Spaces of analytic functions on the disc where the growth of \(M_ p(F,r)\) depends on a weight (Q923761)

The authors consider some function spaces of analytic functions on the disc D, depending on a weight \(\rho\) (t), nonnegative and nondecreasing on [0,1). They treat the following spaces (1\(\leq p\leq \infty)\) \[ HL^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad M_ p(F',r)=O(\frac{\rho (1-r)}{1-r})\}, \] \[ Z^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad M_ p(F'',r)=O(\frac{\rho (1-r)}{(1-r)^ 2})\}, \] \[ B^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad \int^{1}_{0}\frac{\rho (1-r)}{1-r}M_ p(F,r)dr<\infty \}, \] \[ J^ p_{\rho}=\{F:D\to {\mathbb{C}}\quad analytic,\quad \int^{1}_{0}\rho (1- r)M_ p(F',r)dr<\infty \}, \] where \(M_ p(F,r)=(1/2\pi \int^{\pi}_{-\pi}| F(re^{i\theta}|^ pd\theta)^{1/p}\). They find conditions on \(\rho\) to get results analogous to those by Hardy and Littlewood for \(\rho (t)=t^{\alpha}(0<\alpha <1)\) and by Zygmund for \(\rho (t)=u\). Under certain assumptions they show that the boundary value functions of the above function spaces must satisfy some conditions on the first or second difference, and that the predual of \(HL^ q_{\rho}\) is \(J^ q_{\rho}\) and that of \(Z^ q_{\rho}\) is \(B^ p_{\rho}\).