The Fourier Transform On The General Nilpotent Lie Oroups Of Step Two And Its Application

Consider the vector fields in R2n×Rp, j=1,2....,2n.{Yj} generate a nilpotent Lie algebra of step two. We use the group Fourier transform on the associated nilpotent Lie group of step two to get the fundamental solution of sub-Laplacian operator Δ=Σj=12nYj2. Firstly, we show the group Fourier transform of the nilpotent Lie group of step two in the theory of the Plancherel formula and inverse formula, and find the Fourier transform of Δ, i.e., a family of invertible Hilbert-Schmidt operators. Secondly, define a tempered distribution by using the Plancherel formula. Fi-nally, the integral representation of the fundamental solution follows by using the propositions of Hermite functions and Laguerre functions.