  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.
