The extension of $H^{p}$-functions from certain hypersurfaces of a polydisc

Abstract:

Let E be a subvariety of the open unit polydisc ${U^n},n \geqslant 2$, of pure dimension $n - 1$, satisfying the following conditions. There exists an annular domain ${Q^n} = \{ ({z_1}, \ldots ,{z_n}) \in {{\mathbf {C}}^n}:r < |{z_i}| < 1,1 \leqslant i \leqslant n\}$, a continuous function $\eta :[r,1) \to [r,1)$, and a $\delta > 0$, such that (i) $|{z_n}| \leqslant \eta ((|{z_1}| + \cdots + |{z_{n - 1}}|)/(n - 1))$ whenever $({z_1}, \ldots ,{z_n}) \in E \cap {Q^n}$, (ii) $|\alpha - \beta | \geqslant \delta$ whenever $1 \leqslant j \leqslant n$ and $(\zeta ’,\alpha ,\zeta '') \ne (\zeta ’,\beta ,\zeta '')$ are both in $({Q^{j - 1}} \times U \times {Q^{n - j}}) \cap E$. Theorem. Let $0 < p < \infty$, let g be holomorphic on E and let u be the real part of a holomorphic function on E. If $|g(z){|^p} \leqslant u(z)$ for all $z \in E$, then g can be extended to a function in the Hardy space ${H^p}({U^n})$.