A1 Journal article (refereed)
Intrinsic Lipschitz graphs and vertical β-numbers in the Heisenberg group (2019)
Chousionis, V., Fässler, K., & Orponen, T. (2019). Intrinsic Lipschitz graphs and vertical β-numbers in the Heisenberg group. American Journal of Mathematics, 141(4), 1087-1147. https://doi.org/10.1353/ajm.2019.0028
JYU authors or editors
Publication details
All authors or editors: Chousionis, Vasileios; Fässler, Katrin; Orponen, Tuomas
Journal or series: American Journal of Mathematics
ISSN: 0002-9327
eISSN: 1080-6377
Publication year: 2019
Volume: 141
Issue number: 4
Pages range: 1087-1147
Publisher: Johns Hopkins University Press
Publication country: United States
Publication language: English
DOI: https://doi.org/10.1353/ajm.2019.0028
Publication open access: Not open
Publication channel open access:
Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/65300
Publication is parallel published: https://arxiv.org/abs/1606.07703
Abstract
The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group H. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 1990s. The theory in H has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in R3. Our main object of study are the intrinsic Lipschitz graphs in H, introduced by B. Franchi, R. Serapioni, and F. Serra Cassano in 2006. We claim that these 3-dimensional sets in H, if any, deserve to be called quantitatively 3-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical β-numbers. Conversely, extending a result of David and Semmes from Rn, we prove that a 3-Ahlfors-David regular subset in H, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.
Keywords: measure theory; partial differential equations
Contributing organizations
Ministry reporting: Yes
Reporting Year: 2019
JUFO rating: 3