On the theory of linear partial differential operators with analytic coefficients

Abstract

Introduction. Consider a linear partial differential operator of order m, P, having coefficients defined and analytic in some open subset Q of Rn+l (n>O), and let S be a piece of analytic hypersurface in Q, noncharacteristic with respect to P at some point x? (we denote by a/lv the differentiation in the direction of the normal to S). The present article is concerned with the Cauchy problenm: (1) Pu =f in some open neighborhood U of x?; (II) (a1aV)kU =gk on S r) U (k =O, . . ., m1). The main result (see ?4) can be roughly stated as follows: it is possible to determine the neighborhood U so that, for all dataf andgk (O 0}, and the hypersurface S into the hyperplane t=0. Then we may say the following: the right-hand sidef and the solution u are functions of t with values in the spaces of ultradistributions with respect to the variables x, which we introduce under the notation KS; the Cauchy data gk are members of such Ks spaces. What is the space KS? A member v of Ks is, by definition, the Fourier transform of a function v(f) in Rn which is square-integrable with respect to the measure e-2sl 1 d6. For s<0, the elements of Ks are analytic functions, holomorphically extendable to slabs IIm xl < -s of the complex space. In this case, the main theorem reduces more or less to the classical Cauchy-Kovalevska theorem, of which it provides what I think is a new proof (in ?5). For s < 0, the class Ks is quite large: it contains various types of distributions in Rn, among which are all the ones that have compact support. In particular, the main theorem implies that the equation Pu=f, with f say a continuous function with compact support in U, is solvable (in U). Of course, the solution is not going to be a distribution, a fortiori not a function-at least in general. At any event, this very weak solvability result justifies a remark made at the beginning of the article by Nirenberg-Treves [2] (a remark which has been slightly mysterious until now, to the authors of the article among others)(1). In this