The best constant in the usual norm inequality for the centered Hardy-Littlewood maximal function on is obtained for the class of all “peak-shaped” functions. A function on the line is called peak-shaped if it is positive and convex except at one point. The techniques we use include variational methods.
@article{bwmeta1.element.bwnjournal-article-smv134i1p57bwm,
author = {Loukas Grafakos and Stephen Montgomery-Smith and Olexei Motrunich},
title = {A sharp estimate for the Hardy-Littlewood maximal function},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {57-67},
zbl = {0933.42010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i1p57bwm}
}
Grafakos, Loukas; Montgomery-Smith, Stephen; Motrunich, Olexei. A sharp estimate for the Hardy-Littlewood maximal function. Studia Mathematica, Tome 133 (1999) pp. 57-67. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i1p57bwm/
[00000] [Al] J. M. Aldaz, Remarks on the Hardy-Littlewood maximal function, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1-9.
[00001] [Ba] J. Barrionuevo, personal comunication.
[00002] [Br] U. Brechtken-Manderscheid, Introduction to the Calculus of Variations, Chapman & Hall, London, 1991.
[00003] [CG] M. Christ and L. Grafakos, Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc. 123 (1995), 1687-1693. | Zbl 0830.42009
[00004] [DGS] R. Dror, S. Ganguli and R. Strichartz, A search for best constants in the Hardy-Littlewood maximal theorem, J. Fourier Anal. Appl. 2 (1996), 473-486. | Zbl 1055.42502
[00005] [GM] L. Grafakos and S. Montgomery-Smith, Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1997), 60-64. | Zbl 0865.42020