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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hölder domains and Poincaré domains

Authors: Wayne Smith and David A. Stegenga
Journal: Trans. Amer. Math. Soc. 319 (1990), 67-100
MSC: Primary 30C20; Secondary 26D10
MathSciNet review: 978378
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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

$\displaystyle {\int\limits_D {\vert u - {u_D}\vert} ^p}dx \leq M_p^p(D){\int\limits_D {\vert\nabla u\vert} ^p}dx$

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$.

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Keywords: Conformal mapping, quasi-hyperbolic geometry, Poincaré inequality, Sobolev spaces, Whitney decomposition
Article copyright: © Copyright 1990 American Mathematical Society