Distance-Weighted Norm Equivalences for Analytic Functions on John Domains
Abstract
Let Ω⊂ℂ be a bounded John domain and set δ(z)=dist(z,∂Ω). For 1<p<∞ and α>dimA(∂Ω)−2, we establish a norm equivalence between ∫Ω|g|pδα dA and ∫Ω|g'|pδα+p dA for analytic functions g on Ω, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking α=mp−2 yields the corresponding comparison between the m-th and (m+1)-st derivatives. For m≥2 the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between ∫Ω|f'|pδp−2 dA and ∫Ω|f''|pδ2p−2 dA when 1<p<2. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every s>1, an inward-cusp s-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.
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BibTeX
@misc{matsuzaki2026distanceweighted,
title = {{Distance-Weighted Norm Equivalences for Analytic Functions on John Domains}},
author = {Katsuhiko Matsuzaki and Huaying Wei},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.23940},
eprint = {2608.23940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distance-Weighted Norm Equivalences for Analytic Functions on John Domains
AU - Katsuhiko Matsuzaki
AU - Huaying Wei
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2608.23940
UR - https://arxiv.org/abs/2608.23940
AB - Let Ω⊂ℂ be a bounded John domain and set δ(z)=dist(z,∂Ω). For 1<p<∞ and α>dim_A(∂Ω)−2, we establish a norm equivalence between ∫_Ω|g|^pδ^α dA and ∫_Ω|g'|^pδ^α+p dA for analytic functions g on Ω, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking α=mp−2 yields the corresponding comparison between the m-th and (m+1)-st derivatives. For m≥2 the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between ∫_Ω|f'|^pδ^p−2 dA and ∫_Ω|f''|^pδ^2p−2 dA when 1<p<2. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every s>1, an inward-cusp s-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.
ER -