Rearrangement Inequalities of the One-Dimensional Maximal Functions Associated with General Measures

We prove a rearrangement inequality for the uncentered Hardy–Littlewood maximal function M_μ associate to general measure μ on ℝ . This inequality is analogous to the Stein’s result cf^**(t)≤ (Mf)^*(t)≤ C f^**(t) , where f^* is the symmetric decreasing rearrangement function of f and f^**(t)=∫ _0^tf^*(x)dx . Moreover, we compute the best constant of M_μ on L^p,∞(ℝ,dμ ) .