Vector-Valued Inequalities, Factorization, and Extrapolation for a Family of Rough Operators

Abstract

We prove very general weighted norm inequalities for rough maximal and singular integral operators whose kernels satisfy some common Fourier transform decay estimates. Examples include homogeneous singular integral operators with kernels which do not necessarily satisfy a Dini condition as well as several operators whose measures are supported on lower dimensional sets, such as the discrete spherical maximal operator and the Hilbert transform along a homogeneous curve. The weights are seen to satisfy analogues of Jones′ factorization theorem and Rubio de Francia′s extrapolation theorem, and so are complete with respect to these important properties. We also obtain the corresponding weighted vector-valued inequalities for these operators and for the maximal operator associated with a starlike set.