A1 Journal article (refereed)
Singular integrals on regular curves in the Heisenberg group (2021)

Fässler, K., & Orponen, T. (2021). Singular integrals on regular curves in the Heisenberg group. Journal de Mathematiques Pures et Appliquees, 153, 30-113. https://doi.org/10.1016/j.matpur.2021.07.004

JYU authors or editors

Fässler, Katrin
Orponen, Tuomas

Publication details

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

Journal or series: Journal de Mathematiques Pures et Appliquees

ISSN: 0021-7824

eISSN: 1776-3371

Publication year: 2021

Volume: 153

Pages range: 30-113

Publisher: Elsevier BV

Publication country: France

Publication language: English

DOI: https://doi.org/10.1016/j.matpur.2021.07.004

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/77393

Publication is parallel published: https://arxiv.org/abs/1911.03223

Abstract

Let be the first Heisenberg group, and let be a kernel which is either odd or horizontally odd, and satisfies
The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel . We prove that convolution with k, as above, yields an -bounded operator on regular curves in . This extends a theorem of G. David to the Heisenberg group.

As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield bounded operators on Lipschitz flags in . This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.

Free keywords: uniform rectifiability; singular integrals; Heisenberg group

Fields of science: