A Generalized Khintchine Inequality in Rearrangement Invariant Spaces

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
\begin{equation*} \bigg\|\sum_{k=1}^\infty f_k\bigg\|_{X}\le C\bigg\|\bigg(\sum_{k=1}^\infty f_k^2\bigg)^{1/2}\bigg\|_X \end{equation*}
holds for an arbitrary sequence $\{f_k\}_{k=1}^\infty\subset X$ of mean zero independent variables. Moreover, the subspace spanned in a rearrangement invariant space by the Rademacher system with independent vector coefficients is studied.