Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications

In this article, we establish sharp quantitative weighted estimates for the lifted Hardy--Littlewood (resp Sobolev sharp) maximal operators $\{\mathcal M_θ\}_{θ\in(0,\infty)}$ (resp $\{\mathcal M_θ^\sharp\}_{θ\in(0,\infty)}$). Specifically, for $p\in(1,\infty)$, $γ\in\mathbb R\setminus\{0\}$, $\upsi