Multilinear operator-valued Calder\'on-Zygmund theory

TL;DR

This paper develops a comprehensive multilinear operator-valued Calderón-Zygmund theory, establishing boundedness results for singular integrals with operator-valued kernels on UMD Banach spaces, including non-commutative L^p spaces.

Contribution

It introduces a new framework for multilinear operator-valued singular integrals, including a $T(1)$-theorem and a Rademacher maximal function condition applicable to non-commutative spaces.

Findings

Established boundedness of multilinear singular integrals on UMD spaces.

Proved a $T(1)$-theorem for bilinear operator-valued singular integrals.

Extended the theory to non-commutative L^p spaces and multi-parameter settings.

Abstract

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the $\mathcal R$-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a $T(1)$-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative $L^p$-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property $\alpha$.