Semi-classical dispersive estimates

Abstract

We prove dispersive estimates for the wave group $e^{it\sqrt{P(h)}}$ and the Schrödinger group $e^{itP(h)}$, where $P(h)$ is a self-adjoint, elliptic second-order differential operator depending on a parameter $0<h\le 1$, which is supposed to be a short-range perturbation of $-h^2\Delta$, $\Delta$ being the Euclidean Laplacian. In particular, applications are made to non-trapping metric perturbations and to perturbations by a magnetic potential.

Appendix

In this appendix we will sketch the proof of the estimates (4.3)–(4.6). To this end, we will use the fact that the kernels of the operators $e^{it\sqrt{P_0(h)}}\varphi (P_0(h))$ and $e^{itP_0(h)}\varphi (P_0(h))$ are of the form $K_h(|x-y|,t)$ and $\widetilde{K}_h(|x-y|,t)$, respectively, where

$$\begin{aligned} K_h(w,t)=\frac{w^{2-n}}{(2\pi )^{n/2}}\int \limits _0^\infty e^{ith\lambda }\varphi (h^2\lambda ^2){\mathcal {J}}_{\frac{n-2}{2}}(w\lambda )\lambda d\lambda =h^{-n}K_1(w/h,t),\end{aligned}$$

(6.1)

$$\begin{aligned} \widetilde{K}_h(w,t)=\frac{w^{2-n}}{(2\pi )^{n/2}}\int \limits _0^\infty e^{ith^2\lambda ^2}\varphi (h^2\lambda ^2){\mathcal {J}}_{\frac{n-2}{2}}(w\lambda )\lambda d\lambda =h^{-n}\widetilde{K}_1(w/h,t), \end{aligned}$$

(6.2)

where ${\mathcal {J}}_{\frac{n-2}{2}}(z)=z^{(n-2)/2}J_{\frac{n-2}{2}}(z)$, $J_{\frac{n-2}{2}}(z)$ being the Bessel function of order $\frac{n-2}{2}$. In view of the inequality

$$\begin{aligned} \langle x\rangle ^{-\sigma }\langle y\rangle ^{-\sigma }\le \langle x-y\rangle ^{-\sigma },\quad \forall \sigma \ge 0, \end{aligned}$$

it is easy to see that the estimates (4.3)–(4.6) follow from the following

Lemma 6.1

For all $w>0, t\ne 0, 0<h\le 1, s\ge 0$, we have

$$\begin{aligned}&\left| K_h(w,t)\right| \le C|t|^{-s}h^{-s-(n+1)/2}g_s(w),\end{aligned}$$

(6.3)

$$\begin{aligned}&\int \limits _{-\infty }^\infty |t|^{2s}\left| K_h(w,t)\right| ^2dt\le Ch^{-2s-n-1}g_s(w)^2,\end{aligned}$$

(6.4)

$$\begin{aligned}&\left| \widetilde{K}_h(w,t)\right| \le C|t|^{-s-1/2}h^{-s-(n+1)/2}g_s(w),\end{aligned}$$

(6.5)

$$\begin{aligned}&\int \limits _{-\infty }^\infty |t|^{2s}\left| \widetilde{K}_h(w,t)\right| ^2dt\le Ch^{-2s-n-1}g_s(w)^2, \end{aligned}$$

(6.6)

where $g_s(w)=w^{s-(n-1)/2}$ if $s\le (n-1)/2$, $g_s(w)=\langle w\rangle ^{s-(n-1)/2}$ if $s\ge (n-1)/2$.

Proof

In view of the identities (6.1) and (6.2), it is clear that it suffices to prove (6.3)–(6.6) for $h=1$. Let first $w\le 1$. Recall that near $z=0$ the function ${\mathcal {J}}_{\frac{n-2}{2}}(z)$ is equal to $z^{n-2}$ times an analytic function. Using this and integrating by parts, it is easy to see that in this case the functions $K_1$ and $\widetilde{K}_1$ satisfy the bounds

$$\begin{aligned}&\left| K_1(w,t)\right| \le C|t|^{-s},\end{aligned}$$

(6.7)

$$\begin{aligned}&\left| \widetilde{K}_1(w,t)\right| \le C|t|^{-s-1/2}, \end{aligned}$$

(6.8)

for every $s\ge 0$. Clearly, when $w\le 1$ the estimates (6.3)–(6.6) follow from (6.7) and (6.8). Let now $w\ge 1$. In this case we will use the fact that for $z\gg 1$ the function ${\mathcal {J}}_{\frac{n-2}{2}}(z)$ is of the form $e^{iz}b^+(z)+e^{-iz}b^-(z)$, where $b^\pm (z)$ are symbols of order $\frac{n-3}{2}$. Given any integers $k,\ell \ge 0$, set

$$\begin{aligned} b_k^\pm (z)=e^{\mp iz}\frac{d^k}{dz^k}\left( e^{\pm iz}b^\pm (z)\right) ,\quad b_{k,\ell }^\pm (z)=\frac{d^\ell }{dz^\ell }b_k^\pm (z). \end{aligned}$$

Clearly, $b_k^\pm (z)$ are also symbols of order $\frac{n-3}{2}$. Hence

$$\begin{aligned} \left| b_{k,\ell }^\pm (z)\right| \le C_{k,\ell }z^{\frac{n-3}{2}-\ell },\quad \forall z\ge 1. \end{aligned}$$

(6.9)

Let $m,N\ge 0$ be integers. Integrating $m$ times by parts, we can write

$$\begin{aligned} K_1(w,t)=\frac{w^{2-n}}{(2\pi )^{n/2}}(it)^{-m} \sum _\pm \sum _{k=0}^m\int \limits _0^\infty e^{i(t\pm w)\lambda }w^kb_k^\pm (w\lambda )\varphi _{k,m}(\lambda )d\lambda , \end{aligned}$$

with some functions $\varphi _{k,m}\in C_0^\infty ((0,+\infty ))$ independent of $w$ and $t$. We now integrate $N$ times by parts to obtain

$$\begin{aligned} K_1(w,t)=\frac{w^{2-n}}{(2\pi )^{n/2}}(it)^{-m}\sum _\pm \sum _{k=0}^m\sum _{\ell =0}^N(t\pm w)^{-N}\int \limits _0^\infty e^{i(t\pm w)\lambda }w^kb_{k,\ell }^\pm (w\lambda )\varphi _{k,\ell ,m,N}(\lambda )d\lambda . \end{aligned}$$

Hence, in view of (6.9), we get the bound

$$\begin{aligned} \left| K_1(w,t)\right| \le C_{m,N}w^{m-\frac{n-1}{2}}|t|^{-m}\left( |t-w|^{-N}+|t+w|^{-N}\right) . \end{aligned}$$

(6.10)

By interpolation, (6.10) holds for all real $m\ge 0$. It is easy to see now that the estimates (6.3) and (6.4) (with $h=1$) follow from (6.10).

Integrating by parts $m$ times with respect to the variable $\lambda ^2$ we can write the function $\widetilde{K}_1$ as follows

$$\begin{aligned} \widetilde{K}_1(w,t)&= \frac{w^{2-n}}{(2\pi )^{n/2}}(it)^{-m}\sum _{k=0}^m\int \limits _0^\infty e^{it\lambda ^2}\widetilde{\varphi }_{k,m}(\lambda )\frac{d^k}{d(\lambda ^2)^k}{\mathcal {J}}_{\frac{n-2}{2}}(w\lambda )d\lambda \\&= \frac{w^{2-n}}{(2\pi )^{n/2}}(it)^{-m}\sum _{k=0}^m\int \limits _0^\infty e^{it\lambda ^2}f_{k,m}(w,\lambda )d\lambda , \end{aligned}$$

where

$$\begin{aligned} f_{k,m}(w,\lambda )=\widetilde{\varphi }_{k,m}^\sharp (\lambda ) \frac{d^k}{d\lambda ^k}{\mathcal {J}}_{\frac{n-2}{2}}(w\lambda ). \end{aligned}$$

We now apply the inequality

$$\begin{aligned} \left| \int \limits _0^\infty e^{it\lambda ^2}f(\lambda )d\lambda \right| \le C|t|^{-1/2}\left\| \widehat{f}\right\| _{L^1},\quad \forall f\in C_0^\infty (\mathbf{R}), \end{aligned}$$

to get

$$\begin{aligned} \left| \widetilde{K}_1(w,t)\right| \le C_mw^{2-n}|t|^{-m-1/2}\sum _{k=0}^m\left\| \widehat{f}_{k,m}(\cdot ,w)\right\| _{L^1}. \end{aligned}$$

(6.11)

On the other hand, as above one can see that the function $\widehat{f}_{k,m}$ satisfies the bound

$$\begin{aligned} \left| \widehat{f}_{k,m}(\tau ,w)\right| \le C_{N}w^{k+\frac{n-3}{2}}\left( |\tau -w|^{-N}+|\tau +w|^{-N}\right) \end{aligned}$$

(6.12)

for every integer $N\ge 0$. By (6.12)

$$\begin{aligned} \left\| \widehat{f}_{k,m}(\cdot ,w)\right\| _{L^1}\le Cw^{k+\frac{n-3}{2}}. \end{aligned}$$

(6.13)

By (6.11) and (6.13)

$$\begin{aligned} \left| \widetilde{K}_1(w,t)\right| \le C_mw^{m-\frac{n-1}{2}}|t|^{-m-1/2} \end{aligned}$$

(6.14)

for every integer $m\ge 0$, and hence by interpolation for all real $m\ge 0$, which in turn proves (6.5). It is easy also to see that (6.6) (with $h=1$) follows from (6.14). Indeed, applying (6.14) with $m=s-\epsilon$ and $m=s+\epsilon$, we have

$$\begin{aligned}&\int \limits _{-\infty }^\infty |t|^{2s}\left| \widetilde{K}_h(w,t)\right| ^2dt\\&\quad \le Cw^{2s-2\epsilon -n+1}\int \limits _{|t|\le w}|t|^{-1+2\epsilon }dt+ Cw^{2s+2\epsilon -n+1}\int \limits _{|t|\ge w}|t|^{-1-2\epsilon }dt\le Cw^{2s-n+1}. \end{aligned}$$

$\square$