The non-pluripolarity of compact sets in complex spaces and the property \((LB^\infty)\) for the space of germs of holomorphic functions (Q2773425)

If \(E\) is a Fréchet space with topology defined by an increasing sequence of seminorms \(\{\|\cdot\|_k\}_{k\geq 1}\) then a sequence of seminorms \(\{\|\cdot\|_k^*\}_{k\geq 1}\) on the topological dual \(E^*\) is defined by \(\|u\|_k^*=\sup\{|u(x)|; \|x\|_k\leq 1\}\). The space \(E\) is said to have property \((LB^\infty)\) if the following holds: NEWLINE\[NEWLINE \forall\{\rho_n\}\uparrow +\infty \;\forall p \;\exists q \;\forall n_0 \;\exists N_0, C>0 \;\forall u\in E^* \;\exists n_0\leq k\leq N_0 : \|u\|_q^{*1+\rho_n}\leq C\|u\|_k^*\|u\|_p^{*\rho_n}. NEWLINE\]NEWLINE In this paper the authors take \(E\) as the strong dual \([H(K)]^*_\beta\) of the space of germs of holomorphic functions on a compact subset \(K\) of a Stein space \(X\). Their main result is that \([H(K)]^*_\beta\) has property \((LB^\infty)\) if and only if \(K\cap Z\) is not pluripolar in \(Z\) for every irreducible branch \(Z\) of every neighbourhood \(U\) of \(K\) in \(X\) with \(K\cap Z\neq \emptyset\).