The Riesz Transform on Intrinsic Lipschitz Graphs in the Heisenberg Group

We prove that the Heisenberg Riesz transform is unbounded in L 2 L_2 on a family of intrinsic Lipschitz graphs in the first Heisenberg group H \mathbb {H} . We construct this family by combining a method from [Acta Math. 229 (2022), no. 1, 55–200, DOI 10.4310/acta.2022.v229.n1.a2. MR 4460594 ] with a stopping time argument, and we establish the L 2 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 H \mathbb {H} for all exponents in [ 2 , 4 ) [2,4) . Our results contrast with two fundamental results in Euclidean harmonic analysis and geometric measure theory: Lipschitz graphs in R n \mathbb {R}^n satisfy the strong geometric lemma, and the m m –Riesz transform is L 2 L_2 –bounded on m m –dimensional Lipschitz graphs in R n \mathbb {R}^n for m ∈ ( 0 , n ) m\in (0,n) .