Kissing numbers and the centered maximal operator

We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$ 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. Thus, the following characterization is obtained: the centered maximal operator satisfies uniform weak type $(1,1)$ bounds if and only if the space $X$ has the Besicovitch intersection property. In $\mathbb{R}^d$ with any norm, the constants coming from the Besicovitch intersection property are bounded above by the translative kissing numbers. The extensive literature on kissing numbers allows us to obtain, first, sharp estimates on the uniform bounds satisfied by the centered maximal operators defined by arbitrary norms on the plane, second, sharp estimates in every dimension when the $\ell_\infty$ norm is used, and third, improved estimates in all dimensions when considering euclidean balls, as well as the sharp constant in dimension 3. Additionally, we prove that the existence of uniform $L^1$ bounds for the averaging operators associated to arbitrary measures and radii, is equivalent to a weaker variant of the Besicovitch intersection property.

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