Reverse Hölder Class and Two-Parameter Good-λ Inequality over Hausdorff Content
Abstract
Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of ℝn with zero n-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-λ inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class ℛℋq,δ associated with Hausdorff content, and then prove the self-improving property of ℛℋq,δ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman--Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-λ principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.
Related
- The reverse Hölder inequality for 𝒜p(·) weights with applications to matrix weights
- Weak reverse Hölder inequalities for doubling measures
- Unique continuation at infinity for Schrödinger equations with Reverse Hölder Potentials
- A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
- Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials
BibTeX
@misc{huang2026reverse,
title = {{Reverse Hölder Class and Two-Parameter Good-$λ$ Inequality over Hausdorff Content}},
author = {Long Huang and Yangzhi Zhang and Ciqiang Zhuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16405},
eprint = {2610.16405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16405},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Reverse Hölder Class and Two-Parameter Good-λ Inequality over Hausdorff Content
AU - Long Huang
AU - Yangzhi Zhang
AU - Ciqiang Zhuo
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16405
UR - https://arxiv.org/abs/2610.16405
AB - Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of ℝ^n with zero n-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-λ inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class ℛℋ_q,δ associated with Hausdorff content, and then prove the self-improving property of ℛℋ_q,δ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman--Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-λ principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.
ER -