The maximal operator in weighted variable spaces Lp(⋅)
We study the boundedness of the maximal operator in the weighted spaces Lp(⋅)(ρ) over a bounded open set Ω in the Euclidean space ℝn or a Carleson curve Γ in a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expec...
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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/10.1155/2007/914143 |
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| author | Vakhtang Kokilashvili Natasha Samko Stefan Samko |
| author_facet | Vakhtang Kokilashvili Natasha Samko Stefan Samko |
| author_sort | Vakhtang Kokilashvili |
| collection | DOAJ |
| description | We study the boundedness of the maximal operator in the weighted spaces Lp(⋅)(ρ) over a bounded open set Ω in the Euclidean space ℝn or a Carleson curve Γ in a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt class Ap in the case of constant p. In the case of Carleson curves there is also considered another class of weights of radial type of the form ρ(t)=∏k=1mwk(|t-tk|), tk∈Γ, where wk has the property that r1p(tk)wk(r)∈Φ10, where Φ10 is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponent p(t) satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces). |
| format | Article |
| id | doaj-art-80016af2be9043e19bc51338297905b9 |
| institution | Kabale University |
| issn | 0972-6802 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-80016af2be9043e19bc51338297905b92025-02-03T01:22:08ZengWileyJournal of Function Spaces and Applications0972-68022007-01-015329931710.1155/2007/914143The maximal operator in weighted variable spaces Lp(⋅)Vakhtang Kokilashvili0Natasha Samko1Stefan Samko2A.Razmadze Mathematical Institute, Tbilisi, GeorgiaCenter CEMAT, IST, Lisbon, PortugalUniversity of Algarve, PortugalWe study the boundedness of the maximal operator in the weighted spaces Lp(⋅)(ρ) over a bounded open set Ω in the Euclidean space ℝn or a Carleson curve Γ in a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt class Ap in the case of constant p. In the case of Carleson curves there is also considered another class of weights of radial type of the form ρ(t)=∏k=1mwk(|t-tk|), tk∈Γ, where wk has the property that r1p(tk)wk(r)∈Φ10, where Φ10 is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponent p(t) satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).http://dx.doi.org/10.1155/2007/914143 |
| spellingShingle | Vakhtang Kokilashvili Natasha Samko Stefan Samko The maximal operator in weighted variable spaces Lp(⋅) Journal of Function Spaces and Applications |
| title | The maximal operator in weighted variable spaces Lp(⋅) |
| title_full | The maximal operator in weighted variable spaces Lp(⋅) |
| title_fullStr | The maximal operator in weighted variable spaces Lp(⋅) |
| title_full_unstemmed | The maximal operator in weighted variable spaces Lp(⋅) |
| title_short | The maximal operator in weighted variable spaces Lp(⋅) |
| title_sort | maximal operator in weighted variable spaces lp ⋅ |
| url | http://dx.doi.org/10.1155/2007/914143 |
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