We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and...We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and to construct its fundamental solution. Besides, the series representation of the fundamental solution for square of the sub-Laplacian on the Heisenberg group is given and we also get the closed form of the fundamental solution for square of the sub-Laplacian on the Heisenberg group with dimension n = 2, 3, 4.展开更多
In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are ...In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.展开更多
In this paper,we introduce a special class of nilpotent Lie groups defined by hermitian maps,which includes all the groups of affine holomorphic automorphisims of Siegel domains of type Ⅱ,in particular,the Heisenberg...In this paper,we introduce a special class of nilpotent Lie groups defined by hermitian maps,which includes all the groups of affine holomorphic automorphisims of Siegel domains of type Ⅱ,in particular,the Heisenberg group.And we study harmonic analysis on these groups as spectral theory of the associated Sub-Laplacian instead of the group representation theory in usual way.展开更多
An important class of homogeneous groups N(Φ) is introduced, which includes the distin-guishing boundaries of the Siegel domains of Type Ⅱ. For the sub-Laplacian ? on N(Φ),its basic eigenfunctions are obtained by a...An important class of homogeneous groups N(Φ) is introduced, which includes the distin-guishing boundaries of the Siegel domains of Type Ⅱ. For the sub-Laplacian ? on N(Φ),its basic eigenfunctions are obtained by a direct calculation, the foundamental solution of? is derived from those eigenfunctions, and finally, a solution of ? with singularities isgiven.展开更多
In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the ...In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.展开更多
基金Supported by Doctor Special Foundation of Jiangsu Second Normal University(JSNU2015BZ07)
文摘We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and to construct its fundamental solution. Besides, the series representation of the fundamental solution for square of the sub-Laplacian on the Heisenberg group is given and we also get the closed form of the fundamental solution for square of the sub-Laplacian on the Heisenberg group with dimension n = 2, 3, 4.
基金supported by the National Natural Science Foundation of China(10871157)Research Fund for the Doctoral Program of Higher Education of China(200806990032)Keji Chuangxin Jijin of Northwestern Polytechnical University(2007KJ01012)
文摘In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.
文摘In this paper,we introduce a special class of nilpotent Lie groups defined by hermitian maps,which includes all the groups of affine holomorphic automorphisims of Siegel domains of type Ⅱ,in particular,the Heisenberg group.And we study harmonic analysis on these groups as spectral theory of the associated Sub-Laplacian instead of the group representation theory in usual way.
文摘An important class of homogeneous groups N(Φ) is introduced, which includes the distin-guishing boundaries of the Siegel domains of Type Ⅱ. For the sub-Laplacian ? on N(Φ),its basic eigenfunctions are obtained by a direct calculation, the foundamental solution of? is derived from those eigenfunctions, and finally, a solution of ? with singularities isgiven.
基金supported by National Security Agency,United States Army Research Offfice and a Hong Kong RGC Competitive Earmarked Research (Grant No. 600607)
文摘In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.
