### Harmonic Analysis for Dispersive Equations (Winter Semester 2017/18)

- Lecturer: apl. Prof. Dr. Peer Christian Kunstmann
- Classes: Lecture (0105350), Problem class (0105360)
- Weekly hours: 4+2

**Attention:** The 3rd homework class will take place on the 3rd of November 8 a.m in 2.66 in the math building

**Quotes** from the Preface of "Classical and Multilinear Analysis" (Vol. I) by Camil Muscalu and Wilhelm Schlag (Cambridge University Press, 2013):

"Harmonic analysis is an old subject. It originated with the ideas of Fourier in the early nineteenth century (which were preceded by work of Euler, Bernoulli, and others). These ideas were revolutionary at the time and could not be understood by means of the mathematics available to Fourier and his contemporaries. (...)

Research into the precise mathematical meaning of such Fourier series consumed the efforts of many mathematicians during the entire nineteenth century as well as much of the twentieth century. Many ideas that took their beginnings and motivations from Fourier series research became disciplines in their own right. Set theory (Cantor) and measure theory (Lebesgue) are clear examples, but others, such as functional analysis (Hilbert and Banach spaces), the spectral theory of operators, and the theory of compact and locally compact groups and their representations, all exhibit clear and immediate connections with Fourier series and integrals. (...)

Not surprisingly harmonic analysis is therefore a vast discipline of mathematics, which continues to be a vibrant research area to this day.In addition, over the past 60 years Euclidean harmonic analysis, as represented by the schools associated with A. Calderón and A. Zygmund at the University of Chicago as well as these associated with C. Fefferman and E. Stein at Princeton University, has been inextricably linked with partial differentiable equations(PDEs). While applicationsto the theory of elliptic PDEs and pseudodifferentiable operators were a driving force in the development of the the Calderón-Zygmund school from the very beginning, the past 25 years have also seen an influx of harmonic analysis techniques to the theory of nonlinear dispersive equations such as tthe Schrödinger and wave equations. These developments continue to this day."

**Comment**: In this lecture we will meet applications to both classes of PDEs mentioned here.

"The basic "divide and conquer" idea of harmonic analysis can be stated as follows: that we should study those classes of functions that arise in interesting contexts (...) by breaking them into basic consituent parts and that

(a) these basic parts are both as simple as possible and amenable to study and

(b) ideally, reflect some structure inherent to the problem at hand."

**Comment**: We shall see this idea at work in different situations.

"This fundamental idea is ubiquitous in science and engineering." etc

Schedule | ||
---|---|---|

Lecture: | Tuesday 11:30-13:00 | SR 2.66 |

Thursday 9:45-11:15 | SR 2.67 | |

Problem class: | Wednesday 8:00-9:30 | SR 2.66 |

Lecturers | ||
---|---|---|

Lecturer | apl. Prof. Dr. Peer Christian Kunstmann | |

Office hours: Thursday, 13 - 14 Uhr | ||

Room 2.027 Kollegiengebäude Mathematik (20.30) | ||

Email: peer.kunstmann@kit.edu | ||

Problem classes | Dr. Nikolaos Pattakos | |

Office hours: Mon. and Wed. 14:00-16:00 | ||

Room 2.024 Kollegiengebäude Mathematik (20.30) | ||

Email: nikolaos.pattakos@gmail.com |

# Content

Fourier transform, Fourier multipliers, interpolation, singular integral operators, Mihlin's Theorem, Littlewood-Paley decomposition, oscillatory integrals, dispersive estimates, Strichartz estimates, nonlinear equations

# Prerequisite

Functional Analysis (recommended)

# Exercise sheets

Sheet 14

On the 31st of January we will continue from Sheet 13, namely, the last question from Ex. 37 until Ex. 40.

Sheet 13

Sheet 12

Sheet 11

Sheet 10

Sheet 09

Sheet 08

Sheet 07

Sheet 06

Sheet 05

Sheet 04

Sheet 03

Sheet 02

Sheet 01

# Examination

Oral exam (about 25 min).

# References

- Serge Alinhac and Patrick Gérard: Opérateurs pseudo-differentiels et théorème de Nash-Moser, InterEditions/Editions du CNRS, 1991.
- Thierry Cazenave: Semilinear Schrödinger Equations, American Mathematical Society, 2003.
- Loukas Grafakos: Modern Fourier Analysis, 2nd edition, Springer, 2009.
- Yitzhak Katznelson: An introduction to harmonic analysis, 3rd edition, Cambridge University Press, 2004.
- Herbert Koch, Daniel Tataru, and Monica Visan: Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars 45, Birkh"auser, 2014.
- Camil Muscalu and Wilhelm Schlag: Classical and Multilinear Analysis, Vol I & II, Cambridge University Press, 2013.
- Elias M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
- Elias M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.
- Terence Tao: Nonlinear Dispersive Equations, CBMS, Regional Conference Series in Mathematics, Volume 106, AMS, 2006.