# Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures - Mathematics > Classical Analysis and ODEs

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures - Mathematics > Classical Analysis and ODEs - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: As shown in A1, the lowest constants appearing in the weak type 1,1inequalities satisfied by the centered Hardy-Littlewood maximal operatorassociated to certain finite radial measures, grow exponentially fast with thedimension. Here we extend this result to a wider class of radial measures andto some values of p>1. Furthermore, we improve the previously known bounds forp=1. Roughly speaking, whenever p\in 1, 1.03, if \mu is defined by a radial,radially decreasing density satisfying some mild growth conditions, then thebest constants c {p,d,\mu} in the weak type p,p inequalities satisfyc {p,d,\mu} \ge 1.005^d for all d sufficiently large. We also show thatexponential increase of the best constants occurs for certain families ofdoubling measures, and for arbitrarily high values of p.A1 Aldaz, J.M. Dimension dependency of the weak type 1,1 bounds formaximal functions associated to finite radial measures. Bull. Lond. Math. Soc.39 2007 203-208. Also available at the Mathematics ArXiv.

Autor: J. M. Aldaz, J. Pérez Lázaro

Fuente: https://arxiv.org/