Calder\'on-Zygmund estimates for higher order systems with p(x) growth

For weak solutions $u \in W^{m,1}(\Omega;\R^N)$ of higher order systems of the type \int_\Omega < A(x,D^m u),D^m \phi > dx = \int_\Omega < |F|^{p(x)-2}F,D^m \phi> dx, for all $\phi \in C^{\infty}_c(\Omega;\R^N), m > 1$ with variable growth exponent $p:\Omega \to (1,\infty)$ we prove that if $|F|^{p(\cdot)} \in L^q_{loc}(\Omega)$ with $1 < q < \frac{n}{n-2} + \delta$, then $|D^m u|^{p(\cdot)} \in L^q_{loc}(\Omega)$. We should note that we prove this implication both in the non-degenerate and in the degenerate case.