
Course unit
HARMONIC ANALYSIS
SCL1001879, A.A. 2015/16
Information concerning the students who enrolled in A.Y. 2015/16
ECTS: details
Type 
ScientificDisciplinary 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 
Examination board
Board 
From 
To 
Members of the board 
7 Analisi Armonica  a.a. 2018/2019 
01/10/2018 
30/09/2019 
LANZA DE CRISTOFORIS
MASSIMO
(Presidente)
LAMBERTI
PIER DOMENICO
(Membro Effettivo)
ANCONA
FABIO
(Supplente)
CIATTI
PAOLO
(Supplente)
MONTI
ROBERTO
(Supplente)
MUSOLINO
PAOLO
(Supplente)

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)

4 Analisi Armonica  a.a. 2015/2016 
01/10/2015 
30/09/2016 
SJOGREN
STEN OLOF PETER
(Presidente)
LAMBERTI
PIER DOMENICO
(Membro Effettivo)
ANCONA
FABIO
(Supplente)
CIATTI
PAOLO
(Supplente)
MONTI
ROBERTO
(Supplente)

Examination methods:

oral examination 
Course unit contents:

The lecture course is mainly devoted to the theory of singular integrals. Singular integral theory has its roots in the early 20th century and in complex function theory. In the 1950's, it was extended to real Euclidean spaces of arbitrary finite dimension, and linked to the Laplacian and other elliptic operators. It turned out to be a very useful tool to treat many partial differential equations, and this led to more general versions. The theory still relied heavily on Fourier analysis for the basic L^2 estimate. But in the 1980's, other methods were developed to deal with the L^2 case, the socalled T1 theorem and generalizations of it. This meant vast extensions of the theory and its applications.
The course will start with the Hilbert and Riesz transforms, which is the classical theory, related to analytic functions and the Laplacian. These operators are invariant under translation, and given by a
convolution kernel. Necessary notions such as weak L^p spaces, the HardyLittlewood maximal operator and real interpolation will be introduced. Then the CalderÃ³nZygmund decomposition will be given, as a fundamental tool to go from L^2 to L^p estimates. Here the singular integrals
need not be translation invariant, and their kernels will depend on two variables. The space BMO (bounded mean oscillation) will then be defined, studied and applied to the singular integrals. This will allow us to state the important T1 theorem. Its proof requires the development of
some tools, like Cotlar's lemma and Carleson measures.
If time allows, we may move to some other model of harmonic analysis, defined in terms of expansion in classical orthogonal polynomials. These models are quite important in both classical and modern physics. There we shall deal with Riesz transforms and other singular integrals. 
Textbooks (and optional supplementary readings) 

Stein, Elias M., Singular integrals and differentiability properties of functionsElias M. Stein. Princeton (N.J.): Princeton university press, 1970.

Stein, Elias M.; Murphy, Timothy S., Harmonic analysisrealvariable methods, orthogonality, and oscillatory integralsElias M. Steinwith the assistance of Timothy S. Murphy. Princeton (N.J.): Princeton university press, 1993.


