On the boundary Strichartz estimates for wave and Schr\"odinger equations

TL;DR

This paper investigates boundary Strichartz estimates for wave and Schrödinger equations in high dimensions, revealing failures at critical regularity and establishing new averaged and smoothing estimates.

Contribution

It demonstrates the failure of certain boundary estimates at critical regularity and introduces spherically averaged and angular smoothing Strichartz estimates.

Findings

Homogeneous $L_t^2L_x^ abla$ estimates fail at critical regularity in high dimensions.

Spherically averaged $L_t^2L_x^ abla$ estimates hold at critical regularity.

Double $L_t^2$-type estimates are established for inhomogeneous cases.

Abstract

We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schr\"odinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable L\'evy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ 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.