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This is a well-known problem in multilinear harmonic analysis; it is fascinating to me because it lies barely beyond the reach of the best technology we have for these problems (namely, multiscale time-frequency analysis), and because the most recent developments in quadratic Fourier analysis seem likely to shed some light on this problem.

Recall that the Hilbert transform is defined on test functions $f \in {\mathcal S}({\Bbb R})$ (up to irrelevant constants) as

$Hf(x) := p.v. \int_{\Bbb R} f(x+t) \frac{dt}{t},$

where the integral is evaluated in the principal value sense (removing the region $|t| < \epsilon$ to ensure integrability, and then taking the limit as $\epsilon \to 0$.)

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