Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function
- Aldaz, J.M. 1
- Colzani, L. 3
- Pérez Lázaro, J. 2
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1
Universidad Autónoma de Madrid
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2
Universidad de La Rioja
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3
University of Milano-Bicocca
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ISSN: 1050-6926
Año de publicación: 2012
Volumen: 22
Número: 1
Páginas: 132-167
Tipo: Artículo
beta Ver similares en nube de resultadosOtras publicaciones en: Journal of Geometric Analysis
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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 â„ are Lip α(Mf)≤(1+α) -1Lip α(f), α∈(0,1]. On â„, 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, â„“ 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.