This article develops results pointing towards affirmation of Rubio de Francia’s 1985 conjecture Asserting that for ω∈A1R\documentclass[12pt]{minimal}
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\begin{document}$$\omega \in A_{1}\left( {\mathbb {R}}\right) $$\end{document} a Littlewood–Paley type inequality, in the setting of L2R,ω\documentclass[12pt]{minimal}
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\begin{document}$$L^{2}\left( {\mathbb {R}},\omega \right) $$\end{document}, holds for arbitrary disjoint intervals of R\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}$$\end{document}. (The present study’s result does include a subsidiary requirement that ω∈A1R\documentclass[12pt]{minimal}
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\begin{document}$$\omega \in A_{1}\left( {\mathbb {R}}\right) $$\end{document} is an even function on R\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}$$\end{document}.) To treat this conjecture in its real line setting, we first completely establish its periodic counterpart for weighted T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}$$\end{document}. This weighted periodic environment is addressed first because of the greater transparency available in its machinery—in particular, the structural simplicity afforded by the finiteness of wtdt\documentclass[12pt]{minimal}
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\begin{document}$$w\left( t\right) \mathrm{d}t$$\end{document} viewed as a measure on the circle T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}$$\end{document}, when w∈A1T\documentclass[12pt]{minimal}
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\begin{document}$$w\in A_{1}\left( {\mathbb {T}}\right) $$\end{document}. Thereafter, we develop avenues for transferring to the weighted real line aspects of the periodic conjecture using steps analogous to those having just been successfully used to establish the periodic conjecture. Our study of the square functions ∑k≥1SJkf21/2\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ \sum _{k\ge 1}\left| S_{J_{k} }f\right| ^{2}\right\} ^{1/2}$$\end{document} associated with the periodic counterpart of Rubio de Francia’s Conjecture winds up being rooted in the interactions of the class of all even A1T\documentclass[12pt]{minimal}
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\begin{document}$$A_{1}\left( {\mathbb {T}}\right) $$\end{document} weights w with the corresponding classes f∈L2T,w:f^Z⊆R.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ f\in L^{2}\left( {\mathbb {T}},w\right) :{\widehat{f}}\left( {\mathbb {Z}}\right) \subseteq {\mathbb {R}}\right\} . \end{aligned}$$\end{document}
