Weighted and multivariate Johnson--Schechtman inequalities with application to interpolation theory

We prove a weighted version of a classical inequality of Johnson and Schechtman from which we derive a decomposition theorem for $p$-th moments ($0<p\leq 1$) of nonnegative generalized $U$-statistics with constant not dependent on $p$. In particular, for $1\leq p\leq 2$, the norm in the subspace $U^p_{\leq m}\left(\Omega^\infty\right)$ of $L^p\left(\Omega^\infty\right)$ spanned by functions dependent on at most $m$ variables is equivalent to the norm in a suitable interpolation sum of $L^p\left(L^2\right)$ spaces. As a consequence, we obtain some interpolation properties of $U^1_m\left(\Omega^\infty,\ell^p\right)$ that are known to imply cotype 2 of $L^1/U_{\leq m}^1\left(\Omega^\infty\right)$.