Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups
Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups
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对 p.c.f 上拉普拉斯算子的解析核的估计
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发表时间：
2010
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通讯作者：
Luke G. Rogers
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作者：
Luke G. Rogers
后临界有限自相似集（pcfss）分析的一个主要特点是可以根据集的自相似结构来理解拉普拉斯算子及其逆算子绿色算子的行为。事实上，通过Dirichlet形式分析自相似分形的方法中的一个主要步骤是Kigami证明[8，10]，对于自相似Dirichlet形式，绿色核可以显式地写为一个系列，其中每个项都是通过自相似结构的单个表达式的重新标度。在[6]中，这个结果被推广到表明拉普拉斯算子的预解核，即（z − 1）−1的核，对于z ∈ C的适当值，也可以写成自相似级数。这项工作的部分动机是，它给出了一个新的理解功能的拉普拉斯算子（如热运营商等）通过书面他们作为积分的预解式。本工作的目的是确定允许采用上述方法的估计数。我们研究了在[6]的级数分解中出现的函数（参见下面的定理3.3），并给出了它们的衰减估计。由此，我们确定估计的预解核和内核的运营商定义为积分的预解核。特别地，我们恢复了Hambly和Kumagai [4]通过概率方法对pcfss集证明的热核的精确上估计（见定理10.2）（也见[1，13，3]关于较不一般的集合类的这种类型的早期结果）。值得注意的是，前面的作者不仅能够证明热核的上估计，而且还能够证明热核的下界，因此能够证明其边界的清晰度。我们的方法允许尖锐的界限上的正真实的轴的预解核，但我们不知道如何获得这些全球性的复平面或如何获得较低的估计，从他们的热核。因此，在这个方向上，我们的结果不如[4]中得到的结果强。然而，在其他方向，我们获得更多的信息比已知的热核估计，我们希望我们的方法将补充现有的概率方法。特别是，我们能够获得预解的界限以外的任何射线在C的负真实的轴（在那里的频谱），而标准的计算从热核界限只给这些估计在半平面。我们的方法的进一步结果是，我们将[6]的分解扩展（在定理9.7中）到爆破的情况，爆破是具有与底层自相似集的局部结构等价的局部结构的非紧集。pcfss集的爆破与原始集的关系与真实的直线与单位区间的关系相同，详见[17]。本文的结构如下。在第3节中，我们回顾了pcfss集分析的一些基本特征，以及[6]的主要结果，即将预解式分解为分段特征函数的加权和。第4节讨论
One of the main features of analysis on post-critically finite self-similar (pcfss) sets is that it is possible to understand the behavior of the Laplacian and its inverse, the Green operator, in terms of the self-similar structure of the set. Indeed, a major step in the approach to analysis on self-similar fractals via Dirichlet forms was Kigami’s proof [8, 10] that for a self-similar Dirichlet form the Green kernel can be written explicitly as a series in which each term is a rescaling of a single expression via the self-similar structure. In [6] this result was extended to show that the resolvent kernel of the Laplacian, meaning the kernel of (z − ∆)−1, can also be written as a self-similar series for suitable values of z ∈ C. Part of the motivation for that work was that it gives a new understanding of functions of the Laplacian (such as the heat operator et∆) by writing them as integrals of the resolvent. The purpose of the present work is to establish estimates that permit the above approach to be carried out. We study the functions occurring in the series decomposition from [6] (see Theorem 3.3 below for this decomposition) and give estimates on their decay. From this we determine estimates on the resolvent kernel and on kernels of operators defined as integrals of the resolvent kernel. In particular we recover the sharp upper estimates for the heat kernel (see Theorem 10.2) that were proved for pcfss sets by Hambly and Kumagai [4] by probabilistic methods (see also [1, 13, 3] for earlier results of this type on less general classes of sets). It is important to note that the preceding authors were able to prove not just upper estimates but also lower bounds for the heat kernel, and therefore were able to prove sharpness of their bounds. Our methods permit sharp bounds for the resolvent kernel on the positive real axis, but we do not know how to obtain these globally in the complex plane or how to obtain lower estimates for the heat kernel from them. Therefore in this direction our results are not as strong as those obtained in [4]. However in other directions we obtain more information than that known from heat kernel estimates, and we hope that our approach will complement the existing probabilistic methods. In particular we are able to obtain resolvent bounds on any ray in C other than the negative real axis (where the spectrum lies), while standard calculations from heat kernel bounds only give these estimates in a half-plane. A further consequence of our approach is that we extend (in Theorem 9.7) the decomposition from [6] to the case of blowups, which are non-compact sets with local structure equivalent to that of the underlying self-similar sets. The blowup of a pcfss set bears the same relation to the original set as the real line bears to the unit interval, see [17] for details. The structure of the paper is as follows. In Section 3 we recall some basic features of analysis on pcfss sets, as well as the main result of [6], which is the decomposition of the resolvent as a weighted sum of piecewise eigenfunctions. Section 4 then discusses