Uncentered Fractional Maximal functions and mean oscillation spaces associated with dyadic Hausdorff content

TL;DR

This paper investigates how uncentered fractional maximal functions operate on mean oscillation spaces linked to dyadic Hausdorff content, refining known results and establishing boundedness properties for these operators in various function spaces.

Contribution

It refines existing results on the action of fractional maximal functions on BMO and VMO spaces and proves boundedness of these operators on dyadic Hausdorff content-based mean oscillation spaces.

Findings

Refined the action of Euclidean uncentered fractional maximal functions on BMO and VMO.

Established boundedness of $eta_2$-dimensional uncentered maximal functions on $ ext{BMO}^{eta_1}( r^n)$.

Extended the understanding of maximal functions in the context of dyadic Hausdorff content.

Abstract

We study the action of uncentered fractional maximal functions on mean oscillation spaces associated with the dyadic Hausdorff content $\mathcal{H}_{\infty}^{\beta}$ with $0<\beta\leq n$. For $0 < \alpha < n$, we refine existing results concerning the action of the Euclidean uncentered fractional maximal function $\mathcal{M}_{\alpha}$ on the functions of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In addition, for $0 < \beta_1 \leq \beta_2 \leq n$, we establish the boundedness of the $\beta_2$-dimensional uncentered maximal function $\mathcal{M}^{\beta_2}$ on the space $\text{BMO}^{\beta_1}(\mathbb{R}^n)$, where $\text{BMO}^{\beta_1}(\mathbb{R}^n)$ denotes the mean oscillation space adapted to the dyadic Hausdorff content $\mathcal{H}_{\infty}^{\beta_1}$ on $\mathbb{R}^n$.