
Course unit
HARMONIC ANALYSIS
SCL1001879, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2017/18
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
6.0 
Mode of delivery (when and how)
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Organisation of didactics
Type of hours 
Credits 
Hours of teaching 
Hours of Individual study 
Shifts 
Practice 
2.0 
16 
34.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
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)

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 ndimensional 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^pestimates 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 Hausdorffâ€”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 realvariable methods, orthogonality, and oscillatory integrals. Princeton: Princeton university press, 1993.


