On some integral operators on the unit polydisk and the unit ball

Abstract

Let Dn be the unit polydisk and B be the unit ball in Cn respectively. In this paper, we extend the Cesàro operator to the unit polydisk and the unit ball. We prove that the generalized Cesàro operator, is bounded on the Hardy space Hp(Dn) and the mixed norm space, when 0 < q < ∞, p ∈ (0,1] and Re(bj +1) > Recj > 0, j = 1,...,n, or if 0 < q < ∞, p>1 and Re(bj+1) > Re cj ≥ 1, j=1,...,n. Here, and each μj, j ∈ {1,..., n} is a positive Borel measure on the interval [0, 1). We also introduce a new class of averaging integral operators Cb,cζ0 (the generalized Cesàro operators) on B and prove the boundedness of the operator on the Hardy space Hp(B), p ∈ (0, ∞), the mixed-norm space Ap,qμ(B), 0 < p, q < ∞ and the α-Bloch space, when α > 1. Finally, we study the boundedness and compactness of recently introduced Riemann-Stieltjes type operators Tg and Lg, from H∞ and Bergman type spaces to α-Bloch spaces and little α-Bloch spaces on B.