Remarks on Generalized Hardy Algebras

Abstract

For a measure space $(Ω, Σ, μ)$ with a positive finite measure $μ$, and a positive real number $p$, we define the space $L_p^{+}(μ)=L_p^{+}$ of all (equivalence classes of) $Σ$-measurable complex functions $f$ defined on $Ω$ such that the function $\left(\log^+|f|\right)^p$ is integrable with respect to $μ$.We define the metric $d_p$ on $L^{+}_p$ which generalizes the metric introduced by Gamelin and Lumer in [G] for the case $p=1$. It is shown that the space $L^{+}_p$ is a topological algebra. On the other hand, one can define on the space $L_p^{+}$ an equivalent $F$-norm $| \cdot|_p$ that makes $L_p^{+}$ into an Orlicz space. For the case of the normalized Lebesgue's measure $dt/2π$ on $[0,2π)$, it follows that the class $N^p(1<p<\infty)$ introduced by I. I. Privalov in [P], may be considered as a generalization of the Smirnov class $N^+$. Furthermore, $N^p(1<p<\infty)$ with the associated modular becomes an Hardy-Orlicz class. Finally, for a strictly positive and measurable on $[0,2π)$ function $w$, we define the generalized Orlicz space $L_p^{w}(\mathrm{d}t/2π)=L^w_p$ with the modular $ρ^w_p$ given by the function $ψ_w(t,u)=\big(\log(1+uw(t))\big)^p$, with a "weight" $w$. We observe that the space $L^w_p$ is a generalized Orlicz space with respect to the modular $ρ^w_p$. We examine and compare different topologies induced on $L^w_p$ by corresponding "weights" $w$.