Covering lemmas associated to maximal operators of weak type

Abstract

Building on the work of A. Córdoba and R. Fefferman, we provide a necessary and sufficient covering condition on a basis $\mathcal{B}$ on a separable metric space $X$ with a Borel measure $μ$ so that the associated \mbox{geometric} maximal operator $M_\mathcal{B}$ satisfies for all $α> 0$ the weak type estimate $$μ\left(\left\{x \in X : M_\mathcal{B}f(x) > α\right\}\right) \leq \int_{X}Φ\left( γ_\mathcal{B} \left|\frac{f}α\right|\right)\,dμ\;,$$ where $Φ: [0, \infty) \rightarrow [0, \infty)$ is an everywhere finite Young's function.