On some metric topologies on Privalov spaces on the unit disk

Abstract

Let $N^p$ $(1 1$, we consider the class $N^p$ as the space $M^p$ equipped with the topology induced by the metric $\rho_p$ defined as $$ \rho_p(f,g) = \Bigg(\int_0^{2\pi}\log^p(1+M(f-g)(\theta))\, \frac{d\theta}{2\pi}\Bigg)^{1/p},\quad f,g\in M^p,\,\, {\mathrm where}\,\, Mf(\theta) = \sup_{0\leqslant r<1} \big\vert f \big(re^{i\theta})\big\vert.$$ On the other hand, we consider the class $N^p$ with the metric topology introduced by Me\v{s}trovi\'{c}, Pavi\'{c}evi\'{c} and Labudovi\'{c} (1999) which generalizes the Gamelin-Lumer's metric which is generally defined on a measure space $(\Omega, \Sigma, \mu)$ with a positive finite measure $\mu$. The space $N^p$ with the associated modular in the sense of Musielak and Orlicz becomes the Hardy-Orlicz class. It is noticed that the all considered metrics induce the same topology on the space $N^p$.