Some remarks on oscillation inequalities

In this paper, we establish uniform oscillation estimates on Lᵖ(X) with p∈ (1,∞ ) for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990s [Pointwise ergodic theorems via harmonic analysis. Proceedings of Conference on Ergodic Theory (Alexandria, Egypt, 1993) (London Mathematical Society Lecture Notes, 205) . Eds. K. Petersen and I. Salama. Cambridge University Press, Cambridge, 1995, pp. 3–151]. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt, and Wierdl for bounded martingales [Oscillation in ergodic theory. Ergod. Th. & Dynam. Sys. 18 (4) (1998), 889–935]. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for r -variation inequalities.