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

Given a positive Borel measure \(\mu \) $\mu$ on the one-dimensional Euclidean space \(\textbf{R}\) $\mathbf{R}$ , consider the centered Hardy–Littlewood maximal function \(M_\mu \) $M_{\mu }$ acting on a finite positive Borel measure \(\nu \) $\nu$ by \(\begin{aligned} M_{\mu }\nu (x):=\sup _{r>r_0(x)}\frac{\nu (B(x,r))}{\mu (B(x,r))},\quad \hbox { }\ x\in \textbf{R}, \end{aligned}\) $$\begin{array}{c}\hfill M_{\mu }\nu (x):=\underset{r>r_0(x)}{sup}\frac{\nu (B(x,r))}{\mu (B(x,r))},x\in \mathbf{R},\end{array}$$ where \(r_0(x) = \inf \{r> 0: \mu (B(x,r)) > 0\}\) $r_0(x)=inf\{r>0:\mu (B(x,r))>0\}$ and B(x, r) denotes the closed ball with centre x and radius \(r > 0\) $r>0$ . In this note, we restrict our attention to Radon measures \(\mu \) $\mu$ on the positive real line \([0,+\infty )\) $[0,+\infty )$ . 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 \(\textbf{R}\) $\mathbf{R}$ , we examine some criteria for the existence of the weak-type asymptotic properties for \(M_\mu \) $M_{\mu }$ on \(\textbf{R}\) $\mathbf{R}$ . We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures.