Riesz transform and vertical oscillation in the Heisenberg group
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
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2023Copyright
© 2023 MSP (Mathematical Sciences Publishers).
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.
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Fässler was supported by the Swiss National Science Foundation through project 161299 Intrinsic rectifiability and mapping theory on the Heisenberg group. Orponen was supported by the Finnish Academy through the project Quantitative rectifiability in Euclidean and non-Euclidean spaces, grants 309365 and 314172.License
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