Revista Matemática Iberoamericana

Uniform Bounds for the Bilinear Hilbert Transforms, II

Xiaochun Li

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Abstract

We continue the investigation initiated in [Grafakos and Li: Uniform bounds for the bilinear Hilbert transforms, I. Ann. of Math. (2) 159 (2004), 889-933] of uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $$ H_{\alpha,\beta} (f,g)(x) = \text{p.v.} \displaystyle\int_{\mathbb{R}} f(x-\alpha t) g(x-\beta t) \frac{dt}{t} \,. $$ In this work we show that $H_{\alpha,\beta}$ map $L^{p_1}(\mathbb R)\times L^{p_2}(\mathbb R)$ into $L^p(\mathbb R)$ uniformly in the real parameters $\alpha$, $\beta$ satisfying $|\frac{\alpha}{\beta}-1|\ge c > 0$ when $1 < p_1, p_2 < 2$ and $\frac{2}{3} < p= \frac{p_1p_2}{p_1+p_2} < \infty$. As a corollary we obtain $L^p \times L^\infty \to L^p$ uniform bounds in the range $4/3 < p < 4 $ for the $H_{1,\alpha}$'s when $\alpha\in [0,1)$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 1069-1126.

Dates
First available in Project Euclid: 22 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1169480040

Mathematical Reviews number (MathSciNet)
MR2320411

Zentralblatt MATH identifier
1133.42022

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46B70: Interpolation between normed linear spaces [See also 46M35] 47B38: Operators on function spaces (general)

Keywords
time-frequency analysis bilinear Hilbert transform uniform bounds

Citation

Li, Xiaochun. Uniform Bounds for the Bilinear Hilbert Transforms, II. Rev. Mat. Iberoamericana 22 (2006), no. 3, 1069--1126. https://projecteuclid.org/euclid.rmi/1169480040


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