For \(\alpha \in {\mathbb {R}}\), let \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) be the corresponding power weight function on \({\mathbb {R}}\) (respectively, the corresponding periodic power weight function, also defined on \({\mathbb {R}} \)). It is known that \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) belongs to \(A_{1}\left( {\mathbb {R}}\right) \) (respectively, \(A_{1}\left( {\mathbb {T}} \right) \)) if and only if \(-1<\alpha \le 0\), and that if \(1<p<\infty \), then \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) belongs to \(A_{p}\left( {\mathbb {R}}\right) \) (respectively, to \(A_{p}\left( {\mathbb {T}}\right) \)) if and only if \(-1<\alpha <p-1\). In recent times, it has been shown that for \(1<p<\infty \), the power weights \(W_{\alpha }\) can be used in the course of determining the sharpest upper bounds (in terms of exponents of \(A_{p}\) weight norms) for all operator norms of the Hilbert transform H acting on \(L^{p}\left( {\mathbb {R}},\omega \right) \), as \(\omega \) runs through \(A_{p}\left( {\mathbb {R}}\right) \). This note provides the analogous results for the periodic Hilbert transform \({\widetilde{H}}\) acting on all \(A_{p}\left( {\mathbb {T}}\right) \) weighted \(L^{p}\) spaces of \({\mathbb {T}}\). These corresponding results are arrived at through appropriate transference techniques aimed at establishing the corresponding upper estimates for the periodic Hilbert transform \({\widetilde{H}}\) within the framework of all \(A_{p}\left( {\mathbb {T}}\right) \) weights; thereafter appropriate weights \(w_{\alpha }\in A_{p}\left( {\mathbb {T}}\right) \) can be singled out from the mix in order to obtain the desired sharpness results from these upper estimates. To illustrate the flexibility of the methods involved in this process, we describe how they can furnish even tighter forms of upper estimates (involving \(A_{\infty }\left( {\mathbb {R}}\right) \) weight characteristics) for \(\left\| {\widetilde{H}}\right\| _{{\mathfrak {B}} \left( L^{p}\left( {\mathbb {T}},w\right) \right) }\) in the framework of all weights \(w\in A_{p}\left( {\mathbb {T}}\right) \). We close with a section devoted to the fine structure of a general periodization process for even \(A_{p}\left( {\mathbb {R}}\right) \) weights, thereby establishing a ubiquitous presence of \(A_{p}\left( {\mathbb {T}}\right) \) weights. Our methods can also treat Calderón–Zygmund operators in weighted multivariable settings. However, this article focuses its main attention on norms of Hilbert transforms in the framework of weighted spaces based on \({\mathbb {R}}\) and \({\mathbb {T}}\) in order to examine how Calderón–Coifman–G. Weiss transference methods can expand to cover tight \(A_{p}\left( {\mathbb {R}} \right) -A_{\infty }\left( {\mathbb {R}}\right) \) operator bounds in the arena of classical weighted settings.

对于\（\ alpha \ in {\ mathbb {R}} \），令\（W _ {\ alpha} \）（分别为\（w _ {\ alpha} \））是\（{ \ mathbb {R}} \）（分别在\（{\ mathbb {R}} \）上定义了相应的周期性功率权重函数）。众所周知\（W _ {\ alpha} \）（分别是\（w _ {\ alpha} \））属于\（A_ {1} \ left（{\ mathbb {R}} \ right）\）（）分别\（A_ {1} \左（{\ mathbb【T}} \右）\） ）当且仅当\（ - 1 <\阿尔法\文件0 \） ，和，如果\（1 <p < \ infty \），然后是\（W _ {\ alpha} \）（分别是\（w _ {\ alpha} \）属于\（A_ {p} \ left（{\ mathbb {R}} \ right）\）（）分别属于\（A_ {p} \ left（{\ mathbb { T}} \ right）\））仅且仅当\（-1 <\ alpha <p-1 \）时。最近，已经证明对于\（1 <p <\ infty \），功率权重\（W _ {\ alpha} \）可用于确定最尖锐的上限（就指数而言）的\（A_ {p} \）重量规范），用于希尔伯特的所有算子范数变换ħ作用在\（L ^ {p} \左（{\ mathbb {R}}，\欧米加\右）\） ，如\（\ omega \）贯穿\（A_ {p} \ left（{\ mathbb {R}} \ right）\）。本注释提供了对所有加权的（（A_ {p} \ left（{\ mathbb {T}} \ right）\）加权的周期性Hilbert变换\ {{\ widetilde {H}} \}的相似结果\（{\ mathbb {T}} \）的^ {p} \）个空格。这些相应的结果是通过适当的转移技术得出的，旨在为所有\（A_ {p} \ left（{\ mathbb ）的框架中的周期性希尔伯特变换\（{\ widetilde {H}} \}建立相应的上估计{T}} \ right）\）重量；此后适当的权重\（w _ {\ alpha} \在A_ {p} \ left（{\ mathbb {T}} \ right）\）可以从混合中挑选出“粗”，以便从这些较高的估计中获得所需的清晰度。为了说明此过程涉及的方法的灵活性，我们描述了它们如何提供更紧密形式的上限估计值（涉及\（A _ {\ infty} \ left（{\ mathbb {R}} \ right）\）权重特征）代表\（\ left \ | {\ widetilde {H}} \ right \ | _ {{\ mathfrak {B}} \ left（L ^ {p} \ left（{\ mathbb {T}}，w \ right ）\ right）} \）在所有权重的框架\（w_in A_ {p} \ left（{\ mathbb {T}} \ right）\）中。我们以专门介绍一般周期化过程的精细结构的部分作为结尾，以期均匀分配\（A_ {p} \ left（{\ mathbb {R}} \ right）\）权重，从而建立了无处不在的\（A_ {p} \ left（{\ mathbb {T}} \ right）\）权重。我们的方法还可以在加权多变量设置中对待Calderón–Zygmund运算符。但是，本文主要关注基于\（{\ mathbb {R}} \）和\（{\ mathbb {T}} \）的加权空间框架中的希尔伯特变换范数，以便研究Calderón –Coifman–G。Weiss转移方法可以扩展为覆盖\（A_ {p} \ left（{\ mathbb {R}} \ right）-A _ {\ infty} \ left（{\ mathbb {R}} \ right）\）运算符范围在古典加权设置的舞台上。