Title: On the Bilinear Hilbert Transform

In joint work with C. Thiele, the author has shown that A. Calder\'on's bilinear Hilbert transform extends to a bounded operator on certain products of $L^p$ spaces. This article illustrates the method of proof by giving a complete proof of an inequality which is slightly weaker than the original conjecture of Calder\'on concerning this transform.

1991 Mathematics Subject Classification: 42A50

Keywords and Phrases: hilbert transform, time--frequency analysis

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