Rademacher series and decoupling (Q2581084)

Inspired by the work of \textit{V.~H.\ de la Peña} and \textit{S.~J.\ Montgomery--Smith} [Ann.\ Probab.\ 23, No.~2, 806--816 (1995; Zbl 0827.60014)], the author in the present paper defines the decoupling property for a quasi-Banach space. He shows that the spaces \(L_p([0,1])\) for \(0<p<1\) have the decoupling property and the Schatten spaces fail to have this property when \(0<p<1\). He also proves that Pisier's property \((\alpha)\) implies the decoupling property and uses this to show that \(L_p/H_p\) and \(L_p/R\), where \(R\) is a reflexive subspace having property \((\alpha)\), and any minimal extension of \(l_1\) or \(L_1\) has decoupling. Some other properties are given such that, for \(B\) a bilinear operator from \(X \times Y\) into \(Z\), where \(X\), \(Y\) are Banach spaces of type 2 and \(Z\) has decoupling, one gets that \(B\) factors through a Banach space.