Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function

  1. Aldaz, J.M. 1
  2. Colzani, L. 3
  3. Pérez Lázaro, J. 2
  1. 1 Universidad Autónoma de Madrid
    info

    Universidad Autónoma de Madrid

    Madrid, España

    ROR https://ror.org/01cby8j38

  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  3. 3 University of Milano-Bicocca
    info

    University of Milano-Bicocca

    Milán, Italia

    ROR https://ror.org/01ynf4891

Revista:
Journal of Geometric Analysis

ISSN: 1050-6926

Año de publicación: 2012

Volumen: 22

Número: 1

Páginas: 132-167

Tipo: Artículo

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DOI: 10.1007/S12220-010-9190-8 SCOPUS: 2-s2.0-84855425707 WoS: WOS:000298402200008 arXiv: 1009.1359v1 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Journal of Geometric Analysis

Resumen

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals of ℝ are Lip α(Mf)≤(1+α) -1Lip α(f), α∈(0,1]. On ℝ, the best bound for Lipschitz functions is Lip(Mf)≤(√2-1)Lip(f). In higher dimensions, we determine the asymptotic behavior, as d→∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, ℓ p balls for p=1,2,∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Hölder functions, the operator norm of the maximal operator is uniformly bounded by 2 -α/q, where q is the conjugate exponent of p=1,2, and as d→∞ the norms approach this bound. When p=∞, best constants are the same as when p=1. © 2010 Mathematica Josephina, Inc.