Brennan's Conjecture and universal Sobolev inequalities

Brennan's Conjecture states integrability of derivatives of plane conformal homeomorphisms φ:Ω→Dφ:Ω→D that map a simply connected plane domain with non-empty boundary Ω⊂CΩ⊂C to the unit disc D⊂R2D⊂R2. We prove that Brennan's Conjecture leads to existence of compact embeddings of Sobolev spaces W˚p1(Ω) into weighted Lebesgue spaces Lq(Ω,h)Lq(Ω,h) with universal conformal weights h(z):=J(z,φ)=|φ′(z)|2h(z):=J(z,φ)=|φ′(z)|2. For p=2p=2 the number q is an arbitrary number between 1 and ∞ (Gol'dshtein and Ukhlov, in press [12]), for p≠2p≠2 the number q depends on p and the integrability exponent for Brennan's Conjecture.