This note mainly concerns the Cauchy problem associated with the first order hyperbolic operators L(x,D) with the symbol \[ L(x,\xi)=\sum^{1}_{j=0}A_ j(x,\xi ')\xi_ 0^{1-j}, \] where x, \(\xi\), \(\xi\) ’ and D stand for \[ x=(x_ 0,x_ 1,...,x_ d)\equiv (x_ 0,x'),\quad \xi =(\xi_ 0,\xi_ 1,...,\xi_ d)\equiv (\xi_ 0,\xi ') \]
\[ D=(D_ 0,D_ 1,...,D_ d)\equiv (D_ 0,D'),\quad D_ j=-i(\partial /\partial x_ j) \] while \(A_ j(x,\xi ')\) are \(N\times N\) classical pseudo-differential symbols of degree j defined in a conic neighborhood of (0,\({\bar \xi}\)’). It is supposed that \(A_ 0(x,\xi ')\) is equal to the identity matrix of degree N, and the principal part of L(x,\(\xi)\) is effectively hyperbolic at (0,\({\bar \xi}\)). Then the author gives some theorems which claim that
i) in a sufficiently small conic neighborhood of (0,\({\bar \xi}\))’ there is a parametrix of L(x,D) with finite propagation speed of wave front sets, and that
ii) the Cauchy problem for L(x,D), with the data on \(x_ 0=0\), is locally solvable in the \(C^{\infty}\) class in a neighborhood of the origin in \(R^{d+1}.\)
The note includes also some results related to the energy estimates for localized systems. The proofs are not given.