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Uniform bounds for the bilinear Hilbert transforms (2004)
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Xiaochun Li
| Venue: | 889–993. MR2113017 (2006e:42011), Zbl 1071.44004. Xiaochun Li |
| Citations: | 36 - 15 self |
BibTeX
@ARTICLE{Li04uniformbounds,
author = {Xiaochun Li},
title = {Uniform bounds for the bilinear Hilbert transforms},
journal = {889–993. MR2113017 (2006e:42011), Zbl 1071.44004. Xiaochun Li},
year = {2004},
volume = {2},
pages = {159}
}
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Abstract
Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying | α β − 1 | ≥ c> 0 when 1 < p1, p2 < 2 and 2 p1p2 3 < p = < ∞. p1+p2 As a corollary we obtain Lp × L ∞ → Lp uniform bounds in the range 4/3 < p < 4 for the H1,α’s when α ∈ [0, 1). 1.
