On some metric topologies on Privalov spaces on the unit disk

Abstract

Let $N^p$ $(1<p<\infty)$ be the Privalov class $N^p$ of holomorphic functions on the open unit disk $\Bbb D$ in the complex plane. In 1977 M. Stoll proved that the class $N^p$ equipped with the topology given by the metric $\lambda_p$ defined by $$\lambda_p(f,g) = \Bigg(\int_0^{2\pi}\big(\log(1+ \vert f^*(e^{i\theta})-g^*(e^{i\theta})\vert)\big)^p\,\frac{d\theta} {2\pi}\Bigg)^{1/p},\quad f,g\in N^p,$$ becomes an $F$-algebra. In the recent overview paper by Me\v{s}trovi\'{c} and Pavi\'{c}evi\'{c} (2017) a survey of some known results on the topological structures of the Privalov spaces $N^p$ $(1<p<\infty)$ and their Fr\'{e}chet envelopes $F^p$ are presented. In this article we continue a survey of results concerning the topological structures of the spaces $N^p$ $(1(p<\infty)$. In particular, for each $p>1$, 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 $N^p$.