Boundedness and compactness of an integral operator on a weighted space on the polydisc

Abstract

We study operators of the form Tg(f)(z)= ∫0z1 ⋯ ∫0zn f(ζ1,..., ζn)g(ζ1,...ζn) ∏i=1n dζj where g is a fixed holomorphic function on the unit polydisc Un, on the space S α→(Un) defined by S α→(Un) = {f ∈ H(Un)|sup z∈Un|f(z)|∏j=1n(1-|z j|)αj < ∞}, where α = (α1, ⋯, αn), αj > 0, j = 1, ⋯, n. It is shown that the operator Tg is bounded on Sα→(Un) if and only if supz∈Un ∏j=1n(1 - |zj|)|g(z)| < ∞, and compact if and only if lim z→∂Un ∏j=1n(1 - |zj|)|g(z)| = 0. Also, we give a necessary and sufficient condition for a holomorphic function on the polydisc to be a p-Bloch function, when p ∈ [1, 2).