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Geometry and Analysis of Metric Spaces via Weighted Partitions (Paperback, 1st ed. 2020)
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Geometry and Analysis of Metric Spaces via Weighted Partitions (Paperback, 1st ed. 2020)
Series: Lecture Notes in Mathematics, 2265
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The aim of these lecture notes is to propose a systematic framework
for geometry and analysis on metric spaces. The central notion is a
partition (an iterated decomposition) of a compact metric space.
Via a partition, a compact metric space is associated with an
infinite graph whose boundary is the original space. Metrics and
measures on the space are then studied from an integrated point of
view as weights of the partition. In the course of the text: It is
shown that a weight corresponds to a metric if and only if the
associated weighted graph is Gromov hyperbolic. Various relations
between metrics and measures such as bilipschitz equivalence,
quasisymmetry, Ahlfors regularity, and the volume doubling property
are translated to relations between weights. In particular, it is
shown that the volume doubling property between a metric and a
measure corresponds to a quasisymmetry between two metrics in the
language of weights. The Ahlfors regular conformal dimension of a
compact metric space is characterized as the critical index of
p-energies associated with the partition and the weight function
corresponding to the metric. These notes should interest
researchers and PhD students working in conformal geometry,
analysis on metric spaces, and related areas.
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