Abstract
We determine geometric necessary and sufficient conditions on a class of strip-like planar domains in order for them to satisfy the Poincaré inequality with exponentp, where 1≤p<∞. The characterization uses hyperbolic geodesics in the domain and a metric which depends onp and generalizes the quasi-hyperbolic metric in the casep=2. As an application, we show that the Poincaré inequality is preserved under Steiner symmetrization of these domains but not in general.
We also show that our geometric condition is preserved under bounded length distortion (BLD) mappings of a domain and thus extend the class of domains for which our characterization is valid.
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The first author is supported in part by a grant from the National Science Foundation.
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Smith, W., Stanoyevitch, A. & Stegenga, D.A. Planar Poincaré domains: Geometry and Steiner symmetrization. J. Anal. Math. 66, 137–183 (1995). https://doi.org/10.1007/BF02788821
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DOI: https://doi.org/10.1007/BF02788821