Singular Supports of Solutions of Partial Differential Equations in a Slab Domain

In [6], F. John proved that for a differential operator, non-solvability to the non-characteristic Canchy problem for any initial data with compact support is equivalent to rather stringent non-hyperbolicity.* In the present paper, we shall study an analogous question where we shall be interested not in the support but in the singular support of solution. To state more precisely, let D denote the imaginary gradient — z(«— , , « — ] and \dx\ o be a differential operator with constant coefficients obtained from a polynomial P(f) of n variables £=(£i, ..., £n). Our main result is the following