Definition and Properties of the Libera Operator on Mixed Norm Spaces. (20th February 2014) - Pavlovic, Miroslav
Abstract : We consider the action of the operator ℒ g ( z ) = ( 1 - z ) - 1 ∫ z 1 ‍ f ( ζ ) d ζ 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 ], d r / ( 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 ℒ g 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.