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

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 $(T_t{)}_{t\in R}$ acting on the Lebesgue measure space $(\Omega ,A,\mu )$, where $\mu$ is a probability measure and for any $t\in R$, $T_t$ is measure-preserving transformation on measure space $(\Omega ,A,\mu )$ with $T_t\circ T_s=T_{t+s}$, for any $t,s\in R$. Then, for any $f\in L^1\left(\mu \right)$, there is a single null set off which  $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t} dt$ exists for all $\theta \in$$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

References

1. El H. El Abdalaoui: On the spectral type of rank one flows and Banach problem with calculus of generalized Riesz products on the real line, arXiv:2007.03684 [math.DS].
2. I. Assani: Wiener-Wintner property of the helical transform, Ergod. Th. & Dynam. Sys., 10 (1992), 185-194.
3. A. Below, V. Losert: The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc., 288 (1985), 307-345.
4. J. Bourgain: Double recurrence and almost sure convergence , J. reine angew. Math., 404 (1990), 140-161.
5. A. Deljunco, D. Rudolph: On ergodic actions whose self joining are graphs, Ergod. Th. & Dynam. Sys., 7 (1987), 531-557.
6. H. Furstenberg: Stationary process and prediction theory , Ann. Math. Studies, 44 Princeton University Press, Princeton (1960).
7. H. Furstenberg: Disjointness in ergodic theory, minimal sets and problem in diophantine approximation, Math. Sys. Theory, 1 (1960), 1-49.
8. F. Hahn, W. Parry: Some characteristic properties od dynamical system with quasi-discrete spectrum , Math. Sys. Theory, 2 (1968), 179-190.

9. M. Lacey, E. Terwilleger: A Wiener–Wintner theorem for the Hilbert transform, Ark. Mat., 46 (2) (2008), 315-336.
10. E. Lesigne: Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème egodique de Wiener-Wintner, Ergod. Th. & Dynam. Sys., 10 (1990), 513-521.
11. E. Lesigne: Spectre quasi-discret et théorème egodique deWiener-Wintner pour les polynômes, Ergod. Th. & Dynam. Sys., 13 (1993), 767-784.
12. E. Lindenstrauss: Pointwise ergodic theorem for amenable groups, Invent. Math., 146 (2) (2001), 259-295..
13. D. Ornstein, B. Weiss: Subsequences ergodic theorems for amenable groups, Isr. J. Maths., 79 (1992), 113-127.
14. J-P. Thouvenot: Some properties and applications of joinings in ergodic theory, In Ergodic Theory and its connections with Harmonic Analysis (Proc. of Alexandria conference), K. E. Petersen and I. Salama, Eds, L.M.S. lectures notes 205, Cambridge Univ. Press, Cambridge (1995), 207-235.
15. N. Wiener, A. Wintner: Harmonic analysis and ergodic theory , Amer. J. Math., 63 (1941), 415-426.