On singular integrals of Calderón-type in $\mathbb R^n$, and BMO

Abstract

We prove L^p (and weighted L^p ) bounds for singular integrals of the form \rm p.v. \int_{\mathbb R^n} E \lgroup \frac{A(x)–A(y)}{|x–y} \rgroup \frac{\Omega(x–y)}{|x–y|^n} f(y)dy, where E(t) = cos t if \Omega is odd, and E(t) = sin t if \Omega is even, and where \bigtriangledown A \in BMO. Even in the case that \Omega is smooth, the theory of singular integrals with "rough" kernels plays a key role in the proof. By standard techniques, the trigonometric function E can then be replaced by a large class of smooth functions F . Some related operators are also considered. As a further application, we prove a compactness result for certain layer potentials.