A generalized Khintchine inequality in rearrangement invariant spaces

Abstract

Let X be a separable or maximal rearrangement invariant space on [0, 1]. Necessary and sufficient conditions are found under which the generalized Khintchine inequality $$\left\| {\sum\limits_{k = 1}^\infty {f_k } } \right\|_X \leqslant C\left\| {\left( {\sum\limits_{k = 1}^\infty {f_k^2 } } \right)^{1/2} } \right\|_X $$ holds for an arbitrary sequence {ƒk} k=1 ∞ ⊂ X of mean zero independent variables. Moreover, the subspace spanned in a rearrangement invariant space by the Rademacher system with independent vector coefficients is studied.