Sawyer-type inequalities for Lorentz spaces

Abstract

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