BCAM-Severo Ochoa Course: Harmonic Analysis

DATES: January 10th - February 23rd, 2022, and February 11th 2022 (15 sessions)
TIME: From 16:00 to 18:00. (a total of 30 hours)
LOCATION: BCAM and Online

Inugural Lecture will be given by María Jesús Carro (Universidad Complutense de Madrid). More info at: BCAM Severo Ochoa Master Class: What is Harmonic Analysis?

1. Description:
This course aims to cover topics in Harmonic Analysis that we consider fundamental within the current development of the field. It is oriented towards graduate students (and possibly undergraduate students in the final year of the Mathematics' degree). The students are expected to have a knowledge on the basic properties and objects related to real variable methods of Fourier analysis.

The lectures will be held face-to-face and live-streamed, at BCAM facilities, during the period 10th January 2022 - 23rd February 2022, Mondays and Wednesdays from 16:00 to 18:00. It will also take place on 11th February 2022 from 16:00 to 18:00. The total number of hours is 30. The language of the course will be English.

The course will have two extra lectures:
- March 7th (Online via Zoom, projected at BCAM seminar room) - Javier Martínez Perales on Poincaré inequalities and related topics.

- March 8th (This talk will be hybrid: Streamed via Zoom while the speaker will be at BCAM seminar room) - Joao Pedro Ramos on Uncertainty Principles in harmonic analysis.

2. Topics for the course
The course is organised in two parts:
- The first part of the course will be a quick review of the basic background.
- The second part, is meant to introduce the students to a broader set of topics in harmonic analysis.

The goal is to provide them with the basic tools and references one needs to understand
modern research topics in harmonic analysis. In what follows, we provide a sequence of topics meant to be explored in the two parts the course. The contents of the second part of the course could be modi ed depending on time and feedback from the students. The order of the contents could be also modified.

2.1. First part.
(1) Convolutions and approximate identities:
- Young's inequality;
- Maximal operators of convolution type;
- Pointwise and Lp convergence of approximate identities.
(2) Real and Complex Interpolation:
- Riesz-Thorin interpolation;
- Marcinkiewicz interpolation.
(3) Fourier Transform:
- L1 + L2 Theory;
(4) The Schwartz Class and Tempered Distributions.
(5) Hardy-Littlewood maximal function and Lebesgue di erentiation:
- Vitali covering Lemma;
- Hardy-Littlewood-Wiener theorem;
- Lebesgue di erentiation and di erentiation bases.
(6) Calder on-Zygmund decomposition.
(7) Hilbert and Riesz Transform:
- Conjugate harmonic extension;
- Strong and weak Lp boundedness;
- Convergence of Fourier integrals.
(8) Singular Integrals:
- Calder on-Zygmund theorem;
- Littlewood-Paley Theory;
- Multipliers.

2.2 Second Part.

(1) Oscillatory Integrals:
- Phases without critical points;
- Van der Corput Lemma;
- Stationary Phase.
(2) Fourier Restriction:
- Stein-Tomas theorem;
- Knapp Examples;
- The restriction conjecture;
- Strichartz estimates;
- Bilinear estimates.
(3) Sparse Domination:
- Definition and properties of a sparse operator;
- Weighted inequalities via sparse domination.
(4) Weighted inequalities:
- The Ap condition and reverse Hölder Property;
- Strong and weak type weighted inequalities;
- Rubio de Francia's extrapolation.


REFERENCES:
[1] J. Duoandikoetxea, Fourier Analysis. Graduate Studies in Mathematics, 29, AMS, Providence, RI 2001.
[2] L. Grafakos, Classical Fourier Analysis. Graduate Texts in Mathematics, 249, Springer, New York, NY 2008.
[3] L. Grafakos, Modern Fourier Analysis. Graduate Texts in Mathematics, 250, Springer, New York, NY 2008.
[4] C. Muscalu, W. Schlag, Classical and Multilinear Harmonic Analysis. Cambridge University Press, New York NY 2013.
[5] E. Stein, Singular Integrals and Di erentiability Properties of Functions. Princeton University Press, Princeton, NJ 1970.
[6] E. Stein, Harmonic Analysis. Princeton University Press, Princeton, NJ 1993.
[7] E. Stein and G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ 1971.
[8] T. Wol , Lectures on Harmonic Analysis. American Mathematical Society, Providence, RI 2003.

*Registration is free, but mandatory before January 3rd, 2022. To sign-up go to form link and fill the registration form.

• BCAM-SO_Courses_2022_01_10