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Some inequalities between Ahlfors regular conformal dimension and spectral dimensions for resistance forms

Sasaya, Kôhei 京都大学 DOI:10.14989/doctor.k24391

2023.03.23

概要

Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension
dimARC (X, d) of a metric space (X, d) is the infimum over the Hausdorff
dimensions of the Ahlfors regular images of the space by quasisymmetric
transformations. For a given regular Dirichlet form with the heat kernel,
the spectral dimension ds is an exponent which indicates the short-time
asymptotic behavior of the on-diagonal part of the heat kernel. In this paper, we consider the Dirichlet form induced by a resistance form on a set X
and the associated resistance metric R. We prove dimARC (X, R) ≤ ds < 2
for ds , a variation of ds defined through the on-diagonal asymptotics of the
heat kernel. We also give an example of a resistance form whose spectral
dimension ds satisfies the opposite inequality ds < dimARC (X, R) < 2. ...

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