论文标题
AHLFOR的常规保形维和抗电阻形式的光谱尺寸之间的一些不平等
Some inequalities between Ahlfors regular conformal dimension and spectral dimensions for resistance forms
论文作者
论文摘要
Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension $\dim_\mathrm{ARC}(X,d)$ of a metric space $(X,d)$ is the infimum over the Hausdorff dimensions of the Ahlfors regular images of the space by quasisymmetric transformations.对于给定的常规Dirichlet形式，具有加热内核，光谱尺寸$ d_s $是指示器指示热内核的二重奏部分的短时渐近行为。在本文中，我们考虑了一套$ x $和相关的电阻度量$ r $在电阻形式引起的dirichlet形式。我们证明$ \ dim_ \ mathrm {arc}（x，x，r）\ le \ overline {d_s} <2 $ for $ \ overline {d_s} $，这是通过diagonal Angototics of the Heat Kernel定义的$ d_s $的变体。我们还举了一个电阻形式的示例，其光谱尺寸$ d_s $满足相反的不等式$ d_s <\ dim_ \ mathrm {arc}（x，x，r）<2
Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension $\dim_\mathrm{ARC}(X,d)$ 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 $\dim_\mathrm{ARC}(X,R)\le \overline{d_s}<2$ 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 $d_s<\dim_\mathrm{ARC}(X,R)<2.$
