A1 Journal article (refereed)
Riesz transform and vertical oscillation in the Heisenberg group (2023)

Fässler, K., & Orponen, T. (2023). Riesz transform and vertical oscillation in the Heisenberg group. Analysis and PDE, 16(2), 309-340. https://doi.org/10.2140/apde.2023.16.309

JYU authors or editors

Publication details

All authors or editors: Fässler, Katrin; Orponen, Tuomas

Journal or series: Analysis and PDE

ISSN: 2157-5045

eISSN: 1948-206X

Publication year: 2023

Publication date: 03/05/2023

Volume: 16

Issue number: 2

Pages range: 309-340

Publisher: Mathematical Sciences Publishers

Publication country: United States

Publication language: English

DOI: https://doi.org/10.2140/apde.2023.16.309

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/87289

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1810.13122

Abstract

We study the L2-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 oscΩ(B(q,r)). These coefficients quantify the vertical oscillation of a domain Ω⊂H 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 ∫∞0oscΩ(B(q,r))drr≤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 12-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.

Keywords: harmonic analysis (mathematics); potential theory; partial differential equations

Free keywords: singular integrals; Riesz transform; intrinsic Lipschitz graphs; Heisenberg group

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2023

JUFO rating: 3