中文摘要：

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.

英文摘要：

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.