PreprintCC BY-NC-SA 4.0arXiv
A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
Abstract
Let p∈[1,∞) and δ∈(0,n]. Recently, to characterize the weighted boundedness of maximal operator on Choquet integrals based on Hausdorff content ℋ∞δ, a class of Capacity Muckenhoupt weights is introduced, denoted by 𝒜p,δ, and, for any fixed δ∈(0,n], this new class of weights is proved to satisfy the self-improving property with respect to the index p∈(1,∞). In this note, we show that, for every fixed p∈[1,∞), the class of capacity Muckenhoupt weights 𝒜p,δ fails to satisfy the self-improving property with respect to the index δ∈(0,n].
Related
- Muckenhoupt weights for the Hardy-Littlewood maximal operator on hyperbolic spaces
- Properties and applications of Muckenhoupt A1 growth weights
- Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents
- Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains
- Weighted norm inequalities for operators with new Muckenhoupt type weights in the Bessel setting and applications
BibTeX
@misc{zhang2026note,
title = {{A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content}},
author = {Yangzhi Zhang and Ciqiang Zhuo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39313},
eprint = {2609.39313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39313},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
AU - Yangzhi Zhang
AU - Ciqiang Zhuo
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39313
UR - https://arxiv.org/abs/2609.39313
AB - Let p∈[1,∞) and δ∈(0,n]. Recently, to characterize the weighted boundedness of maximal operator on Choquet integrals based on Hausdorff content ℋ_∞^δ, a class of Capacity Muckenhoupt weights is introduced, denoted by 𝒜_p,δ, and, for any fixed δ∈(0,n], this new class of weights is proved to satisfy the self-improving property with respect to the index p∈(1,∞). In this note, we show that, for every fixed p∈[1,∞), the class of capacity Muckenhoupt weights 𝒜_p,δ fails to satisfy the self-improving property with respect to the index δ∈(0,n].
ER -