Harmonic Analysis
A.Y. 2024/2025
Learning objectives
The aim of the course is:
- to bring students to research topics, or to the preparation of the Tesi Magistrale;
- to provide basic notions on analysis of function spaces and differential equations on Riemannian and sub-Riemannian manifolds.
- to bring students to research topics, or to the preparation of the Tesi Magistrale;
- to provide basic notions on analysis of function spaces and differential equations on Riemannian and sub-Riemannian manifolds.
Expected learning outcomes
At the end of the course, students will acquire the basic knowledge of the analysis on differentiable manifols, and they will be able to apply it to exercises that need also computational techniques
Lesson period: Second semester
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
The Spectral Theorem for Unbounded Operators
- Some functional calculus
- Spectrum of a bounded operator
- Square root of a positive operator
- Projection-valued measures
- The spectral theorem for bounded operators
- The spectral theorem for bounded normal operators
- Unbounded operators
- The Cayley transform
- The spectral theorem for unbounded self-adjoint operators
- Unitary invariance
Semigroups of operators
- The Stone's theorem
Pseudodifferential operators
- Various definitions
- Non-local operators
- Boundary values for non-local operators
The fractional Laplacian
- The fundamental solution
- Bessel equations and functions
- The Caffarelli- Silvestre extension property
- Some functional calculus
- Spectrum of a bounded operator
- Square root of a positive operator
- Projection-valued measures
- The spectral theorem for bounded operators
- The spectral theorem for bounded normal operators
- Unbounded operators
- The Cayley transform
- The spectral theorem for unbounded self-adjoint operators
- Unitary invariance
Semigroups of operators
- The Stone's theorem
Pseudodifferential operators
- Various definitions
- Non-local operators
- Boundary values for non-local operators
The fractional Laplacian
- The fundamental solution
- Bessel equations and functions
- The Caffarelli- Silvestre extension property
Prerequisites for admission
The contents of the courses in Mathematical Analysis 1 to 4. Basics in General Topology, in Linear Algebra, and in Real Analysis, in particular is fundamental the familiarity with L^p and Hilbert spaces.
Familiarity with the contents of the class Analisi di Fourier is strongly suggested.
Familiarity with the contents of the class Analisi di Fourier is strongly suggested.
Teaching methods
The course is offered as standard blackboard lectures.
Teaching Resources
-- Course Notes
-- L. Grafakos, Classical Fourier Analysis, Springer-Verlag Ed., New York 2008
-- L. Grafakos, Classical Fourier Analysis, Springer-Verlag Ed., New York 2008
Assessment methods and Criteria
During the semester, the student will be assigned some homework consisting of some exercises. The final examination consists of an oral examination. The student will be required to illustrate and to discuss results presented during the course or directly connected with them, as well as to solve problems in that context, in order to evaluate her/his knowledge and comprehension of the subjects covered as well as the ability in connecting and applying them correctly. The typical duration of the oral exam is about 45 minutes.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Peloso Marco Maria
Shifts:
Turno
Professor:
Peloso Marco MariaProfessor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica