Separately holomorphic functions on compact sets (Q1281588)

Let \(Z\) be a Stein space. Let \({\mathcal H}(Z)\) be the space of holomorphic functions on \(Z\) equipped with the compact-open topology. \({\mathcal H}(Z)\) is said to have the property \[ \text{if }\exists p, \forall q,\;d>0,\;\exists k,\;C>0: \|\cdot\|_q^{1+d}\leq C\|\cdot\|_k \|\cdot \|^d_p. \tag{DN} \] The main aim of the paper is to give a characterization for \({\mathcal H}(Z)\) to have the property (DN) in terms of extension of separately holomorphic functions. It also gives conditions for a compact set \(K\) in a locally irreducible Stein space not to be pluripolar in every irreducible branch of all neighborhoods of \(K\).