Riesz transform and vertical oscillation in the Heisenberg group

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