An Extension of a Theorem of Hamada on the Cauchy Problem with Singular Data

Introduction. Hamada [1] proved the following result about the propagation of singularities in the Cauchy problem for an analytic linear partial differential operator. Assume that the initial data are analytic at the point 0 except for singularities along a submanifold T of the initial surface containing 0. Let K\ • • •, K be the characteristic surfaces of the operator emanating from T. Under the assumption that the K have multiplicity one, he showed that the solution of the Cauchy problem is analytic at 0 except for logarithmic singularities along the K. We extend his result to the case where the K have constant multiplicity. 1. Definitions and theorem. Let C denote the set of (n 4l)-tuples x = (x°,. . . , x) of complex numbers. Let S be an n-dimensional analytic submanifold of C, and let T be an (n — l)-dimensional analytic submanifold of S. Since our results are local, we can assume S = {(0, x , . . . ,x f )eC + }andT = {(0,0, x , . . . ,x")eC" + }. Let Dt = d/dx\D = (D0 , . . . , D„), and let a:x -• a{x;D) be an analytic partial differential operator on a neighborhood of 0 in C. Let h(x;D) be the principal part of a(x ;D). We assume that S is not a characteristic surface of a at 0, so h(0 ; 1,0,..., 0) # 0. Let p = (p0, • • •, Pn) be an (n + l)-tuple of formal variables, so h(x ;p) is a homogeneous polynomial in p with analytic coefficients. We say that the operator a has constant multiplicity at 0 in the direction of Tif we can factor h as