  We prove a sharp higher differentiability result for local minimizers of
functionals of the form $$\mathcal{F}\left(w,\Omega\right)=\int_{\Omega}\left[
F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand
$F(x,\xi)$ which is convex with respect to the gradient variable, under
$p$-growth conditions, with $1<p<2$. The main novelty here is that the results
are obtained assuming that the partial map $x\mapsto D_\xi F(x,\xi)$ has weak
derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong
to a suitable Lebesgue space $L^r$.
  We also prove that it is possible to weaken the assumption on the datum $f$
and on the map $x\mapsto D_\xi F(x,\xi)$, if the minimizers are assumed to be a
priori bounded.
