Orlicz regularity of the gradient of solutions to quasilinear elliptic equations in the plane

Given a planar domain Ω, we study the Dirichlet problem $$\textstyle\begin{cases} {-}\operatorname {div}A(x, \nabla v)= f & \mbox{in } \Omega,\\ v=0 & \mbox{on } \partial\Omega, \end{cases} $$ where the higher-order term is a quasilinear elliptic operator, and f belongs to the Zygmund space $L (\log L)^{\delta} (\log\log\log L)^{\frac{\beta}{2}}(\Omega)$ with $\beta\geq0$ and $\delta\geq \frac{1}{2}$ . We prove that the gradient of the variational solution $v \in W^{1,2}_{0}(\Omega)$ belongs to the space $L^{2} (\log L)^{2\delta-1} (\log \log\log L)^{\beta}(\Omega)$ .