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 $m\in (0,n)$