Super regularity for Beltrami systems

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in planar domain \(\Omega\), \[f_\overline{z} = \mathcal{A}(f_z)\ a.e.\ z\in\Omega,\] when \(\mathcal{A}\) is linear at \(\infty\). Namely \(W^{1,1}_{\operatorname{loc}}(\Omega)\) are \(W^{2,2+\epsilon}_{\operatorname{loc}}(\Omega)\). Here \(\epsilon>0\) depends explicitly on ellipticity bounds of \(\mathcal{A}\). The condition "is \(\infty\)" necessary - result false \(f_\overline{z} k|f_z|\), any \(0<k<1\), (\(k=0\) Weyl's lemma) and improved sharp, but can be further if, instance, smooth. also discuss subsequent implications fully non-linear systems \mathcal{A}(z, f_z)\ z\in\Omega.\] There "linear implies solutions.