Stein–Weiss and Caffarelli–Kohn–Nirenberg inequalities with angular integrability

Abstract

We prove an extension of the Stein–Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtained: in particular we deduce precised versions of weighted Sobolev embeddings, Caffarelli–Kohn–Nirenberg estimates, and Strichartz estimates for the wave equation, which extend the radial improvements to the case of arbitrary functions.