H_(2n+1)上一类非齐次不变微分算子的显式基本解

An Explicit Fundamental Solution for a Nonhomogeneous Differential Operator on the Heisenberg Group

用随机分析方法，构造了Ｈｅｉｓｅｎｂｅｒｇ群Ｈ２ｎ＋１上一类二阶非齐次不变微分算子Ｐ的显式基本解，并讨论了Ｐ的亚椭圆性和局部可解性．这里Ｐ＝１２２ｎｊ，ｋ＝１ａｊｋＮｊＮｋ＋２ｎｊ＝１ｃｊＮｊ＋γＴ．其中：Ａ＝（ａｊｋ）２ｎ×２ｎ是对称正定矩阵，ｃｊ（ｊ＝１，２，…，２ｎ），γ是满足一定条件的复数．约定Ｎｊ＝Ｌｊ，Ｎｊ＋ｎ＝Ｍｊ，ｊ＝１，２，…，ｎ．其中：Ｌ１，Ｌ２，…，Ｌｎ，Ｍ１，Ｍ２，…，Ｍｎ，Ｔ是Ｈ２ｎ＋１的左不变向量场．

By using the stochastic methods, explicit fundamental solutions for one class of nonhomogeneous differential operators P on the Heisenberg group H 2n+1  are constructed and the hypoellipticity and local solvability of P are discussed . HereP=122nj,k=1a jk N jN k+2nj=1c jN j+γT.where A=(a jk ) 2n×2n  is positive matrix, c j,j=1,2,…,2n,γ are complex satisfying certain conditions. L 1,L 2,…,L n, M 1,M 2,…,M n,T are left invariant vector fields on H 2n+1 . We set N j=L j,N j+n =M j,j=1,2,…,n.