Remarks on Generalized Hardy Algebras

Abstract: For a measure space $(\Omega, \Sigma, \mu)$ with a positive finite measure $\mu$, and a positive real number $p$, we define the space $L_p^{+}(\mu)=L_p^{+}$ of all (equivalence classes of) $\Sigma$-measurable complex functions $f$ defined on $\Omega$ such that the function $\left(\log^+|f|\right)^p$ is integrable with respect to $\mu $.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 $dt/2\pi$ on $[0,2\pi)$, 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 $[0,2\pi)$ function $w$, we define the generalized Orlicz space $L_p^{w}(\mathrm{d}t/2\pi)=L^w_p$ with the modular $\rho^w_p$ given by the function $\psi_w(t,u)=\big(\log(1+uw(t))\big)^p$, with a "weight" $w$. We observe that the space $L^w_p$ is a generalized Orlicz space with respect to the modular $\rho^w_p$. We examine and compare different topologies induced on $L^w_p$ by corresponding "weights" $w$.