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

We study the action of uncentered fractional maximal functions on mean oscillation spaces associated with the dyadic Hausdorff content $\mathcal{H}_{\infty}^β$ with $0<β\leq n$. For $0 < α< n$, we refine existing results concerning the action of the Euclidean uncentered fractional maximal function $\mathcal{M}_α$ on the functions of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In addition, for $0 < β_1 \leq β_2 \leq n$, we establish the boundedness of the $β_2$-dimensional uncentered maximal function $\mathcal{M}^{β_2}$ on the space $\text{BMO}^{β_1}(\mathbb{R}^n)$, where $\text{BMO}^{β_1}(\mathbb{R}^n)$ denotes the mean oscillation space adapted to the dyadic Hausdorff content $\mathcal{H}_{\infty}^{β_1}$ on $\mathbb{R}^n$.