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 $$ D^\mu(f)(z)=\sum_{ n=0}^{ \infty} \frac{ \widehat{ f}(n)}{ \mu_{ 2n+1}}z^n, $$ 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 $I^\mu(f)(z)=\sum_{ n=0}^{ \infty} \mu_{ 2n+1}\widehat{ f}(n)z^n$, and the fractional Volterra-type operator $$ V_{ \mu,g}(f)(z)= I^\mu(f\cdot D^\mu(g))(z),\quad f\in\mathcal{ H}(\mathbb{ D}), $$ for any fixed $g\in\mathcal{ H}(\mathbb{ D})$. 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 $\text{ BMOA}$ ($\text{ VMOA}$). 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 $H^p$, $\text{ BMOA}$, $\text{ VMOA}$ and $B_p$ in terms of the fractional derivative $D^\mu$.
Categories: math.CV math.CA math.FA