Boundedness and compactness of an integral operator in a mixed norm space on the polydisk

Abstract

We study the following integral type operator equation presented in the space of analytic functions on the unit polydisk U n in the complex vector space ℂn. We show that the operator is bounded in the mixed norm space [Figure not available: see fulltext.], with p, q [1, ∞) and α = (α1, ⋯, αn), such that αj > -1, for every j = 1, ⋯, n, if and only if supz∈Un ∏j=1n (1 - |z j|)|g(z)| < ∞. Also, we prove that the operator is compact if and only if limz→∂Un ∏ j=1n (1 - |zj|)| g(z)| = 0.