On low-dimensional uniform rectifiability in Heisenberg groups - Part 2
Abstract
Let 1≤ k≤ n. We prove that k-dimensional intrinsic Lipschitz graphs in the Heisenberg group ℍn satisfy a geometric lemma GLem(β2,𝒱k,p) for horizontal β-numbers with an exponent p=p(k). Previously, this result was known only in the case k=1; our proof recovers the sharp exponent p=4 in this setting. For k>1, we adapt an integral geometric approach originally developed by Orponen for Euclidean and parabolic Lipschitz functions. In addition, for k=n, we show how to deduce a geometric lemma with p=4 directly from an isotropic Dorronsoro theorem in ℝ2n using a Poincaré inequality. Building on the new geometric lemmas, we establish a necessary condition for k-regular sets in ℍn to admit corona decompositions by intrinsic Lipschitz graphs. The condition is known to be sufficient by earlier work of the last two authors together with Pinamonti. It involves additional flatness coefficients besides β2,𝒱k. Along the way, we therefore extend the known stability results for geometric lemmas under the ``big pieces'' functor to a larger class of coefficients.
Related
- The Isoperimetric Inequality in Higher-Dimensional Heisenberg Groups
- The Bernstein problem in higher dimensional Heisenberg groups
- Uniform rectifiability of brittle fractures in linear elasticity
- Riesz Transform Bounds Imply Parabolic Uniform Rectifiability of Graphs
- Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
BibTeX
@misc{chen2026lowdimensional,
title = {{On low-dimensional uniform rectifiability in Heisenberg groups - Part 2}},
author = {Yibo Chen and Katrin Fässler and Kilian Zambanini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30269},
eprint = {2608.30269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On low-dimensional uniform rectifiability in Heisenberg groups - Part 2
AU - Yibo Chen
AU - Katrin Fässler
AU - Kilian Zambanini
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.30269
UR - https://arxiv.org/abs/2608.30269
AB - Let 1≤ k≤ n. We prove that k-dimensional intrinsic Lipschitz graphs in the Heisenberg group ℍ^n satisfy a geometric lemma GLem(β_2,𝒱_k,p) for horizontal β-numbers with an exponent p=p(k). Previously, this result was known only in the case k=1; our proof recovers the sharp exponent p=4 in this setting. For k>1, we adapt an integral geometric approach originally developed by Orponen for Euclidean and parabolic Lipschitz functions. In addition, for k=n, we show how to deduce a geometric lemma with p=4 directly from an isotropic Dorronsoro theorem in ℝ^2n using a Poincaré inequality. Building on the new geometric lemmas, we establish a necessary condition for k-regular sets in ℍ^n to admit corona decompositions by intrinsic Lipschitz graphs. The condition is known to be sufficient by earlier work of the last two authors together with Pinamonti. It involves additional flatness coefficients besides β_2,𝒱_k. Along the way, we therefore extend the known stability results for geometric lemmas under the ``big pieces'' functor to a larger class of coefficients.
ER -