JournalsrmiVol. 22, No. 3pp. 1069–1126

Uniform Bounds for the Bilinear Hilbert Transforms, II

  • Xiaochun Li

    University of Illinois at Urbana-Champaign, USA
Uniform Bounds for the Bilinear Hilbert Transforms, II cover
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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 LpL^{p} bounds for the family of bilinear Hilbert transforms

Hα,β(f,g)(x)=p.v.Rf(xαt)g(xβt)dtt.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 Hα,βH_{\alpha,\beta} map Lp1(R)×Lp2(R)L^{p_1}(\mathbb R)\times L^{p_2}(\mathbb R) into Lp(R)L^p(\mathbb R) uniformly in the real parameters α\alpha, β\beta satisfying αβ1c>0|\frac{\alpha}{\beta}-1|\ge c > 0 when 1<p1,p2<21 < p_1, p_2 < 2 and 23<p=p1p2p1+p2<\frac{2}{3} < p= \frac{p_1p_2}{p_1+p_2} < \infty. As a corollary we obtain Lp×LLpL^p \times L^\infty \to L^p uniform bounds in the range 4/3<p<44/3 < p < 4 for the H1,αH_{1,\alpha}'s when α[0,1)\alpha\in [0,1).

Cite this article

Xiaochun Li, Uniform Bounds for the Bilinear Hilbert Transforms, II. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 1069–1126

DOI 10.4171/RMI/483