The generalized Beltrami equation has the form \(\bar \partial f - \mu \partial f -\nu \overline{\partial f} = g\) where \(f,g,\mu,\nu\) are defined in the complex plane and \(\| |\mu|+|\nu|\|_\infty < 1 \). The properties of the Beltrami equation, especially the regularity of its solutions, are important for the theory of quasiconformal maps in the plane. Iwaniec proved that the Beltrami equation \(\bar \partial f - \mu \partial f = g\) with VMO (vanishing mean oscillation) coefficient \(\mu\) has a solution \(f\) such that \(Df\in L^p\), provided that \(g\in L^p\) and \(1<p<\infty\); see [T. Iwaniec, Lect. Notes Math. 1508, 39–64 (1992; Zbl 0785.30010)].
In the present paper this result is extended to weighted \(L^p\) spaces, and to generalized Beltrami equation. Suppose that \(\mu\) and \(\nu\), in addition to the above, are compactly supported VMO functions. Let \(\omega \) be a weight in the Muckenhoupt class \(A_p\) for some \(1<p<\infty\). Then the generalized Beltrami equation has, for any \(g\in L^p(\omega)\), a solution with \(Df \in L^p(\omega)\). Such a solution is unique up to an additive constant. Moreover, one has \(\|Df\|_{L^p(\omega)} \leq C \|g\|_{L^p(\omega)}\) for some \(C>0\) independent of \(g\).