Singular integral operators with kernels associated to negative powers of real-analytic functions

Abstract

Given a real-analytic function b(x) defined on a neighborhood of the origin with b(0) = 0, we consider local convolutions with kernels which are bounded by |b(x)|^(-a), where a > 0 is the smallest number for which |b(x)|^(-a) is not integrable on any neighborhood of the origin. Under appropriate first derivative bounds and a cancellation condition, we prove L^p boundedness theorems for such operators including when the kernel is not integrable. We primarily (but not exclusively) consider the p = 2 situation. The operators considered generalize both local versions of Riesz transforms and some local multiparameter singular integrals. Generalizations of our results to nontranslation-invariant versions as well as singular Radon transform versions are also proven.