Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin{gather*} \min \biggl{ \int_{\Omega} F(x,w,Dw) d x \ : \ w \in $\mathcal{K}${\psi}(\Omega) \biggr}, \end{gather*} with $F$ double phase functional of the form \begin{equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end{equation*} where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^n$, $\psi \in W^{1,p}(\mathrm{\Omega })$ is a fixed function called $\mathit{\text{obstacle}}$ 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. Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property.