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

We prove an extension of the Stein–Weiss weighted estimates for fractional integrals, in the context of L p 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.