The Riesz tranform on intrinsic Lipschitz graphs in the Heisenberg group

TL;DR

This paper demonstrates that the Heisenberg Riesz transform is unbounded on certain intrinsic Lipschitz graphs in the Heisenberg group, contrasting Euclidean cases where such transforms are bounded.

Contribution

It introduces new techniques to analyze singular integrals on intrinsic Lipschitz graphs in the Heisenberg group and constructs examples showing $L_2$--unboundedness.

Findings

Heisenberg Riesz transform is $L_2$--unbounded on specific intrinsic Lipschitz graphs.

The strong geometric lemma fails in $ extbf{H}$ for all exponents in $[2,4)$.

Contrasts with Euclidean harmonic analysis where Lipschitz graphs satisfy the strong geometric lemma.

Abstract

We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time argument, and we establish the $L_2$--unboundedness of the Riesz transform by introducing several new techniques to analyze singular integrals on intrinsic Lipschitz graphs. These include a formula for the Riesz transform in terms of a singular integral on a vertical plane and bounds on the flow of singular integrals that arises from a perturbation of a graph. On the way, we use our construction to show that the strong geometric lemma fails in $\mathbb{H}$ for all exponents in $[2,4)$. Our results are in stark contrast to two fundamental results in Euclidean harmonic analysis and geometric measure theory: Lipschitz graphs in $\mathbb{R}^n$ satisfy the strong geometric lemma, and the $m$--Riesz transform is $L_2$--bounded on $m$--dimensional Lipschitz graphs in $\mathbb{R}^n$ for $m\in (0,n)$.