ON EPIMORPHICITY OF A CONVOLUTION OPERATOR IN CONVEX DOMAINS IN $ \mathbf{C}^l$

Abstract

Let be a convex domain and a convex compact set in ; let be the space of analytic functions in , provided with the topology of uniform convergence on compact sets, and the space of germs of analytic functions on with the natural inductive limit topology; and let be the space dual to . Each functional generates a convolution operator , , , which acts continuously from into . Further let be the Fourier-Borel transform of the functional .In this paper the following theorem is proved:Theorem. Let be a bounded convex domain in with boundary of class or , where the are bounded planar convex domains with boundaries of class , and let . In order that it is necessary and sufficient that 1) for all , and 2) be a function of completely regular growth in in the sense of weak convergence in .Here is the regularized radial indicator of the entire function , and is the support function of the compact set .Bibliography: 29 titles.