On the extension of $H^{p}$-functions in polydiscs

Abstract:

For $N = 2\;{\text {or}}\;3$ it is shown that if $E$ is the zero set of a holomorphic function in ${U^N}$ satisfying the separation condition of Alexander [1], viz., there exist $r \in (0,1)$ and $\delta > 0$ such that $\left | {\alpha - \beta } \right | \geqslant \delta$ whenever $(z’,\alpha ,z'') \ne (z’,\beta ,z'')$ are both in $({Q^{k - 1}} \times U \times {Q^{N - k}}) \cap E$, where $Q = \{ \lambda \in {\mathbf {C}}:r < \left | \lambda \right | < 1\}$, then (a) $E$ is the zero set of some $F \in {H^\infty }({U^N})$, and (b) $0 < p \leqslant \infty$, every $g \in H(E)$ such that ${\left | g \right |^p}$ has a pluriharmonic majorant on $E$ extends to a $G \in {H^p}({U^N})$. This generalizes earlier results of the author [3] and Zarantonello [9].