Last updated
August 27, 2020
|
|
THIS COURSE WILL REMAIN ACTIVE FOR THE DURATION OF THE
A.Y. 2020/21.
- All lectures, syllabus and course notes are uploaded on the webpage
on the
Ariel course website.
- If you wish to receive further information, please send
me an email.
|
Semestre: II
Anno accademico: 2019/20
Docenti: Marco Peloso
Ore di didattica: 42
Periodo delle lezioni: 2 marzo -- 20 maggio 2020.
Orario delle lezioni (Schedule) : Monday 13:30-!5:30,
Wednesday 8:30-10:30, room 6.
Modalità d'esame: prova
orale.
Prerequisits: Real Analysis and Fourier Analysis.
|
|
|
Materiale didattico
|
|
|
List of Topics
This course is designed as continuation of the course
Fourier Analysis, as introduction to various research
topics, and eventually as background material for a
"tesi di laurea magistrale".
|
During this course we will introduce the analysis of several instances
of partial differential operators and equations,
in R^d, and the theory of function spaces modelled to better
describe the regularity properties of such operators. Toward the
end of the course, we will discuss also sub-Laplacians in
some sub-Riemannian context.
- Review of the Fourier trasform in R^d;
singular integrals in R^d, in particular the Riesz transforms
and their L^p boundedness.
- Fractional integrals.
- Riesz and Bessel potentials.
- $L^2$-Sobolev spaces, their basic proprieties, embedding
theorem.
- Bounded Fourier multipliers.
- Complex interpolation method.
- $L^p$-Sobolev spaces, interpolation.
- Vector valued singular integrals.
- Littlewood--Paley theory.
- The Dirichlet problem for the Laplacian in the half-space, Poisson
kernel. Heat equation and heat kernel.
- Littlewood--Paley $g$-functions.
- Semigroups of operators.
- Spectral theorem for bounded self-adjoint operators.
- Elements of Time-Frequency analysis and Donoho--Stark uncertainty
principle.
- Short-Time Fourier Transform.
|
Bibliografy
Alcuni testi di riferimento:
- J. Duoandikoetxea, Fourier Analysis, Graduate Studies in
Mathematics 29, A.M.S., 2001
- L. Grafakos, Modern Fourier Analysis, 2nd Edtion,
Springer-Verlag, New York 2008.
- Karlheinz Gröchenig, Foundations of Time-Frequency Analysis,
Birkhäuser,
Applied and Numerical Harmonic Analysis, Boston 2001.
- F. Linares, G. Ponce, Introduction of Nonlinaear Dispersive
Equations, Second Edition, Springer University Texts, New York
2015.
- M. Peloso, Course Notes..
.
| | |