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 note on the limiting weak-type behavior for maximal operators


Authors: Jiaxin Hu and Xueping Huang
Journal: Proc. Amer. Math. Soc. 136 (2008), 1599-1607
MSC (2000): Primary 42B25
DOI: https://doi.org/10.1090/S0002-9939-08-09313-1
Published electronically: January 3, 2008
MathSciNet review: 2373589
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the following open question raised by Janakiraman in (2006): for $ f\in L^1 (\mathbb{R}^n )\cap L^{\infty} (\mathbb{R}^n )$ and $ \lambda > 0$, what is the limiting behavior of

$\displaystyle \left[m\left(\{x \in \mathbb{R}^n :M(\vert f\vert^p )(x)>\lambda\}\right)\right]^{1/p} $

as $ p\to\infty$? In this note, we give a complete answer to this question.


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

  • 1. J. Heinonen, Lectures on analysis on metric spaces, Springer, 2001. MR 1800917 (2002c:30028)
  • 2. P. Janakiraman, Limiting weak-type behavior for singular integral and maximal operators, Trans. Amer. Math. Soc. 358 (2006), 1937-1952. MR 2197436 (2006m:42023)
  • 3. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 1993. MR 1232192 (95c:42002)
  • 4. E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970. MR 0290095 (44:7280)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B25

Retrieve articles in all journals with MSC (2000): 42B25


Additional Information

Jiaxin Hu
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: hujiaxin@mail.tsinghua.edu.cn

Xueping Huang
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: hxp@mails.thu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-08-09313-1
Received by editor(s): September 3, 2006
Published electronically: January 3, 2008
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society