SUMMER SCHOOL
Harmonic Analysis, Carleson Theorems and Multilinear Analysis

  1. $L\sp p$ estimates on the bilinear Hilbert transform for $2 by M. Lacey and C. Thiele
    Ann. of Math. (2) 146 (1997), no. 3, 693--724.


    The goal is to prove the Lp bounds for the bilinear Hilbert transform in the local L2 region (as in this paper). Much af the tools used here are used in other topics below. Some extra inspriations and simplifications can be drawn from Thiele, Christoph: Wave packet analysis. CBMS Regional Conference Series in Mathematics, 105. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. vi+86 pp. ISBN: 0-8218-3661-7 If (and only if) time allows, presenter may discuss modifications of the proof for the extended region of exponents as in "On Calderon's conjecture" by Lacey and Thiele, Ann. of Math. (2) 149 (1999), no. 2, 475--496.
    [presenter: Reguera]

  2. A proof of boundedness of the Carleson operator.
     by Lacey, M. and Thiele, C.
    Math. Res. Lett. 7 (2000), no. 4, 361--370.


    The goal is to prove weak type L2 bounds for Carleson's operator. Obviously the classical references are by Carleson, Hunt, and Fefferman. The approach in the paper above is the one closest to the language of the other papers in this summer school. Further inspiration/simplification can be drawn from Thiele's lecture notes as in topic 1.
    [presenter: Condori]

  3. The (weak-$L^2$) Boundedness of The Quadratic Carleson Operator by Lie, Victor
    http://arxiv.org/abs/0805.1580


    Prove the main theorem of the title. The paper obviously relates to topic 2, however, it refers much to C. Fefferman's proof of Carleson's theorem.
    [presenter: Gautam]

  4. Double recurrence and almost sure convergence.
    by Bourgain, J.
    J. Reine Angew. Math. 404 (1990), 140--161.


    Prove the main theorem of almost everywhere convergence for ergodic averages formed by iterating two powers of T. Some explanations/simplifications can be drawn from notes by C. Demeter that can be obtained directly from the co-organizer.
    [presenter: Stovall]

  5. Breaking the duality in the return times theorem. (Part 1)
    by Demeter, Lacey, Tao, Thiele
    Duke Math. J. 143 (2008), no. 2, 281--355. or http://arxiv.org/abs/math/0601455


    This paper proves a result from Ergodic Theory using Harmonic Analysis. It relies on a strengthening of the bounds for Carleson's operator, topic 2. The presenter should focus on the first 7 sections of the paper. This includes the transfer from ergodic averages to averages on the real line, the discretization and reduction to a model operator, and a brief discussion on trees.
    [presenter: LaVictoire]

  6. Breaking the duality in the return times theorem. (Part 2)
    by Demeter, Lacey, Tao, Thiele
    Duke Math. J. 143 (2008), no. 2, 281--355. or http://arxiv.org/abs/math/0601455


    This presentation is joint with the previous. It should cover sections 7-11. The emphasis should be on the combinatorics of trees (Sec 10). The proof of the weighted Bourgain's lemma may be sketched, if time permits. A version of this lemma also appears in topic 4 above.
    [presenter: Bateman]
  7. A Carleson type theorem for a Cantor group model of the scattering transform.
    by Muscalu, Tao, Thiele
    Nonlinearity 16 (2003), no. 1, 219--246. or http://arxiv.org/abs/math/0205139


    There is a well known open conjecture on a non-linear variant of Carleson's theorem. This paper proves a Cantor group variant of this conjecture, the main theorem shoudl be presented here. Interesting additional reading is the paper "A counterexample to a multilinear endpoint question of Christ and Kiselev." by the same authors in Math. Res. Lett. 10 (2003), no. 2-3, 237--246, also availabl on arxiv. This refers to the following topic and may be discussed if (and only if) time allows.
    [presenter: Dabkowski]

  8. WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials.
     by Christ M. and Kiselev, A.
    J. Funct. Anal. 179 (2001)


    This relates to the previous two topics in that it is the nonlinear variant of a maximal Hausdorff Young inequality, the conjectured endpoint of which is the above mentioned nonlinear Carleson conjecture. This topic relies on the following topic.
    [presenter: Palsson]

  9. Maximal functions associated to filtrations. by Christ, M. and Kiselev, A.
    J. Funct. Anal. 179 (2001), no. 2, 409--425.


    A beautiful piece of analysis that is needed in the previous topic.
    [presenter: Powder]

  10. On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields by Lacey, M and Li, X.
    http://arxiv.org/abs/0704.0808


    Part I: The main result of this paper can be regarded as a two dimensional analog of Carleson' theorem. The argument here bears some resemblance with the one in topic 2. The first speaker should go over Chapters 1,2 and 3.
    [presenter: Yung]

  11. On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields by Lacey, M and Li, X.
    http://arxiv.org/abs/0704.0808


    Part II: The second speaker should cover Chapter 4. The result in Chapter 5 may be briefly sketched, time permitting.
    [presenter: Kovac]

  12. On the multilinear restriction and Kakeya conjectures. by Bennett, Carbery, Tao
    Acta Math. 196 (2006), no. 2, 261--302. http://arxiv.org/abs/math/0509262


    Prove T 1.15 and T 1.16
    [presenter: Bond]

  13. The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture. by Guth, Larry
    http://arxiv.org/abs/0811.2251


    This paper proves the endpoint of the result from the previous topic. Prove the main Theorem.
    [presenter: Do]

  14. The Brascamp-Lieb inequalities: finiteness, structure and extremals. by Bennett, Carbery, Christ, Tao
    Geom. Funct. Anal. 17 (2008), no. 5, 1343--1415.


    This paper is somewhat long. Extract a 2 hour presentation from it.
    [presenter: Moen]

  15. On certain elementary trilinear operators by Christ, M.
    Math. Res. Lett. 8 (2001), no. 1-2, 43-56.


    This paper proves both positive and negative results for some (mostly single scale) averages. Give a flavor of both. Further applications and questions can also be found in "Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform", see http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.2494v1.pdf
    [presenter: Silva]

  16. On multilinear oscillatory integrals, nonsingular and singular. by Christ, Li, Tao, Thiele
    Duke Math. J. 130 (2005), no. 2, 321--351.


    This paper is mostly concerned with decay estimates for certain multilinear oscillatory integrals.
    [presenter: Oliveira]

  17. Bilinear Hilbert transforms along curves I. The monomial case by Li, Xiaochun
    http://arxiv.org/abs/0805.0107


    Prove Theorem 1.1. The key sections are 2, 3, and especially 7. The speaker should touch on the various methods employed in various regimes, like the TT^* method, estimates for various oscillatory integrals and uniformity.
    [presenter: Lie]

  18. A Variational norm Carleson Theorem notes



    Discuss the variational Carleson theorem and applications to nonlinear problems.
    [presenter: Oberlin]

  19. Sum rules and spectral measures of Schrödinger operators with L2 potentials by Killip, Simon
    http://arxiv.org/abs/0608767


    Outline the proof of the main theorem.
    [presenter: Krueger]

  20. Bi-parameter paraproducts by Muscalu, Pipher, Tao, Thiele
    http://front.math.ucdavis.edu/math.CA/0310367
    Acta Math 193 (2004) no 2 269-296

    Outline the proof of the main theorem.
    [presenter: Ott]