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

We study the following integral type operator

$T_g (f)(z) = intlimits_0^{z_{} } { cdots intlimits_0^{z_n } {f(zeta _1 , ldots ,zeta _n )} g(zeta _1 , ldots ,zeta _n )dzeta _1 , ldots ,zeta _n } $

in the space of analytic functions on the unit polydisk U ⁿ in the complex vector space ?ⁿ. We show that the operator is bounded in the mixed norm space

Open image in new windowp, q ∈ [1, ∞) and α = (α₁, …, α_n), such that α_j > ?1, for every j = 1, …, n, if and only if (sup _{z in U^n } prodnolimits_{j = 1}^n {left( {1 - left| {z_j } right|} right)} left| {g(z)} right| < infty ). Also, we prove that the operator is compact if and only if (lim _{z to partial U^n } prodnolimits_{j = 1}^n {left( {1 - left| {z_j } right|} right)} left| {g(z)} right| = 0).