First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
MATHEMATICS
Course unit
HARMONIC ANALYSIS
SCL1001879, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
MATHEMATICS
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2017/18
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination HARMONIC ANALYSIS
Website of the academic structure http://matematica.scienze.unipd.it/2017/laurea_magistrale
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Branch PADOVA
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Lecturers
Teacher in charge PAOLO CIATTI MAT/05

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/05 Mathematical Analysis 6.0

Course unit organization
Period Second semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
hours
Hours of
Individual study
Shifts
Practice 2.0 16 34.0 No turn
Lecture 4.0 32 68.0 No turn

Calendar
Start of activities 26/02/2018
End of activities 01/06/2018

Examination board
Board From To Members of the board
6 Analisi Armonica - a.a. 2017/2018 01/10/2017 30/09/2018 CIATTI PAOLO (Presidente)
LAMBERTI PIER DOMENICO (Membro Effettivo)
ANCONA FABIO (Supplente)
LANZA DE CRISTOFORIS MASSIMO (Supplente)
MONTI ROBERTO (Supplente)
5 Analisi Armonica - a.a. 2016/2017 01/10/2016 31/12/2017 SJOGREN STEN OLOF PETER (Presidente)
LAMBERTI PIER DOMENICO (Membro Effettivo)
CIATTI PAOLO (Supplente)
LANZA DE CRISTOFORIS MASSIMO (Supplente)
MONTI ROBERTO (Supplente)

Syllabus
Prerequisites: Real Analysis. Some knowledge of Complex Analysis in one variable could be useful.
Target skills and knowledge: The main task of this class will be to present and discuss the Restriction Problem for the Fourier Transform, one of the deepest open problems in Mathematical Analysis.

To accomplish this task we will start from the very definition of the Fourier transform in the n-dimensional Euclidean space E.
Examination methods: Oral exam
Assessment criteria: Check of the learning of the taught notions and on the ability of their application.
Course unit contents: A basic problem in Harmonic Analysis is to determine all of the L^p-estimates that the Fourier transform obey: to find all the inequalities of the form ||F(f)||_p < C ||f||_q, where F(f) is the Fourier transform of a function f. This problem was solved in the early 20th century. The Hausdorff—Young inequalities, that we will prove and discuss, give all the inequalities of this form. The restriction problem concerns a generalisation of this problem where we replace the L^q(E) norm on the right hand side with an L^q norm on a surface S in E. In the sixties Elias M. Stein discovered several interesting things about this question. One important discovery is that the curvature of the surface S matters. The unit sphere in the Euclidean space obeys inequalities that
are qualitative different from the inequalities that obeys a unit flat disk of the same dimension. When the surface S is the unit sphere, Stein made a conjecture about all the restriction estimates associated to the sphere and proved some nontrivial cases. The two dimensional case in this conjecture was then proved by Charles Fefferman. We will prove the theorems of Stein and Fefferman and then state and discuss the conjecture. In dimension three and higher, this conjecture is wide open and looks very difficult.
Planned learning activities and teaching methods: Lectures. Recommended exercises.
Textbooks (and optional supplementary readings)
  • Thomas H. Wolff, Lectures in harmonic analysis. --: --, --. (http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf).
  • Elias M. Stein with the assistance of Timothy S. Murph, Harmonic analysis real-variable methods, orthogonality, and oscillatory integrals. Princeton: Princeton university press, 1993. Cerca nel catalogo