Proof that a form of Rubio de Francia's conjectured Littlewood-Paley type inequality for \(A_1 (\mathbb{R})\)-weighted \(L^2 (\mathbb{R})\) is valid for every even \(A_1 (\mathbb{R})\) weight (Q6611135)

The present paper is devoted to establishing a form of the even-\(A_1(\mathbb R)\) case of Rubio de Francia's Littlewood-Paley type conjecture that was promulgated in 1985 for every \(A_1(\mathbb R)\)-weight [\textit{J. L. Rubio de Francia}, Rev. Mat. Iberoam. 1, No. 2, 1--14 (1985; Zbl 0611.42005)]. The author completes his trilogy on these matters first initiated in [the author, J. Geom. Anal. 32, No. 8, Paper No. 223, 24 p. (2022; Zbl 1498.30029)], and continued in [the author, ibid. 33, No. 8, Paper No. 241, 7 p. (2023; Zbl 1516.30064)] where the periodic counterpart was established for even, \(2\pi\)-periodic \(A_1(\mathbb R)\)-weights and the case of even and decreasing on \([0,\infty)\) were already considered. In the paper, the author shows that \N\[\N\|(\sum_{k} |S_{J_k}f|^2)^{1/2}\|_{L^2(\mathbb R,\omega(t)dt)}\le 8^{3/2}(2C)^{1/2}\|f\|_{L^2(\mathbb R,\omega^*(t)dt)}\N\]\Nwhere \(\omega\) is an even-\(A_1(\mathbb R)\) with \(A_1(\mathbb R)\)-constant \(C\) and \(\omega^*\) stands for the nonsymmetric decreasing rearrangement of \(\omega\).