Hp spaces and integral operators

Let $\scr H(U)$ denote the class of analytic functions in the unit disc $U$ and $g$ be analytic in $U$, normalized by $g(0)=g'(0)-1=0$ and $g(z)\ne0$ for $z\in U\sbs\{0\}$. $H^p$, $0<p\le \infty$, denotes the Hardy class and $H\,\roman{log}^+\,H$ the class for which $\int_0^{2\pi}|f(re^{i\theta})| \roman{log}^+|f(re^{i\theta})|\,d\theta$ is bounded when $r\rightarrow 1^-$. The author considers the integral operator $L_g\colon \scr H(U)\rightarrow \scr H(U)$ defined by $L_g(f)(z)=(z/g(z))\int_0^{z}f(t)g'(t)\,dt$ and shows that: (i) if $zg'/g\in H\,\roman{log}^+\,H$ and $f\in H^p$ then $L_g(f)\in H^p$; (ii) if $zg'/g\in H^q$, $q>1$ and $f\in H^p$ then $L_g(f)\in H^r$ where $r=pq/(p+q-pq)$ for $0<p<q/(q-1)$ and $r=\infty$ for $p\ge\break q/(q-1)$; and (iii) if $zg'/g\in H^\infty$ and $f\in H^p$, then $L_g(f)$ is in $H^r$ where $r=p/(1-p)$ for $0<p<1$ and $r=\infty$ for $p\ge 1$. This result generalizes a result of the reviewer [same journal 29(52) (1987), no. 1, 29--31; MR0939548 (89e:30061)]. An interesting example is given.