Characterization of C^1,α-domains in terms of Carleson-type ε-coefficients

Abstract

Let Ω=Ω⁺⊂ ℝⁿ⁺¹ be a bounded open set with n-Ahlfors regular boundary and let Ω^-=ℝⁿ⁺¹∖Ω⁺̅. For x∈∂Ω, r>0, define ε_n,Ω(x,r):=1/rⁿ inf_H⁺ℋⁿ(((∂ B(x,r)∩ H⁺)∖ Ω⁺) ∪ ((∂ B(x,r)∩ H^-)∖ Ω^-)), where ℋⁿ stands for n-dimensional Hausdorff measure and the infimum is taken over all open affine half-spaces H⁺ such that x∈ ∂ H⁺. Here H^-=ℝⁿ⁺¹∖ H⁺̅. In this work we prove that Ω is a C^1,α open set, with 0<α<1, if and only if there exists some constant C>0 such that ε_n,Ω(x,r)≤ C r^α for all x∈∂Ω and all 0<r≤ diam(∂Ω).

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BibTeX RIS

BibTeX

@misc{tolsa2026characterization,
  title = {{Characterization of $C^1,α$-domains in terms of Carleson-type $\varepsilon$-coefficients}},
  author = {Xavier Tolsa and Tobias Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20207},
  eprint = {2610.20207},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20207},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Characterization of C^1,α-domains in terms of Carleson-type ε-coefficients
AU  - Xavier Tolsa
AU  - Tobias Wang
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20207
UR  - https://arxiv.org/abs/2610.20207
AB  - Let Ω=Ω^+⊂ ℝ^n+1 be a bounded open set with n-Ahlfors regular boundary and let Ω^-=ℝ^n+1∖Ω^+̅. For x∈∂Ω, r>0, define ε_n,Ω(x,r):=1/r^n inf_H^+ℋ^n(((∂ B(x,r)∩ H^+)∖ Ω^+) ∪ ((∂ B(x,r)∩ H^-)∖ Ω^-)), where ℋ^n stands for n-dimensional Hausdorff measure and the infimum is taken over all open affine half-spaces H^+ such that x∈ ∂ H^+. Here H^-=ℝ^n+1∖ H^+̅. In this work we prove that Ω is a C^1,α open set, with 0<α<1, if and only if there exists some constant C>0 such that ε_n,Ω(x,r)≤ C r^α for all x∈∂Ω and all 0<r≤ diam(∂Ω).
ER  -

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