About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
Volume 5 (2007), Issue 1, Pages 49-88
http://dx.doi.org/10.1155/2007/294367

A note on maximal operator on {pn} and Lp(x)()

Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16629 Prague 6, Czech Republic

Received 1 December 2005

Academic Editor: Pankaj Jain

Copyright © 2007 Hindawi Publishing Corporation. 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 a discrete analogue of Hardy-Littlewood maximal operator on the generalized Lebesque space {pn} of sequences defined on . It is known a necessary and sufficient condition P which guarantees an existence of a real number p>1 such that the norms in the space {pn} and in the classical space p are equivalent. Of course, this condition immediately implies the boundedness of maximal operator on {pn} and, moreover, lim|n|pn=p. We construct two examples of sequences {pn} satisfying lim|n|pn=p in this paper. In the first example the maximal operator is unbounded on {pn} and the sequence {pn} from the second example does not satisfy P but the maximal operator is bounded. Moreover, it is known a sufficient integral condition to a behavior of a function p(x) at infinity which guarantees the boundedness of the maximal operator on Lp()(n). As a main result of this paper we construct a function p(x) which does not satisfy this integral condition nevertheless the maximal operator is bounded.