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

Abstract

Given Q > 1, we construct an Ahlfors Q-regular space that admits a weak (1, 1)-Poincaré inequality. IntroductionRecently a lot of analysis has been done in metric measure spaces with controlled geometry. In particular, Ahlfors-regular spaces admitting an appropriate Poincaré inequality have been shown to carry a very Euclideanlike theory. See for example [HK] and [C]. Recall that a metric measure space (X, µ) admits a weak (1, 1)-Poincaré inequality if for any ball Bwhenever u is a bounded continuous function in a ball CB and ρ its upper gradient there. The constant C should be independent of B and u. Ahlfors Q-regularity means that r Q /C ≤ µ(B(x, r)) ≤ Cr Q for some C independent of x and r, whenever r ≤ diam(X). It has nevertheless not been clear, exactly how strong these assumptions are. In [HS] Heinonen and Semmes asked whether there are fractional dimensional Ahlfors regular spaces admitting a weak (1, 1)-Poincaré inequality. Recently Bourdon and Pajot showed the existence of such spaces for a discrete countable set of non-integer dimensions, see [BP]. Thus it is natural to ask, whether there are gaps in the set of possible dimensions. The purpose of this paper is to show that there are no restrictions: For any Q ≥ 1 there exists an Ahlfors Q-regular space admitting a weak (1,1)-Poincaré inequality. Note that an Ahlfors Q-reqular space has always Hausdorffdimension Q.Our method is based on ideas of Semmes in [S]. Semmes proved the Poincaré and Sobolev inequalities from the existence of curve-families that

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

“…This follows from Theorem 1.6 and the equivalence of the Q-Loewner condition with the Q-Poincaré inequality in quasiconvex Ahlfors Q-regular spaces [20]. We remark that the examples of BourdonPajot [6] and Laakso [30] are Q-regular Q-Loewner metric spaces of topological dimension one, however, these examples admit no bi-Lipschitz embedding into any finite-dimensional Euclidean space. Corollary 1.8.…”

“…• compact Riemannian manifolds or noncompact Riemannian manifolds satisfying suitable curvature bounds [9], • Carnot groups and more general sub-Riemannian manifolds equipped with CarnotCarathéodory (CC) metric [20], [19], [22], • boundaries of certain hyperbolic Fuchsian buildings, see Bourdon and Pajot [6], • Laakso's spaces [30], • linearly locally contractible manifolds with good volume growth [34]. These examples fall into two (overlapping) classes: examples for which the underlying topological space is a manifold, and abstract metric examples which admit no bi-Lipschitz embedding into any finite-dimensional Euclidean space.…”

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

“…In [46], Laakso gives an explicit construction of an Ahlfors Q-regular Loewner space for each Q > 1. His examples arise as quotients of the product of the unit interval with a Cantor set under a finite-to-one map.…”

“…Heinonen and Koskela [38] have developed an abstract theory of quasiconformal maps on a particular class of metric measure spaces which recovers much of the classical theory in the case of Euclidean spaces or more general Carnot groups. Other examples of spaces satisfying the assumptions of the Heinonen-Koskela theory have been constructed by Bourdon and Pajot [9], [10] and Laakso [46]; see section 11 of this paper.…”

Metric Structures in L 1: Dimension, Snowflakes, and Average Distortion

“…G 0 is the graph on two vertices with one edge. To construct G i , take six copies of G i−1 and scale their metric by a factor of As shown in [14], the graphs {G i } ∞ i=0 are uniformly doubling (see also [15], for a simple argument showing they are doubling with constant 6). Moreover, since the G i 's are series parallel graphs, they embed uniformly in L 1 (see [8]).…”