Hölder domains and Poincaré domains

Abstract

A domain D ⊂ R d D \subset {R^d} of finite volume is said to be a p p -Poincaré domain if there is a constant M p ( D ) {M_p}(D) so that \[ ∫ D | u − u D | p d x ≤ M p p ( D ) ∫ D | ∇ u | p d x {\int \limits _D {|u - {u_D}|} ^p}dx \leq M_p^p(D){\int \limits _D {|\nabla u|} ^p}dx \] for all functions u ∈ C 1 ( D ) u \in {C^1}(D) . Here u D {u_D} denotes the mean value of u u over D D . Techniques involving the quasi-hyperbolic metric on D D are used to establish that various geometric conditions on D D are sufficient for D D to be a p p -Poincaré domain. Domains considered include starshaped domains, generalizations of John domains and Hàlder domains. D D is a Hàlder domain provided that the quasi-hyperbolic distance from a fixed point x 0 ∈ D {x_0} \in D to x x is bounded by a constant multiple of the logarithm of the euclidean distance of x x to the boundary of D D . The terminology is derived from the fact that in the plane, a simply connected Hàlder domain has a Hàlder continuous Riemann mapping function from the unit disk onto D D . We prove that if D D is a Hàlder domain and p ≥ d p \ge d , then D D is a p p -Poincaré domain. This answers a question of Axler and Shields regarding the image of the unit disk under a Hàlder continuous conformal mapping. We also consider geometric conditions which imply that the imbedding of the Sobolev space W 1 , p ( D ) → L p ( D ) {W^{1,p}}(D) \to {L^p}(D) is compact, and prove that this is the case for a Hàlder domain D D .