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 \(\mathord {\textrm{BMOA}}\) ( \(\mathord {\textrm{VMOA}}\) ). 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 \(H^p\) , \(\mathord {\textrm{BMOA}}\) , \(\mathord {\textrm{VMOA}}\) and \(B_p\) in terms of the fractional derivative \(D^\mu \) .
