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Symmetrization and norm of the Hardy–Littlewood maximal operator on Lorentz and Marcinkiewicz spaces
Author(s) -
Colzani Leonardo,
Laeng Enrico,
Morpurgo Carlo
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm111
Subject(s) - symmetrization , mathematics , pointwise , lorentz transformation , maximal function , hardy space , real line , operator (biology) , pure mathematics , norm (philosophy) , function (biology) , maximal operator , mathematical analysis , physics , quantum mechanics , biochemistry , chemistry , repressor , political science , transcription factor , law , gene , evolutionary biology , biology , bounded function
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 the symmetrization, then ℳf ( x )⩽ ℳf ( x ) for every x , and equality holds for all x if and only if, up to translations, f ( x ) = 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.