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Hölder maps under Pfaffian constraints
Abstract
Given a one form λ in ℝN and f: 𝕊n → ℝN with f∗ λ= 0 we discuss the maximal Hölder regularity of extensions F: 𝔹n+1 → ℝN such that F∗λ= 0 in distributional sense. Our analysis applies to the Heisenberg groups ℍn. It implies in particular that for all n ≥ 1 any smooth horizontal map f: 𝕊n → ℍn can be extended to a Cα-map F: 𝔹n+1 → ℍn for some α> 1/2. Moreover, if n ≥ 3 we find Cα-embeddings from 𝔹n+1 into ℍn for some α> 1/2. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find Cα-embeddings from 𝔹2 into ℍ1 for some α> 1/2, assisted by GPT-6 Astra.
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BibTeX
@misc{schikorra2026hölder,
title = {{Hölder maps under Pfaffian constraints}},
author = {Armin Schikorra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.03667},
eprint = {2607.03667},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03667},
}
RIS
TY - UNPB
TI - Hölder maps under Pfaffian constraints
AU - Armin Schikorra
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2607.03667
UR - https://arxiv.org/abs/2607.03667
AB - Given a one form λ in ℝ^N and f: 𝕊^n → ℝ^N with f^∗ λ= 0 we discuss the maximal Hölder regularity of extensions F: 𝔹^n+1 → ℝ^N such that F^∗λ= 0 in distributional sense. Our analysis applies to the Heisenberg groups ℍ_n. It implies in particular that for all n ≥ 1 any smooth horizontal map f: 𝕊^n → ℍ_n can be extended to a C^α-map F: 𝔹^n+1 → ℍ_n for some α> 1/2. Moreover, if n ≥ 3 we find C^α-embeddings from 𝔹^n+1 into ℍ_n for some α> 1/2. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find C^α-embeddings from 𝔹^2 into ℍ_1 for some α> 1/2, assisted by GPT-6 Astra.
ER -