Spaces defined by the Paley function (Q2863119)

The paper under review deals with the behaviour of the Rademacher and Haar systems in symmetric Banach function spaces. Consider a symmetric space \(E\) on \([0,1]\) and consider its Boyd indices \(\alpha_E\) and \(\beta_E\). If \(0 < \alpha_E \leq \beta_E < 1\), then the Haar system defines an unconditional basis for \(E\). This condition is also equivalent to the fact that \(\|x\|_E \) and \(\|Px\|_E\) are equivalent. Here, \(x(t):= \sum_{n,k} c_{n,k} \chi^k_n(t)\) and \(Px(t):= (\sum_{n,k} (c_{n,k} \chi^k_n(t))^2)^{1/2}\), where \(\chi_n^k(t)\) is a Haar function of indices \(n\) and \(k\). The second equation gives what is called a Paley function. In this setting, \(P(E)\) denotes the space of all functions \(x \in L_1[0,1]\) such that \(Px \in E\) with the norm \(\|x\|_{P(E)} :=\|Px\|_E\).NEWLINENEWLINEThis paper is devoted to the study of the spaces \(P(E)\) and, in particular, of \( P(L_\infty)\). After some preliminary results, the authors prove some precise results on the relation between \(E\) and \(P(E)\) in terms of the Boyd indices of \(E\). In Theorem 2, they prove that the embedding \(P(E) \subseteq E\) holds if and only if \(\alpha_E > 0\), and that the embedding \(E \subseteq P(E)\) holds if and only if \(0 < \alpha_E \leq \beta_E < 1\). The authors analyse also the behavior of the space Rad of the Rademacher functions in \(P(E)\), and characterize the symmetric hull of Rad and \(P(L_\infty)\). Finally, they study the interpolation properties of the Banach couple \((L_\infty, P (L_\infty))\).