Sharp Inequalities for the Haar System and Fourier Multipliers
A classical result of Paley and Marcinkiewicz asserts that the Haar system h=hkk≥0 on 0,1 forms an unconditional basis of Lp0,1 provided 1<p<∞. That is, if 𝒫J denotes the projection onto the subspace generated by hjj∈J (J is an arbitrary subset of ℕ), then 𝒫JLp0,1→Lp0,1≤βp for some universal c...
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Wiley
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/646012 |
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| author | Adam Osȩkowski |
| author_facet | Adam Osȩkowski |
| author_sort | Adam Osȩkowski |
| collection | DOAJ |
| description | A classical result of Paley and Marcinkiewicz asserts that the Haar system h=hkk≥0 on 0,1 forms an unconditional basis of Lp0,1 provided 1<p<∞. That is, if 𝒫J denotes the projection onto the subspace generated by hjj∈J (J is an arbitrary subset of ℕ), then 𝒫JLp0,1→Lp0,1≤βp for some universal constant βp depending only on p. The purpose of this paper is to study related restricted weak-type bounds for the projections 𝒫J. Specifically, for any 1≤p<∞ we identify the best constant Cp such that 𝒫JχALp,∞0,1≤CpχALp0,1 for every J⊆ℕ and any Borel subset A of 0,1. In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established. |
| format | Article |
| id | doaj-art-bc4aa315eb044b5e8c1d7b3f13d6b2e6 |
| institution | DOAJ |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-bc4aa315eb044b5e8c1d7b3f13d6b2e62025-08-20T03:23:18ZengWileyJournal of Function Spaces and Applications0972-68021758-49652013-01-01201310.1155/2013/646012646012Sharp Inequalities for the Haar System and Fourier MultipliersAdam Osȩkowski0Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, PolandA classical result of Paley and Marcinkiewicz asserts that the Haar system h=hkk≥0 on 0,1 forms an unconditional basis of Lp0,1 provided 1<p<∞. That is, if 𝒫J denotes the projection onto the subspace generated by hjj∈J (J is an arbitrary subset of ℕ), then 𝒫JLp0,1→Lp0,1≤βp for some universal constant βp depending only on p. The purpose of this paper is to study related restricted weak-type bounds for the projections 𝒫J. Specifically, for any 1≤p<∞ we identify the best constant Cp such that 𝒫JχALp,∞0,1≤CpχALp0,1 for every J⊆ℕ and any Borel subset A of 0,1. In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established.http://dx.doi.org/10.1155/2013/646012 |
| spellingShingle | Adam Osȩkowski Sharp Inequalities for the Haar System and Fourier Multipliers Journal of Function Spaces and Applications |
| title | Sharp Inequalities for the Haar System and Fourier Multipliers |
| title_full | Sharp Inequalities for the Haar System and Fourier Multipliers |
| title_fullStr | Sharp Inequalities for the Haar System and Fourier Multipliers |
| title_full_unstemmed | Sharp Inequalities for the Haar System and Fourier Multipliers |
| title_short | Sharp Inequalities for the Haar System and Fourier Multipliers |
| title_sort | sharp inequalities for the haar system and fourier multipliers |
| url | http://dx.doi.org/10.1155/2013/646012 |
| work_keys_str_mv | AT adamosekowski sharpinequalitiesforthehaarsystemandfouriermultipliers |