Hölder domains and Poincaré domains

Abstract:

A domain $D \subset {R^d}$ of finite volume is said to be a $p$-Poincaré domain if there is a constant ${M_p}(D)$ so that \[ {\int \limits _D {|u - {u_D}|} ^p}dx \leq M_p^p(D){\int \limits _D {|\nabla u|} ^p}dx\] for all functions $u \in {C^1}(D)$. Here ${u_D}$ denotes the mean value of $u$ over $D$. Techniques involving the quasi-hyperbolic metric on $D$ are used to establish that various geometric conditions on $D$ are sufficient for $D$ to be a $p$-Poincaré domain. Domains considered include starshaped domains, generalizations of John domains and Hàlder domains. $D$ is a Hàlder domain provided that the quasi-hyperbolic distance from a fixed point ${x_0} \in D$ to $x$ is bounded by a constant multiple of the logarithm of the euclidean distance of $x$ to the boundary of $D$. 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 $D$. We prove that if $D$ is a Hàlder domain and $p \ge d$, then $D$ is a $p$-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 ${W^{1,p}}(D) \to {L^p}(D)$ is compact, and prove that this is the case for a Hàlder domain $D$.