Littlewood-Paley theorem in the Ba spaces (Q1812744)

The Ba space \(\{L_{p_ m},a_ m\}\) is a set of functions \(f\in\prod_{m\in\mathbb{N}}L_{p_ m}(\mathbb{R})\) for which there exists \(\lambda\) such that \(\sum_{m=1}^ \infty a_ m\lambda^ m| f|^ m_{p_ m}\) is finite, where \(\{p_ m\}\) and \(\{a_ m\}\) are appropriate sequences of non-negative real numbers. It is shown that the Littlewood-Paley function \[ g(f)(x)=\left(\int_ 0^ \infty|\nabla u_ f(x,y)|^ 2y dy\right)^{1/2} \] where \(u_ f\) is the Poisson integral of \(f\), has the \(L_ p\)-norm, \(1<p<\infty\), equivalent to the \(L_ p\)-norm of \(f\) if and only if \(1<\text{inf}_ mp_ m\leq\sup_ mp_ m<\infty\).