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

Abstract

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.