SUMMER SCHOOL
Carleson theorems and Radon like behavior.

  1. Lp theory for outer measures and two themes of Lennart Carleson united (Part 1)
    by Yen Do, Christoph Thiele
    arXiv:1309.0945 (45 pp.)
    Sections 2, 3 and 4: Outer measures, outer Lp spaces, Carleson embeddings, paraproducts, and the T (1) theorem.
    [presenter: Yumeng Ou, Brown University]

  2. Lp theory for outer measures and two themes of Lennart Carleson united (Part 2)
    by Yen Do, Christoph Thiele
    arXiv:1309.0945 (45 pp.)
    Sections 5 and 6: Embedding theorems and application to the bilinear Hilbert transform.
    [presenter: Polona Durcik, Bonn University]

  3. Single annulus Lp estimates for Hilbert transforms along vector fields
    by M. Bateman
    Rev. Mat. Iberoam. 29 no. 3, 1021 - 1069 (2013).
    This paper introduces Radon-like behavior to the Hilbert transform, and also gives a careful and thorough treatment of two-dimensional time-frequency tools. The main theorem to prove is Theorem 2.1 for 1 < p < ∞. The companion to this paper is Bateman and Thiele [4].
    [presenter: Kevin Hughes, University of Edinburgh]

  4. Lp estimates for the Hilbert transforms along a one-variable vector field
    by M. Bateman and C. Thiele
    arxiv/1109.6396 (25 pp.)
    The introduction covers some history and the relation to [3]; the main theorem to prove is Theorem 1, for 3/2 < p < infinity.
    [presenter: Cristina Benea, Cornell University]

  5. A Calderon Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain
    by F. Nazarov, R. Oberlin, C. Thiele
    Math. Res. Lett. 17 (2010) no. 3, 529-545
    This paper presents a variant of a Calderon-Zygmund decomposition in which the “bad” functions of mean zero are replaced by functions that are orthogonal to a finite collection of fre- quencies. This is then used to prove a variational norm maximal inequality. Prove Theorem 1.1 (fairly quick) and prove Theorem 1.2 (longer).
    [presenter: Ioann Vasiliev, St Petersburg University]

  6. New Uniform bounds for a Walsh model of the bilinear Hilbert transform
    by R. Oberlin and C. Thiele
    arXiv:1004.4019 (19 pp.)
    This applies a discrete version of the multi-frequency Calderon-Zygmund decomposition of [5]. Apply Theorem 1.2 of Nazarov, Oberlin, Thiele (2010), to prove Theorems 1.1 and 1.2.
    [presenter: Luis Lopez, ICMAT Madrid]

  7. A proof of boundedness of the Carleson operator
    by M. Lacey and C. Thiele
    Math.Res. Lett 7 (2000) 361-370
    This paper presents Carleson's theorem, a core ingredient for many of the other presentations.
    [presenter: Guillermo Rey, Michigan State U.]

  8. Weak-type (1,1) bounds for oscillatory singular integrals with rational phases
    M. Folch-Gabayet and J. Wright
    (preprint on Wright's webpage, 17 pp.)
    This paper considers a Hilbert transform with an oscillatory factor with rational phase. It opens the question of whether the Stein-Wainger approach for the polynomial Carleson oper- ator could be pushed to Carleson operators with certain types of rational phase.
    [presenter: Prince Romeo Mensah, Bonn University ]

  9. The (Weak-L2 ) Boundedness of the Quadratic Carleson Operator.
    by V. Lie
    GAFA 19 no. 2 (2009) 457-497.
    Prove Theorem 1 of this paper. Preparation for this paper includes Fefferman (1973); this also relates to the results of Stein and Wainger (2001) and Pierce and Yung (2013), although the methods are completely different.
    [presenter: Gennady Uraltsev, Bonn University]

  10. Oscillatory Integrals Related to Carleson's Theorem
    by E. M. Stein and S. Wainger
    Math. Res. Lett. 8 789-800 (2001)
    Prove Theorem 1 on the auxiliary oscillatory integral operator, and then deduce Theorem 2 for L2 bounds for the polynomial Carleson operator (lacking linear terms). Along the way the speaker will prove van der Corput estimates and small set maximal function estimates that will be very relevant to Pierce and Yung (2013). The speaker can also use the method made explicit in [11] to derive Lp bounds for 1 < p < infinity, so that this can be omitted from the presentation of [11].
    [presenter: Jose Conde, ICMAT Madrid]

  11. Polynomial Carleson operators along the paraboloid
    by L. B. Pierce and P.-L. Yung
    preprint(2014)
    Prove the key theorem, an L2 estimate for an auxiliary oscillatory integral operator, and deduce the L2 result for the polynomial Carleson operator (lacking linear terms and a certain type of quadratic term) via a square function argument.
    [presenter: Tess Anderson, Brown University]

  12. On the maximal ergodic theorem for certain subsets of the integers.
    by J. Bourgain
    Israel Journal of Mathematics, Vol. 61, no. 1 (1988)
    This paper represents one of the earliest systematic treatments of a discrete operator via the circle method of Hardy and Littlewood; the model case L2 bound proved in detail (Theorem 1) is a discrete maximal Radon transform involving the parabola. If time permits, the corresponding pointwise convergence statement of Theorem 5 can be treated.
    [presenter: Bartosz Trojan, Wroclaw University]

  13. Cotlar's ergodic theorem along the prime numbers
    M. Mirek and B. Trojan
    arXiv:1311.7572 (17 pp.)
    This paper illustrates the current state-of-the art of the discrete methods initiated in [12]. The main theorem (Theorem 2) proves the p boundedness (1 < p < infinity) of a maximal truncated 1- dimensional Calderon-Zygmund operator that encodes Radon-type behavior by summing only over prime numbers. Theorem 1 states a corresponding pointwise convergence result for an ergodic truncated Hilbert transform along the primes.
    [presenter: Michal Warchalski, Bonn University]

  14. Pointwise ergodic theorems for arithmetic sets. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein.
    J. Bourgain
    Inst. Hautes Etudes Sci. Publ. Math. No. 69 (1989), 5--45.
    The Lp theory with p other than 2 can be skipped.
    [presenter: Pavel Zorin Kranich, Hebrew University]

  15. Singular and maximal Radon transforms: analysis and geometry. Part 1
    M. Christ, A. Nagel, E. Stein, S. Wainger
    Ann. of Math. (2) 150 (1999), no. 2, 489--577.
    Do introduction and Analysis part (part 3)
    [presenter: Shaoming Guo, Bonn University]

  16. Singular and maximal Radon transforms: analysis and geometry. Part 2
    M. Christ, A. Nagel, E. Stein, S. Wainger
    Ann. of Math. (2) 150 (1999), no. 2, 489--577.
    Do the equivalence of the curvature conditions, Section II. Section I, the background material should only explicitly be discussed as needed.
    [presenter: Joris Roos, Bonn University]

  17. TBA



    [presenter: TBA]