  We establish the higher fractional differentiability of bounded minimizers to
a class of obstacle problems with non-standard growth conditions of the form
\begin{gather*} \min \biggl\{ \displaystyle\int_{\Omega} F(x,Dw)dx \ : \ w \in
\mathcal{K}_{\psi}(\Omega) \biggr\}, \end{gather*} where $\Omega$ is a bounded
open set of $\mathbb{R}^n$, $n \geq 2$, the function $\psi \in W^{1,p}(\Omega)$
is a fixed function called \textit{obstacle} and $\mathcal{K}_{\psi}(\Omega) :=
\{ w \in W^{1,p}(\Omega) : w \geq \psi \ \text{a.e. in} \ \Omega \}$ is the
class of admissible functions. If the obstacle $\psi$ is locally bounded, we
prove that the gradient of solution inherits some fractional differentiability
property, assuming that both the gradient of the obstacle and the mapping $x
\mapsto D_\xi F(x,\xi)$ belong to some suitable Besov space. The main novelty
is that such assumptions are not related to the dimension $n$.
