ON SOME INTEGRAL OPERATORS ON THE UNIT POLYDISK AND THE UNIT BALL

Abstract

Let $\mathbb D^n$ be the unit polydisk and $B$ be the unit ball in $\mathbb C^n$ respectively. In this paper, we extend the Cesàro operator to the unit polydisk and the unit ball. We prove that the generalized Cesàro operator $\mathcal{C}^{\vec b,\vec c}$ is bounded on the Hardy space $H^p(\mathbb D^n)$ and the mixed norm space $A^{p,q}_{\vec \mu}(\mathbb D^n)$, when $0 \lt q \lt \infty$, $p \in (0,1]$ and $Re\, (b_j+1) \gt Re\, c_j \gt 0$, $j = 1,\ldots,n$, or if $0 \lt q \lt \infty$, $p \gt 1$ and $Re\, (b_j+1) \gt Re\, c_j \geq 1$, $j = 1,\ldots,n$. Here $\vec\mu = (\mu_1,\ldots,\mu_n)$ and each $\mu_j$, $j \in \{1,\ldots,n\}$ is a positive Borel measure on the interval $[0,1)$. We also introduce a new class of averaging integral operators $\mathcal{C}^{b,c}_{\zeta_0}$ (the generalized Cesàro operators) on $B$ and prove the boundedness of the operator on the Hardy space $H^p(B)$, $p \in (0,\infty)$, the mixed-norm space $\mathcal{A}^{p,q}_\mu(B)$, $0 \lt p,q \lt \infty$ and the $\alpha$-Bloch space, when $\alpha \gt 1$. Finally, we study the boundedness and compactness of recently introduced Riemann-Stieltjes type operators $T_g$ and $L_g$, from $H^\infty$ and Bergman type spaces to $\alpha$-Bloch spaces and little $\alpha$-Bloch spaces on $B$.