Existence theorems for Beltrami equations with measure constraints (Q2901604)

The authors show that many recent results on the existence of ACL homeomorphic solutions for the Beltrami equation follow from some extension of the well-known Lehto existence theorem. Let \(\mu:D\rightarrow {\mathbb C}\) be a measurable function with \(| \mu(z)| <1\) a.e., let \(K_{\mu}(z)=\frac{1+| \mu(z)| }{1-| \mu(z)| }\in L_{\text{loc}}^1(D)\) and suppose that there exists a nonnegative measurable function \(\varphi(t):[0,\infty]\rightarrow [0,\infty]\) such that the measure of the set of all \(z\in D\) for which \(K_{\mu}(z)>t\) is at most \(\varphi(t)\) for every \(t\in [1,\infty)\) and such that \(\int\limits_{0}^{\delta}\frac{d\tau}{\tau\varphi^{\,-1}(\tau)}=\infty\) for some \(\delta<\varphi(+0)\). Then the Beltrami equation \(f_{\overline{z}} = \mu (z)\cdot f_z\) has a homeomorphic solution in the class ACL. Some existence theorems for Beltrami equations with integral constraints are also proved.