## Endpoint bounds for the bilinear Hilbert transform

HTML articles powered by AMS MathViewer

- 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.

## References

- Dmitriy Bilyk and Loukas Grafakos,
*Distributional estimates for the bilinear Hilbert transform*, J. Geom. Anal.**16**(2006), no. 4, 563–584. MR**2271944**, DOI 10.1007/BF02922131 - Dmitriy Bilyk and Loukas Grafakos,
*A new way of looking at distributional estimates; applications for the bilinear Hilbert transform*, Collect. Math.**Vol. Extra**(2006), 141–169. MR**2264208** - Peter Borwein and Tamás Erdélyi,
*Nikolskii-type inequalities for shift invariant function spaces*, Proc. Amer. Math. Soc.**134**(2006), no. 11, 3243–3246. MR**2231907**, DOI 10.1090/S0002-9939-06-08533-9 - María Carro, Leonardo Colzani, and Gord Sinnamon,
*From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces*, Studia Math.**182**(2007), no. 1, 1–27. MR**2326489**, DOI 10.4064/sm182-1-1 - María Jesús Carro, Loukas Grafakos, José María Martell, and Fernando Soria,
*Multilinear extrapolation and applications to the bilinear Hilbert transform*, J. Math. Anal. Appl.**357**(2009), no. 2, 479–497. MR**2557660**, DOI 10.1016/j.jmaa.2009.04.021 - R. R. Coifman and Yves Meyer,
*On commutators of singular integrals and bilinear singular integrals*, Trans. Amer. Math. Soc.**212**(1975), 315–331. MR**380244**, DOI 10.1090/S0002-9947-1975-0380244-8 - Ciprian Demeter and Francesco Di Plinio,
*Endpoint bounds for the quartile operator*, J. Fourier Anal. Appl.**19**(2013), no. 4, 836–856. MR**3089425**, DOI 10.1007/s00041-013-9275-4 - Francesco Di Plinio,
*Lacunary Fourier and Walsh-Fourier series near $L^1$*, Collect. Math.**65**(2014), no. 2, 219–232. MR**3189278**, DOI 10.1007/s13348-013-0094-3 - Francesco Di Plinio,
*Weak-$L^p$ bounds for the Carleson and Walsh-Carleson operators*, C. R. Math. Acad. Sci. Paris**352**(2014), no. 4, 327–331 (English, with English and French summaries). MR**3186922**, DOI 10.1016/j.crma.2014.02.005 - Yen Q. Do and Michael T. Lacey,
*On the convergence of lacunacy Walsh-Fourier series*, Bull. Lond. Math. Soc.**44**(2012), no. 2, 241–254. MR**2914604**, DOI 10.1112/blms/bdr088 - Yen Do and Christoph Thiele,
*$L^p$ theory for outer measures and two themes of Lennart Carleson united*, Bull. Amer. Math. Soc. (N.S.)**52**(2015), no. 2, 249–296. MR**3312633**, DOI 10.1090/S0273-0979-2014-01474-0 - Loukas Grafakos and Xiaochun Li,
*Uniform bounds for the bilinear Hilbert transforms. I*, Ann. of Math. (2)**159**(2004), no. 3, 889–933. MR**2113017**, DOI 10.4007/annals.2004.159.889 - Loukas Grafakos and Rodolfo H. Torres,
*Multilinear Calderón-Zygmund theory*, Adv. Math.**165**(2002), no. 1, 124–164. MR**1880324**, DOI 10.1006/aima.2001.2028 - Lars Hörmander,
*The analysis of linear partial differential operators. I*, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR**1996773**, DOI 10.1007/978-3-642-61497-2 - A. E. Ingham,
*A Note on Fourier Transforms*, J. London Math. Soc.**9**(1934), no. 1, 29–32. MR**1574706**, DOI 10.1112/jlms/s1-9.1.29 - Carlos E. Kenig and Elias M. Stein,
*Multilinear estimates and fractional integration*, Math. Res. Lett.**6**(1999), no. 1, 1–15. MR**1682725**, DOI 10.4310/MRL.1999.v6.n1.a1 - Michael Lacey and Christoph Thiele,
*$L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$*, Ann. of Math. (2)**146**(1997), no. 3, 693–724. MR**1491450**, DOI 10.2307/2952458 - Michael Lacey and Christoph Thiele,
*On Calderón’s conjecture*, Ann. of Math. (2)**149**(1999), no. 2, 475–496. MR**1689336**, DOI 10.2307/120971 - Michael T. Lacey,
*The bilinear maximal functions map into $L^p$ for $2/3<p\leq 1$*, Ann. of Math. (2)**151**(2000), no. 1, 35–57. MR**1745019**, DOI 10.2307/121111 - Xiaochun Li,
*Uniform bounds for the bilinear Hilbert transforms. II*, Rev. Mat. Iberoam.**22**(2006), no. 3, 1069–1126. MR**2320411**, DOI 10.4171/RMI/483 - Yi Yu Liang, Li Guang Liu, and Da Chun Yang,
*An off-diagonal Marcinkiewicz interpolation theorem on Lorentz spaces*, Acta Math. Sin. (Engl. Ser.)**27**(2011), no. 8, 1477–1488. MR**2822822**, DOI 10.1007/s10114-011-0287-1 - Victor Lie,
*On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences*, J. Funct. Anal.**263**(2012), no. 11, 3391–3411. MR**2984070**, DOI 10.1016/j.jfa.2012.08.013 - Victor Lie,
*On the boundedness of the Carleson operator near $L^1$*, Rev. Mat. Iberoam.**29**(2013), no. 4, 1239–1262. MR**3148602**, DOI 10.4171/RMI/755 - Camil Muscalu, Jill Pipher, Terence Tao, and Christoph Thiele,
*Multi-parameter paraproducts*, Rev. Mat. Iberoam.**22**(2006), no. 3, 963–976. MR**2320408**, DOI 10.4171/RMI/480 - Camil Muscalu, Terence Tao, and Christoph Thiele,
*Multi-linear operators given by singular multipliers*, J. Amer. Math. Soc.**15**(2002), no. 2, 469–496. MR**1887641**, DOI 10.1090/S0894-0347-01-00379-4 - Fedor Nazarov, Richard Oberlin, and Christoph Thiele,
*A Calderón-Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain*, Math. Res. Lett.**17**(2010), no. 3, 529–545. MR**2653686**, DOI 10.4310/MRL.2010.v17.n3.a11 - Richard Oberlin and Christoph Thiele,
*New uniform bounds for a Walsh model of the bilinear Hilbert transform*, Indiana Univ. Math. J.**60**(2011), no. 5, 1693–1712. MR**2997005**, DOI 10.1512/iumj.2011.60.4445 - Christoph Thiele,
*A uniform estimate*, Ann. of Math. (2)**156**(2002), no. 2, 519–563. MR**1933076**, DOI 10.2307/3597197 - Christoph Thiele,
*Wave packet analysis*, CBMS Regional Conference Series in Mathematics, vol. 105, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. MR**2199086**, DOI 10.1090/cbms/105

## 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