Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

  1. Aldaz, J.M. 1
  2. Lazaro, J.P. 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

Revista:
Positivity

ISSN: 1385-1292

Año de publicación: 2010

Volumen: 15

Número: 2

Páginas: 199-213

Tipo: Artículo

DOI: 10.1007/S11117-010-0051-2 SCOPUS: 2-s2.0-79957908859 WoS: WOS:000291060100003 GOOGLE SCHOLAR

Otras publicaciones en: Positivity

Resumen

As shown in Aldaz (Bull. Lond. Math. Soc. 39:203-208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p > 1. Furthermore, we improve the previously known bounds for p = 1. Roughly speaking, whenever p ∈(1,1.03], if μ is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c p,d,μ in the weak type (p, p) inequalities satisfy c p,d,μ ≥ 1.005 d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p. © 2010 Birkhäuser / Springer Basel AG.