Convergence of weighted polynomial multiple ergodic averages

We 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 $\varphi$ on an ergodic system, the sequence $\varphi (T^{n}x)$ is universally good for almost every $x$. The linear case was understood by Host and Kra.