Operator-valued dyadic shifts and the T(1) theorem

TL;DR

This paper extends dyadic shifts and the T(1) theorem to an operator-valued setting, proving boundedness and representation results for Calderón-Zygmund operators with operator-valued kernels in UMD-spaces.

Contribution

It introduces operator-valued dyadic shifts, extends the dyadic representation theorem, and provides new proofs for key boundedness results in the operator-valued context.

Findings

Operator-valued dyadic shifts are bounded.

Representation theorem extended to operator-valued Calderón-Zygmund operators.

New proofs for boundedness of operator-valued paraproducts and related inequalities.

Abstract

In this paper we extend dyadic shifts and the dyadic representation theorem to an operator-valued setting: We first define operator-valued dyadic shifts and prove that they are bounded. We then extend the dyadic representation theorem, which states that every scalar-valued Calder\'on-Zygmund operator can be represented as a series of dyadic shifts and paraproducts averaged over randomized dyadic systems, to operator-valued Calder\'on-Zygmund operators. As a corollary, we obtain another proof of the operator-valued, global T(1) theorem. We work in the setting of integral operators that have R-bounded operator-valued kernels and act on functions taking values in UMD-spaces. The domain of the functions is the Euclidean space equipped with the Lebesgue measure. In addition, we give new proofs for the following known theorems: Boundedness of the dyadic (operator-valued) paraproduct, a variant of Pythagoras' theorem for (vector-valued) functions adapted to a sparse collection of dyadic cubes, and a decoupling inequality for (UMD-valued) martingale differences.