关于共形行走维数：对称扩散的拟对称一致化
On the conformal walk dimension: Quasisymmetric uniformization for symmetric diffusions
摘要：
我们引入共形行走维数的概念，它作为椭圆和抛物Harnack不等式之间的桥梁。这个概念的重要性是由于这样一个事实，即对于一个给定的强局部，正则对称狄利克雷空间，其中每个度量球都有紧闭包（MMD空间），共形行走维数的有限性表征了度量加倍性质和椭圆Harnack不等式的结合。粗略地说，MMD空间的共形行走维数被定义为行走维数的所有可能值上的下确界，通过该下确界，抛物Harnack不等式可以通过相关扩散的时变和度量的准对称变化而成立。证明了满足度量加倍性质和椭圆Harnack不等式的任意MMD空间的共形行走维数为2，并给出了一个必要条件，使一对这样的变换达到定义共形行走维数的下确界.我们还证明了在自相似集上的自相似Dirichlet型的设定下存在这样一对达到下确界的必要条件，并应用它证明了Vicsek集和具有$N\geq 3$的$N$维Sierpienski垫片不能达到下确界，与Kigami [Math.Ann.340（2008），no.4，781- 804]的二维Sierpienski垫圈的实现相反。
英文摘要：
We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which the parabolic Harnack inequality can be made to hold by a time change of the associated diffusion and by a quasisymmetric change of the metric. We show that the conformal walk dimension of any MMD space satisfying the metric doubling property and the elliptic Harnack inequality is two, and provide a necessary condition for a pair of such changes to attain the infimum defining the conformal walk dimension when it is attained by the original pair. We also prove a necessary condition for the existence of such a pair attaining the infimum in the setting of a self-similar Dirichlet form on a self-similar set, and apply it to show that the infimum fails to be attained for the Vicsek set and the $N$-dimensional Sierpiński gasket with $N\geq 3$, in contrast to the attainment for the two-dimensional Sierpiński gasket due to Kigami [Math. Ann. 340 (2008), no. 4, 781--804].
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