Singular integrals on $C^{1,\alpha}$ intrinsic graphs in step 2 Carnot groups

Abstract

We study singular integral operators induced by Calder\'on-Zygmund kernels in any step-$2$ Carnot group $\mathbb{G}$. We show that if such an operator satisfies some natural cancellation conditions then it is $L^2$ bounded on all intrinsic graphs of $C^{1,\alpha}$ functions over vertical hyperplanes that do not have rapid growth at $\infty$. In particular, the result applies to the Riesz operator $\mathcal{R}$ induced by the kernel $$ \mathsf{R}(z)= \nabla_{\mathbb{G}} \Gamma(z), \quad z\in \mathbb{G}\backslash \{0\}, $$ the horizontal gradient of the fundamental solution of the sub-Laplacian. The $L^2$ boundedness of $\mathcal{R}$ is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that closed subsets of the intrinsic graphs mentioned above are non-removable.