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Given a real-analytic function b(x) defined on a neighborhood of the originwith 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 notintegrable on any neighborhood of the origin. Under appropriate firstderivative bounds and a cancellation condition, we prove L^p boundednesstheorems for such operators including when the kernel is not integrable. Weprimarily (but not exclusively) consider the p = 2 situation. The operatorsconsidered generalize both local versions of Riesz transforms and some localmultiparameter singular integrals. Generalizations of our results tonontranslation-invariant versions as well as singular Radon transform versionsare also proven.

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