On an improved restricted reverse weak-type bound for the maximal operator

We obtain an improved lower bound for the restricted reverse weak-type estimate of the Hardy-Littlewood maximal operator $M$. This result is applied to the $\lambda$-median maximal operator $m_{\lambda }$ acting on a Banach function space $X$. We show that under certain assumptions on $X$, the boundedness properties of $m_{\lambda }$ and $M$ are equivalent.