Weighted Lorentz spaces: sharp mixed $A_p-A_{\infty}$ estimate for maximal functions

Abstract: We prove the sharp mixed $A_{p}-A_{\infty}$ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely [ |M|{L^{p,q}(w)} \lesssim{p,q,n} [w]^{\frac1p}{A_p}[\sigma]^{\frac1{\min(p,q)}}{A_{\infty}}, ] where $\sigma=w^{\frac{1}{1-p}}$. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for $M$ in a dual setting.