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 $\mathcal{H}_{\infty}^{\beta}$ 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 $\mathbb{R}^n$.