SUMMER SCHOOL
Carleson theorems and Radon like behavior.

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]

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]

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 Radonlike behavior to the Hilbert transform, and also gives a careful
and thorough treatment of twodimensional timefrequency 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]

Lp estimates for the Hilbert transforms along a onevariable 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]

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, 529545
This paper presents a variant of a CalderonZygmund 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]

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 multifrequency CalderonZygmund
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]

A proof of boundedness of the Carleson operator
by M. Lacey and C. Thiele
Math.Res. Lett 7 (2000) 361370
This paper presents Carleson's theorem, a core ingredient for many of
the other presentations.
[presenter: Guillermo Rey, Michigan State U.]

Weaktype (1,1) bounds for oscillatory singular integrals with rational phases
M. FolchGabayet 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 SteinWainger 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 ]

The (WeakL2 ) Boundedness of the Quadratic Carleson Operator.
by V. Lie
GAFA 19 no. 2 (2009) 457497.
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]

Oscillatory Integrals Related to Carleson's Theorem
by E. M. Stein and S. Wainger
Math. Res. Lett. 8 789800 (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]

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]

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]

Cotlar's ergodic theorem along the prime numbers
M. Mirek and B. Trojan
arXiv:1311.7572 (17 pp.)
This paper illustrates the current stateofthe 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 CalderonZygmund operator that encodes Radontype 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]

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), 545.
The Lp theory with p other than 2 can be skipped.
[presenter: Pavel Zorin Kranich, Hebrew University]

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, 489577.
Do introduction and Analysis part (part 3)
[presenter: Shaoming Guo, Bonn University]

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, 489577.
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]

TBA
[presenter: TBA]