We consider the Lt 2Lx r estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show Lt 2Lx ∞ estimates fail at the critical regularity in high dimensions by using stable Lévy process in Rd. Moreover, we show that some spherically averaged Lt 2Lx ∞ 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 Lt 2-type estimates.
- Nonlinear Schrödinger equation
- Nonlinear wave equation
- Strichartz estimates