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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