论文标题
在保形步行维度：对称扩散的准对称均匀化
On the conformal walk dimension: Quasisymmetric uniformization for symmetric diffusions
论文作者
论文摘要
我们介绍了共形步行维度的概念，该概念是椭圆形和抛物线harnack不平等之间的桥梁。这个概念的重要性是由于以下事实：对于给定的局部，规则的对称的迪里奇空间，每个度量球都具有紧凑的闭合（MMD空间），因此，形式的步行尺寸的有限性表征了度量倍增属性和椭圆形harnack平等性的共同点。粗略地说，MMD空间的共形步行维度定义为步行维度的所有可能值，可以通过将相关扩散的时间变化和度量标准的准对称变化来使抛物线抛物线不平等的所有可能值保持。我们表明，满足度量倍增属性和椭圆形harnack不平等的任何MMD空间的共形步行尺寸为两个，并且在原始对完成时，为一对此类变化提供了必要的条件，以实现结构性步行尺寸。我们还证明，在自相似集合的自我相似的dirichlet形式下，在自我相似的设置中获得了最小的dimimum的存在是必要条件由于kigami [数学。安。 340（2008），否。 4，781--804]。
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].
