Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations

Abstract

This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy–Littlewood–Sobolev (HLS) inequalities. In the same way, we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schrödinger equation.

The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy–Littlewood maximal inequality in the weighted Lebesgue space due to Muckenhoupt. Under radial symmetry, we get significant gains over the usual HLS inequality and Strichartz estimate.