  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.
