EXISTENCE OF C 1 NULL-SOLUTIONS FOR SOME DIFFERENTIAL OPERATORS WITH NON-ANALYTIC COEFFICIENTS TAKESHI MANDAI Abstract. We give a su±cient condition for an operator with C 1 coe±- cients to have a C 1 null-solution with respect to a characteristic hypersurface. This condition is stated using a coordinate invariant which is de¯ned by the prin- cipal symbol and a de¯ning function of the surface. We need no perturbations of lower order terms although the coe±cients are not assumed to be analytic. 1. Introduction We are concerned with the existence of C 1 null-solutions for partial di®erential operators with respect to characteristic surfaces. Let P be a di®erential operator in a neighborhood - of 0 in R n and Á(x) be a real-valued C 1 function in - such that Á(0) = 0, dÁ(x) 6= 0. A C 1 function in a neighborhood of 0 is called a C 1 null-solution for P at x = 0 with respect to § (or rather § + := fx 2 - j Á(x) ¸ 0g), if Pu = 0 in a neighborhood of 0 and 0 2 supp u ½ § + , where supp u denotes the support of u. The existence of a C 1 null-solution means the non-uniqueness of the prolongation of C 1 solutions beyond §. Assume that the hypersurface § = fx 2 - j Á(x)=0g is characteristic for P . It is known that if P is Fuchsian in the sense of Baouendi-Goulaouic [1] and if Á and the coe±cients of P are analytic, then there exist no C 1 null-solutions. Therefore, we consider non-Fuchsian operators. When the coe±cients of P are analytic, there are many results of the existence of C 1 null-solutions. We are, however, concerned with the case when the coe±cients are assumed only to be of class C 1 . In most results in this case, the existence of C 1 null-solutions are shown only for operators whose lower order terms are suitably perturbed. That is, the conclusion of these results is that there exists a di®erential operator Q of lower order than P such that P + Q has a C 1 null-solutions. In many examples, such perturbations are unnecessary unlike in the case of Fuchsian operators. In [6], the author showed the existence of C 1 null-solutions without any perturbation of the lower order terms under some conditions for the principal symbol. 1991 Mathematics Subject Classi¯cation. Primary 35A07; Secondary 35A30,35B60. Key words and phrases. C 1 null-solution, characteristic surface, non-Fuchsian. The research is supported in part by Grant-in-Aid for Scienti¯c Research (No.05640168), Ministry of Education, Science and Culture. 1