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Uniform bounds for the bilinear Hilbert transforms. I. (English) Zbl 1071.44004

The family of bilinear Hilbert transforms \[ H_{\alpha,\beta}(f_1, f_2)(x)= \text{p.v.\;}\int_{\mathbb{R}} f_1(x-\alpha t) f_2(x-\beta t){dt\over t},\;\alpha,\beta\in\mathbb{R}, \] was introduced by Calderòn.
In this article the authors obtain bounds for \(H_{i,\alpha}\) from \(L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\) into \(L^p(\mathbb{R})\), uniformly in \(\alpha\) when \(2< p_1\), \(p_2<\infty\) and \(1< p= {p_1p_2\over p_1+ p_2}< 2\).
In the second part bounds are obtained for \(H_{1,\alpha}\) from \(L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\) into \(L^p(\mathbb{R})\), uniformly in a satisfying \(|\alpha-1|\geq c> 0\) when \(1< p_1\), \(p_2< 2\) and \({2\over 3}< p= {p_1p_2\over p_1+ p_2}< 1\).
Some interpolation results are studied.
The main result is the following.
Theorem. Let \(2< p_1\), \(p_2<\infty\) and \(1< p={p_1p_2\over p_1+ p_2}< 2\).
Then there is a constant \(C= C(p_1, p_2)\) such that for all \(f_1\), \(f_2\) Schwartz functions on \(\mathbb{R}\) \[ \sup_{\alpha,\beta\in\mathbb{R}}\| H_{\alpha,\beta}(f_1, f_2)\|_p\leq c\| f_1\|_{p_1}\| f_2\|_{p_2}. \]

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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