Higher-order Volterra-type integral operator on Hardy and Bergman spaces

We investigate the higher-order Volterra-type integral operator $T_{g,n}$ on the unit disk, defined for $n\in \mathbb{N}$ by $$T_{g,n}[f](z):=\underset{n\text{ times}}{\underbrace{{\int }_0^z{\int }_0^{t_1}\cdots {\int }_0^{t_{n-1}}}}f(t_n)g^{\mathrm{\prime }}(t_n)d{t}_n\cdots d{t}_1,z\in \mathbb{D},$$ where $f$ and $g$ are analytic in the unit disk $\mathbb{D}$. We establish sharp norm and essential norm estimates, and give complete characterizations of boundedness and compactness of $T_{g,n}$ on Hardy spaces $H^p$ and weighted Bergman spaces $A_{\alpha }^p$, in terms of (vanishing) Carleson measure conditions determined by $\mathrm{\mid }{g}^{\mathrm{\prime }}\mathrm{\mid }$.