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Journal of Function Spaces and Applications
Volume 2012, Article ID 610649, 9 pages
http://dx.doi.org/10.1155/2012/610649
Research Article

Boundedness of Weighted Hardy Operator and Its Adjoint on Triebel-Lizorkin-Type Spaces

Canqin Tang and Ruohong Zhou

Department of Mathematics, Dalian Maritime University, Liaoning, Dalian, 116026, China

Received 26 October 2010; Accepted 18 November 2010

Academic Editor: Hans Triebel

Copyright © 2012 Canqin Tang and Ruohong Zhou. 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

Let p∈[1,∞], q∈[1,∞), τ∈(0,∞), and α∈(0,1) such that τ>1/p-1/q and α≤n(1/p-τ), let Uψ be the weighted Hardy operator and Vψ its adjoint operator with respect to the weight function ψ. In this paper, the authors establish a sufficient and necessary condition on weight function ψ to ensure the boundedness of Uψ and Vψ on the Triebel-Lizorkin-type spaces Ḟp,qα,τ(ℝn) and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on Q-type spaces.