On the local behavior of certain homeomorphisms. II

In the previous paper by the authors, it is shown that a sense-preserving homeomorphism, absolutely continuous on lines (ACL), under certain integral conditions on the complex dilation is conformal at a point. This result includes the Teichmüller-Wittich-Belinski theorem. In the present paper, it is shown that under weakened conditions the local behavior of such a homeomorphism is similar to that of a quasiconformal mapping. Bibliography: 2 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 11–20.Dedicated to the 90th anniversary of G. M. Goluzin's birth.     

Extremal functions for modules of systems of measures

We extend a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in R², to R ⁿ making use of Fuglede’s p-module of systems of measures. The extremal functions are identified and the p-module of systems of measures is computed in condensers of rather general type and in their images under homeomorphisms of certain regularity. At the beginning, we discuss and apply Rodin’s Theorem in order to obtain estimates for the conformal modules of parallelograms and ring domains in terms of directional dilatations.     

On solutions of the Beltrami equation

In this paper we study the existence and uniqueness of solutions of the Beltrami equationf _-z (z) =Μ(z)f _z (z), whereΜ(z) is a measurable function defined almost everywhere in a plane domain ‡ with ‖ΜΜ∞ = 1-Here the partialsf _z andf _z of a complex valued functionf _z exist almost everywhere. In case ‖Μ‖∞ ≤9 < 1, it is well-known that homeomorphic solutions of the Beltrami equation are quasiconformal mappings. In case ‖Μ‖∞= 1, much less is known. We give sufficient conditions onΜ(z) which imply the existence of a homeomorphic solution of the Beltrami equation, which isACL and whose partial derivativesf _z andf _z are locally inL ^q for anyq < 2. We also give uniqueness results. The conditions we consider improve already known results.