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The Scientific World Journal
Volume 2014 (2014), Article ID 590656, 15 pages
http://dx.doi.org/10.1155/2014/590656
Research Article

Definition and Properties of the Libera Operator on Mixed Norm Spaces

Faculty of Mathematics, University of Belgrade, Studentski Trg 16, P.O. Box 550, 11001 Beograd, Serbia

Received 7 August 2013; Accepted 23 October 2013; Published 20 February 2014

Academic Editors: H. Jafari and Y. Wang

Copyright © 2014 Miroslav Pavlovic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the action of the operator on a class of “mixed norm” spaces of analytic functions on the unit disk, , defined by the requirement , where , , , and is a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expression need not be defined for analytic in the unit disk, even for . A sufficient, but not necessary, condition is that . We identify the indices , , , and for which is well defined on , acts from to , the implication holds. Assertion extends some known results, due to Siskakis and others, and contains some new ones. As an application of we have a generalization of Bernstein’s theorem on absolute convergence of power series that belong to a Hölder class.