Singular integrals on regular curves in the Heisenberg group

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 $$\mathrm{\mid }{\mathrm{\nabla }}_{\mathbb{H}}^{n}k(p)\mathrm{\mid }\le C_n\mathrm{\parallel }p{\mathrm{\parallel }}^{-1-n},p\in \mathbb{H}\setminus \{0\},n\ge 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{$\mathbb{H}$} \log |p|$\mathrm{.}Weprovethatconvolutionwith$k$, 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.