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

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