On the extension of -functions in polydiscs

Author:
P. S. Chee

Journal:
Proc. Amer. Math. Soc. **88** (1983), 270-274

MSC:
Primary 32A35; Secondary 32D15, 46J15

MathSciNet review:
695257

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Abstract: For it is shown that if is the zero set of a holomorphic function in satisfying the separation condition of Alexander [**1**], viz., there exist and such that whenever are both in , where , then (a) is the zero set of some , and (b) , every such that has a pluriharmonic majorant on extends to a . This generalizes earlier results of the author [**3**] and Zarantonello [**9**].

**[1]**Herbert Alexander,*Extending bounded holomorphic functions from certain subvarieties of a polydisc*, Pacific J. Math.**29**(1969), 485–490. MR**0244508****[2]**Aldo Andreotti and Wilhelm Stoll,*The extension of bounded holomorphic functions from hypersurfaces in a polycylinder*, Rice Univ. Studies**56**(1971), no. 2, 199–222 (1971). MR**0277752****[3]**P. S. Chee,*Zero sets and extensions of bounded holomorphic functions in polydiscs*, Proc. Amer. Math. Soc.**60**(1976), 109–115 (1977). MR**0422678**, 10.1090/S0002-9939-1976-0422678-5**[4]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[5]**Walter Rudin,*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841****[6]**Yum-tong Siu,*Sheaf cohomology with bounds and bounded holomorphic functions*, Proc. Amer. Math. Soc.**21**(1969), 226–229. MR**0237827**, 10.1090/S0002-9939-1969-0237827-8**[7]**E. L. Stout,*The second Cousin problem with bounded data*, Pacific J. Math.**26**(1968), 379–387. MR**0235155****[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**0355092**, 10.1090/S0002-9947-1974-0355092-4**[9]**Sergio E. Zarantonello,*The extension of 𝐻^{𝑝}-functions from certain hypersurfaces of a polydisc*, Proc. Amer. Math. Soc.**78**(1980), no. 4, 519–524. MR**556624**, 10.1090/S0002-9939-1980-0556624-1**[10]**Sergio E. Zarantonello,*The sheaf of 𝐻^{𝑝}-functions in product domains*, Pacific J. Math.**93**(1981), no. 2, 479–493. MR**623579**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0695257-6

Keywords:
Polydiscs,
Hardy spaces,
zero sets,
extensions of -functions

Article copyright:
© Copyright 1983
American Mathematical Society