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

Author:
Sergio E. Zarantonello

Journal:
Proc. Amer. Math. Soc. **78** (1980), 519-524

MSC:
Primary 32A35

DOI:
https://doi.org/10.1090/S0002-9939-1980-0556624-1

MathSciNet review:
556624

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Abstract | References | Similar Articles | Additional Information

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* ${H^p}({U^n})$.

- Herbert Alexander,
*Extending bounded holomorphic functions from certain subvarieties of a polydisc*, Pacific J. Math.**29**(1969), 485–490. MR**244508** - P. S. Chee,
*Zero sets and extensions of bounded holomorphic functions in polydiscs*, Proc. Amer. Math. Soc.**60**(1976), 109–115 (1977). MR**422678**, DOI https://doi.org/10.1090/S0002-9939-1976-0422678-5 - Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - Walter Rudin,
*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841** - Yum-tong Siu,
*Sheaf cohomology with bounds and bounded holomorphic functions*, Proc. Amer. Math. Soc.**21**(1969), 226–229. MR**237827**, DOI https://doi.org/10.1090/S0002-9939-1969-0237827-8 - Sergio E. Zarantonello,
*The multiplicative Cousin problem and a zero set for the Nevanlinna class in the polydisc*, Trans. Amer. Math. Soc.**200**(1974), 291–313. MR**355092**, DOI https://doi.org/10.1090/S0002-9947-1974-0355092-4
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*The sheaf of*${H^p}$-

*functions in product domains*, Pacific J. Math. (to appear).

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Additional Information

Keywords:
Polydisc,
Hardy space,
subvariety

Article copyright:
© Copyright 1980
American Mathematical Society