Quasiconformal and BMO-quasiconformal homeomorphisms

Abstract

We expose a proof of the existence and uniqueness of homeomorphic solutions to the Beltrami equation $\bar{\partial }f=\mu \partial f$ a.e. in two cases. First, when the $L^{\infty }$ norm of $\mu$ is $<1$: the homeomorphic solution is quasiconformal. Then when $\frac{1}{1-|\mu |}$ is bounded above by a function which is locally in BMO. In this case, the homeomorphic solution is BMO-quasiconformal.