On the end-point of Stein-Weiss inequality

TL;DR

This paper extends the Stein-Weiss inequality to the endpoint case p=1 and further generalizes it to a multi-parameter setting involving fractional integral operators with singular kernels.

Contribution

It proves the Stein-Weiss inequality at p=1 and introduces a multi-parameter extension with singular kernels, advancing understanding of fractional integral inequalities.

Findings

Stein-Weiss inequality holds at p=1.

Extension to multi-parameter fractional integrals.

Operators have kernels singular on coordinate subspaces.

Abstract

This paper has two purposes. First, we show that the classical Stein-Weiss inequality is true for p=1. Second, by considering a family of strong fractional integral operators whose kernels have singularity on every coordinate subspace, we extend this end-point result to the multi-parameter setting.