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

Let D-n be the unit polydisk and B be the unit ball in C-n respectively. In this paper, we extend the Cesaro operator to the unit polydisk and the unit ball. We prove that the generalized Cesaro operator C-(b) over right arrow,C-(c) over right arrow is bounded on the Hardy space H-P(D-n) and the mixed norm space A((mu) over right arrow)(p,q)(D-n), when 0 < q < infinity, 1, p is an element of (0, 1] and Re (b(j) + 1) > Re c(j) > 0, j = 1,..., n, or if 0 < q < infinity, p > 1 and Re (b(j) + 1) > Re c(j) >= 1, j = 1, ..., n. Here (mu) over right arrow = (mu(1),..., mu(n)) and each mu(j), j is an element of {1,..., n} is a positive Borel measure on the interval [0, 1). We also introduce a new class of averaging integral operators C-zeta 0(b,c) (the generalized Cesaro operators) on B and prove the boundedness of the operator on the Hardy plq space H-p (B), p is an element of (0, infinity), the mixed-norm space A(mu)(p,q) (B), 0 < p, q < infinity and the alpha-Bloch space, when alpha > 1. Finally, we study the boundedness and compactness of recently introduced Riemann-Stieltjes type operators T-g and L-g, from H-infinity and Bergman type spaces to alpha-Bloch spaces and little alpha-Bloch spaces on B.