First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
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
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
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
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

No lecturer assigned to this course unit

ECTS: details
Type Scientific-Disciplinary 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
Hours of
Individual study
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 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 so-called 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 Hardy-Littlewood maximal operator and real interpolation will be introduced. Then the Calderón-Zygmund 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)