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It is well-known that for some weakly hyperbolic operators, the regularityloss of solutions depends largely on lower order terms. For example, if P has the principal part 3|—tdx and the Cauchy problem for P with the initial surface £=0 is well-posed, then the lower order terms have the form adt+ t~bdx+c where a, b, c are C°°-functions. And the regularity-loss depends on \Reb(Q, x)\. In this paper, we pay attention to the behavior of the principal part with respect to the time variable t, and we give a result of the following type (cf. [2], [3] and their references).

Necessary conditions for well-posedness of the flat Cauchy problem and the regularity-loss of solutions