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

TL;DR

This paper investigates how the constants in weak type inequalities for high-dimensional maximal operators grow exponentially with dimension for a broad class of radial measures, extending previous results and improving bounds.

Contribution

It extends exponential growth results of weak type bounds to more radial measures and higher p-values, and improves existing bounds for p=1.

Findings

Constants grow exponentially with dimension for certain radial measures.

Exponential growth occurs for doubling measures and high p-values.

Improved bounds for weak type (1,1) inequalities.

Abstract

As shown in [A1], 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\in (1, 1.03], if \mu is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c_{p,d,\mu} in the weak type (p,p) inequalities satisfy c_{p,d,\mu} \ge 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. [A1] Aldaz, J.M. Dimension dependency of the weak type (1,1) bounds for maximal functions associated to finite radial measures. Bull. Lond. Math. Soc. 39 (2007) 203-208. Also available at the Mathematics ArXiv.