Singular convolution integrals with operator-valued kernel

Abstract

We study operators \(f\mapsto Kf\) of the form \((Kf)(t)=\int_{{\bf R}^{n}} k(t-s)f(s) {\rm d}s\), where f is a vector-valued function and k an operator-valued singular kernel. Sufficient conditions for boundedness on Lp-spaces of UMD-valued functions are given in terms of a Hormander-type condition involving R-boundedness. The results cover large classes of kernels and also provide new proofs of recent operator-valued Fourier multiplier theorems. Moreover, they give new information about families of singular integral operators.