Skip Navigation



Journal of the London Mathematical Society Advance Access published online on February 7, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm111
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
77/2/349    most recent
jdm111v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Colzani, L.
Right arrow Articles by Morpurgo, C.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2008 London Mathematical Society

Symmetrization and norm of the Hardy–Littlewood maximal operator on Lorentz and Marcinkiewicz spaces

Leonardo Colzani

Dipartimento di Matematica
Università di Milano – Bicocca
Edificio U5, via R.Cozzi 53
20125 Milano Italy
leonardo.colzani{at}unimib.it

Enrico Laeng

Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci 32
20133 Milano
Italy

Carlo Morpurgo

Department of Mathematics
University of Missouri–Columbia
202 Mathematical Sciences Building
Columbia, MO 65211
USA
morpurgo{at}math.missouri.edu

We prove that when a function on the real line is symmetrically rearranged, the distribution function of its uncentered Hardy–Littlewood maximal function increases pointwise, while it remains unchanged only when the function is already symmetric. Equivalently, if M is the maximal operator and S the symmetrization, then SMf(x)≤MSf(x) for every x, and equality holds for all x if and only if, up to translations, f(x)=S f(x) almost everywhere. Using these results, we then compute the exact norms of the maximal operator acting on Lorentz and Marcinkiewicz spaces, and we determine extremal functions that realize these norms.

2000 Mathematics Subject Classification 42B25.

Received February 15, 2006; revised January 4, 2007;
Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.