Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality

By T. Laakso. 2000

220 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/80ac16228e57c6638b5d795fcbe1eefe402d8aac

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Bruce Kleiner
2. Anders Bjorn
3. Assaf Naor
4. Jana Björn
5. S. Keith
6. P. Hajłasz
7. Yuval Peres
8. Scott Sheffield

In Geometry and Topology

1. Assaf Naor
2. Bruce Kleiner
3. Yuval Peres
4. M. Bonk
5. P. Hajłasz
6. Scott Sheffield
7. Ilya Kapovich
8. Nadia Benakli

Citations in the corpus

• Bilipschitz Embeddings of Metric Spaces into Space Forms (2001, methodology): “A metric space which is doubling with constant L has Hausdorff dimension at most log2 L.”
• Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group (2009, background): “Some of the more exotic examples of doubling metric measure spaces with a Poincaré inequality are the boundary of hyperbolic buildings (see [4]), some Cantor-like sets with worm-holes (see [14] and the erratum [15]) and the Heisenberg group (see [29]).”
• On the differentiability of Lispschitz maps from metric measure spaces to Banach spaces (2006, background): “Examples of PI spaces include Gromov-Hausdorff limits of sequences of Riemannian manifolds with a uniform lower Ricci curvature bound, Bourdon-Pajot spaces, Laakso spaces and Carnot groups such as the Heisenberg group H; see [CC97], [BP99], [Laa00], [Gro96].”
• Sobolev Mappings between Manifolds and Metric Spaces (2009, background): “There are, however, many examples of spaces supporting Poincaré inequalities which carry some mild geometric structure, but do not resemble Riemannian manifolds [7, 45, 46, 59, 60, 75].”
• Geometric embeddings of metric spaces (2003, unknown): “Read more in [61], [62], [65], [48], [45], [46].”
• The Perron method for p-harmonic functions in metric spaces (2003, methodology): “Secondly, and perhaps more importantly, we extend the Perron method to the setting of proper metric spaces endowed with a doubling measure supporting a weak Poincaré type inequality, thus unifying the theory developed in the Euclidean setting and the theory developed in Markina–Vodop’yanov [31,32] f”
• Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces (2004, background): “The standard example showing that in both embedding results we must pass to a power of the metric is the family of graphs known as the Laakso graphs [37, 38], which are planar graphs whose graph metric is uniformly doubling, yet they do not uniformly embed into Hilbert space.”
• Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property (2008, background): “ L N ∩ ({c }× [0, 1]) =0 , where L denotes Lebesgue measure on [0, 1]. Clearly, a countable union of negligible sets is negligible.”
• Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary (2002, background): “nd quasisymmmetric homeomorphisms to the setting of Q–regular Q–Loewner spaces. By now there is a substantial body of literature about metric spaces satisfying Poincar´e inequalities; see for example [12, 10, 18, 17, 15, 14]. These spaces play a central role in Cheeger’s theory of differentiability o”
• Metric and geometric quasiconformality in Ahlfors regular Loewner spaces (2001, methodology): “Moreover, the measure onIK descends to a measure (still denoted )o nX which is Ahlfors Q-regular [ 46, Section 1 ]; this fact is easily proved by noting that the quotient map s is regular in the sense of David and Semmes [22, Denition 12.1] and using [22, Lemma 12.7].”