Extension of solutions of convolution equations in spaces of holomorphic functions with polynomial growth in convex domains

Abstract

In this paper we consider a problem of extension of solutions to homogeneous convolution equations defined by operators acting from a space A−∞(D+K)A−∞(D+K) of holomorphic functions with polynomial growth near the boundary of D+KD+K into another space of such a type A−∞(D)A−∞(D) (D and K being a bounded convex domain and a convex compact set in CC, respectively). We show that under some exact conditions each such solution can be extended as A−∞(Ω+K)A−∞(Ω+K)-solution, where Ω⊃DΩ⊃D is a certain convex domain.