# Schedule of Spring/Summer School: Carleson theorems and Radon type behaviour

## Monday, May 26

08:45 - 09:30 |
Shaoming Guo: Singular and Maximal Radon Transform: Analysis and Geometry - Part I |

09:45 - 10:30 |
Joris Roos: Singular and maximal Radon transforms: analysis and geometry. Part 2 |

10:30 - 11:00 |
Coffee break |

11:00 - 11:45 |
Yumeng Ou: L^{p} theory for outer measures |

11:45 - 13:15 |
Lunch break |

13:15 - 14:00 |
Polona Durcik: L^{p} theory for outer measures and two themes of Lennart Carleson united (Part 2) |

14:00 - 14:45 |
Ioann Vasilyev: Extention of a multi-frequency maximal inequality of Bourgain |

15:00 - 16:00 |
Senior lecture - Tuomas Hytönen: Recent advances on weighted estimates in harmonic analysis |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:15 |
Luis Daniel Lopez-Sanchez: Uniform bounds for a Walsh model of the bilinear Hilbert transform |

## Tuesday, May 27

08:45 - 09:30 |
Guillermo Rey: A proof of boundedness of the Carleson operator |

09:45 - 10:30 |
Kevin Hughes: Single Annulus L^{p} Estimates for Hilbert Transforms along Vector Fields |

10:30 - 11:00 |
Coffee break |

11:00 - 11:45 |
Cristina Benea: L^{p} Estimates for the Hilbert Transform along a one-variable Vector Field |

11:45 - 13:15 |
Lunch break |

13:15 - 14:00 |
Prince Romeo Mensah: Weak-type (1 ; 1) bounds for oscillatory singular integrals with rational phases |

14:00 - 14:45 |
Jose M. Conde-Alonso: Oscillatory integrals related to Carleson's theorem |

15:00 - 16:00 |
Senior lecture - Detlef Müller: On L^{p} -multipliers for subelliptic operators |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:15 |
Theresa C. Anderson: A polynomial Carleson operator along the paraboloid |

## Wednesday, May 28

08:45 - 09:30 |
Bartosz Trojan: On maximal ergodic theorem for certain subsets of the integers |

09:45 - 10:30 |
Michal Warchalski: Cotlar's ergodic theorem along the prime numbers |

10:30 - 11:00 |
Coffee break |

11:00 - 11:45 |
Pavel Zorin-Kranich: Pointwise ergodic theorems for arithmetic sets |

11:45 - 16:00 |
- |

16:00 - 16:30 |
Tea and cake |

## Thursday, May 29

## Friday, May 30

08:45 - 09:30 (EG) |
Bartosz Trojan: On maximal ergodic theorem for certain subsets of the integers |

08:45 - 09:30 (UG) |
Jose M. Conde-Alonso: Oscillatory integrals related to Carleson's theorem |

09:45 - 10:30 (EG) |
Theresa C. Anderson: A polynomial Carleson operator along the paraboloid |

09:45 - 10:30 (UG) |
Michal Warchalski: Cotlar's ergodic theorem along the prime numbers |

10:30 - 11:00 |
Coffee break |

11:00 - 11:45 (EG) |
Pavel Zorin-Kranich: Pointwise ergodic theorems for arithmetic sets |

11:00 - 11:45 (UG) |
Gennady Uraltsev: The (weak- L2) boundedness of the quadratic Carleson operator |

15:00 - 16:00 |
Senior lecture - Tanja Eisner: A generalisation of the Wiener-Wintner theorem |

16:00 - 16:30 |
Tea and cake |

# Abstracts

#### Theresa C. Anderson: A polynomial Carleson operator along the paraboloid

We prove an L^{p} bound for a polynomial Carleson operator integrated along a parabolloid. This integration introduces Radon-type behavior which leads to new innovation in the techniques and style of proof.

#### Cristina Benea: L^p Estimates for the Hilbert Transform along a one-variable Vector Field

Boundedness of the Hilbert transform along a non-vanishing vector field was proved in [2], under the assumptions that the vector field only depends on one variable and 2/3 < p < ∞. Here we present the main ideas of the proof.

#### Jose M. Conde-Alonso: Oscillatory integrals related to Carleson's theorem

The authors of [3] prove an (almost) generalization of the famous Carleson theorem on oscillatory integrals of the second kind, hence initiating the study of variants of the Carleson operator. We summarize their results.

#### Polona Durcik: L^p theory for outer measures and two themes of Lennart Carleson united (Part 2)

We discuss an application of outer L^{p} spaces in time-frequency analysis. Using a generalized Carleson embedding theorem we reprove bounds for the bilinear Hilbert transform.

#### Luis Daniel Lopez-Sanchez: Uniform bounds for a Walsh model of the bilinear Hilbert transform

We study the L^{p} boundedness behaviour of Walsh analogues of bilinear Hilbert transforms at the known region of exponents and beyond. The main tool at our disposal for exponents close to 1 is a multi-frequency Calderon-Zygmund decomposition.

#### Prince Romeo Mensah: Weak-type (1 ; 1) bounds for oscillatory singular integrals with rational phases

This paper considers singular integral operators on R with an oscillatory factor that has a rational phase R(x) = P(x) = Q(x). Relying only on the degrees of P and Q, the paper derives weak-type (1 ; 1) bounds for such operators and establishes conditions for which these operators map the Hardy norm H^{1} into itself or into L^{1.}

#### Yumeng Ou: L^p theory for outer measures

We develop a theory of L^{p} spaces based on outer measures, which includes as a special case the classical L^{p} theory on Euclidean spaces. As an application, we rephrase several classical results concerning Carleson embedding, paraproducts and the T(1) theorem in the language of outer measure spaces.

#### Guillermo Rey: A proof of boundedness of the Carleson operator

We give a short summary of the proof of the weak-type (2,2) boundedness of the Carleson operator in [5].

#### Joris Roos: Singular and maximal Radon transforms: analysis and geometry. Part 2

The authors of [1] show L^{p} boundedness of a class of singular Radon transforms and their corresponding maximal operators under some curvature assumption. This condition can be formulated in essentially three dierent ways which are all equivalent.

#### Bartosz Trojan: On maximal ergodic theorem for certain subsets of the integers

We prove L^{2}-boundedness of a maximal function for averages along the squares (n^{2} : n ∈ N).

#### Gennady Uraltsev: The (weak- L 2 ) boundedness of the quadratic Carleson operator

[1] provides a proof of weak L^{2} boundedness of the Carleson operator with both linear and quadratic modulation terms. The proof is similar to [3] and it extends the time-frequency tile approach to be able to study the quadratic Carleson operator in terms of its vaster space of symmetries.

#### Ioann Vasilyev: Extention of a multi-frequency maximal inequality of Bourgain

We give a proof of a strenghtened version of Bourgain's multi-frequency maximal inequality. Proof contains one nice version of Calderon-Zygmund decomposition.

#### Michal Warchalski: Cotlar's ergodic theorem along the prime numbers

We prove Cotlar's ergodic theorem modeled on the set of primes.