401-4463-62L  Fourier Analysis in Function Space Theory

SemesterAutumn Semester 2018
LecturersT. Rivière
Periodicitynon-recurring course
Language of instructionEnglish


401-4463-62 VFourier Analysis in Function Space Theory2 hrs
Thu13:15-15:00HG G 43 »
01.11.13:15-15:00HG G 26.1 »
T. Rivière

Catalogue data

AbstractIn the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
ContentDuring the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces.
In the second part of the course we will study fundamental pro­perties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occa­sion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings.
In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course.
If time permits we shall present the notion of Paraproduct, Para­compositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE.
Literature1) Elias M. Stein, "Singular Integrals and Differentiability Proper­ties of Functions" (PMS-30) Princeton University Press.
2) Javier Duoandikoetxea, "Fourier Analysis" AMS.
3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer.
4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer.
Prerequisites / NoticeNotions from ETH courses in Measure Theory, Functional Analysis I and II (Fun­damental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions)

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits4 credits
ExaminersT. Rivière
Typesession examination
Language of examinationEnglish
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examinationoral 30 minutes
This information can be updated until the beginning of the semester; information on the examination timetable is binding.

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Offered in

Doctoral Department of MathematicsGraduate SchoolWInformation
Mathematics MasterSelection: AnalysisWInformation