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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Endpoint bounds for the bilinear Hilbert transform
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by Francesco Di Plinio and Christoph Thiele PDF
Trans. Amer. Math. Soc. 368 (2016), 3931-3972 Request permission

Abstract:

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
  • Email: fradipli@math.brown.edu
  • 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
  • Email: thiele@math.uni-bonn.de
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 3931-3972
  • MSC (2010): Primary 42B20
  • DOI: https://doi.org/10.1090/tran/6548
  • MathSciNet review: 3453362