Optimality in a Multilinear Extension of Kwapień's Theorem

Bayart, Pellegrino and Rueda proved that every continuous $m$-linear operator from $({\mathrm{\ell }}_1{)}^m$ into ${\mathrm{\ell }}_p$ is absolutely $(r_{m,p};1)$-summing, for explicit exponents $r_{m,p}$. The optimality of these exponents was known for $2\le p\le \mathrm{\infty }$. 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 ${\mathrm{\ell }}_p^n$ uniformly. As applications, we recover the sharp exponent $2\mathrm{/}m$ for $L_1[0,1]$-valued mappings and show that cotype alone does not determine the optimal absolute $(r;1)$-coincidence exponent.