Banach-valued multilinear singular integrals

We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis’s operator-valued Hörmander-Mihlin linear multiplier theorem [51]. Furthermore, we derive from our main result a wide range of mixed L p -norm estimates for multiparameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rules of Muscalu et al. [40]. Our approach works just as well for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform, extending results of Silva [48]. We also prove several operator-valued T(1)-type theorems both in one parameter, and of multi-parameter, mixed-norm type. A distinguishing feature of our T(1) theorems is that the usual explicit assumptions on the distributional kernel of T are replaced with testing-type conditions. Our proofs rely on a newly developed Banach-valued version of the outer L p space theory of Do and Thiele [11].