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 script S sign the symmetrization, then script S signℳf(x)≤ℳscript S signf(x) for every x, and equality holds for all x if and only if, up to translations, f(x) = script S sign 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. © 2008 London Mathematical Society.

Colzani, L., Laeng, E., Morpurgo, C. (2008). Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY, 77, 349-362 [10.1112/jlms/jdm111].

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

COLZANI, LEONARDO;
2008

Abstract

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 script S sign the symmetrization, then script S signℳf(x)≤ℳscript S signf(x) for every x, and equality holds for all x if and only if, up to translations, f(x) = script S sign 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. © 2008 London Mathematical Society.
Articolo in rivista - Articolo scientifico
Maximal function; symmetrization
English
349
362
Colzani, L., Laeng, E., Morpurgo, C. (2008). Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY, 77, 349-362 [10.1112/jlms/jdm111].
Colzani, L; Laeng, E; Morpurgo, C
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/8452
Citazioni
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 3
Social impact