Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Maximal multilinear operators


Authors: Ciprian Demeter, Terence Tao and Christoph Thiele
Journal: Trans. Amer. Math. Soc. 360 (2008), 4989-5042
MSC (2000): Primary 42B25; Secondary 37A45
DOI: https://doi.org/10.1090/S0002-9947-08-04474-7
Published electronically: April 21, 2008
MathSciNet review: 2403711
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish multilinear $ L^p$ bounds for a class of maximal multilinear averages of functions of one variable, reproving and generalizing the bilinear maximal function bounds of Lacey (2000). As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system.


References [Enhancements On Off] (What's this?)

  • 1. I. Assani, Pointwise convergence of ergodic averages along cubes, preprint.
  • 2. I. Assani, Multiple recurrence and almost sure convergence for weakly mixing dynamical systems, Israel J. Math. 103 (1998), 111-124. MR 1613556 (99f:28021)
  • 3. J. Barrionuevo and M. Lacey, A weak-type orthogonality principle, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1763-1769. MR 1955263 (2004f:42025)
  • 4. J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. Math. IHES 69 (1989), 5-45. MR 1019960 (90k:28030)
  • 5. J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math. 404 (1990), 140-161. MR 1037434 (91d:28029)
  • 6. M. Christ, On certain elementary trilinear operators, Math. Res. Lett. 8 (2001), no. 1-2, 43-56. MR 1825259 (2002e:47077)
  • 7. C. Demeter, Divergence ofcombinatorial averages, preprint.
  • 8. C. Demeter, Pointwise convergence of the ergodic bilinear Hilbert transform, accepted for publication in the Illinois Journal of Mathematics.
  • 9. C. Demeter, T. Tao and C. Thiele, A trilinear maximal function via arithmetic combinatorics, work in progress.
  • 10. C. Demeter, M. Lacey, T. Tao and C. Thiele, Breaking the duality in the Return Times Theorem, preprint.
  • 11. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • 12. H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerdi on arithmetic progressions, J. Analyze Math. 31 (1977), 204-256. MR 0498471 (58:16583)
  • 13. B. J. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, preprint.
  • 14. B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), no. 1, 397-488. MR 2150389 (2007b:37004)
  • 15. M. Lacey, The bilinear maximal functions map into $ L\sp p$ for $ 2/3<p\leq1$, Ann. of Math. (2) 151 (2000), no. 1, 35-57. MR 1745019 (2001b:42015)
  • 16. E. Lesigne, Sur la convergence ponctuelle de certaines moyennes ergodiques, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 17, 425-428. MR 765266 (86d:28019)
  • 17. J-M. Derrien and E. Lesigne, Un théorème ergodique polynomial ponctuel pour les endomorphismes exacts et les $ K$-systèmes, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 6, 765-778. MR 1422310 (98k:28023)
  • 18. C. Muscalu, T. Tao and C. Thiele, Multilinear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), no. 2, 469-496. MR 1887641 (2003b:42017)
  • 19. C. Muscalu, T. Tao and C. Thiele, $ L^p$ estimates for the ``Biest'' I. The Walsh case, Math. Ann. 329 (2004), no. 3, 401-426. MR 2127984 (2005k:42053)
  • 20. C. Muscalu, T. Tao and C. Thiele, $ L^p$ estimates for the ``Biest'' II. The Fourier model, Math. Ann. 329 (2004), no. 3, 427-461. MR 2127985 (2005k:42054)
  • 21. E. M. Stein, On limits of sequences of operators, Ann. of Math. 74 (1961): 140-170. MR 0125392 (23:A2695)
  • 22. E. Szemerédi, On sets of integers containing no $ k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199-245. MR 0369312 (51:5547)
  • 23. T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc. 20 (2007), 53-97. MR 2257397

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B25, 37A45

Retrieve articles in all journals with MSC (2000): 42B25, 37A45


Additional Information

Ciprian Demeter
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: demeter@math.ucla.edu

Terence Tao
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: tao@math.ucla.edu

Christoph Thiele
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: thiele@math.ucla.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04474-7
Keywords: Maximal operators, multilinear averages
Received by editor(s): November 30, 2005
Received by editor(s) in revised form: October 27, 2006
Published electronically: April 21, 2008
Additional Notes: The first author was supported by NSF Grant DMS-0556389
The second author was supported by NSF Grant CCF-0649473 and a grant from the McArthur Foundation
The third author was supported by NSF Grants DMS-0400879 and DMS-0701302
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society