一般二步幂零群上Laplacian算子的基本解

Fundamental solution of Laplacian on the general nilpotent groups of step two

考虑(2n+p)维空间R^(2n)×R^p上的向量场X_j,j=1,…,2n.通过构造二步幂零Lie群,利用群上的Fourier变换的方法得到了△=1/2∑_(j=1)^(2n) X_j^2的基本解.首先由二步幂零群的Fourier变换理论得到了群上的Plancherel公式,逆公式以及△的表示,即△通过群上的Fourier变换转化为一个可逆的Hilbert-Schmidt算子,其次,通过群上的Plancherel公式得到的逆算子定义一个缓增分布,最后,利用Heimite函数和Laguerre函数的性质得到了基本解的积分表达式.

Consider the vector fieldsX_j in R^（2n）×R^p,j = 1,...,2n.By constructing the nilpotent Lie group of step two,the fundamental solution of△=1/2∑_（j=1）~n X_j^2 is got.First,by using the group Fourier transform of the nilpotent Lie group of step two,the Plancherel formula and inverse formula are got and the Fourier transform of△is also found,i.e.,an invertible Hilbert-Schmidt operator.Secondly, a tempered distribution is defined by using the Plancherel formula.Finally,the integral form of the fundamental solution is followed by using the related propositions of Hermite function and Laguerre function.