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 $\mu$ so that the associated \mbox{geometric} maximal operator $M_{\mathcal{B}}$ satisfies for all $\alpha >0$ the weak type estimate $$\mu \left(\left\{x\in X:M_{\mathcal{B}}f(x)>\alpha \right\}\right)\le {\int }_X\Phi \left({\gamma }_{\mathcal{B}}\left|\frac{f}{\alpha }\right|\right)d\mu ,$$ where $\Phi :[0,\infty )\to [0,\infty )$ is an everywhere finite Young's function.