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

Given a positive Borel measure \(\mu \) on the one-dimensional Euclidean space \(\textbf{R}\), consider the centered Hardy–Littlewood maximal function \(M_\mu \) acting on a finite positive Borel measure \(\nu \) by

$$\begin{aligned} M_{\mu }\nu (x):=\sup _{r>r_0(x)}\frac{\nu (B(x,r))}{\mu (B(x,r))},\quad \hbox { }\ x\in \textbf{R}, \end{aligned}$$

where \(r_0(x) = \inf \{r> 0: \mu (B(x,r)) > 0\}\) 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 \(\mu \) on the positive real line \([0,+\infty )\). 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 \(\textbf{R}\), we examine some criteria for the existence of the weak-type asymptotic properties for \(M_\mu \) on \(\textbf{R}\). We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures.