Heisenberg群上不变微分算子的特征值问题

讨论了Ｈｅｉｓｅｎｂｅｒｇ群Ｈｎ上一类不变微分算子Ｐ＝ｍｌ＝０ａｌＬｌ的离散特征值的存在性．这里ａｌ＞０，ｌ＝０，１，…，ｍ，ｍ≥２．Ｌ为Ｈｎ上的ｓｕｂ－Ｌａｐｌａｃｅ算子．我们通过建立向量场的Ｐｏｉｎｃａｒｅ型不等式，结合Ｆｒｉｅｄｒｉｃｈｓ对欧氏空间上Ｌａｐｌａｃｅ算子的方法。

二步幂零李群H_(2n+1)×R^(2d)上一类非齐次非横截椭圆不变微分算子的显式基本解

用随机分析方法，我们构造出了二步幂零李群Ｈ２ｎ＋１×Ｒ２ｄ上的一类非齐次非横截椭圆不变微分算子的显式基本解，并在此基础上讨论了它们的局部可解性和亚椭圆性．

幂零李群上非齐次左不变微分算子局部可解的必要条件

得到了幂零李群上非齐次左不变微分算子局部可解的一些必要条件，推广了ＣｏｒｗｉｎＬ，Ｌｅｖｙ－ＢｒｕｈＬ，ＭｕｌｌｅｒＤ，ＣｕｉＳｈａｎｓｂｉｎ等人有关文献中的已有结果。

幂零李群上一类非齐次左不变微分算子局部可解的必要条件

本文讨论了幂零李群上一类非齐次左不变微分算子的局部可解性，给出了此类算子局部可解的必要条件和表示理论准则，这些条件较以前的结果都弱．

Heisenberg群上的一类奇性左不变微分算子的初值问题和基本解

In this paer we will study the wellposed of the inital value problem associated to the homogeneous left invariant differential operator on the Heisenberg group by the method of group Forier anasis L=(-1)^k/2^k ∑j=1^1 aj(Zj^k Zj^-k+Zj^-kZj^k)+(-i)^3k rS^k(where aj>0,r∈R)on the Heisenboerg group,then obtain the explicit expression of the fundamental solutions for the generalized heat operator δ/δt+L and the operator L.

幂零左群G^2n＋R上的左不变微分算子次椭圆性的一个充要条件

In this paper,we use the representation of nilpotent Lie group method to give a sufficient and necessary condition for subellipticity of Left invarient differental operators on nilpotent Lipgroup G^2n+k.

二步幂零李群上—类左不变微分算子亚椭圆的充要条件

本文我们用幂零李群表示的方法，给出了二步幂零李群上一类左不变微分算子是亚椭圆的充要条件．

A Left Invariant Differential Operator not of Transitive Elliptic Type

The left invariant differential operators on a nilpotent Lie group which is elliptic on the generating direction have been widely studied, but little has been performed on the other kinds of differential operators [8]. UP to now, no example which is nonhomogenerous differential operators is available. In this paper. One left invariant differential operator not of transitive elliptic type is studied, and one of its fundamental solutions is given by the method of representations of nilpotent Lie groups.

Boundeness of a Class of Maximal Operators on H^p Spaces on Compact Lie Groups

In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we obtain the correspoding convergent rosults.

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

自七十年代以来,人们对Heisenberg群乃至一般幂零李群上左(右)不变微分算子的局部可解性、亚椭圆性等性质做了大量研究。在亚椭圆性方面,目前最好的结果是由B.Helffer和J.Nourrigat所得到的一个充分条件,这个条件在算子为齐次的情形下也是必要的。在局部可解性方面。

Joint eigenfunctions of invariant differential operators on the quaternion Heisenberg group

Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.

Differential operators for Siegel-Jacobi forms

For any positive integers n and m, H_(n,m):= H_n× C^(m,n) is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. We compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we construct a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for H_(n,m) are obtained.

Forward-backward stochastic differential equation with subdifferential operator and associated variational inequality

We study the existence and uniqueness of the solution to a forward-backward stochastic differential equation with subdifferential operator in the backward equation. This kind of equations includes, as a particular case, multi-dimensional forward-backward stochastic differential equation where the backward equation is reflected on the boundary of a closed convex(time-independent) domain. Moreover, we give a probabilistic interpretation for the viscosity solution of a kind of quasilinear variational inequalities.