Construction of the fundamental solution for degenerate parabolic systems and its application to construction of a parametrix of \(\square _ b\) (Q1071175)

The author studies the fundamental solution E(t) of a degenerate parabolic system of pseudo-differential operators: \(\{D_ t+p(x,D)\}E(t)=0\) for \(t>0\), \(x\in R^ n\) and \(E(0)=I\), where p(x,\(\xi)\) is a \(k\times k\) matrix having the following expansion: \[ p(x,\xi)=p_ m(x,\xi)+p_{m-1}(x,\xi)+p_{m-2}(x,\xi),\quad p_{m-j}(x,\xi)\in S^{m-j}_{1,0}\quad (j=0,1,2), \] \[ p_{m-j}(x,\lambda \xi)=\lambda^{m-j} p_{m-j}(x,\xi),\quad \lambda >0,\quad \xi \neq 0\quad (j=0,1)\quad with\quad m>1. \] The main result of this paper is that one can construct the fundamental solution E(t) is the class \(S^ 0_{,}\) of pseudo-differential operators with parameter t provided the symbol p satisfies a suitable condition. The author constructs directly the symbol of E(t) in the form \(e^{\phi}\cdot f\); the function \(\phi\) is expressible in terms of an explicit function of the principal symbol \(p_ m\), its derivatives of the first order, the subprincipal symbol near the characteristic set \(\Sigma =\{(x,\xi)\in R^ n\times R^ n:\quad q_ m(x,\xi)=0\}\) and the fundamental matrix A. Here \(q_ m\) is a non negative scalar symbol such that \(p_ m=q_ m\cdot I\) and \(A=iJH_{q_ m}\) where \(H_{q_ m}\) is the Hessian matrix of \(q_ m\) and \(J=\left( \begin{matrix} 0\\ - I\end{matrix} \begin{matrix} I\\ 0\end{matrix} \right).\) The exact form of E(t) is available to obtain the asymptotic behavior of \(\sum^{\infty}_{j=1}\exp (-t\lambda_ j)\) as t tends to zero, where the \(\lambda_ j's\) are the eigenvalues of p(x,D), if p(x,D) is a self- adjoint operator on a bundle over a compact manifold and has double characteristic. An application of this theorem gives an explicit construction of a parametrix for \(\square_ b\) under a suitable condition for the Levi form which does not imply nondegeneracy. Finally the asymptotic behavior of the eigenvalues of \(\square_ b\) is given.