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 Ces\`aro operator to the unit polydisk and the unit ball. We prove that the generalized Ces\`aro operator ${\cal C}^{\vec b,\vec c}$ 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 $0Re\, c_j>0,$ $j=1,\ldots,n$, or if $01$ and $Re\, (b_j+1)>Re\, c_j\geq 1,$ $j=1,\ldots,n$. 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 ${\cal C}^{b,c}_{\zeta_0}$ (the generalized Ces\`aro 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 ${\cal A}^{p,q}_\mu(B),\,01.$ 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 $B$.