Abstract
We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T t ) and f∈L p(X,μ), there is a set X f ⊂X of probability one, so that for all x∈X f ,
The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Assani, I., Caractérisation spectrale des systèmes dynamiques du type Wiener–Wintner, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 321–324.
Assani, I., Wiener Wintner Ergodic Theorems, World Scientific, River Edge, NJ, 2003.
Auscher, P. and Carro, M. J., On relations between operators on R N, T N and Z N, Studia Math. 101 (1992), 165–182.
Bourgain, J., Temps de retour pour les systèmes dynamiques, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 483–485.
Bourgain, J., Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45.
Calderón, A., Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353.
Campbell, J., Jones, R. L., Reinhold, K. and Wierdl, M., Oscillation and variation for the Hilbert transform, Duke Math. J. 105 (2000), 59–83.
Campbell, J. and Petersen, K., The spectral measure and Hilbert transform of a measure-preserving transformation, Trans. Amer. Math. Soc. 313 (1989), 121–129.
Carleson, L., On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.
Demeter, C., Lacey, M., Tao, T. and Thiele, C., Breaking the duality in the return times theorem, Duke Math. J. 143(2) (2008), 281–355
Grafakos, L., Tao, T. and Terwilleger, E., L p bounds for a maximal dyadic sum operator, Math. Z. 246 (2004), 321–337.
Hunt, R. A., On the convergence of Fourier series, in Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, IL, 1967), pp. 235–255, Southern Illinois Univ. Press, Carbondale, IL, 1968.
Jones, R. L., Kaufman, R., Rosenblatt, J. M. and Wierdl, M., Oscillation in ergodic theory, Ergodic Theory Dynam. Systems 18 (1998), 889–935.
Lacey, M., Carleson’s theorem: proof, complements, variations, Publ. Mat. 48 (2004), 251–307.
Lacey, M. and Thiele, C., On Calderón’s conjecture, Ann. of Math. 149 (1999), 475–496.
Lacey, M. and Thiele, C., A proof of boundedness of the Carleson operator, Math. Res. Lett. 7 (2000), 361–370.
Máté, A., Convergence of Fourier series of square integrable functions, Mat. Lapok 18 (1967), 195–242.
Muscalu, C., Tao, T. and Thiele, C., Multi-linear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), 469–496.
Pramanik, M. and Terwilleger, E., A weak L 2 estimate for a maximal dyadic sum operator on ℝn, Illinois J. Math. 47 (2003), 775–813.
Prestini, E. and Sjölin, P., A Littlewood–Paley inequality for the Carleson operator, J. Fourier Anal. Appl. 6 (2000), 457–466.
Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30, Princeton University Press, Princeton, NJ, 1970.
Stein, E. M. and Weiss, G., An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263–284.
Wiener, N. and Wintner, A., On the ergodic dynamics of almost periodic systems, Amer. J. Math. 63 (1941), 794–824.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lacey, M., Terwilleger, E. A Wiener–Wintner theorem for the Hilbert transform . Ark Mat 46, 315–336 (2008). https://doi.org/10.1007/s11512-008-0080-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-008-0080-2