Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis
Let \(X\) be a rearrangement invariant (r.i.)\ function space on \([0,1]\) and \(\mathrm{Radi}\,X\) be the subspace of \(X\) consisting of all functions of the form \(f=\sum_{k=1}^{\infty}x_kr_k\), where \(x_k\) are arbitrary (stochastically) independent functions from \(X\) and \(r_k\) the usual Rademacher functions independent of \((x_k)\). The author investigates when \(\mathrm{Radi}\,X\) is complemented in \(X\). Namely, he proves that \(\mathrm{Radi}\,X\) is complemented in \(X\) if and only if both \(X\) and its Köthe dual \(X'\) possess the Kruglov property. An r.i.\ space on \([0,1]\) has the Kruglov property if the relation \(f\in X\) implies \(\pi(f)\in X\). Here, \(\pi(f)=\sum_1^Nf_i\), where \(f_i\) are independent copies of \(f\) and \(N\) is a random variable independent of the sequence \((f_i)\) and having the Poisson distribution with parameter 1. Every r.i.\ space with positive lower Boyd index has this property. As a consequence, the author shows that the Kruglov property of \(X\) and \(X'\) guarantees that \(X\) is isomorphic to an r.i.\ function space on \((0,\infty)\); more exactly, to the space \(Z_X^2\) consisting of all measurable on \((0,\infty)\) functions \(f\) with \(\|f^*\chi_{[0,1]}\|_X+\|f^*\chi_{[1,\infty)}\|_{L_2[1,\infty)}<\infty \), where \(f^*\) is the non-increasing rearrangement of \(f\). This strengthens the analogous result of [\textit{W. B.~Johnson, B.~Maurey, G.~Schechtman} and \textit{L.~Tzafriri}, ``Symmetric structures in Banach spaces'' (Mem. Am. Math. Soc. 217) (1979; Zbl 0421.46023)] proved under the stronger condition when \(X\) has nontrivial Boyd indices.
scientific article; zbMATH DE number 5825480
scientific article