Convergence of weighted polynomial multiple ergodic averages

In this article we study 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 $\phi$ on an ergodic system, the sequence $\phi(T^{n}x)$ is universally good for almost every $x$. The linear case was covered by Host and Kra.