Sierpiński地毯上的一阶Sobolev空间、自相似能量与能量测度
First-order Sobolev spaces, self-similar energies and energy measures on the Sierpiński carpet
AI总结：
本文在平面Sierpiński地毯上对所有p构造一阶Sobolev空间、p-能量及能量测度，利用Loewner型估计证明Poincaré不等式，并应用于Ahlfors正则共形维数的可达性。
摘要：
我们构造并研究了平面Sierpiński地毯上对所有$p \in (1, \infty)$的$(1, p)$-Sobolev空间、$p$-能量以及相应的$p$-能量测度。我们的方法基于Kusuoka和Zhou [Probab. Theory Related Fields $\textbf{93}$ (1992), no. 2, 169--196]的思想，其中在包括平面Sierpiński地毯在内的自相似集上构造了布朗运动（即$p = 2$的情形）。与这一早期工作类似，我们使用一系列离散图逼近以及相应的离散$p$-能量来定义Sobolev空间和$p$-能量。然而，我们需要一种新方法以确保当$p$小于Ahlfors正则共形维数时，我们的$(1, p)$-Sobolev空间具有稠密的连续函数集。新的要素是使用组合模量的Loewner型估计来在一系列逼近图上获得Poincaré不等式和椭圆Harnack不等式。我们Sobolev空间的一个重要特征是$p$-能量的自相似性，这使我们能够在平面Sierpiński地毯上定义相应的$p$-能量测度。我们证明我们的Sobolev空间也可以被视为Korevaar-Schoen型空间。我们将结果应用于Sierpiński地毯的Ahlfors正则共形维数的可达性问题。特别地，我们证明如果Ahlfors正则共形维数（记为$\dim_{\mathrm{ARC}}$）可达，那么任何达到$\dim_{\mathrm{ARC}}$的最优测度必须与某个函数在我们的$(1, \dim_{\mathrm{ARC}})$-Sobolev空间中的$\dim_{\mathrm{ARC}}$-能量测度相差一个乘法常数。在这种情况下，我们还证明对应于任何最优测度和度量的Newton-Sobolev空间可以被等同于我们的自相似$(1, \dim_{\mathrm{ARC}})$-Sobolev空间。
英文摘要：
We construct and investigate $(1, p)$-Sobolev space, $p$-energy, and the corresponding $p$-energy measures on the planar Sierpiński carpet for all $p \in (1, \infty)$. Our method is based on the idea of Kusuoka and Zhou [Probab. Theory Related Fields $\textbf{93}$ (1992), no. 2, 169--196], where Brownian motion (the case $p = 2$) on self-similar sets including the planar Sierpiński carpet were constructed. Similar to this earlier work, we use a sequence of discrete graph approximations and the corresponding discrete $p$-energies to define the Sobolev space and $p$-energies. However, we need a new approach to ensure that our $(1, p)$-Sobolev space has a dense set of continuous functions when $p$ is less than the Ahlfors regular conformal dimension. The new ingredients are the use of Loewner type estimates on combinatorial modulus to obtain Poincaré inequality and elliptic Harnack inequality on a sequence of approximating graphs. An important feature of our Sobolev space is the self-similarity of our $p$-energy, which allows us to define corresponding $p$-energy measures on the planar Sierpiński carpet. We show that our Sobolev space can also be viewed as a Korevaar-Schoen type space. We apply our results to the attainment problem for Ahlfors regular conformal dimension of the Sierpiński carpet. In particular, we show that if the Ahlfors regular conformal dimension, say $\dim_{\mathrm{ARC}}$, is attained, then any optimal measure which attains $\dim_{\mathrm{ARC}}$ should be comparable with the $\dim_{\mathrm{ARC}}$-energy measure of some function in our $(1, \dim_{\mathrm{ARC}})$-Sobolev space up to a multiplicative constant. In this case, we also prove that the Newton-Sobolev space corresponding to any optimal measure and metric can be identified as our self-similar $(1, \dim_{\mathrm{ARC}})$-Sobolev space.
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