Limiting Weak-Type Behavior of the Centered Hardy–Littlewood Maximal Function of General Measures on the Positive Real Line.

Given a positive Borel measure μ on the one-dimensional Euclidean space R , consider the centered Hardy–Littlewood maximal function M μ acting on a finite positive Borel measure ν by M μ ν (x) : = sup r > r 0 (x) ν (B (x , r)) μ (B (x , r)) , x ∈ R , <graphic href="11785_2024_1533_Article_Equ9.gif"></graphic> where r 0 (x) = inf { r > 0 : μ (B (x , r)) > 0 } and B(x, r) denotes the closed ball with centre x and radius r > 0 . In this note, we restrict our attention to Radon measures μ on the positive real line [ 0 , + ∞) . We provide a complete characterization of measures having weak-type asymptotic properties for the centered maximal function. Although we don’t know whether this fact can be extended to measures on the entire real line R , we examine some criteria for the existence of the weak-type asymptotic properties for M μ on R . We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures. [ABSTRACT FROM AUTHOR]