van der Corput inequality for real line andWiener-Wintner theorem for amenable groups

Abstract

ABSTRACT. We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the R-action which assert that for any family of maps (Tt)t2R acting on the Lebesgue measure space ( ;A; ), where  is a probability measure and for any t 2 R, Tt is measure-preserving transformation on measure space ( ;A; ) with TtTs = Tt+s, for any t; s 2 R. Then, for any f 2 L1(), there is a single null set off which lim T!+1 1 T Z T 0 f(Tt!)e2itdt exists for all  2 R. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss .