On Grothendieck type duality for the space of holomorphic functions of several variables

Abstract

We describe the strong dual space $({\mathcal O} (D))^*$ for the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables over a bounded Lipschitz domain $D$ with connected boundary $\partial D$ (as usual, ${\mathcal O} (D)$ is endowed with the topology of the uniform convergence on the compact subsets of $D$). We identify the dual space with a closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, with elements vanishing at the infinity and satisfying the tangential Cauchy-Riemann equations on $\partial D$. In particular, we extend in a way the classical Grothendieck-K{\"o}the-Sebasti\~{a}o e Silva duality for the space of holomorphic functions of one complex variable to the multi-dimensional situation. We use the Bochner-Martinelli kernel ${\mathfrak U}_n$ in ${\mathbb C}^n$, $n>1$, instead of the Cauchy kernel over the complex plane ${\mathbb C}$ and we prove that the duality holds true if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of the Sobolev holomorphic functions over $D$ is dense in ${\mathcal O} (D)$.