Singular integrals on regular curves in the Heisenberg group

Abstract

Let H be the first Heisenberg group, and let k∈C∞(H∖{0}) be a kernel which is either odd or horizontally odd, and satisfies|∇Hnk(p)|≤Cn‖p‖−1−n,p∈H∖{0},n≥0. The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel k(p)=∇Hlog⁡‖p‖. We prove that convolution with k, as above, yields an L2-bounded operator on regular curves in H. 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 L2 bounded operators on Lipschitz flags in H. 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.