Sawyer-type inequalities for Lorentz spaces
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The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate ∥Mfv∥L1,∞(uv)≤Cu,v‖f‖L1(u),where u∈ A1 and uv∈ A∞. We prove a novel extension of this result to the general restricted weak type case. That is, for p> 1 , u∈ApR, and uvp∈ A∞, ∥Mfv∥Lp,∞(uvp)≤Cu,v‖f‖Lp,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∞ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of ApR. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator M, denoted by APR, 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, ApR and APR weights, and Lorentz spaces.