A sharp estimate for the Hardy-Littlewood maximal function

Loukas Grafakos; Stephen Montgomery-Smith; Olexei Motrunich

Studia Mathematica (1999)

  • Volume: 134, Issue: 1, page 57-67
  • ISSN: 0039-3223

Abstract

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The best constant in the usual L p norm inequality for the centered Hardy-Littlewood maximal function on ℝ 1 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.

How to cite

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Grafakos, Loukas, Montgomery-Smith, Stephen, and Motrunich, Olexei. "A sharp estimate for the Hardy-Littlewood maximal function." Studia Mathematica 134.1 (1999): 57-67. <http://eudml.org/doc/216622>.

@article{Grafakos1999,
abstract = {The best constant in the usual $L^p$ norm inequality for the centered Hardy-Littlewood maximal function on $ℝ^1$ 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.},
author = {Grafakos, Loukas, Montgomery-Smith, Stephen, Motrunich, Olexei},
journal = {Studia Mathematica},
keywords = {peak-shaped function; Hardy-Littlewood maximal function; best constant},
language = {eng},
number = {1},
pages = {57-67},
title = {A sharp estimate for the Hardy-Littlewood maximal function},
url = {http://eudml.org/doc/216622},
volume = {134},
year = {1999},
}

TY - JOUR
AU - Grafakos, Loukas
AU - Montgomery-Smith, Stephen
AU - Motrunich, Olexei
TI - A sharp estimate for the Hardy-Littlewood maximal function
JO - Studia Mathematica
PY - 1999
VL - 134
IS - 1
SP - 57
EP - 67
AB - The best constant in the usual $L^p$ norm inequality for the centered Hardy-Littlewood maximal function on $ℝ^1$ 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.
LA - eng
KW - peak-shaped function; Hardy-Littlewood maximal function; best constant
UR - http://eudml.org/doc/216622
ER -

References

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  1. [Al] J. M. Aldaz, Remarks on the Hardy-Littlewood maximal function, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1-9. 
  2. [Ba] J. Barrionuevo, personal comunication. 
  3. [Br] U. Brechtken-Manderscheid, Introduction to the Calculus of Variations, Chapman & Hall, London, 1991. 
  4. [CG] M. Christ and L. Grafakos, Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc. 123 (1995), 1687-1693. Zbl0830.42009
  5. [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. Zbl1055.42502
  6. [GM] L. Grafakos and S. Montgomery-Smith, Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1997), 60-64. Zbl0865.42020

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