  We establish some higher differentiability results of integer and fractional
order for solution to non-autonomous obstacle problems of the form
  \begin{equation*}
  \min \left\{\int_{\Omega}f(x, Dv(x))\,:\, v\in
  \mathcal{K}_\psi(\Omega)\right\},
  \end{equation*} where the function $f$ satisfies $p-$growth conditions with
respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_\psi(\Omega)$
is the class of admissible functions $v\in u_0+W^{1, p}_0(\Omega)$ such that
$v\ge\psi$ a. e. in $\Omega$, where $u_0\in W^{1,p}(\Omega)$ is a fixed
boundary datum.
  Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the
gradient of the obstacle $\psi$ transfers to the gradient of the solution,
provided the partial map $x\mapsto D_\xi f(x,\xi)$ belongs to a suitable
Sobolev or Besov space. The novelty here is that we deal with subquadratic
growth conditions with respect to the gradient variable, i. e. $f(x,
\xi)\approx a(x)|\xi|^p$ with $1<p<2,$ and where the map $a$ belongs to a
Sobolev or Besov-Lipschitz space.
