Inequalities with angular integrability and applications

TL;DR

This paper extends weighted fractional integral estimates to spaces with mixed radial and angular integrability, leading to new regularity criteria for Navier-Stokes solutions based on these refined norms.

Contribution

It introduces a unified framework for weighted estimates in mixed radial-angular spaces, improving classical inequalities and deriving new regularity conditions for Navier-Stokes equations.

Findings

Extended Stein-Weiss estimates for fractional integrals in mixed spaces

Derived new weighted Sobolev and Strichartz estimates with radial-angular improvements

Established novel regularity criteria for Navier-Stokes solutions using these estimates

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. Then we apply this technology in order to give new a priori assumptions on weak solutions of the Navier-Stokes equation so as to be able to conclude that they are smooth. The regularity criteria are given in terms of mixed radial-angular weighted Lebesgue space norms.