Re: Positive resolution of Rubio de Francia's Littlewood-Paley conjecture for arbitrary disjoint intervals in the context of \(A_1\)-weighted \(L^2\) (Q2145687)

The author presents a partial solution to the \textit{J. L. Rubio de Francia}'s conjecture [Rev. Mat. Iberoam. 1, No. 2, 1--14 (1985; Zbl 0611.42005)] which asserts that the Littlewood-Paley type inequality, in the setting of \(L^2(\mathbb R, \omega)\), holds for arbitrary disjoint intervals of \(\mathbb R\) for weights \(\omega\in A_1(\mathbb R)\). The approach in the paper leads to the solution only when the weights are assumed to be even functions in \(\mathbb R\). The result is first proved for its periodic counterpart. In the paper the class \(A_p(\mathbb T)\) consists on those weights in \(A_p(\mathbb R)\) which are \(2\pi\)-periodic. The main result in the paper establishes that if \(\omega \in A_1(\mathbb T)\) is an even function on \(\mathbb R\) with \(A_1\)-constant \(C\) and \(f\in L^2(\mathbb T,\omega)\) then, for any sequence of disjoint intervals \({V_k}\) in \(\mathbb Z\) one has \[ \left\| \left(\sum_{k\ge 1} |S_{V_k}(f)|^2\right)^{1/2}\right\|_{L^2(\mathbb T,\omega)}\le 4C^{1/2}\|f\|_{L^2(\mathbb T,\omega)} \] where for an interval \(V\) in \(\mathbb Z\) we denote \(S_Vf(t)= \sum_{k\in V}\hat f(k) e^{ikt}\). The author first covers the cases where the cardinality of the intervals is bounded and where the functions satisfy \(\hat f(\mathbb Z)\subset \mathbb R\). To deal with the case of the real line, even assuming that \(\omega_0\in A_1(\mathbb R)\) is an even functions, the author needs some extra limitations such that \(\mathrm{essinf\,} \omega_0>0\). In particular it is shown that if \(\omega_0 \in A_1(\mathbb R)\) is an even function on \(\mathbb R\) such that \(\operatorname{essinf} \omega_0>0\) then, for any sequence of disjoint intervals \({J_k}\) in \(\mathbb R\) one has \[ \left\| \left(\sum_{k\ge 1} |S_{J_k}(f)|^2\right)^{1/2}\right\|_{L^2(\mathbb R,\omega_0)}\le C\|f\|_{L^2(\mathbb R,\omega_0)} \] where the constant depends on the \(A_1\)-constant of \(\omega_0\) and \(\operatorname{essinf} \omega_0\) and for an interval \(J\) in \(\mathbb R\) we denote \(S_Jf(t)= \int_J\hat f(x) e^{ixt}dx\).