On singular integrals of Calderón-type in \(\mathbb{R}^ n\), and BMO
Summary: 0e prove \(L^p\) (and weighted \(L^p\)) bounds for singular integrals of the form \[ \text{p.v. } \int_{\mathbb{R}^n} E\Biggl({A(x)- A(y)\over|x-y|}\Biggr) {\Omega(x- y)\over|x-y|^n} f(y)dy, \] where \(E(t)=\cos t\) if \(\Omega\) is odd, and \(E(t)=\sin t\) if \(\Omega\) is even, and where \(\nabla A\in\text{BMO}\). 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.
scientific article; zbMATH DE number 722305