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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The traces of holomorphic functions on real submanifolds

Author: Gary Alvin Harris
Journal: Trans. Amer. Math. Soc. 242 (1978), 205-223
MSC: Primary 32C05
MathSciNet review: 0477120
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Abstract: Suppose M is a real-analytic submanifold of complex Euclidean n = space and consider the following question: Given a real-analytic function f defined on M, is f the restriction to M of an ambient holomorphic function? If M is a C.R. submanifold the question has been answered completely. Namely, f is the trace of a holomorphic function if and only if f is a C.R. function. The more general situation in which M need not be a C.R. submanifold is discussed in this paper.

A complete answer is obtained in case the dimension of M is larger than or equal to n and M is generic in some neighborhood of each point off its C.R. singularities. The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function f and a holomorphic mapping $ \Phi$, when does there exist a holomorphic mapping F such that $ f = F \circ \Phi $?

References [Enhancements On Off] (What's this?)

  • [E-H] Paul Eakin and Gary Harris, When $ \Phi (f)$ convergent implies f is convergent, Math. Ann. 229 (1977), 211-221. MR 0444651 (56:3001)
  • [G-R] Hans Grauert and Reinhold Remmert, Analytische Stellenalgebren, Springer-Verlag, New York, 1970. MR 0316742 (47:5290)
  • [T] Giuseppe Tomassini, Trace delle funzioni olomorfe sulle sottovarieta analitiche reali d'una varieta complessa, Ann. Scuola Norm. Sup. Pisa 20 (1966), 31-43. MR 0206992 (34:6808)

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Keywords: Holomorphic trace, real-analytic submanifold of $ {{\textbf{C}}^n}$
Article copyright: © Copyright 1978 American Mathematical Society

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