The Riesz tranform on intrinsic Lipschitz graphs in the Heisenberg group

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 $m\in (0,n)$.