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 $L_p^w(\mathrm{d}t/2\pi )=L_p^w$ with the modular ${\rho }_p^w$ given by the function ${\psi }_w(t,u)=(\log (1+uw(t)){)}^p$, with a "weight" $w$. We observe that the space $L_p^w$ is a generalized Orlicz space with respect to the modular ${\rho }_p^w$. We examine and compare different topologies induced on $L_p^w$ by corresponding "weights" $w$.

18 pages, no figures, Journal-ref: Mathematica Montisnigri, vol. 11 (1999), pp. 25-42; Mathematical Reviews: MR1781340 (2001h:46040)