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Riesz transform and vertical oscillation in the Heisenberg group

Katrin Fässler and Tuomas Orponen

Vol. 16 (2023), No. 2, 309–340

We study the L2-boundedness of the 3-dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group . Inspired by the notion of vertical perimeter, recently defined and studied by Lafforgue, Naor, and Young, we first introduce new scale and translation invariant coefficients  osc Ω(B(q,r)). These coefficients quantify the vertical oscillation of a domain Ω around a point q Ω, at scale r > 0. We then proceed to show that if Ω is a domain bounded by an intrinsic Lipschitz graph Γ, and

0osc Ω(B(q,r)) dr r C < ,q Γ,

then the Riesz transform is L2-bounded on Γ. As an application, we deduce the boundedness of the Riesz transform whenever the intrinsic Lipschitz parametrisation of Γ is an 𝜖 better than 1 2-Hölder continuous in the vertical direction.

We also study the connections between the vertical oscillation coefficients, the vertical perimeter, and the natural Heisenberg analogues of the β-numbers of Jones, David, and Semmes. Notably, we show that the Lp-vertical perimeter of an intrinsic Lipschitz domain Ω is controlled from above by the p-th powers of the L1-based β-numbers of Ω.

Singular integrals, Riesz transform, intrinsic Lipschitz graphs, Heisenberg group
Mathematical Subject Classification 2010
Primary: 42B20
Secondary: 28A78, 31C05, 32U30, 35R03
Received: 29 December 2019
Revised: 19 May 2021
Accepted: 24 June 2021
Published: 3 May 2023
Katrin Fässler
Department of Mathematics and Statistics
University of Jyväskylä
Tuomas Orponen
Department of Mathematics and Statistics
University of Jyväskylä

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