Weighted inequalities for product fractional integrals

TL;DR

This paper studies weighted inequalities for product fractional integrals, extending classical results to two parameters and identifying conditions under which operator norms are controlled, with new techniques for nonproduct weights.

Contribution

It extends one-parameter theory to two parameters in the one weight case and introduces new methods for nonproduct weights in the two weight case.

Findings

Most one-parameter theory extends to two parameters in the one weight case.

Rectangle characteristic does not control the operator norm in the two weight case without side conditions.

Stein-Weiss inequality extends to two parameters with nonproduct weights using sandwiching techniques.

Abstract

We investigate one and two weight norm inequalities for product fractional integrals. We show that in the one weight case, most of the 1 parameter theory carries over to the 2 parameter setting. However, in the two weight case, apart from the trivial case of product weights, the rectangle characteristic never controls the operator norm without side conditions. The Stein-Weiss extension of the classical Hardy-Littlewood-Sobolev inequality carries over to the setting of 2 parameters with nonproduct power weights using a sandwiching technique.