On the weak-type (1,1) of the uncentered Hardy–Littlewood maximal operator associated with certain measures on the plane

Abstract

Suppose μ is a positive measure on $\mathbb{R}^{2}$ given by μ=ν×λ, where ν and λ are Radon measures on $\mathcal{S}^{1}$ and $\mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}$, respectively, which do not vanish on any open interval. We prove that if for either ν or λ there exists a set of positive measure A in its domain for which the upper and lower s-densities, 0< s≤1, are positive and finite for every x∈A then the uncentered Hardy–Littlewood maximal operator M_μ is weak-type (1,1) if and only if ν is doubling and λ is doubling away from the origin. This generalizes results of Vargas concerning rotation-invariant measures on $\mathbb{R}^{n}$ when n=2.