The Davies Method for Heat Kernel Upper Bounds of Non-Local Dirichlet Forms on Ultra-Metric Spaces
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE DAVIES METHOD FOR HEAT KERNEL UPPER BOUNDS OF NON-LOCAL DIRICHLET FORMS ON ULTRA-METRIC SPACES∗
pA)
Jin GAO (
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail : [email protected] Abstract We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space. Key words
heat kernel; ultra-metric; Davies method
2010 MR Subject Classification
1
35K08; 28A80; 60J35
Introduction
We are concerned with the heat kernel estimate for the non-local Dirichlet form on the ultrametric space. Let (M, d, µ) be an ultra-metric measure space; that is, M is locally compact and separable, d is an ultra-metric, and µ is a Radon measure with full support in M . Recall that a metric d is called an ultra-metric if, for any points x, y, z ∈ M , d(x, y) ≤ max{d(x, z), d(z, y)}. For any x ∈ M and r > 0 the metric ball B(x, r) is defined by B(x, r) := {y ∈ M, d(x, y) ≤ r}. It is known that any two metric balls are either disjoint or that one contains the other (see [1, 2]). Thus any ball is both closed and open so that its boundary is empty. An ultra-metric space is totally disconnected, so the process described by the heat kernel is a pure jump process. In this article, we consider the the Dirichlet form E only having a non-local part without the killing part in the Beurling-Deny decomposition (see [6, p.120]). Let E be the energy form given by ZZ E(f, g) = (f (x) − f (y))(g(x) − g(y))dj(x, y), (1.1) M×M\diag
∗ Received March 12, 2019; revised September 14, 2019. The author was supported by National Natural Science Foundation of China (11871296).
No.5
J. Gao: HEAT KERNEL UPPER BOUNDS
1241
where j is a symmetric Radon measure on (M × M )\ diag. For simplicity, we denote E(f, f ) by E(f ). We need to specify the domain of E. Let D be the space defined by ( n ) X D= ci 1Bi : n ≥ 1, ci ∈ R, Bi are compact disjoint balls . (1.2) i=0
Let the space F be determined by
q 2 F = the closure of D under norm E(u) + kuk2 .
(1.3)
Then (E, F ) is a regular Dirichlet form in L2 := L2 (M, µ) if the measure j satisfies that j(B, B c ) < ∞ for any ball B (see [2, Theorem 2.2]). Recall that the indicator function 1B for any ball B belongs to the space F if j(B, B c ) < ∞, because E(1B ) = 2j(B, B c ) < ∞ (see [2, formula (4.1)]). In the sequel, we will fix some numbers α > 0 and β > 0, and some value R0 ∈ (0, diam M ], which contains the case when R0 = diam M = ∞. The letter C is universal positive constant which may vary at each occurrence. We list some conditions to be used later on. • Condition (TJ): there exists a transition function J(x, dy) such that dj(x, y) =
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