In this paper we give quantitative bounds for the norms of different kinds of singular integral operators on weighted Hardy spaces \(H_w^p\), where \(0<p\le 1\) and w is a weight in the Muckenhoupt \(A_{\infty }\) class. We deal with Fourier multiplier operators, Calderón–Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood–Paley functions and the grand maximal function. We also establish explicit quantitative bounds depending on the weight w when switching between the \(H^p_w\)-norms defined by the Littlewood–Paley–Stein square function and its discrete version, and also by applying the mixed bound \(A_q-A_\infty \) result for the vector-valued extension of the Hardy–Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1):253–272, 1993).

中文翻译：

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加权Hardy空间上奇异积分的混合估计

在本文中，我们给出了加权Hardy空间\（H_w ^ p \）上不同奇异积分算子范数的范数界，其中\（0 <p \ le 1 \）和w是Muckenhoupt \（ A _ {\ infty} \）类。我们处理傅立叶乘法算子，齐次类型的Calderón–Zygmund算子（它们是第一个算子的特殊情况），并且更广泛地讲，我们研究卷积类型的奇异积分。为了证明加权Hardy空间设置中的混合估计，我们需要通过平方函数，Littlewood-Paley函数和极大函数来介绍加权Hardy空间的几种特征。我们还根据权重确定明确的定量范围w在由Littlewood–Paley–Stein平方函数定义的\（H ^ p_w \）-范数与其离散版本之间进行切换时，以及通过对向量应用混合边界\（A_q-A_ \ infty \）结果时w Buckley中给出的Hardy–Littlewood最大值算子的有效扩展（Trans Am Math Soc 340（1）：253–272，1993）。