Guanhua Liu
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Guanhua Liu has a total of 2 co-authors. Here are the top co-authors by number of shared publications:
- Mathav Murugan
Department of Mathematics The University of British Columbia 1984 Mathematics Road V6T 1Z2 Vancouver Columbia British Canada
2 shared publications - Jiaxin Hu
Department of Mathematical Sciences, Tsinghua University, Beijing, China
1 shared publication
Latest Publications
- Open Access
The parabolic Harnack inequality on non-local Dirichlet spaces in the view of pure analysis
This paper provides the general theory on parabolic Harnack inequalities (PHI, for short) for regular Dirichlet forms without killing part. We prove PHI by pure analytic methods, using both Nash and Moser approaches, and yield some important properties contained in PHI. Combining our recent result on weak Harnack inequalities, we greatly enlarge the list of equivalent characterizations of PHI.
Open AccessThe parabolic Harnack inequality on non-local Dirichlet spaces in the view of pure analysis
This paper provides the general theory on parabolic Harnack inequalities (PHI, for short) for regular Dirichlet forms without killing part. We prove PHI by pure analytic methods, using both Nash and Moser approaches, and yield some important properties contained in PHI. Combining our recent result on weak Harnack inequalities, we greatly enlarge the list of equivalent characterizations of PHI.
- Open Access
- 3 May 2024
- Published by Elsevier in Journal of Mathematical Analysis and Applications
- Vol. 539 (1)
Heat kernel estimates on local and non-local Dirichlet spaces satisfying a weak chain condition
- 3 May 2024
- Published by Elsevier in Journal of Mathematical Analysis and Applications
- Vol. 539 (1)
Open AccessHeat kernel estimates on local and non-local Dirichlet spaces satisfying a weak chain condition
- 9 November 2023
- Published by Springer Nature in manuscripta mathematica
- Vol. 174 (1-2) , 597-647
Existence of self-similar Dirichlet forms on post-critically finite fractals in terms of their resistances
- 9 November 2023
- Published by Springer Nature in manuscripta mathematica
- Vol. 174 (1-2) , 597-647
Existence of self-similar Dirichlet forms on post-critically finite fractals in terms of their resistances
- Open Access
Heat kernel estimates on local and non-local Dirichlet spaces satisfying a weak chain condition
In this paper, we focus on the heat kernel estimates for diffusions and jump processes on metric measure spaces satisfying a weak chain condition, where the length of a nearly shortest $\varepsilon$-chain between two points $x,y$ is comparable with a function of $d(x,y)$ and $\varepsilon$. For a diffusion, the best estimate is already given by Grigor'yan and Telcs, and we make it explicit in our particular case. For jump processes, especially those where the scale of the process is different with that of the jump kernel, we improve the results by Bae, Kang, Kim and Lee. Uniformity of the coefficients (or parameters) in the known estimates and metric transforms play the key role in our proof. We also show by examples how the weak chain condition is valid in practice.Open AccessHeat kernel estimates on local and non-local Dirichlet spaces satisfying a weak chain condition
In this paper, we focus on the heat kernel estimates for diffusions and jump processes on metric measure spaces satisfying a weak chain condition, where the length of a nearly shortest $\varepsilon$-chain between two points $x,y$ is comparable with a function of $d(x,y)$ and $\varepsilon$. For a diffusion, the best estimate is already given by Grigor'yan and Telcs, and we make it explicit in our particular case. For jump processes, especially those where the scale of the process is different with that of the jump kernel, we improve the results by Bae, Kang, Kim and Lee. Uniformity of the coefficients (or parameters) in the known estimates and metric transforms play the key role in our proof. We also show by examples how the weak chain condition is valid in practice. - 1 December 2021
- Published by Walter de Gruyter GmbH in Forum Mathematicum
- Vol. 34 (1) , 225-277
Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces
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 -norm estimate of the survival function for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic mean-value inequality is borrowed to use.- 1 December 2021
- Published by Walter de Gruyter GmbH in Forum Mathematicum
- Vol. 34 (1) , 225-277
Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces
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 -norm estimate of the survival function for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic mean-value inequality is borrowed to use.- 5 March 2021
- Published by Springer Nature in Potential Analysis
- Vol. 57 (1) , 155-166
Parabolic Harnack Inequality Implies the Existence of Jump Kernel
- 5 March 2021
- Published by Springer Nature in Potential Analysis
- Vol. 57 (1) , 155-166
Parabolic Harnack Inequality Implies the Existence of Jump Kernel
- Open Access
Parabolic Harnack inequality implies the existence of jump kernel
We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the L\'evy system formula and a near-diagonal heat kernel lower bound.
Open AccessParabolic Harnack inequality implies the existence of jump kernel
We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the L\'evy system formula and a near-diagonal heat kernel lower bound.