# MA8106 Harmonic Analysis Spring Semester 2016

There will be a first meeting in room 922 in SB2 on January 5 at 10:15 a.m. A main purpose of this meeting will be to fix 4 weekly hours for lectures and exercises.

Book: Camil Muscalu and Wilhelm Schlag, “Classical and Multilinear Harmonic Analysis", Cambridge Studies in Advanced Mathematics Vol. 137, Cambridge University Press, Cambridge, 2013

We might consider topics from Volume 2 as well, but this will be decided later. The first volume starts with classical one-dimensional topics such as Fourier series, harmonic functions, and the Hilbert transform. Then the higher-dimensional Calderón–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as well as applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to Weyl calculus.

Auxiliary texts:

- E.M.Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press

- A.Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc.

- E.M. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, 1970 (old but very nice).

## Teaching Hours

- Wednesday, 9:15-11, in room 656 SB2. First lecture on January 13.
- Friday, 13:15-15, in room 734 SB2.

## Syllabus and requirements for the examination

- The syllabus for the course is as defined by the lectures described in detail below.

## Contents of the lectures

- January 13: Most of Chapter 1.1-1.3, basics from 1.4 in M&S.
- January 15: Results about embeddings in 1.4, then 1.5-1.6.
- January 20: 2.1-2.4 in M&S.
- January 22: Problem session (see below).
- January 27: 2.4 continued (almost everywhere convergence), 2.5 in M&S.
- January 29: 3.1-3.2 in M&S.
- February 3: 3.2 (continued), 3.3-3.4, 3.6 in M&S.
- February 5: Problem session (see below).
- February 10: 3.5, 4.1 in M&S. (We discuss also Riemann's proof of the functional equation for the zeta function using the Fourier transform of the Gaussian and Poisson's summation formula.)
- February 12: 4.1 ctd., 4.2 in M&S.
- February 17: 5.1, 5.2 (up to CLT) in M&S.
- February 19: Problem session (chapters 4-5; see below).
- February 24: 5.2 (ctd.) in M&S
- February 26: 5.3 in M&S.
- March 2: NO LECTURE
- March 4: 7.1, 7.2 in M&S.
- March 9: NO LECTURE
- March 11: 7.3, 7.4 in M&S.
- March 16: NO LECTURE
- March 18: NO LECTURE
- March 30: Problem session (see below)
- April 1: Lecture cancelled
- April 6: 7.5, 7.6 in M&S.
- April 8: 8.1, 8.2 in M&S.
- April 13: 8.4 (first part) in M&S.
- April 15: 8.4 (final part), 8.5 in M&S.
- April 20: 9.1 in M&S.
- April 22: Problem session (see below).

## Exercises

- January 22: Problems 1.1, 1.5, 1.6, 1.7, 1.9, 1.10, 1.13, 2.3, 2.4, 2.6, 2.8, 2.9, 2.10, 2.11.
- February 5: Problems 3.1, 3.2, 3.5, 3.6, 3.7, 3.9, 3.10, 3.11.
- February 19: Exercises 4.4, 4.5, 5.7; Problems 4.1, 4.5 (See this note for some remarks related to this problem), 4.6, 4.7, (5.2, 5.3. 5.7 postponed until March 15).
- March 30: Problems 5.2, 5.3. 5.7, (7.1, 7.2, 7.3, 7.4 postponed).
- April 23: Problems 7.1, 7.2, 7.3, 7.4, 8.3, 8.5, 8.7, 9.2.