Singular integrals on regular curves in the Heisenberg group

Abstract

Let be the first Heisenberg group, and let be a kernel which is either odd or horizontally odd, and satisfies The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel . We prove that convolution with k, as above, yields an -bounded operator on regular curves in . This extends a theorem of G. David to the Heisenberg group.

As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield bounded operators on Lipschitz flags in . This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.