Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Weighted $ L^p$ boundedness of Carleson type maximal operators


Authors: Yong Ding and Honghai Liu
Journal: Proc. Amer. Math. Soc. 140 (2012), 2739-2751
MSC (2010): Primary 42B20, 42B25; Secondary 42B99
DOI: https://doi.org/10.1090/S0002-9939-2011-11110-9
Published electronically: December 8, 2011
MathSciNet review: 2910762
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 2001, E. M. Stein and S. Wainger gave the $ L^p$ boundedness of the Carleson type maximal operator $ \mathcal {T}^\ast $, which is defined by

$\displaystyle \mathcal {T}^\ast f(x)=\sup _\lambda \bigg \vert\int _{{\mathbb{R}}^n}e^{iP_\lambda (y)}K(y)f(x-y)dy\bigg \vert.$

In this paper, the authors show that if $ K$ is a homogeneous kernel, i.e. $ K(y)=\Omega (y')\vert y\vert^{-n}$, then Stein-Wainger's result still holds on the weighted $ L^p$ spaces when $ \Omega $ satisfies only an $ L^q$-Dini condition for some $ 1<q\le \infty $.

References [Enhancements On Off] (What's this?)

  • 1. A. P. Calderón, M. Weiss and A. Zygmund, On the existence of singular integrals, Proc. Symp. Pure Math., 10, AMS (1967), 56-73. MR 0338709 (49:3473)
  • 2. L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 111 (1966), 361-370. MR 0199631 (33:7774)
  • 3. L. Colzani, Hardy spaces on spheres, Ph.D. Thesis, Washington University, St. Louis, 1982.
  • 4. Y. Ding and H. Liu, $ L^p$ boundedness of Carleson type maximal operators with nonsmooth kernels, Tohoku Math. J. (2), 63 (2011), 255-267. MR 2812453
  • 5. J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869-880. MR 1089418 (93f:42030)
  • 6. J. Garcfa-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Mathematics Studies, 116, North-Holland, 1985. MR 807149 (87d:42023)
  • 7. R. A. Hunt, On the convergence of Fourier series, Orthogonal expansions and their continuous analogues, Proceedings of the Conference held at Southern Illinois University, Edwardsville, 1967. Southern Illinois Univ. Press, Carbondale, IL. MR 0238019 (38:6296)
  • 8. R. A. Hunt and W. Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274-277. MR 0338655 (49:3419)
  • 9. D. Kurtz and R. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343-362. MR 542885 (81j:42021)
  • 10. E. Prestini and P. Sjölin, A Littlewood-Paley inequality for the Carleson operator, J. Fourier Anal. Appl. 6 (2000), 457-466. MR 1781088 (2001m:42017)
  • 11. P. Sjölin, Convergence almost everywhere of certain singular integral and multiple Fourier series, Ark. Mat. 9 (1971), 65-90. MR 0336222 (49:998)
  • 12. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • 13. E. M. Stein and S. Wainger. Oscillatory integrals related to Carleson's theorem, Math. Res. Lett. 8 (2001), 789-800. MR 1879821 (2002k:42038)
  • 14. E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159-172. MR 0092943 (19:1184d)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B20, 42B25, 42B99

Retrieve articles in all journals with MSC (2010): 42B20, 42B25, 42B99


Additional Information

Yong Ding
Affiliation: School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Beijing Normal University, Ministry of Education of China, Beijing 100875, People’s Republic of China
Email: dingy@bnu.edu.cn

Honghai Liu
Affiliation: School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, People’s Republic of China
Email: hhliu@hpu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2011-11110-9
Keywords: Carleson operator, homogeneous kernel, $L^{q}$-Dini condition, $A_{p}$ weight
Received by editor(s): May 29, 2010
Received by editor(s) in revised form: March 6, 2011
Published electronically: December 8, 2011
Additional Notes: The first author was supported by the NSF of China (Grant 10931001), SRFDP of China (Grant 20090003110018) and Program for Changjiang Scholars and Innovative Research Team in University.
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society