Optimality in a Multilinear Extension of Kwapie\'n's Theorem

Abstract

Bayart, Pellegrino and Rueda proved that every continuous $m$-linear operator from $(\ell_1)^m$ into $\ell_p$ is absolutely $(r_{m,p};1)$-summing, for explicit exponents $r_{m,p}$. The optimality of these exponents was known for $2\le p\le\infty$. We prove optimality in the remaining range $1\le p<2$. Our argument is finite-dimensional and is based on convolution on $\mathbb F_2^d$ and the Walsh character system. The same construction yields a local obstruction theorem for range spaces containing $\ell_p^n$ uniformly. As applications, we recover the sharp exponent $2/m$ for $L_1[0,1]$-valued mappings and show that cotype alone does not determine the optimal absolute $(r;1)$-coincidence exponent.