PLANE WITH A∞ -WEIGHTED METRIC NOT BILIPSCHITZ EMBEDDABLE TO Rⁿ

A planar set G ⊂ R2 is constructed that is bilipschitz equivalent to (G,dz), where (G, d) is not bilipschitz embeddable to any uniformly convex Banach space. Here, Z ∈ (0, 1) and dz denotes the zth power of the metric d. This proves the existence of a strong A∞ weight in R2, 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 R2. 2000 Mathematics Subject Classification 54E40 (primary); 30C62, 30C65, 28A80 (secondary).