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.