Оn a space of holomorphic functions on a bounded convex domain of ${\mathbb C}^n$ and smooth up to the boundary and its dual space

A locally convex space of holomorphic functions in a convex bounded domain of multidimensional complex space and smooth up to the boundary is considered in the article. The topology of this space is defined by a countable family of norms constructed with a help of some special logarithmically convex sequences. Due to conditions on the indicated sequences this space is a Fréchet–Schwartz space. The problem of description of the strong dual for this space in terms of the Laplace transforms of functionals is studied in the article. Interest in the problem is connected with the researches by B. A. Derjavets devoted to classical problems of theory of linear differential operators with constant coefficients and the researches by A. V. Abanin, S. V. Petrov and K. P. Isaev of modern problems of the theory of absolutely representing systems in various spaces of holomorphic functions with given boundary smoothness in convex domains of complex space with a help of obtained by them Paley–Wiener–Schwartz type theorems. The main result of the article is Theorem 1. It states that the Laplace transformation establishes an isomorphism between the strong dual for functional space under consideration and some space of entire functions of exponential type in ${\mathbb C}^n$ which is an inductive limit of weighted Banach spaces of entire functions. Note that in this case an analytic representation of the strong dual space is obtained under the less restrictions on the family ${\mathfrak M}$ than in an article of the author published in 2002. In the proof of Theorem 1 we apply the scheme taken from M. Neymark and B. A. Taylor. Also some previous results of the author are essentially used.