Capacitary Muckenhoupt Weight, BMO and BLO Spaces with Hausdorff Content, Factorization Theorems and Applications

Abstract

Let $\delta\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^\delta$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,\delta}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,\delta}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$ spaces for all dimension $\delta\in(0,n]$, and further to comprehend the structure of these two spaces. Our main result shows that $\mathcal A_{p,\delta}$ for $p\in(1,\infty)$ is equivalent to the BMO spaces, while $\mathcal A_{1,\delta}$ is equivalent to the BLO spaces, and consequently yields the factorization theorems for these BMO and BLO spaces via capacitary Hardy--Littlewood maximal operators, which essentially extend main results of Coifman and Rochberg in 1980 beyond measure theory. As applications, by establishing some capacitary weighted John--Nirenberg inequalities, we obtain the equivalence between capacitary weighted BMO or BLO spaces and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^{\delta})$ respectively. These results reveal deep connections between $\mathcal A_{p,\delta}$ and BMO or BLO spaces with Hausdorff content, beyond the classical measure-theoretic settings. We develop some approaches in the proofs and using a new observation, that is, the additivity of measures and linearity of integrals are superfluous for the corresponding classical theory.