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\(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty\). (English) Zbl 0914.46034

Let \(f,g\in S(\mathbb{R})\), where \(S(\mathbb{R})\) is the Schwartz space of smooth and rapidly decaying functions \(f\) on the real line. The bilinear Hilbert transform of \(f\) and \(g\) is defined by \[ H(f, g)(x):= \text{p.v. }\int f(x- t)g(x+ t){1\over t} dt. \] Calder√≥n studied the above transform in connection with Cauchy integral of Lipschitz curves and posed the question whether it satisfies any \(L^p\) estimates [P. Jones, “Bilinear singular integrals and maximal functions” in: V. Havin and N. Nikolski (ed.), “Linear and complex analysis problem book” 3, Part I, Lect. Notes Math. 1573 (1994; Zbl 0893.30036)]. The following theorem established by the authors answers this question:
Theorem: Let \(\Lambda: S(\mathbb{R})\times S(\mathbb{R})\times S(\mathbb{R})\to \mathbb{C}\) be the trilinear form defined by \[ \Lambda(f_1, f_2, f_3):= \int \Biggl[\text{p.v. }\int f_1(x- t)f_2(x+ t) {1\over t} dt\Biggr] f_3(x)dx. \] Then for all \(2< p_1,p_2,p_3< \infty\) satisfying \({1\over p_1}+{1\over p_2}+ {1\over p_3}= 1\), there is a constant \(C\) such that \(\forall f_1,f_2,f_3\in S(\mathbb{R})\) \[ |\Lambda(f_1, f_2, f_3)|\leq C\| f_1\|_{p_1}\| f_2\|_{p_2}\| f_3\|_{p_3}. \] {}.

MSC:

46F12 Integral transforms in distribution spaces
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42A50 Conjugate functions, conjugate series, singular integrals

Citations:

Zbl 0893.30036
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