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
A planar set G ⊂ R 2 is constructed that is bilipschitz equivalent to (G, d z ), where (G, d) is not bilipschitz embeddable to any uniformly convex Banach space. Here, z ∈ (0, 1) and d z denotes the zth power of the metric d. This proves the existence of a strong A ∞ weight in R 2 , such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of R 2 .
Let α ≥ 1 and let (X, d, µ) be an α-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, µ) with uniformly big 1-modulus.
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