On Libera-type transforms on the unit disc, polydisc and the unit ball

Abstract

We investigate the boundedness and compactness of the following generalization of the Libera operator Λz0γ (f)(z) = γ+1/(z-z0)γ+1 ∫z0z f(ζ)(z-ζ)γdζ where ℛγ > -1, f(z) = Σk=0∞akz k, and z0 belongs to the closure of the unit disk double-struck D. Among other results, it is shown that if p≥1,α≥-1/p and z0 ∈ ∂ double-struck D, the operator is bounded on the mixed norm space Aαp,q(double-struck D) = {f ∈ H(double-struck D)|∫01 Mpq(f,r)(1-r)αdr < ∞}, if and only if 1/p+(α+1)/q<1. The compactness of the operator is also investigated. We introduce two Libera-type transforms on the unit ball B ⊂ ℂn. For one of these operators we give some sufficient conditions to be compact on the mixed norm space on the unit ball, and for the other we show that the operator is bounded on the weighted Bergman space A αp on the unit ball.