The vector-valued non-homogeneous Tb theorem

TL;DR

This paper extends the non-homogeneous Tb theorem to Banach space-valued functions, establishing boundedness of singular integral operators on UMD spaces using advanced decoupling and Haar function estimates.

Contribution

It provides a Banach space-valued extension of the non-homogeneous Tb theorem with a new proof valid for all Lebesgue exponents, utilizing decoupling inequalities and Haar function estimates.

Findings

Boundedness of singular integral operators on UMD Banach spaces.

Extension of the Tb theorem to Banach space-valued functions.

New proof valid for all 1<p<infinity, using decoupling inequalities.

Abstract

The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces of functions with values in a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all Lebesgue exponents p in the range 1<p<infinity, and relies on delicate estimates for the non-homogenous "Haar" functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences.