Strichartz estimate away from the obstacle

PDF p. 7: Proposition 2.3; (2.4)

Assume the Axiom of Choice. Let $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, $n\ge 2$, be the connected smooth exterior of a nonempty compact non-trapping obstacle, in the sense of Exterior and truncated obstacle domains and Non-trapping obstacle. Let ${\mathrm{\Delta }}_D$ be the Dirichlet Laplacian, $H_D^s(\mathrm{\Omega })$ its spectral scale, and $W(t)=e^{it{\mathrm{\Delta }}_D}$ as in Dirichlet spectral Sobolev scale and duality and Scale isometries and free Schrödinger evolution.

Let $T>0$, $0\le s\le 1$, and let $2<p\le \mathrm{\infty }$ and $2\le q<\mathrm{\infty }$ satisfy $2\mathrm{/}p+n\mathrm{/}q=n\mathrm{/}2$, with $1\mathrm{/}\mathrm{\infty }=0$. Let $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ equal one near $\mathrm{\Theta }$. For this statement use $W^{0,q}(\mathrm{\Omega })=L^q(\mathrm{\Omega })$, the usual first-order weak Sobolev space $W^{1,q}(\mathrm{\Omega })$, and, for $0<s<1$, the complex interpolation space $W^{s,q}(\mathrm{\Omega })=[L^q(\mathrm{\Omega }),W^{1,q}(\mathrm{\Omega }){]}_s$ in the convention of An explicit common interpolation model for the fractional Strichartz targets. Then there is a constant $C$ depending on the fixed data such that, for every $u_0\in H_D^s(\mathrm{\Omega })$, $\mathrm{\parallel }(1-\chi )W(t)u_0{\mathrm{\parallel }}_{L^p(0,T;W^{s,q}(\mathrm{\Omega }))}\le C\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H_D^s(\mathrm{\Omega })}\mathrm{.}$ The time-space norm is the Bochner norm, with essential supremum when $p=\mathrm{\infty }$, as in Continuous and mixed time-space function spaces.

The article is cited for attribution of Proposition 2.3 and (2.4), not as proof of its own estimate. The independent supplier records the imported endpoint estimates and its local interpolation argument. The fractional target convention is made explicit here; this does not assert that the article uniquely specified that convention.