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.