Definition and Properties of the Libera Operator on Mixed Norm Spaces - Miroslav Pavlovic (The Scientific World Journal)
We consider the action of the operator ℒg(z)=(1-z)-1∫z1‍f(ζ)dζ on a class of “mixed norm” spaces of analytic functions on the unit disk, X=Hα,νp,q, defined by the requirement g∈X⇔r↦(1-r)αMp(r,g(ν))∈Lq([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 ∑n=0∞‍|g^(n)|/(n+1)<∞. 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 g∈X⇒∑n=0∞‍|g^(n)|/(n+1)<∞ 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.