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

Abstract

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 HardyLittlewood 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.

On high dimensional maximal operators

“…It was shown there that the best constants c 1,d satisfy c 1,d ≥ Θ (1) 2/ √ 3 d/6 , in strong contrast with the linear O(d) upper bounds known for Lebesgue measure. Exponential increase was also shown for the same measures and small values of p > 1 in [Cri]; shortly after (and independently) these results were improved in [AlPe5], as they applied to larger exponents p and to a wider class of measures. It was also shown in [AlPe5] that exponential increase could occur for arbitrarily large values of p and suitably chosen doubling measures.…”

“…Exponential increase was also shown for the same measures and small values of p > 1 in [Cri]; shortly after (and independently) these results were improved in [AlPe5], as they applied to larger exponents p and to a wider class of measures. It was also shown in [AlPe5] that exponential increase could occur for arbitrarily large values of p and suitably chosen doubling measures. Together with the results for hyperbolic spaces mentioned before, this shows that the doubling condition is neither necessary nor sufficient to have "good bounds" for maximal inequalities in terms of the dimension.…”

“…In fact, by the 1-homogeneity of the operator, we can just take χ B(0,r) , since constants cancel out. We note that the proofs of exponential growth of the weak and strong type constants in the papers [A1], [AlPe5], [Cri], [CriSjo], all use this method of considering δ 0 or χ B(0,r) , and then estimating how shifting balls away from the origin reduces their measure (the differences between these papers lie in the values of r > 0 selected, the shifted balls chosen, and how their sizes are controlled).…”

The best constant for the centered maximal operator on radial decreasing functions

“…[A4]) rather than being uniformly bounded by 1. In fact, exponential growth can be shown to hold for some (sufficiently small) values of p > 1, simply by using, instead of δ 0 , the characteristic function of a small ball centered at 0, and then arguing as in [A4]; a step in this direction is carried, for weak type (p, p) inequalities, in [AlPe3].…”

Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

“…The constants appearing in the weak and strong type inequalities satisfied by the Hardy-Littlewood maximal operator, in its different variants, have been subject to considerable scrutiny. We mention, for instance, [CF], [Al1], [Me1], [Me2], [GMM], [GM], [GK], [BD], [CLM], [St1], [St2], [St3], [Bou1], [Bou2], [Bou3], [Ca], [Mu], [StSt], [Al2], [Al3], [AlPe4], [AlPe5], [NaTa], and the references contained therein. Interest lies not only in determining sharp inequalities, which in general are hard to come by (in fact, no best constants are known for dimensions larger than one) but also in finding out how constants change as certain parameters (for instance, the dimension) vary, or when the type of set one is averaging over is modified, or the space of functions one is considering is changed.…”