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 $\phi \in L_p^1(\Omega )$, for $p>n$ we show the domain $\phi (\Omega )$ has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. {\bf The Inverse Brennan's conjecture} states that for any simply connected plane domain ${\Omega }^{\prime }\subset \mathbb{C}$ with nonempty boundary and for any conformal homeomorphism $\phi$ from the unit disc $\mathbb{D}$ onto ${\Omega }^{\prime }$ the complex derivative ${\phi }^{\prime }$ is integrable in the degree $s$, $-2<s<2/3$. If ${\Omega }^{\prime }$ is bounded than $-2<s\le 2$. We prove that integrability in the degree $s>2$ is not possible for domains ${\Omega }^{\prime }$ with infinite geodesic diameter.

8 pages

References in corpus (1)

• Brennan's conjecture for composition operators on Sobolev spaces