Proof that a Form of Rubio de Francia’s Conjectured Littlewood-Paley Type Inequality for $$A_{1}\left( {\mathbb {R}}\right) $$ -Weighted $$L^{2}\left( {\mathbb {R}}\right) $$ is Valid for Every Even $$A_{1}\left( {\mathbb {R}}\right) $$ Weight

It is demonstrated that a form of Rubio de Francia’s hitherto unresolved Littlewood-Paley Type Conjecture from the year 1985 is valid for the weighted-\(L^{2}\left( {\mathbb {R}}\right) \) space corresponding to any even \(A_{1}\left( {\mathbb {R}}\right) \) weight. Otherwise expressed, we show that if \(\omega \) is any even \(A_{1}\left( {\mathbb {R}}\right) \) weight, C is an \(A_{1}\left( {\mathbb {R}}\right) \) weight constant for \(\omega \), \(\ f\in \) \(L^{2}\left( {\mathbb {R}},\omega \left( t\right) dt\right) \), and \(\left\{ J_{k}\right\} _{k\ge 1}\) is any finite or infinite sequence of disjoint intervals of \({\mathbb {R}}\), then the following estimate holds for the corresponding Littlewood-Paley Type square function defined by \(\left\{ S_{J_{k}}\left( f\right) \right\} _{k\ge 1}\)(where the symbol \(S_{_{J_{k}} }\) denotes the indicated partial sum projection for the context of \({\mathbb {R}}\)):

$$\begin{aligned} \left\| \left\{ \sum \limits _{k\ge 1}\left| S_{J_{k}}\left( f\right) \right| ^{2}\right\} ^{1/2}\right\| _{L^{2}\left( {\mathbb {R}},\omega \left( t\right) dt\right) }\le 2^{5}C^{1/2}\left\| f\right\| _{L^{2}\left( {\mathbb {R}},\omega ^*\left( t\right) dt\right) }, \end{aligned}$$

where \(\omega ^*\) is the decreasing rearrangement of \(\omega \). A corollary of this even \(A_{1}\left( {\mathbb {R}}\right) \)-weighted theorem is obtained which provides a related variant thereof in the setting of any (not necessarily even) \(A_{1}\left( {\mathbb {R}}\right) \) weight.