The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate

where \(u\in A_1\) and \(uv\in A_{\infty }\). 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 \(uv^p \in A_\infty \),

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 \(A_\infty \) 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.

Hardy-Littlewood 极大值算子M满足经典的 Sawyer 类型估计

其中\(u\in A_1\)和\(uv\in A_{\infty }\)。我们证明了这个结果对一般受限弱类型情况的新颖扩展。也就是说，对于\(p>1\)、\(u\in A_p^{{\mathcal {R}}}\)和\(uv^p \in A_\infty \)，

从这些估计中，我们为 Hardy-Littlewood 极大算子的m倍积推导出新的加权受限弱类型边界和 Sawyer 类型不等式。我们还提出了一种创新技术，使我们能够将此类估计转移到一大类多变量算子，包括m -线性 Calderón-Zygmund 算子，避免\(A_\infty \)外推定理并产生许多没有的估计之前出现在文献中。特别是，我们获得了\(A_p^{{\mathcal {R}}}\)的新特征。此外，我们引入了一类权重，它表征了多（子）线性极大算子\({\mathcal {M}}\)的受限弱类型边界，表示为\(A_{\mathbf {P}}^{{\mathcal {R}}}\)，建立稀疏算子和m -线性 Calderón-Zygmund 算子的类比界限，并研究相应的多变量 Sawyer 型不等式这样的运算符和权重。我们的结果结合了混合受限弱类型范数不等式，\(A_p^{{\mathcal {R}}}\)和\(A_{\mathbf {P}}^{{\mathcal {R}}}\)权重，和洛伦兹空间。