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)

 
 

 

Continuity of the maximal operator in Sobolev spaces


Author: Hannes Luiro
Journal: Proc. Amer. Math. Soc. 135 (2007), 243-251
MSC (2000): Primary 42B25, 46E35, 47H99
DOI: https://doi.org/10.1090/S0002-9939-06-08455-3
Published electronically: June 30, 2006
MathSciNet review: 2280193
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces $ W^{1,p}(\mathbb{R}^n)\,$, $ 1<p<\infty\,$. As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.


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

  • [AL] F.J. Almgren and E.H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773. MR 1002633 (90f:49038)
  • [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983. MR 0737190 (86c:35035)
  • [HO] P. Haj\lasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29 (2004), 167-176. MR 2041705 (2005a:42010)
  • [K] J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev-function, Israel J. Math. 100 (1997), 117-124. MR 1469106 (99a:30029)
  • [KL] J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161-167. MR 1650343 (99j:42027)
  • [KS] J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529-535. MR 1979008 (2004e:42035)
  • [Ko] S. Korry, Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces, Rev. Mat. Complut. 15 (2002), 401-416. MR 1951818 (2004a:42020)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B25, 46E35, 47H99

Retrieve articles in all journals with MSC (2000): 42B25, 46E35, 47H99


Additional Information

Hannes Luiro
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 University of Jyväskylä, Finland
Email: haluiro@maths.jyu.fi

DOI: https://doi.org/10.1090/S0002-9939-06-08455-3
Keywords: Maximal function, Sobolev spaces, continuity, regularity
Received by editor(s): June 4, 2004
Received by editor(s) in revised form: August 8, 2005
Published electronically: June 30, 2006
Additional Notes: The author was supported by the Academy of Finland, project 201015
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society