2 01 3 Inverse Limit Spaces Satisfying a Poincaré Inequality
نویسنده
چکیده
We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [HK96]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [CK09], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, this was noted in [CK09]. However according to [CK13] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [CK10a].
منابع مشابه
ec 2 01 3 INVERSE LIMIT SPACES SATISFYING A POINCARÉ INEQUALITY
We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [HK96]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of exam...
متن کاملar X iv : m at h / 07 01 69 3 v 2 [ m at h . D G ] 1 4 Fe b 20 07 WEIGHTED POINCARÉ INEQUALITY AND RIGIDITY OF COMPLETE MANIFOLDS
Abstract. We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green’s function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields a criterion of parabolicity of connected components at infinity in ...
متن کاملCharacterizing Spaces Satisfying Poincaré Inequalities and Applications to Differentiability
We show that a proper metric measure space is a RNP-differentiability space if and only if it is rectifiable in terms of doubling metric measure spaces with some Poincaré inequality. This result characterizes metric measure spaces that can be covered by spaces admitting Poincaré inequalities, as well as metric measure spaces that admit a measurable differentiable structure which permits differe...
متن کاملMathematische Zeitschrift Modulus and the Poincaré inequality on metric measure spaces
The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging se...
متن کاملSelf-improving Properties of Generalized Poincaré Type Inequalities through Rearrangements
We prove, within the context of spaces of homogeneous type, L and exponential type selfimproving properties for measurable functions satisfying the following Poincaré type inequality: inf α ( (f − α)χB )∗ μ ( λμ(B) ) ≤ cλa(B). Here, f ∗ μ denotes the non-increasing rearrangement of f , and a is a functional acting on balls B, satisfying appropriate geometric conditions. Our main result improves...
متن کامل