A note on maximal operator on ℓ{pn} and Lp(x)(ℝ)
We consider a discrete analogue of Hardy-Littlewood maximal operator on the generalized Lebesque space ℓ{pn} of sequences defined on ℤ. 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 ℓ{pn} and in the classical...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
|
| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2007/294367 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We consider a discrete analogue of Hardy-Littlewood maximal operator on the generalized Lebesque space ℓ{pn} of sequences defined on ℤ. 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 ℓ{pn} and in the classical space ℓp are equivalent. Of course, this condition immediately implies the boundedness of maximal operator on ℓ{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 ℓ{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(⋅)(ℝn). 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. |
|---|---|
| ISSN: | 0972-6802 |