Some optimal pointwise ergodic theorems with rate

Let T be a Dunford-Schwartz operator on the probability space (X, Sigma, mu) and p > 1. For f in the range of suitable operators of L(p)(X, Sigma, mu), we obtain pointwise ergodic theorems with rate, using a method of Derriennic and Lin (2001). When T is induced by a it-preserving transformation, these results are shown to be optimal. The proof of the optimality is inspired from a construction of Daniel (1989). To cite this article: C. Cuny, C. R. Acad Sci. Paris, Ser. I 347 (2009). (c) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.