Capacitary Muckenhoupt Weight Characterizations of BMO and BLO Spaces with Hausdorff Content and Applications

TL;DR

This work extends the classical BMO/BLO framework to spaces defined via the δ-Hausdorff content by introducing capacitary Muckenhoupt weights ${\mathcal A}_{p,\delta}$. It proves sharp characterizations: for p>1, $f\in {\rm BMO}(\mathbb R^n, {\mathcal H}_{\infty}^{\delta})$ iff $e^{\alpha f}\in {\mathcal A}_{p,\delta}$, and $f\in {\rm BLO}(\mathbb R^n, {\mathcal H}_{\infty}^{\delta})$ iff $e^{\beta f}\in {\mathcal A}_{1,\delta}$, establishing weighted John--Nirenberg inequalities and a capacitary weighted John--Nirenberg inequality that yield equivalences between weighted and unweighted BMO spaces for weights in ${\mathcal A}_{p,\delta}$. The authors introduce a dyadic integral average $f_{Q,\delta}$ and prove Jensen-type inequalities in the Choquet-analytic setting, enabling a robust norm equivalence between ${\rm BMO}$ and its dyadic counterpart. They also develop two factorization theorems for BMO/BLO via Hardy–Littlewood maximal operators, revealing deep connections between capacitary Muckenhoupt weights and BMO/BLO spaces beyond classical measure theory. Overall, the results generalize classical $A_p$-weight characterizations of BMO/BLO to a non-additive, capacity-based framework and provide new tools for capacitary harmonic analysis with Hausdorff content.

Abstract

Let $δ\in(0,n]$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content defined on subsets of $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class with $p\in[1,\infty)$. For the space ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ of bounded $δ$-dimensional mean oscillation defined with respect to $\mathcal H_{\infty}^δ$, we establish its equivalent characterizations via the capacitary Muckenhoupt $\mathcal A_{p,δ}$-weight for any $p\in(1,\infty)$, that is, we show that \[f\in {\rm BMO}(\mathbb R^n, \mathcal H_{\infty}^δ)~~~\text{ if and only if} ~~~e^{αf}\in \mathcal A_{p,δ}\] for some non-negative constant $α$. As a subset of ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$, the space ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ of bounded $δ$-dimensional lower oscillation is characterized in terms of the capacitary Muckenhoupt $\mathcal A_{1,δ}$-weight by establishing a John--Nirenberg inequality for the space $\rm{BLO}(\mathbb R^n,\mathcal H_{\infty}^δ)$, namely, we obtain \[f\in {\rm BLO}(\mathbb R^n, \mathcal H_{\infty}^δ)~~~\text{ if and only if}~~~e^{βf}\in \mathcal A_{1,δ}\] for some non-negative constant $β$. As applications, we explore the capacitary weighted $\rm{BMO}$ space, and discover that it coincides with the unweighted space for any $w\in\mathcal A_{p,δ}$ by establishing a capacitary weighted John--Nirenberg inequality. Finally, we build two factorization theorems of BMO/BLO spaces with Hausdorff content via Hardy--Littlewood maximal operators, respectively. These results reveal connections between capacitary Muckenhoupt weights and BMO/BLO spaces with Hausdorff content, beyond the classical measure-theoretic settings.

Theorems & Definitions (51)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Proposition 1.8
• Theorem 1.9
• Remark 1.10