On the boundary Strichartz estimates for wave and Schrödinger equations

We consider the L_t² L_x^r estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show L_t² L_x^∞ estimates fail at the critical regularity in high dimensions by using stable Lévy process in R^d. Moreover, we show that some spherically averaged L_t² L_x^∞ 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_t² -type estimates.