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
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 investigate a Dirichlet boundary value problem for a class of fractional degenerate elliptic equations on homogeneous Carnot groups G=(R^(n),o),namely{(-△_(G))^(s)u=f(x,u)+g(x,u)inΩ;u∈H_(0)^(s)(Ω)...In this paper,we investigate a Dirichlet boundary value problem for a class of fractional degenerate elliptic equations on homogeneous Carnot groups G=(R^(n),o),namely{(-△_(G))^(s)u=f(x,u)+g(x,u)inΩ;u∈H_(0)^(s)(Ω),where s∈(0,1),Ω■G is a bounded open domain,(-△_(G))^(s)is the fractional sub-Laplacian,H_(0)^(s)(Ω)denotes the fractional Sobolev space,f(x,u)∈C(Ω×R),g(x,u)is a Carath′eodory function on Ω×R.Using perturbation methods and Morse index estimates in conjunction with fractional Dirichlet eigenvalue estimates,we establish the existence of multiple solutions to the problem.展开更多
In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 a...In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.展开更多
We build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral.Then we give the FeynmanKac formula.
In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the resc...In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic oscillator.Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin.Our results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.展开更多
A Pohozaev-Rellich type identity for the p-sub-Laplacian on groups of Heisenberg type, G, is given. A Carleman estimate for the sub-Laplacian on G is established and, as a consequence, a unique continuation result is ...A Pohozaev-Rellich type identity for the p-sub-Laplacian on groups of Heisenberg type, G, is given. A Carleman estimate for the sub-Laplacian on G is established and, as a consequence, a unique continuation result is proved.展开更多
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.
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.
基金supported by the NSFC(12131017,12221001)the National Key R&D Program of China(2022YFA1005602)。
文摘In this paper,we investigate a Dirichlet boundary value problem for a class of fractional degenerate elliptic equations on homogeneous Carnot groups G=(R^(n),o),namely{(-△_(G))^(s)u=f(x,u)+g(x,u)inΩ;u∈H_(0)^(s)(Ω),where s∈(0,1),Ω■G is a bounded open domain,(-△_(G))^(s)is the fractional sub-Laplacian,H_(0)^(s)(Ω)denotes the fractional Sobolev space,f(x,u)∈C(Ω×R),g(x,u)is a Carath′eodory function on Ω×R.Using perturbation methods and Morse index estimates in conjunction with fractional Dirichlet eigenvalue estimates,we establish the existence of multiple solutions to the problem.
基金partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University
文摘In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.
基金supported by National Natural Science Foundation of China (Grant No. 10990012)50-Class General Financial Grant from the China Postdoctoral Science Foundation (Grant No. 2011M501317)
文摘We build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral.Then we give the FeynmanKac formula.
基金Supported by the National Natural Science Foundation of China (Grant Nos.12101546,11771385)。
文摘In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic oscillator.Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin.Our results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.
基金The project supported by National Natural Science Foundation of China, Grant No. 10371099.
文摘A Pohozaev-Rellich type identity for the p-sub-Laplacian on groups of Heisenberg type, G, is given. A Carleman estimate for the sub-Laplacian on G is established and, as a consequence, a unique continuation result is proved.
