We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power -2/3 (modulo some logarithmic factors). These results yield all known L p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L p1 × L p2 when either 1/p 1 + 1/p 2 = 3/2 or one of p 1, p 2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.
Bibliographical noteFunding Information:
Math Subject Classifications. 46B70, 42B99. Key Words and Phrases. Multilinear operators, distributional estimates. Acknowledgements andNotes. Both authors have been supported by the National Science Foundation under grant DMS 0400387 and by the University of Missouri Research Council.
- Multilinear operators
- distributional estimates