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

Suppose μ is a positive measure on R² given by μ=ν×λ, where ν and λ are Radon measures on S¹ and R^ .17ex +.17ex +.1ex ++, 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 Mμ 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 Rⁿ when n=2.