Kissing numbers and the centered maximal operator

Abstract

We prove that in a metric measure space X, if for some p ∈ (1, ∞) there are uniform bounds (independent of the measure) for the weak type (p, p) of the centered maximal operator, then X satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type (1, 1) of the centered operator.In R d with any norm, the constants coming from the Besicovitch intersection property are bounded above by the translative kissing numbers. This leads to improved estimates on the uniform bounds satisfied by the centered maximal operator defined using euclidean balls, as well as the sharp constants in dimensions 2 and 3. For the centered maximal operator defined using ℓ ∞ -balls (cubes) we obtain the sharp uniform bounds 2 d .