A two weight local $Tb$ theorem for $n$-dimensional fractional singular integrals

TL;DR

This paper establishes a local two weight $Tb$ theorem for higher-dimensional fractional Calderón-Zygmund operators, extending one-dimensional results and addressing new challenges from higher-dimensional complexities.

Contribution

It introduces a novel local two weight $Tb$ theorem with an energy side condition for higher-dimensional fractional singular integrals, building upon and extending previous one-dimensional theorems.

Findings

Proves a local two weight $Tb$ theorem for higher dimensions.

Addresses challenges from the failure of one-dimensional inequalities in higher dimensions.

Provides a framework for analyzing fractional Calderón-Zygmund operators in multiple dimensions.

Abstract

We obtain a local two weight $Tb$ theorem with an energy side condition for higher dimensional fractional Calder\'{o}n-Zygmund operators. The proof follows the general outline of the proof for the corresponding one-dimensional $Tb$ theorem in [SaShUr12], but encountering a number of new challenges, including several arising from the failure in higher dimensions of T. Hyt\"{o}nen's one-dimensional two weight $A_{2}$ inequality [Hyt].