Fractional Volterra-type operator induced by radial weight acting on Hardy space

Abstract

Given a radial doubling weight $\mu$ on the unit disc $\mathbb{D}$ of the complex plane and its odd moments $\mu_{2n+1}=\int_0^1 s^{2n+1}\mu(s)\, ds$, we consider the fractional derivative $$ D^\mu(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{\mu_{2n+1}}z^n, %\quad z\in \D, $$ of a function $ f(z)=\sum_{n=0}^{\infty}\widehat{f}(n)z^n$ analytic in $\mathbb{D}$.

We also consider the fractional integral operator $ I^\mu(f)(z)=\sum_{n=0}^{\infty} \mu_{2n+1}\widehat{f}(n)z^n, %\, z\in\D, $ and the fractional Volterra-type operator $$ V_{\mu,g}(f)(z)= I^\mu(f\cdot D^\mu(g))(z),\quad f\in\H(\D),%,\quad z\in \D. $$ for any fixed $g\in\H(\D)$. We prove that $V_{\mu,g}$ is bounded (compact) on a Hardy space $H^p$, $0<p<\infty$, if and only if $g$ belongs to $\BMOA$ ($\VMOA$). Moreover, if $\int_0^1 \frac{\left(\int_r^1 \mu(s)\, ds\right)^p}{(1-r)^2}\,dr=+\infty$, we prove that $V_{\mu,g}$ belongs to the Schatten class $S_p(H^2)$ if and only if $g=0$. On the other hand, if $\frac{\left(\int_r^1 \mu(s)\, ds\right)^p}{(1-r)^2}$ is a radial doubling weight it is proved that $V_{\mu,g} \in S_p(H^2)$ if and only if $g$ belongs to the Besov space $B_p$. En route, we obtain descriptions of $H^p$, $\BMOA$, $\VMOA$ and $B_p$ in terms of the fractional derivative $D^\mu$.