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

Authors : El Abdalaoui EL HOUCEİN

Pages : 420-427

Doi:10.33205/cma.1029202

View : 17 | Download : 8

Publication Date : 2021-12-13

Article Type : Research Paper

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 R -action which assert that for any family of maps insert ignore into journalissuearticles values( T t ); t ∈ R insert ignore into journalissuearticles values(Tt);t∈R acting on the Lebesgue measure space insert ignore into journalissuearticles values( Ω , A , μ ); insert ignore into journalissuearticles values(Ω,A,μ); , where μ μ is a probability measure and for any t ∈ R t∈R , T t Tt is measure-preserving transformation on measure space insert ignore into journalissuearticles values( Ω , A , μ ); insert ignore into journalissuearticles values(Ω,A,μ); with T t ∘ T s = T t + s Tt∘Ts=Tt+s , for any t , s ∈ R t,s∈R . Then, for any f ∈ L 1 insert ignore into journalissuearticles values( μ ); f∈L1insert ignore into journalissuearticles values(μ); , there is a single null set off which  $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} finsert ignore into journalissuearticles values(T_t\omega); e^{2 i \pi \theta t} dt$ limT→+∞1T∫0Tfinsert ignore into journalissuearticles values(Ttω);e2iπθtdt exists for all θ ∈ θ∈\R R R . We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss.
Keywords : van der Corput inequality, Wiener Wintner theorem, joinings, amenable group