ESTIMATES OF AN INTEGRAL OPERATOR ON FUNCTION SPACES

Abstract

In this paper, we shall study the family of operators of the form $$ T_{\vec g}(f)(z)=\int_0^{z_1}\cdots\int_0^{z_n}f(\zeta_1,\cdots,\zeta_n)\prod_{j=1}^ng'_j(\zeta_j)d\zeta_j $$ on Hardy $H^p(D_n)$, the generalized weighted Bergman ${\cal A}_\mu^{p,q}(D_n),$ $p\in (0,\infty),$ and $\alpha$-Bloch ${\cal B}^\alpha (D_n)$ spaces on the polydisk $D_n=\{(z_1,\dots, z_n)\in {\bf C}^n:\,|z_j|\lt 1,\,\,j=1,\dots, n\}.$