Rearrangement inequalities of the one-dimensional maximal functions associated with general measures

Abstract

We prove a rearrangement inequality for the uncentered Hardy-Littlewood maximal function $M_{\mu }$ associate to general measure $\mu$ on $\mathbb{R}$. This inequality is analogous to the Stein's result $c{f}^{\ast \ast }(t)\le (Mf{)}^{\ast }(t)\le C{f}^{\ast \ast }(t)$, where $f^{\ast }$ is the symmetric decreasing rearrangement function of $f$ and $f^{\ast \ast }(t)={\int }_0^t{f}^{\ast }(x)dx$. Moreover, we compute the best constant of $M_{\mu }$ on $L^{p,\infty }(\mathbb{R},d\mu )$.