Multiple ergodic averages for tempered functions

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying corresponding tempered functions, a class which also contains that oscillate and is in general more restrictive deal with. Our main result existence explicit expression \begin{document}$ L^2 $\end{document}-norm limit aforementioned averages, turns out, as case, be "expected" one. The ingredients are use of, now classical, PET induction (introduced Bergelson), covering namely "nice" (developed Chu-Frantzikinakis-Host polynomials Frantzikinakis functions) some equidistribution results nilmanifolds (analogous ones case).