Sobolev Homeomorphisms and Brennan’s Conjecture

Abstract

Let $\Omega \subset {\mathbb {R}}^n$ be a domain that supports the $p$-Poincaré inequality. Given a homeomorphism $\varphi \in L^1_p(\Omega )$, for $p>n$ we show that the domain $\varphi (\Omega )$ has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. The Inverse Brennan’s conjecture states that for any simply connected plane domain $\Omega ' \subset {\mathbb {C}}$ with non-empty boundary and for any conformal homeomorphism $\varphi$ from the unit disc ${\mathbb {D}}$ onto $\Omega '$ the complex derivative $\varphi '$ is integrable in the degree $s, -2<s<2/3$. If $\Omega '$ is bounded then $-2<s\le 2$. We prove that integrability in the degree $s> 2$ is not possible for domains $\Omega '$ with infinite geodesic diameter.