Geodesic
Independent functions in rearrangement invariant
Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 945-963 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $X$ be a separable or maximal rearrangement invariant space on $[0,1]$. It is shown that the inequality \begin{equation*} \biggl\|\,\sum_{k=1}^\infty f_k\biggr\|_{X} \le C\biggl\|\biggl(\,\sum_{k=1}^\infty f_k^2\biggl)^{1/2}\biggr\|_X \end{equation*} holds for an arbitrary sequence of independent functions $\{f_k\}_{k=1}^\infty\subset X$, $\displaystyle\int_0^1f_k(t)\,dt=0$, $k=1,2,\dots$, if and only if $X$ has the Kruglov property. As a consequence, it is proved that the same property is necessary and sufficient for a version of Maurey's well-known inequality for vector-valued Rademacher series with independent coefficients to hold in $X$. Bibliography: 24 titles. 
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     author = {S. V. Astashkin},
     title = {Independent functions in rearrangement invariant},
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S. V. Astashkin. Independent functions in rearrangement invariant. Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 945-963. http://geodesic.mathdoc.fr/item/SM_2008_199_7_a0/

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