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On the bilinear Hilbert transform. (English) Zbl 0963.42007
The author and C. Thiele in their highly acclaimed paper “On Calderón’s conjecture” [Ann. Math. (2) 149, No. 2, 475-496 (1999; Zbl 0934.42012)] have proved that the bilinear Hilbert transform $Hfg(x):= \lim_{\varepsilon \rightarrow 0}\int_{|y|>\varepsilon}{f(x+y)g(x-y) {\frac{dy}{y}}}$ extends to a bounded operator on $$L^p\times L^p$$ into $$L^r$$ if $$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$$, $$1<p,q\leq{\infty}$$ and $$\frac{2}{3}<r< \infty$$. This proves, in particular, a conjecture of Calderón from 1964 that $$H$$ maps $$L^2 \times L^2$$ into $$L^1$$. The current highly readable article illustrates the method of proof of the general result by giving a complete proof of the weaker result that the bilinear Hilbert transform maps $$L^2 \times L^2$$ into weak $$L^1$$.

##### MSC:
 42A50 Conjugate functions, conjugate series, singular integrals
Zbl 0934.42012
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