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Properties and applications of Muckenhoupt A1 growth weights
Abstract
For β≥0, let A1(β) be the class of weights satisfying [w]A1(β):= sup0<ε≤1 εβ[w]A1+ε<∞. These classes satisfy A1subsetneq A1(β)subsetneq⋂p>1Ap for β>0. For every α≥0, we prove that w∈ A1(β) if and only if w({ML(log L)αf>λ}) ≤ C∫ℝn |f(x)|/λ [ log(e+|f(x)|/λ) ]α+β w(x) dx. The least constant C is comparable to [w]A1(β). Also, we obtain corresponding characterizations for Calderón--Zygmund operators and their commutators.
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BibTeX
@misc{wang2026properties,
title = {{Properties and applications of Muckenhoupt $A_1$ growth weights}},
author = {Dinghuai Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32653},
eprint = {2610.32653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32653},
}
RIS
TY - UNPB
TI - Properties and applications of Muckenhoupt A_1 growth weights
AU - Dinghuai Wang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32653
UR - https://arxiv.org/abs/2610.32653
AB - For β≥0, let A_1^(β) be the class of weights satisfying [w]_A_1^(β):= sup_0<ε≤1 ε^β[w]_A_1+ε<∞. These classes satisfy A_1subsetneq A_1^(β)subsetneq⋂_p>1A_p for β>0. For every α≥0, we prove that w∈ A_1^(β) if and only if w({M_L(log L)^αf>λ}) ≤ C∫_ℝ^n |f(x)|/λ [ log(e+|f(x)|/λ) ]^α+β w(x) dx. The least constant C is comparable to [w]_A_1^(β). Also, we obtain corresponding characterizations for Calderón--Zygmund operators and their commutators.
ER -