Brennan's Conjecture and universal Sobolev inequalities Academic Article uri icon

abstract

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

publication date

  • January 1, 2014