Singular convolution operators on the Heisenberg group

Abstract

Statement of results and outline of method. The purpose of this note is to announce results dealing with convolution operators on the Heisenberg group. As opposed to the well-known situation where the kernels are homogeneous and C away from the origin, the kernels we study are homogeneous but have singularities on a hyperplane. Convolution operators with such kernels arise in the study of the 3-Neumann problem, as we indicate below. The main feature of our study will be an analysis which has no direct analogue in the case of the usual (abelian) convolution operators, but is instead based on the noncommutative Fourier analysis of the Heisenberg group.