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

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 \({p\in (1, 1.03]}\), 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.