Division Theorems in Spaces of Entire Functions with Growth Conditions and their Applications to PDE of Infinite Order

Abstract

It is shown that every entire function /eExp0(C) , /(0)*0 is slowly decreasing. As an application of this property, a theorem on analytic continuation of solutions of infinite order differential equations with constant coefficients is proven. §0. Introduction In 1969 C. O. Kiselman [12] proved the following theorem: Given an open, convex set U a , any holomorphic solution u in U of the linear partial differential equation with constant coefficients P(D)u = £ aaD u = 0, can be analytically con\a 0 there exists a positive constant C£ so that |/(z)| 0 exists Re > 0 aeN so that V|z| > R£ and \f(z) | I, we will denote the n 1 space of n-tuples z = (zl,...,zn),zl e C, equipped with the norm |z| = (£ z t^) 2 .The 1=1 bracket { , ) will denote the bilinear product of two elements in C given by Definition 0.1. An entire function f:C — > C given by f ( z ) = S aaz a is of aeN exponential type zero (infraexponential type) if and only if for every £ > 0 there exists Ce such that for every z e we have |/(z)| , //(£(0,l/fc)) be the space of germs of holomorphic functions near the origin. We know that every operator L: & -> & is continuous if and DIVISION AND ENTIRE FUNCTIONS 747 only if L\H(B(Q,l/ k)) is continuous for every k. We will be interested in a special class of continuous operators. Assume that we have a function a e N, £ e , of exponential type zero. Then we can define a continuous operator (homomorphism), which in turn gives an element of the dual :/ € 0 ^ Iaa/ (0) e C, space 0 *