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 $\frac 1{1-\vert \mu \vert }$ is bounded above by a function which is locally in BMO. In this case, the homeomorphic solution is BMO-quasiconformal.