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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform
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by Vjekoslav Kovač
Proc. Amer. Math. Soc. 140 (2012), 915-926
DOI: https://doi.org/10.1090/S0002-9939-2011-11078-5
Published electronically: July 13, 2011
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Abstract:

Analogues of Hausdorff-Young inequalities for the Dirac scattering transform (a.k.a. the $\mathrm {SU}(1,1)$ nonlinear Fourier transform) were first established by Christ and Kiselev. Later Muscalu, Tao, and Thiele raised a question whether the constants can be chosen uniformly in $1\leq p\leq 2$. Here we give a positive answer to that question when the Euclidean real line is replaced by its Cantor group model.
References
  • Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI 10.1006/jfan.2000.3687
  • Michael Christ and Alexander Kiselev, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal. 179 (2001), no. 2, 426–447. MR 1809117, DOI 10.1006/jfan.2000.3688
  • Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
  • D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686
  • Camil Muscalu, Terence Tao, and Christoph Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity 16 (2003), no. 1, 219–246. MR 1950785, DOI 10.1088/0951-7715/16/1/314
  • F. L. Nazarov and S. R. Treĭl′, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), no. 5, 32–162 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 5, 721–824. MR 1428988
  • R. Oberlin, A. Seeger, T. Tao, C. Thiele, and J. Wright, A variation norm Carleson theorem, to appear in J. Eur. Math. Soc.
  • T. Tao and C. Thiele, Nonlinear Fourier analysis, IAS/Park City Math. Institute, Graduate Summer School, 2003, Lecture Notes.
  • S. Verblunsky, On positive harmonic functions, Proc. London Math. Soc., 34 (1934), 125–157.
  • S. Verblunsky, On positive harmonic functions II. Proc. London Math. Soc., 40 (1935), 290–320.
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Bibliographic Information
  • Vjekoslav Kovač
  • Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555
  • MR Author ID: 962691
  • Email: vjekovac@math.ucla.edu
  • Received by editor(s): December 14, 2010
  • Published electronically: July 13, 2011
  • Communicated by: Michael T. Lacey
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 915-926
  • MSC (2010): Primary 34L25; Secondary 42A38
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11078-5
  • MathSciNet review: 2869075