Determine the sharp quantitative exponent for Lorentz–Muckenhoupt maximal estimates

Determine the optimal exponent \(\Gamma_s\) in the quantitative bound \(\|M_\alpha\|_{L_w^{p,p}\to L_w^{q,s}}\lesssim [w]_{\mathcal A_{p,q}^{[s]}}^{\Gamma_s}\) for \(s>q\), thereby resolving the gap between the known lower bound \(1+p'/s\) and upper bound \(1+p'/q\).

Background

For $q\le s\le\infty$, the paper defines the optimal exponent $\Gamma_s$ governing the dependence of the maximal-operator norm on the Lorentz–Muckenhoupt characteristic. The authors establish the bounds $1+p'/s\le\Gamma_s\le1+p'/q$. These bounds coincide when $s=q$, recovering the classical sharp estimate, but leave a gap for every $s>q$.