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 \(\mathbb R^n\) 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 $\mathcal R\mathcal H_{q,δ}$ associated with Hausdorff content, and then prove the self-improving property of $\mathcal R\mathcal H_{q,δ}$ 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.