Abstract
Abstract We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.
Highlights
A homeomorphism f ∶ X → Y between two metric spaces X, Y is said to be quasiconformal if there is a constantH ≥ such that for all x ∈ X, lim sup r→ +supy∈B(x,r) dY (f (x), f (y)) infy∈X∖B(x,r) dY (f (x), f (y)) ≤ H.In metric measure spaces satisfying suitable conditions such as Ahlfors regularity and a Poincaré inequality, the study of quasiconformal mappings was begun by Heinonen and Koskela in [12] and the literature is extensive, see for example [3, 11, 13, 21, 25]
We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality
We apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces
Summary
A homeomorphism f ∶ X → Y between two metric spaces X, Y is said to be quasiconformal if there is a constantH ≥ such that for all x ∈ X, lim sup r→ +supy∈B(x,r) dY (f (x), f (y)) infy∈X∖B(x,r) dY (f (x), f (y)) ≤ H.In metric measure spaces satisfying suitable conditions such as Ahlfors regularity and a Poincaré inequality, the study of quasiconformal mappings was begun by Heinonen and Koskela in [12] and the literature is extensive, see for example [3, 11, 13, 21, 25]. Let < p < ∞ and suppose X is a complete metric space equipped with a doubling measure and supporting a -Poincaré inequality.
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