Multilinear operator-valued calderón-zygmund theory
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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 R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of mul- tilinear, 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 nec- essary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lat- tice cases, our RMF condition covers suitable tuples of non-commutative Lp-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 α.