Definition and Properties of the Libera Operator on Mixed Norm Spaces

Abstract

We consider the action of the operatorℒg(z)=(1-z)-1∫z1‍f(ζ)dζon a class of “mixed norm” spaces of analytic functions on the unit disk,X=Hα,νp,q, defined by the requirementg∈X⇔r↦(1-r)αMp(r,g(ν))∈Lq([0,1],dr/(1-r)), where1≤p≤∞,0<q≤∞,α>0, andνis a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expressionℒgneed not be defined forganalytic in the unit disk, even forg∈X. A sufficient, but not necessary, condition is that∑n=0∞‍|g^(n)|/(n+1)<∞. We identify the indicesp,q,α, andνfor which1∘ℒis well defined onX,2∘ℒacts fromXtoX,3∘the implicationg∈X⇒∑n=0∞‍|g^(n)|/(n+1)<∞holds. Assertion2∘extends some known results, due to Siskakis and others, and contains some new ones. As an application of3∘we have a generalization of Bernstein’s theorem on absolute convergence of power series that belong to a Hölder class.