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

Abstract: Let $\delta\in(0,n]$, $\mathcal H_{\infty}^\delta$ denote the Hausdorff content defined on subsets of $\mathbb R^n$, and $\mathcal A_{p,\delta}$ be the capacitary Muckenhoupt weight class with $p\in[1,\infty)$. For the space ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$ of bounded $\delta$-dimensional mean oscillation defined with respect to $\mathcal H_{\infty}^{\delta}$, we establish its equivalent characterizations via the capacitary Muckenhoupt $\mathcal A_{p,\delta}$-weight for any $p\in(1,\infty)$, that is, we show that [f\in {\rm BMO}(\mathbb R^n, \mathcal H_{\infty}^{\delta})~~~\text{ if and only if} ~e^{\alpha f}\in \mathcal A_{p,\delta}] for some non-negative constant $\alpha$. As a subset of ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$, the space ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$ of bounded $\delta$-dimensional lower oscillation is characterized in terms of the capacitary Muckenhoupt $\mathcal A_{1,\delta}$-weight by establishing a John--Nirenberg inequality for the space $\rm{BLO}(\mathbb R^n,\mathcal H_{\infty}^{\delta})$, namely, we obtain [f\in {\rm BLO}(\mathbb R^n, \mathcal H_{\infty}^{\delta})~\text{ if and only if}~~~e^{\beta f}\in \mathcal A_{1,\delta}] for some non-negative constant $\beta$. As applications, we explore the capacitary weighted $\rm{BMO}$ space, and discover that it coincides with the unweighted space for any $w\in\mathcal A_{p,\delta}$ 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.