On the boundary Strichartz estimates for wave and Schrödinger equations

Abstract

We consider the $L_t^2{L}_x^r$ estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2{L}_x^{\infty }$ estimates fail at the critical regularity in high dimensions by using stable Lévy process in ${\mathbb{R}}^d$. Moreover, we show that some spherically averaged $L_t^2{L}_x^{\infty }$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.