Harmonic Analysis

Timetable

Module aim (aims)

The aim of the module is to introduce the main ideas of harmonic analysis. We concentrate on harmonic analysis of functions defined on the Euclidean spaces $R^d$. The main concepts of the harmonic analysis on $\R^d$ will be defined, the relation among them will be described as well as some of their applications.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

It is assume that the student has knowledge congruent with the traditional course of mathematical analysis of functions of one and several real variables.

Syllabus

The proposal consists of 30 hours of lectures and 30 hours of exercises. The following topics will be discussed:
1. Spaces of functions integrable with p-power – completeness of the spaces, dual spaces , convolutions, regularization
2. Operators of strong and weak types, the Riesz-Thorin interpolation Theorem, Marcinkiewicz interpolation Theorem
3. The Fourier transform of integrable functions –elementary properties, the Riemann-Lebesgue Lemma.
4 The Hardy-Littlewood maxima function, Calderon-Zygmund decomposition,
5. The Schwartz spaces of rapidly decreasing functions and it Fourier transform, inversion formula
6. The Fourier transform of p-integrable functions , convolution, Hausdorff-Young inequality
7. The Fourier transform of functions from $L^2$ - the Plancherela theorem,
8. The Hilbert transform.
9. The Fourier trans form and smoothness and the compactness of the support – the Paley-Wiener Theorem
11. Singular integral operators: almost orthogonal, the Calderona-Zygmunda operators.
12. The Littlewood-Paley Theorem and quadratic functions,
13. Fourier multipliers - the Hormander theorem and the Feffermann theorem.
14. Hardy spaces and BMO spaces

Reading list

1. J. Duoandikoetxea, Fourier Analysis, AMS 2001
2. E.Stein, R.Shakarchi, Functional Analysis. Introduction to further topics in Analysis. Princeton Univ. Press 2011.
3. E.Stein, R.Shakarchi, Fourier Analysis. An introduction. Princeton Univ. Press 2003.
4. Th. Wolff, Lectures on Harmonic Analysis, AMS 2003