Multipliers and linear functionals in Privalov's spaces of holomorphic functions in the disk (Q1594349)

Suppose \(q>1\). Let \(N^q\) be the Privalov space in the open unit disc \(D\) and \(S_q\) the set of all functions \(g(z)=\sum^\infty_{n=0} b_nz^n\) holomorphic in \(D\) and such that for each of these functions, \(\log|b_n |\leq A-\varepsilon n^{1/(1+q)}\), \(n\in\mathbb{Z}_+\) for some real numbers \(A\) and \(\varepsilon >0\). The authors show that the following: Each c.l.f. \(\Phi\) on \(N^q\) is determined by a unique function \(g\in S_q\) where \(g(z)= \sum^\infty_{n= 0} b_nz^n\), and acts according to the formula \(\Phi(f)= \sum^\infty_{n=0} a_n b_n\) for arbitrary function \(f\in N^q\) of the form \(f(z)= \sum^\infty_{n=0} a_n z^n\). Conversely each function \(g(z)= \sum^\infty_{n=0} b_nz^n\) from the space \(S_q\) determines c.l.f. on \(N^q\). This gives a positive answer to the queston in [\textit{M. Stoll}, Ann. Pol. Math. 35, 139-158 (1977; Zbl 0377.30036)].