The extension of -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 , of pure dimension , satisfying the following conditions. There exists an annular domain , a continuous function , and a , such that

(i) whenever ,

(ii) whenever and are both in .

Theorem. *Let* , *let g be holomorphic on E and let u be the real part of a holomorphic function on E. If* *for all* , *then g can be extended to a function in the Hardy space* .

**[1]**H. Alexander,*Extending bounded holomorphic functions from certain subvarieties of a polydisc*, Pacific J. Math.**29**(1969), 485-490. MR**0244508 (39:5822)****[2]**P. S. Chee,*Zero sets and extensions of bounded holomorphic functions in polydiscs*, Proc. Amer. Math. Soc.**60**(1975), 109-115. MR**0422678 (54:10664)****[3]**P. L. Duren,*Theory of*-*spaces*, Pure and Applied Mathematics, vol. 38, Academic Press, New York and London, 1970. MR**0268655 (42:3552)****[4]**W. Rudin,*Function theory in polydiscs*, Benjamin, New York, 1969. MR**0255841 (41:501)****[5]**Y. T. Siu,*Sheaf cohomology with bounds and bounded holomorphic functions*, Proc. Amer. Math. Soc.**21**(1969), 226-229. MR**0237827 (38:6108)****[6]**S. 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 (50:7569)****[7]**-,*The sheaf of*-*functions in product domains*, Pacific J. Math. (to appear).

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Additional Information

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

Keywords:
Polydisc,
Hardy space,
subvariety

Article copyright:
© Copyright 1980
American Mathematical Society