2025/11/03 by Long Huang, Zhuo, Ciqiang, Yangzhi Zhang +3
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2511.01161
openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28
Let δ∈(0,n], p∈[1,∞), \mathcal H∞δ denote the Hausdorff content on \mathbb Rn, and \mathcal Ap,δ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class \mathcal Ap,δ and \rmBMO(\mathbb Rn, \mathcal H∞δ) or \rmBLO(\mathbb Rn, \mathcal H∞δ) spaces for all dimension δ∈(0,n], and further to comprehend the structure of these two spaces. Our main result shows that \mathcal Ap,δ for p∈(1,∞) is equivalent to the BMO spaces, while \mathcal A1,δ 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 \rmBMO(\mathbb Rn, \mathcal H∞δ) or \rmBLO(\mathbb Rn, \mathcal H∞δ) respectively. These results reveal deep connections between \mathcal Ap,δ 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.