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 $$T_{g,n}[f](z) := \underbrace{\int_{0}^{z}\int_{0}^{t_1}\cdots\int_{0}^{t_{n-1}}}_{n\ \text{times}} f(t_n)g'(t_n)\,dt_n\cdots dt_1,\quad z\in\mathbb D,$$ 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 $|g'|$.