Foundations of vector-valued singular integrals revisited---with random dyadic cubes

TL;DR

This paper simplifies key results in vector-valued harmonic analysis, specifically the $T(1)$ theorem and square function estimates, by employing the method of random dyadic cubes to streamline proofs.

Contribution

It introduces a simplified approach to fundamental theorems in vector-valued harmonic analysis using random dyadic cubes, making the proofs more accessible.

Findings

Simplified proof of the vector-valued $T(1)$ theorem.

Streamlined square function estimates for UMD spaces.

Enhanced understanding of harmonic analysis techniques in vector-valued settings.

Abstract

The vector-valued $T(1)$ theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a simplified approach to these results is presented, exploiting Nazarov, Treil and Volberg's method of random dyadic cubes, which allows to circumvent the most subtle parts of the original arguments.