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• W2205812074 abstract "Let V be a complex analytic set and p e V. Let ((V), ((V), and Ck(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 Ck(V)r)((V)=C(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. Let V be a complex analytic set and p E V. Let (9(V), (9(V), and Ck(V) denote respectively the rings of germs of holomorphic, weakly holomorphic, and k-times continuously differentiable functions on V. In [6], Spallek proved that there exists sufficiently large k such that Ck(V) n (9(V)=(9( 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. An element u E (9 is said to be a universal denominator if u(9c (9. Let I be the ideal of (9 of all functions vanishing on Sing(V) and J be the ideal of universal denominators. Then locus(J) c Sing V [3, p. 56], so by the Hilbert Nullstellensatz there is a positive integer N such that IN c J. The main result of this paper is that k can be chosen so that k_ N. Siu has proven [5] that if k(p) is the minimal value of k such that Ck r(9)= (9 at the point p E V, then the function k(p) is bounded on compact subsets of V. This result also follows from the above estimate, by the coherence of the ideal sheafs of I and J. (The ideal sheaf of J is coherent [2, Theorem 22] because it is the kernel of 0--Hom,((9, (91/0).) This estimate is, in general, not the best possible. In an earlier work [1], for the example of a curve in C2 normalized by a map t__>_ tPu(t)) where u(t) is a unit, p>q, and p and q are relatively prime, it was shown that Received by the editors June 30, 1972. AMS (MOS) subject classifications (1970). Primary 32K15." @default.
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• W2205812074 date "1973-01-01" @default.
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• W2205812074 title "$Csp{k}$, weakly holomorphic functions on analytic sets" @default.
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• W2205812074 doi "https://doi.org/10.1090/s0002-9939-1973-0322214-5" @default.
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