Harmonic Analysis
A.Y. 2019/2020
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 cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Singular integrals in Rᵐ. Recall of the theory of the Fourier tranform in Rᵐ: L¹ and L² theory, Schwartz functions and tempered distributions. Singular integrals in Rᵐ and their L^p boundedness.
Sobolev spaces. Definition and first properties. Littlewood—Paley decomposition. Embedding theorems.
Classical differential operators from mathematical physics. Laplace operator, heat and wave equations. Functions of the Laplacian, multipliers. Elliptic and hypoelliptic operators.
Subelliptic operators. Hormander's theorem of sums of squares. Pseudodifferendial operators. The Heisenberg group and the sub-Laplacian. Fundamental solution and the heat semigroup.
Extension of the theory to Riemannian and sub-Riemannian manifolds.
Sobolev spaces. Definition and first properties. Littlewood—Paley decomposition. Embedding theorems.
Classical differential operators from mathematical physics. Laplace operator, heat and wave equations. Functions of the Laplacian, multipliers. Elliptic and hypoelliptic operators.
Subelliptic operators. Hormander's theorem of sums of squares. Pseudodifferendial operators. The Heisenberg group and the sub-Laplacian. Fundamental solution and the heat semigroup.
Extension of the theory to Riemannian and sub-Riemannian manifolds.
Prerequisites for admission
Real Analysis, and Fourier Analysis.
Teaching methods
Traditional blackboard lectures.
Teaching Resources
-J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics 29, A.M.S., 2001
-F. Linares, G. Ponce, Introduction of Nonlinaear Dispersive Equations, Second Edition, Springer University Texts, New York 2015.
-M. Peloso, Course Notes.
-F. Linares, G. Ponce, Introduction of Nonlinaear Dispersive Equations, Second Edition, Springer University Texts, New York 2015.
-M. Peloso, Course Notes.
Assessment methods and Criteria
The final examination consists of an oral exam.
- During the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
- The final mark is given using the numerical range 0-30, and will be communicated immediately after the oral examination.
- During the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
- The final mark is given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Peloso Marco Maria
Shifts:
-
Professor:
Peloso Marco MariaProfessor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica