摘要
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。
In this paper it is proved that local fundamental solution exists in some space Wm(Hn) (m∈Z), if the left invariant differential operator on the Heisenberg group Hn satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on Hn . If there exist R≥0, r,s∈R and operators {Bλ|λ∈ΓR} ∈Vs(ΓR, Mr) such that, for almost all λ∈ΓR, Bλ is the right inverse of Ⅱλ(L), then there exists E∈Wm(Hn) (when m≥0 or m even) or E∈Wm-1(Hn) (when m<0 and odd) such thatLE =δ(near the origie) Where m = min(〔r〕, -〔2s〕-n-2);2 . Let L(W, T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ |≥R ,Cλ=inf(1+ |a |rL(-|λ|· (2a + e),iλ)|>0,a∈ Z+nandthen the same conclusion as above holds with m = min (-〔2r〕-n-2, -〔-2s〕-n-2.
