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.     

四元素Heisenberg群上次Laplace算子的平均值定理和不确定原理

给出四元素Heisenberg群上次Laplace算子的平均值定理,并用其导出Hardy不等式和不确定原理.     

关于Heisenberg群上次Laplace算子的唯一延拓性

在Heisenberg群Hn中对微分不等式|△Hnu|≤C/d(z,t)2ψ|u|的非负解证明了某个唯一延拓性结果.     

H-型群上多重次拉普拉斯算子的基本解及其估计

通过热核方法得到了H-型群上多重次拉普拉斯算子基本解的精确表达式,并且得到了该基本解的几类估计.     

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.     

Heisenberg群上广义波算子基本解的奇性分析

本文借助于Heisenberg群H。上热算子乡,,,,,一___、_。,一二,、r一_{不-十之尸于阴量今胖址明J灯二上数异寸妙卜菜了 牙二基本解的一个奇性分析定理,t尹‘中牙二是H。上关于CR结构在一般的Hermite度量下的(广义)K。五n一Laplace算子. Heisenberg群是一特殊的CR流形,其底流形为C。xR,群运算如下定义:丫)z,幻,(z‘,s‘)任c,x尺,有(z,:)·(:’,s,)=(z z,,s s‘ Zlm习z声二).H。上左不变问量场有 夕一1基底:z,=口归z, 乞乞刃/日s,忍,=乡厂旅,一乞z刃/口s,l镇声毛n,S=口归S- 著名的Kohn一Laplace算子口。任leoi形式所对应的Hermite度量…     

Heisenberg群上齐次左不变偏微分算子的相对基本解

运用映 S(R~n)到 S′(R~n)的连续线性算子的 Hermite 表示理论和 Heisenberg 群的酉表示理论,证明了当 Heisenberg 群上齐次左不变偏微分算子的群 Fourier 变换满足一定条件时,必存在相对基本解,并给出了相对基本解的计算公式.本文结果把 Greiner-Kohn-Stein 和Geller 等人关于齐次横截椭圆算子的相应结果推广到一般齐次算子.     

Heisenberg群上齐次左不变偏微分算子的相对基本解

运用映S(R~n)到S′(R~N)的连续线性算子的Hermite表示理论和Hedsenberg群的酉表示理论,证明了当Heisenberg群上齐次左不变偏微分算子的群Fourier变换满足一定条件时,必存在相对基本解,并给出了相对基本解的计算公式。本文结果把Greiner-Kohn-Stein和Geller等人关于齐次横截椭圆算子的相应结果推广到一般齐次算子。     

Heisenberg群上次椭圆p-Laplace方程的边界估计

研究次椭圆p-Laplace方程(P＞1)解的边界性质,通过建立Heisenberg群上带有区域内点到边界Carnot-Carath閛dory距离函数的Hardy型不等式,给出了有界域上次椭圆p-Laplace方程以及带非平凡位势的次椭圆p-Laplace方程的解在边界附近的若干估计.     

Heisenberg群上算子■_0~′的显式基本解

本文力图解决[3]中提出的问题,给出了n为偶数情形(?)_0′的显式基本解及有关的证明。     

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.     

On the Initial Value Problem and Fundamental Solution for a Class of Homogeneous Left Invariant Differential Operators on the Heisenberg Group

ＯｎｔｈｅＩｎｉｔｉａｌＶａｌｕｅＰｒｏｂｌｅｍａｎｄＦｕｎｄａｍｅｎｔａｌＳｏｌｕｔｉｏｎｆｏｒａＣｌａｓｓｏｆＨｏｍｏｇｅｎｅｏｕｓＬｅｆｔＩｎｖａｒｉａｎｔＤｉｆｆｅｒｅｎｔｉａｌＯｐｅｒａｔｏｒｓｏｎｔｈｅＨｅｉｓｅｎｂｅｒｇＧｒｏｕｐＮｉｕ...     

A Restriction Theorem on the Reduced Product Heisenberg Group

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔΩｂｅａｒｅｇｕｌａｒｃｏｎｅｉｎＲｎ,Φ：Ｃｍ×Ｃｍ→Ｃｎ＝Ｒｎ＋ｉＲｎａｎΩ－ｐｏｓｉｔｉｖｅＨｅｒｍｉｔｉａｎｍａｐ．ＴｈｅＳｉｅｇｅｌｄｏｍａｉｎＤΩ,ΦｏｆｔｙｐｅｔｗｏｉｎＣｎ×ＣｍｉｓｄｅｆｉｎｅｄｂｙＤΩ,Φ＝｛（ｚ,ｗ）∈：Ｃｎ×Ｃｍ：Ｉｍｚ－Φ（ｗ,ｗ）∈Ω｝（１）（ｓｅｅ［６］）．Ｓｐｅｃｉａｌｌｙ,ｗｅａｓｓｕｍｅｔｈａｔｎ＝ｍ,Ω＝｛ｔ＝（ｔ１,ｔ２,…,ｔｎ）∈Ｒｎ：ｔｉ＞０,ｉ＝１,２,…,ｎ｝,Φ（ｕ,ｖ）＝ｕ·ｖ＝（ｕ１ｕ…     

Hilbert Transforms Along Curves in the Heisenberg Group

We consider Hilbert transforms along curves on the Heisenberggroup. In particular, necessary and sufficient conditions forthe L²-boundedness are given for (t)=(t,(t),0) when is evenor odd and convex on R⁺. 1991 Mathematics Subject Classification:42B20, 42B25.     

Heisenberg群上Folland—Stein定理的注记

该文在G．B．Folland与E．M．Stein研究的算子的基础上．拓展考虑了算子，其中λ＋μ≠0且λ≠α／2n，μ≠—α／2n），证明了：如果，使得有限（其中ψa，b，1（z，t）＝—4（λ＋μ）ab（｜z｜2＋1—it）a-1（｜z｜2＋1＋it）b-1，那么在分布的意义下将有．特别，当λ＝μ＝—1／2时，此结果即原来的Folland－Stein定理．     

Positive definite temperature functions on the Heisenberg group

We characterize positive definite temperature functions, i.e., positive definite solutions of the heat equation, on the Heisenberg group in terms of the initial values. We also obtain an integral representation for positive definite and U(n)-invariant temperature functions with polynomial growth, where U(n) is the group of all n× n unitary matrices.     

Asymptotics for Certain Harmonic Functions and the Martin Compactification on the Quaternionic Heisenberg Group

In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.