The local non-homogeneous Tb theorem for vector-valued functions

TL;DR

This paper extends the local non-homogeneous Tb theorem to vector-valued functions with operator-valued kernels, advancing the theory for UMD spaces and impacting applications in harmonic analysis.

Contribution

It introduces a local non-homogeneous Tb theorem for vector-valued functions in UMD lattices, expanding previous scalar and UMD space results.

Findings

Extended the local non-homogeneous Tb theorem to vector-valued functions.

Identified new challenges and questions in the limits of vector-valued harmonic analysis.

Provided techniques applicable to operator-valued kernels in UMD lattices.

Abstract

We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, `vector-valued' means `taking values in a function lattice with the UMD (unconditional martingale differences) property'. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.