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.