The general problem is the following: Given a quasiconformal mapping on the plane, what conditions on its dilatation coefficient guarantee certain geometric properties of the quasicircle? In this talk we will mainly consider the case where the quasicircle is a chord-arc curve. We will show that the invertivility of a Beurling-type operator on a particular weighted L^2 space characterizes such curves. (Joint work with K. Astala)
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