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

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.