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Distributional estimates for the bilinear Hilbert transform

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Abstract

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 Lp bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on Lp1 × Lp2 when either 1/p1 + 1/p2 = 3/2 or one of p1, p2 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO

    Dmitriy Bilyk & Loukas Grafakos

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  1. Dmitriy Bilyk
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  2. Loukas Grafakos
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Correspondence to Dmitriy Bilyk.

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Bilyk, D., Grafakos, L. Distributional estimates for the bilinear Hilbert transform. J Geom Anal 16, 563–584 (2006). https://doi.org/10.1007/BF02922131

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  • DOI: https://doi.org/10.1007/BF02922131

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