A two weight local Tb theorem for the Hilbert transform

TL;DR

This paper establishes a new two weight local Tb theorem for fractional singular integral operators on the real line, including the Hilbert transform, enhancing previous T1 theorems with broader applicability.

Contribution

It introduces a novel two weight local Tb theorem for elliptic and gradient elliptic fractional singular integrals, extending the theoretical framework for these operators.

Findings

Proves a two weight local Tb theorem for the Hilbert transform.

Includes elliptic and gradient elliptic fractional singular integrals.

Improves upon previous T1 theorems.

Abstract

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and in a sense improves on the T1 theorem by the authors and M. Lacey.