Cancellation for the multilinear Hilbert transform

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• 作者：Terence Tao
• 关键词：Singular integrals ; Multilinear harmonic analysis ; Arithmetic regularity lemma ; Time ; frequency analysis
• 刊名：Collectanea Mathematica
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：67
• 期：2
• 页码：191-206
• 全文大小：491 KB
• 参考文献：1.Christ, M.: On certain elementary trilinear operators. Math. Res. Lett. 8(12), 43–56 (2001)MathSciNet CrossRef MATH
  2.Demeter, C.: Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform. Ergod. Theory Dyn. Syst. 28(5), 1453–1464 (2008)MathSciNet CrossRef MATH
  3.Green, B., Tao, T.: The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. Math. (2) 175(2), 465–540 (2012)
  4.Green, B., Tao, T.: An arithmetic regularity lemma, an associated counting lemma, and applications. An irregular mind, 261–334, Bolyai Soc. Math. Stud., 21, János Bolyai Math. Soc., Budapest (2010)
  5.Gowers, W.T.: A new proof of Szemerédi’s theorem for progressions of length four. GAFA 8(3), 529–551 (1998)MathSciNet MATH
  6.Gowers, W.T.: A new proof of Szemerédi’s theorem. GAFA 11, 465–588 (2001)MathSciNet MATH
  7.Kovač, V.: Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(3), 813–846 (2011)MathSciNet CrossRef MATH
  8.Lacey, M., Thiele, C.: \(L^p\) estimates on the bilinear Hilbert transform for \(2 < p < \infty \) . Ann. Math. (2) 146(3), 693–724 (1997)
  9.Lacey, M., Thiele, C.: On Calderón’s conjecture. Ann. Math. (2) 149(2), 475–496 (1999)
  10.Muscalu, C.: Some remarks on the \(n\) -linear Hilbert transform for \(n\ge 4\) (preprint). arXiv:​1209.​6391
  11.Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)
  12.Szemerédi, E.: On sets of integers containing no \(k\) elements in arithmetic progression, collection of articles in memory of Juriǐ. Vladimirovič Linnik. Acta Arith. 27, 199–245 (1975)MATH
  13.Tao, T., Vu, V.: Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)
  14.Zorin-Kranich, P.: Cancellation for the simplex Hilbert transform (preprint). arXiv:​1505.​06479
  
• 作者单位：Terence Tao (1)
  
  1. UCLA Department of Mathematics, Los Angeles, CA, 90095-1555, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  Analysis
  Applications of Mathematics
  Geometry
  
• 出版者：Springer Milan
• ISSN：2038-4815

文摘

For any natural number k, consider the k-linear Hilbert transform $$\begin{aligned} H_k( f_1,\dots ,f_k )(x) := {\text {p.v.}} \int _\mathbb {R}f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} \end{aligned}$$for test functions \(f_1,\dots ,f_k: \mathbb {R}\rightarrow \mathbb {C}\). It is conjectured that \(H_k\) maps \(L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})\) whenever \(1 < p_1,\dots ,p_k,p < \infty \) and \(\frac{1}{p} = \frac{1}{p_1} + \dots + \frac{1}{p_k}\). This is proven for \(k=1,2\), but remains open for larger k. In this paper, we consider the truncated operators $$\begin{aligned} H_{k,r,R}( f_1,\dots ,f_k )(x) := \int _{r \leqslant |t| \leqslant R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} \end{aligned}$$for \(R > r > 0\). The above conjecture is equivalent to the uniform boundedness of \(\Vert H_{k,r,R} \Vert _{L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})}\) in r, R, whereas the Minkowski and Hölder inequalities give the trivial upper bound of \(2 \log \frac{R}{r}\) for this quantity. By using the arithmetic regularity and counting lemmas of Green and the author, we improve the trivial upper bound on \(\Vert H_{k,r,R} \Vert _{L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})}\) slightly to \(o( \log \frac{R}{r} )\) in the limit \(\frac{R}{r} \rightarrow \infty \) for any admissible choice of k and \(p_1,\dots ,p_k,p\). This establishes some cancellation in the k-linear Hilbert transform \(H_k\), but not enough to establish its boundedness in \(L^p\) spaces. Keywords Singular integrals Multilinear harmonic analysis Arithmetic regularity lemma Time-frequency analysis Mathematics Subject Classification 11B30 42B20 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (14) References1.Christ, M.: On certain elementary trilinear operators. Math. Res. Lett. 8(12), 43–56 (2001)MathSciNetCrossRefMATH2.Demeter, C.: Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform. Ergod. Theory Dyn. Syst. 28(5), 1453–1464 (2008)MathSciNetCrossRefMATH3.Green, B., Tao, T.: The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. Math. (2) 175(2), 465–540 (2012)4.Green, B., Tao, T.: An arithmetic regularity lemma, an associated counting lemma, and applications. An irregular mind, 261–334, Bolyai Soc. Math. Stud., 21, János Bolyai Math. Soc., Budapest (2010)5.Gowers, W.T.: A new proof of Szemerédi’s theorem for progressions of length four. GAFA 8(3), 529–551 (1998)MathSciNetMATH6.Gowers, W.T.: A new proof of Szemerédi’s theorem. GAFA 11, 465–588 (2001)MathSciNetMATH7.Kovač, V.: Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(3), 813–846 (2011)MathSciNetCrossRefMATH8.Lacey, M., Thiele, C.: \(L^p\) estimates on the bilinear Hilbert transform for \(2 < p < \infty \). Ann. Math. (2) 146(3), 693–724 (1997)9.Lacey, M., Thiele, C.: On Calderón’s conjecture. Ann. Math. (2) 149(2), 475–496 (1999)10.Muscalu, C.: Some remarks on the \(n\) -linear Hilbert transform for \(n\ge 4\) (preprint). arXiv:​1209.​6391 11.Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)12.Szemerédi, E.: On sets of integers containing no \(k\) elements in arithmetic progression, collection of articles in memory of Juriǐ. Vladimirovič Linnik. Acta Arith. 27, 199–245 (1975)MATH13.Tao, T., Vu, V.: Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)14.Zorin-Kranich, P.: Cancellation for the simplex Hilbert transform (preprint). arXiv:​1505.​06479 About this Article Title Cancellation for the multilinear Hilbert transform Journal Collectanea Mathematica Volume 67, Issue 2 , pp 191-206 Cover Date2016-05 DOI 10.1007/s13348-015-0162-y Print ISSN 0010-0757 Online ISSN 2038-4815 Publisher Springer Milan Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Applications of Mathematics Algebra Geometry Analysis Keywords Singular integrals Multilinear harmonic analysis Arithmetic regularity lemma Time-frequency analysis 11B30 42B20 Authors Terence Tao (1) Author Affiliations 1. UCLA Department of Mathematics, Los Angeles, CA, 90095-1555, USA Continue reading... To view the rest of this content please follow the download PDF link above.