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Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform

Published online by Cambridge University Press:  01 October 2008

CIPRIAN DEMETER
CIPRIAN DEMETER*
Affiliation:
Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA (email: [email protected])
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Abstract

We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of Lp spaces. This contrasts with the positive behavior exhibited by these averages in a different range, as proved in Demeter et al [Maximal multilinear operators. Trans. Amer. Math. Soc.360(9) (2008), 4989–5042]. We also prove that the trilinear Hilbert transform is unbounded in a similar range of Lp spaces. The principle underlying these constructions is stated, setting the stage for more general results.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1453 - 1464
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Assani, I.. Pointwise convergence of ergodic averages along cubes. Preprint.Google Scholar
[2]Assani, I.. Multiple recurrence and almost sure convergence for weakly mixing dynamical systems. Israel J. Math. 103 (1998), 111–124.CrossRefGoogle Scholar
[3]Bourgain, J.. Double recurrence and almost sure convergence. J. Reine Angew. Math. 404 (1990), 140–161.Google Scholar
[4]Christ, M.. On certain elementary trilinear operators. Math. Res. Lett. 8(1–2) (2001), 43–56.CrossRefGoogle Scholar
[5]Demeter, C., Tao, T. and Thiele, C.. Maximal multilinear operators. Trans. Amer. Math. Soc. 360(9) (2008), 4989–5042.CrossRefGoogle Scholar
[6]Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerdi on arithmetic progressions. J. Analyze Math. 31 (1977), 204–256.CrossRefGoogle Scholar
[7]Gowers, W. T.. A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11(3) (2001), 465–588.CrossRefGoogle Scholar
[8]Halmos, P.. Lectures in Ergodic Theory. Chelsea Publishing Co., New York, 1956.Google Scholar
[9]Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397–488.CrossRefGoogle Scholar
[10]Lacey, M. and Thiele, C.. L p estimates on the bilinear Hilbert transform for . Ann. of Math. (2) 146(3) (1997), 693–724.CrossRefGoogle Scholar
[11]Lacey, M. and Thiele, C.. On Caldern’s conjecture. Ann. of Math. (2) 149(2) (1999), 475–496.CrossRefGoogle Scholar
[12]Lacey, M.. The bilinear maximal functions map into L p for 2/3<p≤1. Ann. of Math. (2) 151(1) (2000), 35–57.CrossRefGoogle Scholar
[13]Lesigne, E.. Sur la convergence ponctuelle de certaines moyennes ergodiques. C. R. Acad. Sci. Paris Sér. I Math. 298(17) (1984), 425–428.Google Scholar
[14]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 53–97.CrossRefGoogle Scholar