The aim of this paper is to establish the property $L{B}^{\mathrm{\infty }}$ on $[H(K_{\epsilon }){]}^{\prime }$ under the assumption that $E$ is a Frechet space with an absolute basis and $K_{ϵ}$ is a balanced convex compact subset of $E$. At the same time, the properties $(\tilde{\mathrm{\Omega }},\bar{\mathrm{\Omega }})$ for the Hartogs domains associated to an open polydisc in a nuclear Frechet space with a basis will be also proved.