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 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.

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.

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

scholarly journals The subelliptic heat kernel on the three-dimensional solvable Lie groups

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Fabrice Baudoin ◽  
Matthew Cecil
Keyword(s):  
Heat Kernel ◽  
Lie Groups ◽  
Three Dimensional ◽  
Heat Kernels ◽  
Heat Semigroup ◽  
Solvable Lie Groups ◽  
New Type ◽  
Gradient Bounds

AbstractWe study the subelliptic heat kernels of the CR three-dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three-dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gradient bounds using a new type of curvature-dimension inequality.

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 UNIQUENESS OF SOLUTIONS TO SCHRÖDINGER EQUATIONS ON -TYPE GROUPS

2013 ◽  
Vol 95 (3) ◽  
pp. 297-314 ◽  
Author(s):  
SALEM BEN SAÏD ◽  
SUNDARAM THANGAVELU ◽  
VENKU NAIDU DOGGA
Keyword(s):  
Initial Data ◽  
Heisenberg Group ◽  
Heat Kernel ◽  
Schrodinger Equation ◽  
Analogous Result ◽  
General Context ◽  
Uniqueness Of Solutions ◽  
Grushin Operator ◽  
Alpha Beta ◽  
Formula Type

AbstractThis paper deals with the Schrödinger equation $i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where $ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where ${q}_{s} $ is the heat kernel associated to $ \mathcal{L} . $ If in addition $\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some ${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that $u(\mathbf{z} , t; s)= 0$ for all $s\in \mathbb{R} $ whenever $\alpha \beta \lt { s}_{0}^{2} . $ This result holds true in the more general context of $H$-type groups. We also prove an analogous result for the Grushin operator on ${ \mathbb{R} }^{n+ 1} . $

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