Higher-Order Volterra-Type Integral Operators on Hardy Spaces

Abstract: We study the $n$-fold iterated Volterra-type integral operator $T_{g,n}$ acting on Hardy spaces, defined by [ T_{g,n}f :=\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 functions in the unit disk $\mathbb{D}$. We show that the boundedness and compactness of $T_{g,n}$ are independent of the order $n$ and depend solely on the symbol $g$. In particular, we obtain sharp norm estimates and characterize boundedness and compactness of $T_{g,n}$ in terms of classical function space conditions on $g$.