Potential theory and quasisymmetric maps between compact Ahlfors regular metric measure spaces via Besov functions: preliminary

Abstract: We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space. This approach is applicable even if the space does not support any Poincar\'e inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric measure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map.