Hardy-Littlewood-Sobolev Inequality on Mixed-Norm Lebesgue Spaces

TL;DR

This paper characterizes the boundedness of the Riesz potential on mixed-norm Lebesgue spaces, extending the Hardy-Littlewood-Sobolev inequality to these spaces including endpoint cases.

Contribution

It provides a complete characterization of indices for boundedness of Riesz potentials on mixed-norm Lebesgue spaces, including endpoint cases.

Findings

Complete characterization of indices for boundedness

Extension of Hardy-Littlewood-Sobolev inequality to mixed norms

Inclusion of all endpoint cases

Abstract

We consider the Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. We give a complete characterization of indices $\vec p$ and $\vec q$ such that the Riesz potential is bounded from $L^{\vec p}$ to $L^{\vec q}$, including all the endpoint cases. As a result, we get the mixed-norm Hardy-Littlewood-Sobolev inequality.