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)

 
 

 

A characterization of maximal operators associated with radial fourier multipliers


Author: Jongchon Kim
Journal: Proc. Amer. Math. Soc. 145 (2017), 1077-1085
MSC (2010): Primary 42B15, 42B25
DOI: https://doi.org/10.1090/proc/13445
Published electronically: November 18, 2016
MathSciNet review: 3589308
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a simple necessary and sufficient condition for maximal operators associated with radial Fourier multipliers to be bounded on $ L^p_{rad}$ and $ L^p$ for certain $ p$ greater than $ 2$. The range of exponents obtained for the $ L^p_{rad}$ characterization is optimal for the given condition. The $ L^p$ characterization is derived from an inequality of Heo, Nazarov, and Seeger regarding a characterization of radial Fourier multipliers.


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

  • [1] Anthony Carbery, José L. Rubio de Francia, and Luis Vega, Almost everywhere summability of Fourier integrals, J. London Math. Soc. (2) 38 (1988), no. 3, 513-524. MR 972135
  • [2] Leonardo Colzani, Giancarlo Travaglini, and Marco Vignati, Bochner-Riesz means of functions in weak-$ L^p$, Monatsh. Math. 115 (1993), no. 1-2, 35-45. MR 1223243, https://doi.org/10.1007/BF01311209
  • [3] Gustavo Garrigós and Andreas Seeger, Characterizations of Hankel multipliers, Math. Ann. 342 (2008), no. 1, 31-68. MR 2415314, https://doi.org/10.1007/s00208-008-0221-8
  • [4] Gustavo Garrigós and Andreas Seeger, A note on maximal operators associated with Hankel multipliers, Rev. Un. Mat. Argentina 50 (2009), no. 2, 137-148. MR 2656531
  • [5] Yaryong Heo, Fëdor Nazarov, and Andreas Seeger, On radial and conical Fourier multipliers, J. Geom. Anal. 21 (2011), no. 1, 96-117. MR 2755678, https://doi.org/10.1007/s12220-010-9171-y
  • [6] Yaryong Heo, Fëdor Nazarov, and Andreas Seeger, Radial Fourier multipliers in high dimensions, Acta Math. 206 (2011), no. 1, 55-92. MR 2784663, https://doi.org/10.1007/s11511-011-0059-x
  • [7] Yūichi Kanjin, Convergence almost everywhere of Bochner-Riesz means for radial functions, Ann. Sci. Kanazawa Univ. 25 (1988), 11-15. MR 964084
  • [8] Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger, Square functions and maximal operators associated with radial Fourier multipliers, Advances in analysis: the legacy of Elias M. Stein, Princeton Math. Ser., vol. 50, Princeton Univ. Press, Princeton, NJ, 2014, pp. 273-302. MR 3329855
  • [9] Sanghyuk Lee and Andreas Seeger, On radial Fourier multipliers and almost everywhere convergence, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 105-126. MR 3338611, https://doi.org/10.1112/jlms/jdu066
  • [10] Richard O'Neil, Convolution operators and $ L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129-142. MR 0146673
  • [11] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [12] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • [13] Terence Tao, The weak-type endpoint Bochner-Riesz conjecture and related topics, Indiana Univ. Math. J. 47 (1998), no. 3, 1097-1124. MR 1665753, https://doi.org/10.1512/iumj.1998.47.1544

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B15, 42B25

Retrieve articles in all journals with MSC (2010): 42B15, 42B25


Additional Information

Jongchon Kim
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: jkim@math.wisc.edu

DOI: https://doi.org/10.1090/proc/13445
Received by editor(s): November 17, 2014
Published electronically: November 18, 2016
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society