Uniform bounds for the bilinear Hilbert transforms. II. (English) Zbl 1133.42022

Li, Xiaochun

Rev. Mat. Iberoam. 22, No. 3, 1069-1126 (2006).

Author’s abstract: We continue the investigation initiated in [L. Grafakos and X. Li, Ann. of Math. (2) 159, 889–933 (2004; Zbl 1071.44004)] of uniform \(L^{p}\) bounds for the family of bilinear Hilbert transforms
\[ H_{\alpha,\beta} (f,g)(x) = \text{p.v.} \int_{\mathbb{R}} f(x-\alpha t) g(x-\beta t) \frac{dt}{t} \,. \]
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 \(|\frac{\alpha}{\beta}-1|\geq c > 0\) 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 \(\alpha\in [0,1)\).

Reviewer: Xuan Thinh Duong (Sydney)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory
46B70	Interpolation between normed linear spaces
47B38	Linear operators on function spaces (general)

Citations:

Zbl 1071.44004

References:

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