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

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 Calderón-Zygmund operator can be represented as a series of dyadic shifts and paraproducts averaged over randomized dyadic systems, to operator-valued Calderón-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.

29 pages, 1 figure

References in corpus (1)

• Boundedness of the square function and rectifiability

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