Periodization, Transference of Muckenhoupt Weights, and Automatic Tight Norm Estimates for the Periodic Hilbert Transform

Abstract

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.