Hadamard product in \(Q_{p}\) spaces (Q1773314)

Let \(H(\Delta)\) denote the class of all functions holomorphic in the unit disk \(\Delta=\{z:| z| <1\}\). The Hadamard product of \(f(z)=\sum_{n=0}^{\infty}a_nz^n\in H(\Delta)\) and \(g(z)=\sum_{n=0}^{\infty}b_nz^n\in H(\Delta)\) is the function \(f\ast g\in H(\Delta)\) defined by \((f\ast g)(z)=\sum_{n=0}^{\infty}a_nb_nz^n\). Let \(X\) and \(Z\) be nonempty subclasses of \(H(\Delta)\). A function \(g\in H(\Delta)\) is said to be a (Hadamard) multiplier from \(X\) to \(Z\) if \(f\ast g\in Z\) whenever \(f\in X\). The set of all multipliers from \(X\) to \(Z\) is denoted by \((X,Z)\). In this interesting paper the author considers multiplier spaces in cases of the Hardy space \(H^1\), mean Lipschitz spaces \(\Lambda(q,\alpha)\), the Bloch space \(\mathcal B\), the \(Q_p\)-spaces and obtains the following main theorem: For \(0<p<1\), (i) \(\Lambda(q,1/q)\subseteq(\Lambda(1,0),Q_p)\) for \(q<2/(1-p)\) (ii) \(\Lambda(2/(1-p),(1-p)/2)\subseteq(H^1,Q_p)\) (iii) \(Q_p\subseteq(\Lambda(2/(2-p),0),Q_p)\) and (iv) \(Q_p\subseteq(\mathcal B,Q_p)\). The case (iv) improves one result of \textit{D. Girela, H. Wulan} and the reviewer [J. Math. Anal. Appl. 258, No. 2, 415--428 (2001; Zbl 0980.30024)]. Also some earlier known theorems on lacunary and random series related to \(Q_p\) are reproved by using the methods developed in this paper.