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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Sergio E. Zarantonello
Journal: Proc. Amer. Math. Soc. 78 (1980), 519-524
MSC: Primary 32A35
MathSciNet review: 556624
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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 < \vert{z_i}\vert < 1,1 \leqslant i \leqslant n\} $, a continuous function $ \eta :[r,1) \to [r,1)$, and a $ \delta > 0$, such that

(i) $ \vert{z_n}\vert \leqslant \eta ((\vert{z_1}\vert + \cdots + \vert{z_{n - 1}}\vert)/(n - 1))$ whenever $ ({z_1}, \ldots ,{z_n}) \in E \cap {Q^n}$,

(ii) $ \vert\alpha - \beta \vert \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 $ \vert g(z){\vert^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})$.

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Keywords: Polydisc, Hardy space, subvariety
Article copyright: © Copyright 1980 American Mathematical Society