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. We find a necessary condition and show that for any bounded measurable function φ on an ergodic system, the sequence φ(Tnx) is universally good for almost every x. The linear case was covered by Host and Kra.