For $v$ a positive continuous weight function on an open subset $G$ of $\Bbb C\sp N$, let $Hv(G)$ and $Hv\sb 0(G)$ denote the Banach spaces, under the norm $\Vert f \Vert = \sup \{v(z) \vert f(z) \vert : z \in G\}$, of holomorphic functions $f$ on $G$ such that $vf$ is bounded and $vf$ vanishes at infinity on $G$, respectively. The principal question addressed in the paper under review is whether $Hv(G)$ is (isometrically isomorphic to) the bidual of $Hv\sb 0 (G)$. This problem may be viewed as an analogue to weighted function spaces because of the fact that $\ell\sb \infty$ is the bidual of $c\sb 0$.
An affirmative result (that $Hv(G)$ is the bidual of $Hv\sb 0 (G))$ was obtained in 1970 by Rubel and Shields for the case where $G$ is the open unit disc in $\Bbb C$ and $v$ is a radial weight function vanishing at the boundary; that is, $v(z) = v( \vert z \vert)$ for all $z$ and $\lim\sb{\vert z \vert \to 1\sp -} v(z) = 0$. This result, cited by the authors here, occurs as a special case of the main result of this paper. Specifically, Theorem 1.1 shows that $Hv(G)$ is always (isometrically isomorphic to) the dual space of a Banach space and that $Hv(G)$ is the bidual of $Hv\sb 0 (G)$ if and only if the unit ball of $Hv\sb 0 (G)$ is dense in the unit ball of $Hv(G)$ in the compact-open topology. This density criterion is then used to obtain the Rubel-Shields result as well as other more general examples. Theorem 2.3 shows that, if the set $G$ is balanced and if $v$ is radial, then $Hv(G)$ is the bidual of $Hv\sb 0 (G)$ whenever the latter space contains all polynomials.
The final section of the paper concerns the predual of $Hv(G)$ and the structure of $Hv(G)$ when equipped with various topologies, most notably the bounded weak star topology.