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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Endpoint bounds for the bilinear Hilbert transform

Authors: Francesco Di Plinio and Christoph Thiele
Journal: Trans. Amer. Math. Soc. 368 (2016), 3931-3972
MSC (2010): Primary 42B20
Published electronically: November 20, 2015
MathSciNet review: 3453362
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We study the behavior of the bilinear Hilbert transform $(\mathrm {BHT})$ at the boundary of the known boundedness region $\mathcal H$. A sample of our results is the estimate \begin{equation*} |\langle \mathrm {BHT}(f_1,f_2),f_3 \rangle | \leq \textstyle C |F_1|^{\frac 34}|F_2| ^{\frac 34} |F_3|^{-\frac 12} \log \log \Big (\mathrm {e}^{\mathrm {e}} +\textstyle \frac {|F_3|}{\min \{|F_1|,|F_2|\}} \Big ),\end{equation*} valid for all tuples of sets $F_j\subset \mathbb {R}$ of finite measure and functions $f_j$ such that $|f_j| \leq \boldsymbol {1}_{F_j}$, $j=1,2,3$, with the additional restriction that $f_3$ be supported on a major subset $F_3’$ of $F_3$ that depends on $\{F_j:j=1,2,3\}$. The use of subindicator functions in this fashion is standard in the given context. The double logarithmic term improves over the single logarithmic term obtained by D. Bilyk and L. Grafakos. Whether the double logarithmic term can be removed entirely, as is the case for the quartile operator, remains open.

We employ our endpoint results to describe the blow-up rate of weak-type and strong-type estimates for $\mathrm {BHT}$ as the tuple $\vec \alpha$ approaches the boundary of $\mathcal H$. We also discuss bounds on Lorentz-Orlicz spaces near $L^{\frac 23}$, improving on results of M. Carro et al. The main technical novelty in our article is an enhanced version of the multi-frequency Calderón-Zygmund decomposition.

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Additional Information

Francesco Di Plinio
Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912

Christoph Thiele
Affiliation: Hausdorff Institute for Mathematics, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany — and — Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555

Keywords: Bilinear Hilbert transform, multi-frequency Calderón-Zygmund decomposition, endpoint bounds
Received by editor(s): March 24, 2014
Published electronically: November 20, 2015
Additional Notes: The first author was partially supported by the National Science Foundation under the grant NSF-DMS-1206438
The second author was partially supported by the grant NSF-DMS-1001535
Article copyright: © Copyright 2015 American Mathematical Society