Nontrapping cutoff resolvent estimates

Assume the Axiom of Choice. Let $n\ge 2$, let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty smooth compact obstacle with connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, and suppose that the obstacle is nontrapping. Let ${\mathrm{\Delta }}_D$ be the Dirichlet Laplacian on $\mathrm{\Omega }$, and let $H_D^s(\mathrm{\Omega })$ be the spectral scale of $I-{\mathrm{\Delta }}_D$. Let $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ be real and nonnegative, acting by multiplication after restriction to $\mathrm{\Omega }$.

Write $B=-{\mathrm{\Delta }}_D$, let $\rho (B)$ denote its resolvent set, and use the library convention $R_B(z)=(zI-B{)}^{-1}$ for $z\in \rho (B)$. Thus the inverse in the estimates below is $(B-zI{)}^{-1}=-R_B(z)$, initially acting on $L^2(\mathrm{\Omega })$. All spectral parameters used below belong to $\rho (B)$.

For every $s\in [-1,1]$ there are constants $C_s>0$ and ${ϵ}_0>0$ such that, for all $\lambda \in \mathbb{R}$ and $0<ϵ\le {ϵ}_0$, the cutoff resolvent $T_{\lambda ,ϵ}^{\pm }=\chi (-{\mathrm{\Delta }}_D-(\lambda \pm iϵ{)}^2{)}^{-1}\chi$ extends boundedly from $H_D^s(\mathrm{\Omega })$ to $H_D^{s+1}(\mathrm{\Omega })$ and satisfies $\mathrm{\parallel }{T}_{\lambda ,ϵ}^{\pm }{\mathrm{\parallel }}_{H_D^s\to H_D^{s+1}}\le C_s$. The constants may depend on the obstacle, cutoff and $s$, but not on $\lambda$ or $ϵ$.

For the same cutoff and sufficiently small $ϵ>0$, the $L^2$ operator norm of $T_{\lambda ,ϵ}^{\pm }$ is at most $C\mathrm{\mid }\lambda {\mathrm{\mid }}^{-1}$ when $\mathrm{\mid }\lambda \mathrm{\mid }\ge 1$ and at most $C$ when $\mathrm{\mid }\lambda \mathrm{\mid }\le 1$, uniformly in those parameters. Also, for every $s\in [-1,1]$ and every complex $z$ with $\mathrm{Re}z<0$, $\mathrm{\parallel }\chi (-{\mathrm{\Delta }}_D-z{)}^{-1}\chi {\mathrm{\parallel }}_{H_D^s\to H_D^{s+1}}\le C_s(1+\mathrm{\mid }\mathrm{Re}z\mathrm{\mid }{)}^{-1\mathrm{/}2}$, uniformly in $z$.

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

The Axiom of Choice

Resolvent and spectrum of an unbounded operator

At the start of Section 2.2, Burq, Gérard and Tzvetkov give the uncut resolvent of the positive self-adjoint operator $B=-{\mathrm{\Delta }}_D$ on $\mathbb{C}\setminus [0,\mathrm{\infty })$; this spectral-domain assertion is imported. The stated parameter families lie in that domain: $(\lambda \pm iϵ{)}^2={\lambda }^2-{ϵ}^2\pm 2i\lambda ϵ$ is nonreal if $\lambda \mathrm{\ne }0$ and equals $-{ϵ}^2<0$ if $\lambda =0$, while $\mathrm{Re}z<0$ excludes $[0,\mathrm{\infty })$. Hence the displayed $L^2$ inverses exist before taking their cutoff extensions.

The high- and low-frequency $L^2$ bounds are imported from Propositions 2.2 and 2.4 of Burq, Gérard and Tzvetkov, under the geometry and parameter restrictions stated above. The low-frequency statement requires a nonempty obstacle.

The uniform $H_D^s\to H_D^{s+1}$ bound is imported from their Proposition 2.5. Its printed range is $s\ge -1$; this item uses only $-1\le s\le 1$. The nonnegative cutoff hypothesis is retained.

The negative-real-part bound is imported from their Remark 2.6, equation (2.10), again only for $-1\le s\le 1$. The source states that an integration by parts gives the estimate; no local proof of that input is claimed here.

The source defines the positive Dirichlet scale using the functional calculus of $I-{\mathrm{\Delta }}_D$, and the negative range through duality and interpolation. The present item uses the corresponding frozen spectral-scale convention. The cited statements are independent of the article under study.