Harmonic Analysis
A.Y. 2022/2023
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
Specific information on the delivery modes of the teaching activities for the academic year 2021/22 will be provided during the coming months, depending on the evolution of the public health.
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.
Interpolation of operators: the complex interpolation method.
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.
Interpolation of operators: the complex interpolation method.
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.
Prerequisites for admission
There are no mandatory prerequisit. However, a good knowledge of (many of) the topics of the courses Analisi Reale and Analisi di Fourier are strongly suggested.
Teaching methods
Classroom lessons with the use of a blackboard. Lecture notes provided.
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, Notes of the course
F. Linares, G. Ponce, Introduction of Nonlinaear Dispersive Equations, Second Edition, Springer University Texts, New York 2015.
M. Peloso, Notes of the course
Assessment methods and Criteria
The final examination consists of an oral exam.
- In the oral exam, the student will be required to illustrate concepts, examples and results presented during the course and will be required to solve problems quite similar at those presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
- In the oral exam, the student will be required to illustrate concepts, examples and results presented during the course and will be required to solve problems quite similar at those presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
Final marks are 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
Educational website(s)
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica