On space of holomorphic functions with boundary smoothness and its dual

Abstract

We consider a Fréchet-Schwartz space $A_{{\mathcal H}}(\Omega)$ of functions holomorphic in a bounded convex domain $\Omega$ in a multidimensional complex space and smooth up to the boundary with the topology defined by means of a countable family of norms. These norms are constructed via some family ${\mathcal H}$ of convex separately radial weight functions in ${\mathbb{R}}^n$. We study the problem on describing a strong dual space for this space in terms of the Laplace transforms of functionals. An interest to such problem is motivated by the researches by B.A. Derzhavets 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 in complex space; these problems were solved by Paley-Wiener-Schwartz type theorems. Our main result states that the Laplace transform is an isomorphism between the strong dual of our functional space 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. This result generalizes the corresponding result of the second author in 2020. To prove this theorem, we apply the scheme proposed by M. Neymark and B.A. Taylor. On the base of results from monograph by L. Hörmander (L. Hörmander. An Introduction to complex analysis in several variables, North Holland, Amsterdam (1990)), a problem of solvability of systems of partial differential equations in $A_{{\mathcal H}}^m (\Omega)$ is considered. An analogue of a similar result from monograph by L. Hörmander is obtained. In this case we employ essentially the properties of the Young-Fenchel transform of functions in the family ${\mathcal H}$.