Abstract: In this paper, we investigate the connection between the uniqueness and the continuous dependence for the Cauchy problem of linear partial differential equations with everywhere characteristic initial hypersurface. The local Carleman's estimate used by S. Alinhac and M. S. Baouendi to prove the uniqueness for the characteristic Cauchy problem is improved to the global estimate with a specific peaking weight factor. Using this estimate, we can prove similarly the uniqueness of the characteristic Cauchy problem. In particular, we derive also the continuous dependence for the solution of this problem in the function class that a certain norm is bounded by the aid of the same inference of proving the global estimate.

Key words: Pseudo-differential operators, Factorization, Estimation