Singular integrals on regular curves in the Heisenberg group

TL;DR

This paper proves that certain singular integral operators with smooth, odd, or horizontally odd kernels are bounded on regular curves in the Heisenberg group, extending classical results to this non-commutative setting.

Contribution

It extends G. David's theorem to the Heisenberg group, showing $L^{2}$ boundedness of singular integrals on regular curves and Lipschitz flags with new classes of kernels.

Findings

Boundedness of singular integrals on regular curves in $ abla_{ ext{Heisenberg}}$

All 3D horizontally odd kernels are bounded on Lipschitz flags

New results on non-negative kernels by Chousionis and Li

Abstract

Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. 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 $L^{2}$ bounded operators on Lipschitz flags in $\mathbb{H}$. This was known earlier for only one specific operator, the $3$-dimensional Riesz transform. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.