Definition and Properties of the Libera Operator on Mixed Norm Spaces

We consider the action of the operator ℒg(z) = (1 − z)(−1)∫(z) (1)‍f(ζ)dζ on a class of “mixed norm” spaces of analytic functions on the unit disk, X = H (α,ν) (p,q), defined by the requirement g ∈ X⇔r ↦ (1 − r)(α) M (p)(r, g ((ν))) ∈ L (q)([0,1], dr/(1 − r)), where 1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 0, and ν is a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expression ℒg need not be defined for g analytic in the unit disk, even for g ∈ X. A sufficient, but not necessary, condition is that [Formula: see text]. We identify the indices p, q, α, and ν for which 1°ℒ is well defined on X, 2°ℒ acts from X to X, 3° the implication [Formula: see text] holds. Assertion 2° extends some known results, due to Siskakis and others, and contains some new ones. As an application of 3° we have a generalization of Bernstein's theorem on absolute convergence of power series that belong to a Hölder class.