{\mathbf{x},\xi}$ the unit speed geodesics originating at

$\mathbf{x}$ along the direction

$\xi $. Then,

${\gamma}_{\mathbf{x},\xi}$ is a characteristics curve of the above transport equation; that is, (

12) reduces to a homogeneous ODE on each geodesic curve.

We then write

Each operator

${\mathbf{W}}_{\pm}$ is a Fourier integral operator (FIO) with the canonical relation given by the pairs

$({\mathbf{y}}_{\pm},\lambda {\eta}_{\pm};\mathbf{x},\lambda \xi )$ for any

$\lambda \in \mathbb{R}$,

$\xi ,\eta $ unit vectors,

${\mathbf{y}}_{\pm}={\gamma}_{\mathbf{x},\xi}(\pm T)$, and

${\eta}_{\pm}={\dot{\gamma}}_{\mathbf{x},\xi}(\pm T)$. Let

${\mathbb{R}}^{3}$ be equipped with the metrics

${c}^{-2}\left(\mathbf{x}\right)\phantom{\rule{3.33333pt}{0ex}}d{\mathbf{x}}^{2}$. Then,

$({\mathbf{y}}_{\pm},{\eta}_{\pm})$ is obtained by translating

$(\mathbf{x},\xi )$ on the geodesic

${\gamma}_{\mathbf{x},\pm \xi}$ by the distance

T. From the initial condition of

${\varphi}_{\pm}$ and

${a}_{0,\pm}$ we see that, up to lower order terms,

Heuristically, under Equations (1)–(3), each singularity of

f at

$(\mathbf{x},\xi )$ is broken into two equal parts. They propagate along the geodesic

${\gamma}_{\mathbf{x},\xi}$ in the opposite directions

$\pm \xi $ to generate a singularity of

${\mathbf{W}}_{T}\left(f\right)$ at

$({\mathbf{y}}_{\pm},{\eta}_{\pm})$.

From the standard theory of FIOs (see [

50]), the adjoint

${\mathbf{W}}_{\pm}^{*}$ translates

$({\mathbf{y}}_{\pm},{\eta}_{\pm})$ back to

$(\mathbf{x},\xi )$ and

${\mathbf{W}}_{\pm}^{*}{\mathbf{W}}_{\pm}$ is a pseudo differential operator. On the other hand,

${\mathbf{W}}_{\mp}^{*}{\mathbf{W}}_{\pm}$ is a FIO whose canonical relation consists of the pairs

$(\mathbf{y},\eta ;\mathbf{x},\xi )$ given by

$\mathbf{y}={\gamma}_{\mathbf{x},\xi}(\pm 2T)$, and

$\eta ={\dot{\gamma}}_{\mathbf{x},\xi}(\pm 2T)$. That is,

${\mathbf{W}}_{\pm}^{*}{\mathbf{W}}_{\mp}$ is an infinitely smoothing operator on

B. Therefore, microlocally, we can write

${\mathbf{W}}_{T}^{*}{\mathbf{W}}_{T}f={\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}\left(f\right)+{\mathbf{W}}_{-}^{*}{\mathbf{W}}_{-}\left(f\right).$ We will show that the principal symbol

${\theta}_{\pm}(\mathbf{x},\xi )$ of

${\mathbf{W}}_{\pm}^{*}{\mathbf{W}}_{\pm}$ satisfies

${\theta}_{\pm}(\mathbf{x},\xi )=1/4$. This result can be intuitively understood as follows. Let us consider

${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}$ and a singularity of

f at

$(\mathbf{x},\xi )$. Under Equations (1)–(3), half of this singularity propagates into the direction

$\xi $ (corresponding to the function

${p}_{+}$). At the moment

$t=T$, it is transformed to a singularity of

${\mathbf{W}}_{+}\left(f\right)={p}_{+}\left(T\right)$ at

$({\mathbf{y}}_{+},{\eta}_{+})$. Under the adjoint Equation (

15), half of this singularity propagates back to

$(\mathbf{x},\xi )$ at

$t=0$ to generate a singularity of

${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}\left(f\right)$. It is natural to believe that this recovered singularity is

$1/4$ of the original singularity of

f (due to twice splitting, as described). The proof below verifies this intuition.

Indeed, denote by

${q}_{+}$ the solution of the time-reversed wave equation, e.g., Equation (

15), with the initial condition given by

${g}_{+}={\mathbf{W}}_{+}\left(f\right)$. Then, by definition (see Theorem 2)

${\mathbf{W}}_{T}^{*}{g}_{+}={q}_{+}{(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)|}_{B}$. The solution

${q}_{+}$ can be decomposed into the sum

${q}_{+}={q}_{0}+{q}_{1}$. Here,

${q}_{0},{q}_{1}$, up to smooth terms, are solutions of the wave equations in

${\mathbb{R}}^{3}\times (0,T)$ and satisfy

${q}_{0}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)={\mathbf{W}}_{+}^{*}\left({g}_{+}\right)$,

${q}_{1}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)={\mathbf{W}}_{-}^{*}\left({g}_{+}\right)$. We are only concerned with

${q}_{0}$ since it defines

${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}f={q}_{0}(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},0)$. We can write

Let

${b}_{0}$ be the principal part of

b. Then, the principal symbol

${\theta}_{+}$ of

${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}$ is given by

${\theta}_{+}(\mathbf{x},\xi )={b}_{0}(\mathbf{x},0,\xi )$. We note that

${b}_{0}$ satisfies the same equation as

${a}_{0,+}$ (see (

12)). Therefore, on each bicharacteristic curve the ratio

${b}_{0}/{a}_{0,+}$ is constant which implies

${b}_{0}(\mathbf{x},0,\xi )={a}_{+,0}(\mathbf{x},0,\xi ){b}_{0}({\mathbf{y}}_{+},T,{\eta}_{+})/{a}_{+,0}({\mathbf{y}}_{+},T,{\eta}_{+})$. Similar to the argument below Equation (

13), up to lower order terms, we have

This and Equation (

14) implies that

${b}_{0}({\mathbf{y}}_{+},T,\xi )={a}_{+,0}({\mathbf{y}}_{+},T,\xi )/2$. Therefore, we obtain

${b}_{0}(\mathbf{x},0,\xi )={a}_{+,0}(\mathbf{x},0,\xi )/2=1/4$. Combining with a similar argument for

${\mathbf{W}}_{-}^{*}{\mathbf{W}}_{-}$, we obtain that the principal symbol of

${\mathbf{W}}_{T}^{*}{\mathbf{W}}_{T}$ is

$\theta (\mathbf{x},\xi )=1/2$. That is,

${\mathbf{W}}_{T}^{*}{\mathbf{W}}_{T}=\mathbf{I}/2+{\mathbf{K}}_{T}$, where

${\mathbf{K}}_{T}$ is a pseudodifferential operator of order at most

$-1$ and

$\mathbf{I}$ is the identity. We have

$({\mathbf{W}}_{T}f,{\mathbf{W}}_{T}f)=({\mathbf{W}}_{T}^{*}{\mathbf{W}}_{T}f,f)=(f,f)/2+({\mathbf{K}}_{T}f,f)$ and therefore conclude

${\parallel f\parallel}_{{L}^{2}}^{2}\le 2(\parallel {\mathbf{W}}_{T}{f\parallel}_{{L}^{2}}^{2}+\parallel {\mathbf{K}}_{T}f{\parallel}_{{L}^{2}}^{2})$. □

We are now ready to prove Theorem 1.