Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces

Jiaxin Hu; Guanhua Liu

doi:10.1515/forum-2021-0096

Article

Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces

Jiaxin Hu and Guanhua Liu

Published/Copyright: January 6, 2022

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From the journal Forum Mathematicum Volume 34 Issue 1

Abstract

In this paper, we present a new approach to obtaining the off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space. One of the novelties is that we have obtained the weighted L2-norm estimate of the survival function 1-PtB⁢1B for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic L2 mean-value inequality is borrowed to use.

Keywords: Heat kernel; Dirichlet form; Markov semigroups

MSC 2010: 35K08; 31C25; 47D07

Communicated by Maria Gordina

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871296

Funding statement: Supported by NSFC No. 11871296.

A Appendix

In this appendix, we first give the following result and then collect the known results that have been used in this paper.

Proposition A.1.

Let (E,F) be a regular Dirichlet form in L2 on a metric measure space (M,d,μ). If Ω is a non-empty open compact subset of M, then the indicator 1Ω is in F⁢(Ω). In particular, if M is bounded and every metric ball is assumed to be precompact, then cutoff⁡(M,M)={1}.

Proof.

Since Ω is open and compact, the indicator function 1Ω is in C0⁢(M), the space of all continuous functions with compact support in M. By [16, Lemma 1.4.2 on p. 29], there exists a sequence {un} of functions from ℱ∩C0⁢(M) with supp⁡[un]⊂Ω such that un→1Ω uniformly on M as n→∞. Therefore, there is some integer n such that un≥12 in Ω, and thus showing that 1Ω=2⁢(12∧un)∈ℱ. Since the function 1Ω vanishes outside Ω, we see that 1Ω∈ℱ⁢(Ω) by using [16, Corollary 2.3.1 on p. 98]).

If M is bounded, then M is compact, since M is the closure of a ball B and every metric ball is assumed to be precompact. Thus 1∈ℱ, and so cutoff⁡(M,M)={1}. ∎

The following results are known.

Lemma A.2 ([24, Proposition 4.6]).

Let Ω be a non-empty open set in M and let f∈L2∩L∞ be nonnegative in M. Then for any t>0 and μ-almost every x∈Ω,

|PtΩ⁢f⁢(x)-QtΩ⁢f⁢(x)|≤2⁢t⁢∥f∥∞⁢esupx∈M∫B⁢(x,ρ)cJ⁢(x,y)⁢𝑑μ⁢(y).

Lemma A.3 ([24, Theorem 3.1]).

Let {Qt≔Qt(ρ)}t≥0 be the heat semigroup of some ρ-local Dirichlet form (E(ρ),F(ρ)) in L2. Let ϕ⁢(r,⋅) be a non-decreasing function in (0,∞) for any r>0. Assume that for any ball B≔B⁢(x,r) and for any t∈(0,T0) where T0∈(0,∞],

1-QtB⁢1B≤ϕ⁢(r,t) in ⁢14⁢B.

Then for any ball B⁢(x,r) with r>ρ and t∈(0,T0), and for any integer k≥1,

Qt⁢1B⁢(x,k⁢r)c≤ϕ⁢(r-ρ,t)k-1 in ⁢B⁢(x,r).

Lemma A.4 ([21, Lemma 4.18]).

Assume that (E,F) is a regular Dirichlet form in L2. Then for any two open subsets U⊂Ω of M, for any compact set K⊂U, for any 0≤f∈L2⁢(M) and all t>0,

(A.1)esupΩ(PtΩ⁢f-PtU⁢f)≤sups∈(0,t]⁡esupΩ∖KPsΩ⁢f.

In particular, when Ω=M, U=B for any metric ball B, for any t>0 we have

PtB⁢f⁢(x)≥Pt⁢f⁢(x)-sups∈(0,t]⁡esupx∈(12⁢B¯)cPs⁢f⁢(x).

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Received: 2021-04-22

Revised: 2021-08-20

Published Online: 2022-01-06

Published in Print: 2022-01-01

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https://doi.org/10.1515/forum-2021-0096

Keywords for this article

Heat kernel; Dirichlet form; Markov semigroups