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

Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension ${\dim }_{\mathrm{A}\mathrm{R}\mathrm{C}}(X,d)$ 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 $d_s$ 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 ${\dim }_{\mathrm{A}\mathrm{R}\mathrm{C}}(X,R)\le \overline{d_s}<2$ for $\overline{d_s}$, a variation of $d_s$ defined through the on-diagonal asymptotics of the heat kernel. We also give an example of a resistance form whose spectral dimension $d_s$ satisfies the opposite inequality $d_s<{\dim }_{\mathrm{A}\mathrm{R}\mathrm{C}}(X,R)<2.$