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

Abstract

We prove a weighted version of a classical inequality of Johnson and Schechtman from which we derive a decomposition theorem for $p$-th moments ($0<p\le 1$) of nonnegative generalized $U$-statistics with constant not dependent on $p$. In particular, for $1\le p\le 2$, the norm in the subspace $U_{\le m}^p\left({\Omega }^{\infty }\right)$ 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 $U_m^1\left({\Omega }^{\infty },{\ell }^p\right)$ that are known to imply cotype 2 of $L^1/U_{\le m}^1\left({\Omega }^{\infty }\right)$.