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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 0972-6802 1758-4965 |