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We establish some higher differentiability results for solution tonon-autonomous obstacle problems of the form  \begin{equation*}  \min \left\{\int_{\Omega}f\left(x, Dv(x)\right)dx\,:\, v\in  \mathcal{K}_\psi(\Omega)\right\},  \end{equation*} where the function $f$ satisfies $p-$growth conditions withrespect to the gradient variable, for $1<p<2$, and $\mathcal{K}_\psi(\Omega)$is the class of admissible functions. Here we show that, if the obstacle $\psi$is bounded, then a Sobolev regularity assumption on the gradient of theobstacle $\psi$ transfers to the gradient of the solution, provided the partialmap $x\mapsto D_\xi f(x,\xi)$ belongs to a Sobolev space, $W^{1, p+2}$. Thenovelty here is that we deal with subquadratic growth conditions with respectto the gradient variable, i.e. $f(x, \xi)\approx a(x)|\xi|^p$ with $1<p<2,$ andwhere the map $a$ belongs to a Sobolev space.

Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions