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

We describe the strong dual space (O(D))& lowast; of the space O(D) of holomorphic functions of several complex variables in a bounded domain D with Lipschitz boundary and connected complement (as usual, 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 C (n) \ D , n > 1 , whose elements vanish at the point at infinity and satisfy the Cauchy-Riemann tangential conditions on partial derivative D. In particular, we generalize classical Grothendieck-Ko<spacing diaeresis>the-Sebastiao 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 O(D) boolean AND H-1(D) of Sobolev-class holomorphic functions in D is dense in O(D). Bibliography: 35 titles.