HARMONIC ANALYSIS

Second cycle degree in MATHEMATICS

Campus: PADOVA

Language: English

Teaching period: Second Semester

Lecturer: -- --

Number of ECTS credits allocated: 6

Syllabus
 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 CalderÃ³n-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.