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...
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| Format: | Article |
| Language: | English |
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Wiley
2007-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2007/294367 |
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| author | Aleš Nekvinda |
| author_facet | Aleš Nekvinda |
| author_sort | Aleš Nekvinda |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-2c982f6db837463abd9f87b810ab4246 |
| institution | OA Journals |
| issn | 0972-6802 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-2c982f6db837463abd9f87b810ab42462025-08-20T02:24:04ZengWileyJournal of Function Spaces and Applications0972-68022007-01-0151498810.1155/2007/294367A note on maximal operator on ℓ{pn} and Lp(x)(ℝ)Aleš Nekvinda0Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16629 Prague 6, Czech RepublicWe 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.http://dx.doi.org/10.1155/2007/294367 |
| spellingShingle | Aleš Nekvinda A note on maximal operator on ℓ{pn} and Lp(x)(ℝ) Journal of Function Spaces and Applications |
| title | A note on maximal operator on ℓ{pn} and Lp(x)(ℝ) |
| title_full | A note on maximal operator on ℓ{pn} and Lp(x)(ℝ) |
| title_fullStr | A note on maximal operator on ℓ{pn} and Lp(x)(ℝ) |
| title_full_unstemmed | A note on maximal operator on ℓ{pn} and Lp(x)(ℝ) |
| title_short | A note on maximal operator on ℓ{pn} and Lp(x)(ℝ) |
| title_sort | note on maximal operator on l pn and lp x r |
| url | http://dx.doi.org/10.1155/2007/294367 |
| work_keys_str_mv | AT alesnekvinda anoteonmaximaloperatoronlpnandlpxr AT alesnekvinda noteonmaximaloperatoronlpnandlpxr |