四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解，该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解，并且给出了基本解的积分表示.     

四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解,该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解,并且给出了基本解的积分表示.     

On fundamental solution for powers of the sub-Laplacian on the Heisenberg group

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.     

Carnot Geometry and the Resolvent of the Sub-Laplacian for the Heisenberg Group

Abstract

We obtain an explicit representation formula for the sub-Laplacian on the isotropic, three-dimensional Heisenberg group. Using the formula we obtain themeromorphic continuation of the resolvent to the logarithmic plane, the existence of boundary values in the continuous spectrum, and semiclassical asymptotics of the resolvent kernel. The asymptotic formulas show the contribution of each Hamiltonian path in Carnot geometry to the spatial and high-energy asymptotics of the resolvent (convolution) kernel for the sub-Laplacian.     

A Note on Hermite and Subelliptic Operators

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.     

Roe–Strichartz theorem on two-step nilpotent Lie groupsa

Strichartz characterized eigenfunctions of the Laplacian on Euclidean spaces by boundedness conditions which generalized a result of Roe for the one-dimensional case. He also proved an analogous statement for the sub-Laplacian on the Heisenberg groups. In this paper, we extend this result to connected, simply connected two-step nilpotent Lie groups.     

The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold

The main technical result of the paper is a Bochner type formula for the sub-Laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz’s theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.     

Large time estimates for heat kernels in nilpotent Lie groups

The aim of this paper is to obtain some estimate for large time for the Heat kernel corresponding to a sub-Laplacian with drift term on a nilpotent Lie group. We also obtain a uniform Harnack inequality for a “bounded” family of sub-Laplacians with drift in the first commutator of the Lie algebra of the nilpotent group.     

The Heat Kernel and Green Functions of Sub-Laplacians on the Quaternion Heisenberg Group

We introduce the quaternion Heisenberg group and show that it is a special case of the model step two nilpotent Lie group studied by Beals, Gaveau and Greiner. Using the heat kernel, we give formulas for Green functions of sub-Laplacians on the quaternion Heisenberg group. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.     

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre. We also prove an energy identity that yields a sharp trace Sobolev embedding.     

Positive Curvature Property for Sub-Laplacian on Nilpotent Lie Group of Rank Two

In this note, we concentrate on the sub-Laplace operator on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by n Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplacian on the Heisenberg group ?. In order to study contraction properties of the associated heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motions model, the restriction of the sub-Laplace operator acting on radial functions (see Definition 3.5) satisfies a positive Ricci curvature condition (more precisely a CD(0,?∞?) inequality), see Theorem 4.5, whereas the operator itself does not satisfy any CD(r,?∞?) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. It can be seen a generalization of the paper (Qian, Bull Sci Math 135:262–278, 2011).     

Product set estimates for non-commutative groups

We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre. T. Tao is supported by a grant from the Packard Foundation.     

Heisenberg群上次Laplace算子的Carleman型估计与唯一延拓性

本文在适当条件下给出了Heisenberg群上次Laplace算子的Carleman型估计, 并由此建立了唯一延拓性.     

Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang (Pac J Math 208(2):325–345, 2003 [26]) for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known Payne–Pólya–Weinberger and Yang universal inequalities.     

Wiener measure for Heisenberg group

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 Feynman-Kac formula.     

Sub-Laplacians on Sub-Riemannian Manifolds

We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in Gordina and Laetsch (Trans. Amer. Math. Soc., 2015). This operator is canonical with respect to the horizontal Brownian motion; we are able to define this sub-Laplacian without some a priori choice of measure. The other operator is \(\operatorname {div}^{\omega } \operatorname {grad}_{\mathcal {H}}\) for some volume form ω on M. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: SU(2), the Heisenberg group and the affine group.     

THE HEAT KERNEL ON THE CAYLEY HEISENBERG GROUP

1IntroductionThe aim of the present paper is to construct an explicit expression of an heat kernel forthe Cayley Heisenberg group of order n.Hulanicki[1]and Gaveau[2]constructed the explicit expression of the heat kernel for theHeisenberg group by using p…     

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ${\Bbb G}We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ? be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a q-dimensional center. Let and Then, with A _Q as the sharp constant, where ∇_? denotes the subellitpic gradient on ? This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18]. Received March 15, 2001, Accepted September 21, 2001     

A Brylinski filtration for affine Kac–Moody algebras

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac–Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac–Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac–Moody algebras.