Sawyer-type inequalities for Lorentz spaces

The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$ Mf v L 1 , ∞ ( u v ) ≤ C u , v ‖ f ‖ L 1 ( u ) , where $$u\in A_1$$ u ∈ A 1 and $$uv\in A_{\infty }$$ u v ∈ A ∞ . We prove a novel extension of this result to the general restricted weak type case. That is, for $$p>1$$ p > 1 , $$u\in A_p^{{\mathcal {R}}}$$ u ∈ A p R , and $$uv^p \in A_\infty $$ u v p ∈ A ∞ , $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}. \end{aligned}$$ Mf v L p , ∞ ( u v p ) ≤ C u , v ‖ f ‖ L p , 1 ( u ) . From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type 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 $$A_\infty $$ A ∞ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $$A_p^{{\mathcal {R}}}$$ A p R . Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $${\mathcal {M}}$$ M , denoted by $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P 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, $$A_p^{{\mathcal {R}}}$$ A p R and $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R weights, and Lorentz spaces.