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

Let $δ\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ spaces for all dimension $δ\in(0,n]$, and further to comprehend the structure of these two spaces. Our main result shows that $\mathcal A_{p,δ}$ for $p\in(1,\infty)$ is equivalent to the BMO spaces, while $\mathcal A_{1,δ}$ 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 ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ respectively. These results reveal deep connections between $\mathcal A_{p,δ}$ 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.

💡 Research Summary

This paper presents a profound and systematic extension of classical harmonic analysis from the setting of the Lebesgue measure to that of low-dimensional Hausdorff content. The central objects of study are the capacitary Muckenhoupt weight class A_{p,δ} and the spaces BMO(ℝ^n, H_δ^∞) and BLO(ℝ^n, H_δ^∞), all defined with respect to the Hausdorff content H_δ^∞ for dimensions δ ∈ (0, n]. The Hausdorff content is a non-additive capacity, and integration against it is performed via the non-linear Choquet integral, moving beyond the traditional measure-theoretic framework reliant on additivity and linearity.

The authors’ primary achievement is establishing precise equivalences between these capacitary weight classes and function spaces. Their first main theorem (Theorem 1.1) proves that for p ∈ (1, ∞), a function f belongs to BMO(ℝ^n, H_δ^∞) if and only if e^(αf) belongs to A_{p,δ} for some α ≥ 0. This characterizes the BMO space completely in terms of the capacitary Muckenhoupt weights. The second main theorem (Theorem 1.3) establishes an analogous equivalence: f ∈ BLO(ℝ^n, H_δ^∞) if and only if e^(βf) ∈ A_{1,δ} for some β ≥ 0. These results are direct generalizations of the seminal work by Coifman and Rochberg (1980) to a non-measure-theoretic context.

Building upon these equivalences, the paper derives powerful factorization theorems (Theorems 1.7 and 1.8). They show that any function in BMO(ℝ^n, H_δ^∞) can be decomposed as a linear combination of logarithms of the capacitary Hardy-Littlewood maximal operator M_H_δ^∞ applied to two locally integrable functions, plus an L∞ function. Similarly, any function in BLO(ℝ^n, H_δ^∞) can be written as the logarithm of such a maximal function plus an L∞ function. These theorems elucidate the intrinsic structure of these spaces.

As significant applications, the authors establish new capacitary weighted John-Nirenberg inequalities. Using these, they demonstrate the equivalence between capacitary weighted BMO (or BLO) spaces and their unweighted counterparts, BMO(ℝ^n, H_δ^∞) (or BLO(ℝ^n, H_δ^∞)). Furthermore, the paper shows that the classes A_{p,δ}, and consequently the spaces BMO(ℝ^n, H_δ^∞) and BLO(ℝ^n, H_δ^∞), are strictly increasing as the dimension δ increases, providing a nuanced hierarchy of spaces for analyzing sets of different fractal dimensions.

The overarching significance of this work lies in its demonstration that the fundamental relationships between weights, maximal functions, and function spaces like BMO and BLO are not reliant on the additivity of measures or the linearity of integrals. By successfully developing the theory using the Hausdorff content, the paper opens new avenues for harmonic analysis on low-dimensional and fractal sets, offering tools with potential applications in geometric measure theory, partial differential equations on irregular domains, and analysis on non-doubling metric spaces.