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

Abstract

We study 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 \(L^{\vec q}\). In particular, all the endpoint cases are studied. As a result, we get the mixed-norm Hardy–Littlewood–Sobolev inequality.