## S4B1 - Graduate Seminar on Analysis - Harmonic Analysis (winter term 2017/2018)

### Instructor:

Dr. Błażej Wróbel

### Date:

### Contents:

In this seminar we will survey various results from both classical and modern harmonic analysis. Some emphasis will be put on dimension-free estimates and sharpness of the constants.

### Bibliography:

- Loukas Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics Volume 249 (2008)
- Javier Duoandikoetxea, Fourier Analysis, Graduate Texts in Mathematics Volume 29 (2001)

### Requirements:

### List of seminar topics

- Introduction: definition of the Fourier transform and its basic properties (Inversion formula, Plancherel formula, etc.).
- Tempered distributions.
- Basics of interpolation of Lp spaces. The Marcinkiewicz and the Riesz-Thorin interpolation theorems.
- Hilbert transform.
- Calderón-Zygmund theory. Dimension-free estimates for the Riesz transforms.
- Hardy-Littlewood maximal operator. Dimension-free estimates in the ball case.
- Convolution operators and multiplier operators. Multiplier theorems.
- Fefferman's counterexample for the ball multiplier problem and the Bochner-Riesz multipliers.
- Spectral theorem and spectral multipiers.
- Basics of the theory of strongly continuous semigroups.
- Symmetric contraction semigroups. The Lp boundedness of their maximal functions.
- Semigroups with Gaussian bounds and spectral multiplier theorems.
- Dimension-free estimates in harmonic analysis 1.
- Dimension-free estimates in harmonic analysis 2.

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