Definition and Properties of the Libera Operator on Mixed Norm Spaces
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...
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/590656 |
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| author | Miroslav Pavlovic |
| author_facet | Miroslav Pavlovic |
| author_sort | Miroslav Pavlovic |
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| description | 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. |
| format | Article |
| id | doaj-art-2f7c3bcce30c41c680e6e27b9d9de40a |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
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| series | The Scientific World Journal |
| spelling | doaj-art-2f7c3bcce30c41c680e6e27b9d9de40a2025-02-03T05:44:16ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/590656590656Definition and Properties of the Libera Operator on Mixed Norm SpacesMiroslav Pavlovic0Faculty of Mathematics, University of Belgrade, Studentski Trg 16, P.O. Box 550, 11001 Beograd, SerbiaWe 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.http://dx.doi.org/10.1155/2014/590656 |
| spellingShingle | Miroslav Pavlovic Definition and Properties of the Libera Operator on Mixed Norm Spaces The Scientific World Journal |
| title | Definition and Properties of the Libera Operator on Mixed Norm Spaces |
| title_full | Definition and Properties of the Libera Operator on Mixed Norm Spaces |
| title_fullStr | Definition and Properties of the Libera Operator on Mixed Norm Spaces |
| title_full_unstemmed | Definition and Properties of the Libera Operator on Mixed Norm Spaces |
| title_short | Definition and Properties of the Libera Operator on Mixed Norm Spaces |
| title_sort | definition and properties of the libera operator on mixed norm spaces |
| url | http://dx.doi.org/10.1155/2014/590656 |
| work_keys_str_mv | AT miroslavpavlovic definitionandpropertiesoftheliberaoperatoronmixednormspaces |