Multipliers, linear functionals and the Fréchet envelope of the Smirnov class $N_ *(\textbf {U}^ n)$

Abstract:

Linear topological properties of the Smirnov class ${N_{\ast }}({\mathbb {U}^n})$ of the unit polydisk ${\mathbb {U}^n}$ in ${\mathbb {C}^n}$ are studied. All multipliers of ${N_{\ast }}({\mathbb {U}^n})$ into the Hardy spaces ${H_p}({\mathbb {U}^n}),\;0 < p \leqslant \infty$, are described. A representation of the continuous linear functionals on ${N_{\ast }}({\mathbb {U}^n})$ is obtained. The Fréchet envelope of ${N_{\ast }}({\mathbb {U}^n})$ is constructed. It is proved that if $n > 1$, then ${N_{\ast }}({\mathbb {U}^n})$ is not isomorphic to ${N_{\ast }}(\mathbb {U}{^1})$.