Weak reverse Hölder inequalities for doubling measures
Abstract
Let μ be a doubling Radon measure in ℝd with support X. We prove a local and quantitative form of the Bennewitz--Lewis theorem in this setting: if a Radon measure ν satisfies the Bennewitz--Lewis condition on a bounded relatively open set V⊂ X with a small enough constant, then ν|V≪μ|V and its density satisfies a reverse Hölder inequality on every compact subset K⊂ V. Along the way we develop the dyadic toolkit these arguments run on --- triple cubes defined through neighbours, the basis 𝒟∪3𝒟, and a Whitney decomposition --- and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for p=1 and p>1.
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BibTeX
@misc{prats2026weak,
title = {{Weak reverse Hölder inequalities for doubling measures}},
author = {Martí Prats},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05320},
eprint = {2610.05320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05320},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weak reverse Hölder inequalities for doubling measures
AU - Martí Prats
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05320
UR - https://arxiv.org/abs/2610.05320
AB - Let μ be a doubling Radon measure in ℝ^d with support X. We prove a local and quantitative form of the Bennewitz--Lewis theorem in this setting: if a Radon measure ν satisfies the Bennewitz--Lewis condition on a bounded relatively open set V⊂ X with a small enough constant, then ν|_V≪μ|_V and its density satisfies a reverse Hölder inequality on every compact subset K⊂ V. Along the way we develop the dyadic toolkit these arguments run on --- triple cubes defined through neighbours, the basis 𝒟∪3𝒟, and a Whitney decomposition --- and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for p=1 and p>1.
ER -