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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 mea- sure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map.
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On Carrasco Piaggio's theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces
In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space.
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- Award ID(s):
- 2054960
- PAR ID:
- 10349399
- Editor(s):
- Xiao, Jie; Lenhart, Suzanne
- Date Published:
- Journal Name:
- Potentials and Partial Differential Equations - The Legacy of David R. Adams (series: Advances in Analysis and Geometry )
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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