On the existence of RUC systems in rearrangement invariant spaces (Q2792247)

Let \(X\) be a Banach space and consider a biorthogonal system \((x_i,x_i^*)\) in it, \(x_i \in X\), \(x_i^* \in X^*\). It is said to be a randomly unconditionally convergent system (RUC system for short) in \(X\) if, for every \(x\) from the closed linear span of \([x_i]\), the series \(\sum_i x_i^*(x) x_i\) is randomly unconditionally convergent, that is, \(\sum_i x_i^*(x) r_i(t) x_i\) converges in \(X\) for almost all \(t \in [0,1]\). Here, the \(r_i\)'s are the Rademacher functions.NEWLINENEWLINEIt is known that this condition is equivalent to the following type-class domination property NEWLINE\[NEWLINE \int_0^1 \Big\| \sum_{i=1}^n c_i r_i(t) x_i \Big\| _X \, dt \leq K \, \Big\| \sum_{i=1}^n c_i x_i \Big\| _X NEWLINE\]NEWLINE for arbitray scalars \(c_1,\dots,c_n\), \(n \in \mathbb N\). In the present paper, an extension of a theorem on the existence of fundamental orthonormal uniformly bounded RUC systems in rearrangement invariant spaces due to \textit{P. G. Dodds} et al. [Stud. Math. 151, No.~2, 161--173 (2002; Zbl 1031.46017)] is given. The idea is to relax the uniform boundedness requirement, which is given in the \(L^\infty\)-norm in the original result, by asking for uniform boundedness in a weaker norm. The larger space for which this can be done is the Marcinkiewicz space \(M_{\varphi_\alpha}\) associated to the function \(\log^{-1/\alpha} (e/t)\), for \(\alpha >0\).NEWLINENEWLINEThe main result is the following.NEWLINENEWLINETheorem 1.1. Let \(X\) be a separable rearrangement invariant space on \([0,1]\) and let \(\alpha >0\), \(\beta= 2 \alpha/(\alpha + 2)\). The following conditions are equivalent. {\parindent=0.7cm\begin{itemize}\item[(i)] Every sequence \((f_i)\) such that \(\sup_i \| f_i\| _{M_{\varphi_\alpha}}< \infty\) and NEWLINE\[NEWLINE \Big\| \sum_i c_i f_i \Big\| _{L^2} \geq C \, \| (c_i)\| _2 \quad \text{for all} \quad (c_i) \in \ell^2 NEWLINE\]NEWLINE is an RUC system in \(X\). \item[(ii)] Every orthonormal sequence \((f_i)\) such that \(\sup_i \| f_i\| _{M_{\varphi_\alpha}}< \infty\) is an RUC system in \(X\). \item[(iii)] The continuous embeddings \(M_{\varphi_\beta}^0 \subseteq X \subseteq L^2\) hold. NEWLINENEWLINE\end{itemize}} Here, \(M_{\varphi_\beta}\) denotes the closure of \(L^\infty\) in the Orlicz space generated by the function \(\exp (t^\beta)-1\).NEWLINENEWLINEThe paper is rather technical; the interested reader can find in it some useful tools on integral dominations for concave functions, mainly related with dominations of integral expressions in which Rademacher averages appear.