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Journal of the London Mathematical Society Advance Access originally published online on February 7, 2008
Journal of the London Mathematical Society 2008 77(2):349-362; doi:10.1112/jlms/jdm111
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© 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@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@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; published online February 7, 2008.


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