Analytical realization of the strong dual of a space of holomorphic functions with boundary smoothness. Applications

We consider the space of functions holomorphic in a bounded convex domain of a multidimensional complex space and infinitely differentiable up to its boundary, with given estimates of all derivatives. For this space a Polya-Martineau-Ehrenpreis type theorem is established. This result allows to study the classical problems of the theory of linear differential operators with constant coefficients in the space under consideration, in particular, to obtain analogs of Theorems 7.6.13 and 7.6.14 from the book by L. Hörmander (L. Hörmander. An Introduction to Complex Analysis in Several Variables, Van Nostrand, London, 1966).