Weighted Lorentz spaces: Sharp mixed Ap − A∞ estimate for maximal functions

Abstract

We prove the sharp mixed Ap−A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely‖M‖Lp,q(w)≲p,q,n[w]Ap1p[σ]A∞1min⁡(p,q), where σ=w11−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.