Sawyer-type inequalities for Lorentz spaces

The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} f \Vert_{L^{1}(u)}, where $u\in A_1$ and $uv\in A_{\infty}$. We prove a novel extension of this result to general restricted weak type case. That is, for $p>1$, A_p^{\mathcal R}$, $uv^p \in A_\infty$, \Vert_{L^{p,\infty}(uv^p)} \Vert_{L^{p,1}(u)}. From these estimates, we deduce new weighted bounds inequalities $m$-fold product operators. also present an innovative technique that allows us transfer such estimates large class multi-variable operators, including $m$-linear Calder\'on-Zygmund avoiding $A_\infty$ extrapolation theorem producing many have not appeared in literature before. In particular, obtain characterization $A_p^{\mathcal R}$. Furthermore, introduce weights characterizes multi(sub)linear $\mathcal M$, denoted by $A_{\vec P}^{\mathcal establish analogous sparse operators m-linear study corresponding weights. Our results combine mixed norm inequalities, R}$ weights, Lorentz spaces.