A new way of looking at distributional estimates; applications for the bilinear Hilbert transform

• Autores: Loukas Grafakos , Dmitriy Bilyk
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. Extra 1, 2006 (Ejemplar dedicado a: Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations), págs. 141-169
• Idioma: español
• Enlaces
  • Texto completo
• Resumen

  • Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt [12]. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially integrable over compact sets. They also provide restricted type endpoint results on products of Lebesgue spaces where one exponent is 1 or the sum of the reciprocal of the exponents is 3/2. The proof of the distributional estimates for the bilinear Hilbert transform rely on an improved energy estimate for characteristic functions with respect to sets of tiles from which appropriate exceptional subsets have been removed.

• Referencias bibliográficas
  • N.Y. Antonov, Convergence of Fourier series, East J. Approx. 2 (1996), 187–196.
  • D. Bilyk and L. Grafakos, Interplay between distributional estimates and boundedness in harmonic analysis, Bull. London Math. Soc. 37 (2005),...
  • D. Bilyk and L. Grafakos, Distributional estimates for the bilinear Hilbert transforms, (preprint).
  • L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.
  • D. Fan and S. Sato, Transference on certain multilinear multiplier operators, J. Aust. Math. Soc. 70 (2001), 37–55.
  • C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. (2) 98 (1973), 551–571.
  • L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, Pearson Education, Upper Saddle River NJ, 2003.
  • L. Grafakos and P. Honzík, Maximal transference and summability of multilinear Fourier series, J. Aust. Math. Soc., to appear.
  • L. Grafakos and C. Lennard, Characterization of $L^p(\mathbb{R}^n)$ using Gabor frames, J. Fourier Anal. Appl. 7 (2001), 101–126.
  • L. Grafakos and T. Tao, Multilinear interpolation between adjoint operators, J. Funct. Anal. 199 (2003), 379–385.
  • L. Grafakos, T. Tao, and E. Terwilleger, $L^p$ bounds for a maximal dyadic sum operator, Math. Z. 246 (2004), 321–337.
  • R.A. Hunt, On the convergence of Fourier Series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967),...
  • M.T. Lacey, On the bilinear Hilbert transform, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math....
  • M.T. Lacey and C. Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2 < p < \infty$, Ann. of Math. (2) 146 (1997), 693–724.
  • M.T. Lacey and C. Thiele, On Calderón's conjecture, Ann. of Math. (2) 149 (1999), 475–496.
  • M.T. Lacey and C. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7 (2000), 361–370.
  • C. Muscalu, T. Tao, and C. Thiele, Multi-linear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), 469–496.
  • P. Sjölin and F. Soria, Remarks on a theorem by N. Yu. Antonov, Studia Math. 158 (2003), 79–97.