2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174689We study here weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in $L^{2}$. We find a necessary condition and show that for any bounded measurable function $ϕ$ on an ergodic system, the sequence $ϕ(T^{n}x)$ is universally good for almost every $x$. The linear case was understood by Host and Kra.Dynamical Systems37A05, 37A30text