A T1 theorem for weakly singular integral operators

TL;DR

This paper extends the T1 theorem framework to establish boundedness conditions for weakly singular integral operators like the Riesz potential on L^p spaces, based on kernel size, regularity, and BMO conditions.

Contribution

It introduces natural size and regularity conditions, along with BMO criteria, to guarantee boundedness of abla T on L^p, generalizing classical T1 theorem results.

Findings

Established boundedness of abla T on L^p under new conditions.

Identified sharp conditions involving T1, T^t1 in ext{BMO}.

Provided example with Riesz potential satisfying these conditions.

Abstract

We establish conditions in the spirit of the T1 theorem of David and Journ\'e which guarantee the boundedness of \nabla T on L^p(\R^n), where T is an integral transformation and 1<p<\infty. These are natural size and regularity conditions for the kernel of the integral transformation, along with the sharp condition T1,T^t1\in\mathcal{I}^1(\mathrm{BMO}). A simple example satisfying these conditions is the Riesz potential denoted by \mathcal{I}^1.