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Sawyer-type inequalities for Lorentz spaces
Mathematische Annalen ( IF 1.3 ) Pub Date : 2021-07-21 , DOI: 10.1007/s00208-021-02240-4
Carlos Pérez 1 , Eduard Roure-Perdices 2
Affiliation  

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

$$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$

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 \(uv^p \in A_\infty \),

$$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}. \end{aligned}$$

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.



中文翻译:

洛伦兹空间的 Sawyer 型不等式

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

$$\begin{对齐} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$

其中\(u\in A_1\)\(uv\in A_{\infty }\)。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\(p>1\)\(u\in A_p^{{\mathcal {R}}}\)\(uv^p \in A_\infty \)

$$\begin{对齐} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}。\end{对齐}$$

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,\(A_p^{{\mathcal {R}}}\)\(A_{\mathbf {P}}^{{\mathcal {R}}}\)权重,和洛伦兹空间。

更新日期:2021-07-22
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