Riesz transform and vertical oscillation in the Heisenberg group

Fässler, Katrin; Orponen, Tuomas

doi:10.2140/apde.2023.16.309

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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

DOI: 10.2140/apde.2023.16.309

Abstract

We study the $L^{2}$ -boundedness of the $3$ -dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group $ℍ$ . 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 $Ω \subset ℍ$ around a point $q \in \partial Ω$ , at scale $r > 0$ . We then proceed to show that if $Ω$ is a domain bounded by an intrinsic Lipschitz graph $Γ$ , and

\int_{0}^{\infty} {osc}_{Ω} (B (q, r)) \frac{d r}{r} \leq C < \infty, q \in Γ,

then the Riesz transform is $L^{2}$ -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 $\frac{1}{2}$ -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 $L^{p}$ -vertical perimeter of an intrinsic Lipschitz domain $Ω$ is controlled from above by the $p$ -th powers of the $L^{1}$ -based $β$ -numbers of $\partial Ω$ .

Keywords

Singular integrals, Riesz transform, intrinsic Lipschitz graphs, Heisenberg group

Mathematical Subject Classification 2010

Primary: 42B20

Secondary: 28A78, 31C05, 32U30, 35R03

Milestones

Received: 29 December 2019

Revised: 19 May 2021

Accepted: 24 June 2021

Published: 3 May 2023

Authors

Katrin Fässler
Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland

Tuomas Orponen
Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland

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Vol. 16, No. 2, 2023