广义3G定理及其对非光滑开集上相对论稳定过程的应用
Generalized 3G theorem and application to relativistic stable process on non-smooth open sets
摘要：
设G（x，y）和GD（x，y）分别是R^d和开集D中旋转不变对称α-稳定过程的格林函数，其中0<α< 2.不等式G_D（x，y）G_D（y，z）/G_D（x，z）\le c（G（x，y）+G（y，z））是研究（局部）Schrodinger算子的一个非常有用的工具。当上述不等式成立且常数c=c（D）>0时，我们说3G定理在D中成立。当D是包含有界John整环的有界κ-胖开集时，本文建立了3G定理的一个推广.我们所考虑的3G具有G_D（x，y）G_D（z，w）/G_D（x，w）的形式，其中y可以不同于z。当y=z时，我们恢复通常的3G。在非定域Schrodinger算子理论中出现了3G形式G_D（x，y）G_D（z，w）/G_D（x，w）。利用推广的3G定理，我们给出了一类具体的函数，它们属于Chen和Song在κ-fat开集上引入的非局部Kato类.作为应用，我们讨论了κ-fat开集上的相对论性α-稳定过程（α=1时的相对论性Hamilton）.证明了相对论性α-稳定过程在κ-fat开集上的Martin边界和极小Martin边界与欧氏边界.进一步证明了相对Fatou型定理对κ-fat开集上的相对论稳定过程成立.本文的主要结果适用于一大类对称马尔可夫过程，如本文的最后一节所示。我们还讨论了一大类对称稳定Levy过程的广义3G定理。
英文摘要：
Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant symmetric α-stable process in R^d and in an open set D respectively, where 0<α< 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrodinger operators. When the above inequality is true with a constant c=c(D)>0, then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local Schrodinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Levy processes.
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