The inheritance of the linear topological invariant \(D(N)\) (Q1373063)

Let \(F\) be a Fréchet space with a fundamental system of semi-norms \(\{\|\cdot\|_k\}_{k\geq1}\). We say that \(E\) has the property \((DN)\) (shortly write \(E \in (DN))\) if \[ \exists p \forall q, d>0 \exists k, c>0: \|\cdot\|_q^{1+d}\leq c \|\cdot\|_k \|\cdot\|_p^d \] The property \((DN)\) and other properties were introduced and investigated by \textit{D. Vogt} [Arch. Math. 45, 255-266 (1985; Zbl 0621.46001); J. Reine Angew. Math. 345, 182-200 (1983; Zbl 0514.46003)]. The aim of the paper is to study the inheritance of the property \((DN)\). The main results are two theorems: Let \(X\) be a locally irreducible Stein space and \(S\) a closed subset of \(X\) such that \(H_{2\dim X-1}(S) = 0\). Then \(H(X) \in (DN)\) iff \(H(X\backslash S)\in (DN)\). Here \(H_{2\dim X-1}(S)\) denotes the \((2\dim X-1)\)-Hausdorff measure of \(S\cap R(X)\) where \(R(X)\) is the regular locus of \(X\). Let \(\theta:X\to Y\) be a holomorphic surjection between locally irreducible Stein spaces with connected fibres. Assume that \(H(\theta^{-1}(y))\in(DN)\) for all \(y\in Y\). Then \(H(X) \in (DN)\) iff \(H(Y)\in (DN)\). For other linear topological invariants, these theorems are not true.