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$ C\sp{k}$, weakly holomorphic functions on analytic sets


Author: Joseph Becker
Journal: Proc. Amer. Math. Soc. 39 (1973), 89-93
MSC: Primary 32K15; Secondary 32B10
DOI: https://doi.org/10.1090/S0002-9939-1973-0322214-5
MathSciNet review: 0322214
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Abstract: Let $ V$ be a complex analytic set and $ p \in V$. Let $ \mathcal{O}(V),\tilde{\mathcal{O}}(V)$, and $ {C^k}(V)$ denote respectively the rings of germs of holomorphic, weakly holomorphic, and $ k$-times continuously differentiable functions on $ V$. Spallek proved that there exists sufficiently large $ k$ such that $ {C^k}(V) \cap \tilde{\mathcal{O}}(V) = \mathcal{O}(V)$. In this paper I give a new proof of this result for curves and hypersurfaces which also establishes that the conduction number of the singularity is an upper bound for $ k$. This estimate also holds for any pure dimensional variety off of a subvariety of codimension two.


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0322214-5
Keywords: $ {C^k}$, weakly holomorphic, conduction number
Article copyright: © Copyright 1973 American Mathematical Society

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