Measure data systems with Orlicz growth

We study the existence of very weak solutions to a system

$$\begin{aligned} {\left\{ \begin{array}{ll}-{\pmb {\textsf{div}}}{{\mathcal {A}}}(x,{D{\pmb {\textsf{u}}}})=\pmb {\mathsf {\mu }}\quad \text {in }\ \Omega ,\\ \pmb {\textsf{u}}=0\quad \text {on }\ \partial \Omega \end{array}\right. } \end{aligned}$$

with a datum \({\pmb {\mathsf {\mu }}}\) being a vector-valued bounded Radon measure and \({{\mathcal {A}}}:\Omega \times {{\mathbb {R}}^{n\times m}}\rightarrow {{\mathbb {R}}^{n\times m}}\) having measurable dependence on the spacial variable and Orlicz growth with respect to the second variable. We are not restricted to the superquadratic case. For the solutions and their gradients we provide regularity estimates in the generalized Marcinkiewicz scale. In addition, we show a precise sufficient condition for the solution to be a Sobolev function.