On Grothendieck-type duality for spaces of holomorphic functions of several variables

Yu. A. Khoryakova ; A. A. Shlapunov

Sbornik. Mathematics, Tome 215 (2024) no. 8, pp. 1114-1133

Résumé

We describe the strong dual space $({\mathcal O} (D))^*$ of the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables in a bounded domain $D$ with Lipschitz boundary and connected complement (as usual, ${\mathcal O} (D)$ is endowed with the topology of local uniform convergence in $D$). We identify the dual space with the closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, whose elements vanish at the point at infinity and satisfy the Cauchy–Riemann tangential conditions on $\partial D$. In particular, we generalize classical Grothendieck–Köthe–Sebastião e Silva duality for holomorphic functions of one variable to the multivariate situation. We prove that the duality we produce holds if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of Sobolev-class holomorphic functions in $D$ is dense in ${\mathcal O} (D)$. Bibliography: 35 titles.

Détail

Keywords: duality, spaces of holomorphic functions of several variables.