Heat Kernel for the Kohn Laplacian on the Heisenberg Group

2010 ◽  
pp. 349-358
Author(s):  
Ovidiu Calin ◽  
Der-Chen Chang ◽  
Kenro Furutani ◽  
Chisato Iwasaki
Keyword(s):  
Heisenberg Group ◽  
Heat Kernel ◽  
Kohn Laplacian

The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary

2004 ◽  
Vol 56 (3) ◽  
pp. 590-611
Author(s):  
Yilong Ni
Keyword(s):  
Riemannian Manifold ◽  
Heisenberg Group ◽  
Green’S Function ◽  
Heat Kernel ◽  
Green's Function ◽  
Beltrami Operator ◽  
Small Time ◽  
Time Estimates

AbstractWe study the Riemannian Laplace-Beltrami operator L on a Riemannian manifold with Heisenberg group H1 as boundary. We calculate the heat kernel and Green's function for L, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of H1. We also restrict L to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary.

Estimates of the Green’s Function and its Regular Part on Heisenberg Group Domains

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Najoua Gamara ◽  
Habiba Guemri
Keyword(s):  
Heisenberg Group ◽  
Green’S Function ◽  
Green's Function ◽  
Forthcoming Paper ◽  
Regular Part ◽  
Kohn Laplacian ◽  
Preliminary Work ◽  
Characteristic Points ◽  
The Given ◽  
Euclidean Domains

AbstractThis paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y.Y. Li, O. Rey in [1], and O. Rey in [16] for Euclidean domains. While the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

scholarly journals A note on the heat kernel on the Heisenberg group

2002 ◽  
Vol 65 (1) ◽  
pp. 115-120
Author(s):  
Adam Sikora ◽  
Jacek Zienkiewicz
Keyword(s):  
Heisenberg Group ◽  
Analytic Continuation ◽  
Heat Kernel ◽  
Smooth Function ◽  
Convolution Kernel

We describe the analytic continuation of the heat kernel on the Heisenberg group ℍn(ℝ). As a consequence, we show that the convolution kernel corresponding to the Schrödinger operatereisLis a smooth function on ℍn(ℝ) \Ss, whereSs= {(0, 0, ±sk) ∈ ℍn(ℝ) :k=n,n+ 2,n+ 4,…}. At every point ofSsthe convolution kernel ofeisLhas a singularity of Calderón–Zygmund type.

scholarly journals On gradient bounds for the heat kernel on the Heisenberg group

2008 ◽  
Vol 255 (8) ◽  
pp. 1905-1938 ◽  
Author(s):  
Dominique Bakry ◽  
Fabrice Baudoin ◽  
Michel Bonnefont ◽  
Djalil Chafaï
Keyword(s):  
Heisenberg Group ◽  
Heat Kernel ◽  
Gradient Bounds

THE HEAT KERNEL ON THE CAYLEY HEISENBERG GROUP

2005 ◽  
Vol 25 (4) ◽  
pp. 687-702 ◽  
Author(s):  
Jingwen Luan ◽  
Fuliu Zhu
Keyword(s):  
Heisenberg Group ◽  
Heat Kernel

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

2013 ◽  
Vol 286 (13) ◽  
pp. 1337-1352 ◽  
Author(s):  
R. Radha ◽  
S. Thangavelu ◽  
D. Venku Naidu
Keyword(s):  
Heisenberg Group ◽  
Heat Kernel ◽  
Sobolev Spaces

scholarly journals Heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140943
Author(s):  
Der-Chen Chang ◽  
Yutian Li
Keyword(s):  
Stationary Phase ◽  
Heisenberg Group ◽  
Heat Kernel ◽  
Asymptotic Expansions ◽  
Small Time ◽  
Heat Kernels ◽  
Grushin Operator ◽  
Geodesic Structure ◽  
Laplace Type ◽  
Method Of Steepest Descent

The sub-Laplacian on the Heisenberg group and the Grushin operator are typical examples of sub-elliptic operators. Their heat kernels are both given in the form of Laplace-type integrals. By using Laplace's method, the method of stationary phase and the method of steepest descent, we derive the small-time asymptotic expansions for these heat kernels, which are related to the geodesic structure of the induced geometries.

scholarly journals On the nonlocal Schr\”{o}dinger-Poisson type system in the Heisenberg group

2021 ◽  
Author(s):  
Zeyi Liu ◽  
Min Zhao ◽  
Deli Zhang ◽  
Sihua Liang
Keyword(s):  
Heisenberg Group ◽  
Bounded Domain ◽  
Type System ◽  
Smooth Bounded Domain ◽  
Kohn Laplacian ◽  
Poisson Type

This paper is concerned with the following nonlocal Schr\”{o}dinger-Poisson type system: \begin{equation*} \begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}dx\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u, &\mbox{in} \ \Omega,\\ -\Delta_{H}\phi=u^2 & \mbox{in}\ \Omega,\\ u=\phi=0 & \mbox{on}\ \partial\Omega, \end{cases} \end{equation*} where $a, b>0$ and $\Delta_H$ is the Kohn-Laplacian on the first Heisenberg group $\mathbb{H}^1$, $\Omega\subset \mathbb{H}^1$ is a smooth bounded domain, $\lambda>0$, $\mu\in \mathbb{R}$ are some real parameters and $1“”

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