Definition and Properties of the Libera Operator on Mixed Norm Spaces

Abstract EN

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.