Riesz transform and vertical oscillation in the Heisenberg group

Abstract

We study the $L^2$-boundedness of the $3$-dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. Inspired by the notion of vertical perimeter, recently defined and studied by Lafforgue, Naor, and Young, we first introduce new scale and translation invariant coefficients ${\mathrm{osc}}_{\Omega }(B(q,r))$. These coefficients quantify the vertical oscillation of a domain $\Omega \subset \mathbb{H}$ around a point $q\in \partial \Omega$, at scale $r>0$. We then proceed to show that if $\Omega$ is a domain bounded by an intrinsic Lipschitz graph $\Gamma$, and $${\int }_0^{\infty }{\mathrm{osc}}_{\Omega }(B(q,r))\frac{dr}{r}\le C<\infty ,q\in \Gamma ,$$ then the Riesz transform is $L^2$-bounded on $\Gamma$. As an application, we deduce the boundedness of the Riesz transform whenever the intrinsic Lipschitz parametrisation of $\Gamma$ is an $\epsilon$ better than $\frac{1}{2}$-Hölder continuous in the vertical direction. We also study the connections between the vertical oscillation coefficients, the vertical perimeter, and the natural Heisenberg analogues of the $\beta$-numbers of Jones, David, and Semmes. Notably, we show that the $L^p$-vertical perimeter of an intrinsic Lipschitz domain $\Omega$ is controlled from above by the $p^{th}$ powers of the $L^1$-based $\beta$-numbers of $\partial \Omega$.

30 pages, 1 figure. v2: expanded Sections 3 and 6, and updated references

References in corpus (1)

• Nonnegative kernels and $1$-rectifiability in the Heisenberg group

Cited by in corpus (3)

• Metric rectifiability of $\mathbb{H}$-regular surfaces with Hölder continuous horizontal normal
• Rickman rugs and intrinsic bilipschitz graphs
• Necessary condition for the $L^2$ boundedness of the Riesz transform on Heisenberg groups