# ADVANCED ANALYSIS - mod. B (529SM-2)

#### A.Y. 2020 / 2021

Professor
Period
Second semester
Credits
6
Duration/Length
48
Type of Learning Activity
Programme specific subjects
Study Path
[PDS0-2018 - Ord. 2018] common
Mutuazione
Mutuato: SM34 - 529SM-2 - ADVANCED ANALYSIS - MOD. B
Syllabus
Teaching language

English

Learning objectives

By the end of the course the student will know the fundamental results of functional analysis, of the theory of AC and BV functions, of distribution theory and of the theory of Sobolev Spaces.
He/she will be able to solve easy exercises and to prove autonomously easy theorems.
The student will be able to recognise the classical techniques in functional analysis (duality, weak form of the
differential problems, approximation) and will be able to discuss with a certain competence on such topics.
The student will be able to read the scientific books on the discipline.

Prerequisites

Differential and integral calculus. Linear algebra. Basic topology. Topics in
functional analysis: Hahn-Banach theorem, weak topologies, L^P spaces,

Contents

1. AC and BV functions. Nowhere differentiable functions. Lebesgue theorem on differentiability a.e. of monotone functions. Functions with bounded variation (BV). Absolutely continuous functions (AC), fundamental theorem of calculus in Lebesgue framework.

2. Differentiation of measures. Hanh decomposition theorem, Radon-Nikodim Theorem. Hardy-Littlewood maximal function. Symmetric derivative of a measure, Lebesgue points theorem for L^1 functions.

3. Distributions. Test functions. Distributions of finite and infinite order. Derivatives in the sense of distributions. The "Théoème de structure". Distributon with compact support. Convolution of distributions. Fourier transform of L^1 functions. Schwartz space, Temperate distributions. Fourier transform of temperate distributions. Plancherel theorem. Fourier-Laplace transform of a distribution with compact support. Paley-Wiener theorem.

4. Sobolev spaces in 1 D. Definition and characterization of Sobolev Spaces in 1 D. Results of extension and density. Sobolev embeddings, Rellich theorem. Poincaré inequality. Boundary value problems in 1D. Maximum principle in 1D.

5. Sobolev spaces in N D. Friedrichs lemma. BV functions in N D. Extension of Sobolev functions defined in open sets. Sobolev-Gagliardo-Nirenberg theorem, Morrey theorem. Rellich theorem. Poincaré inequality in N D. Boundary value problems in n D.

Teaching format

Lectures and exercises in class. A part of the didactical matherial will be put on Moodle. Some homeworks will be proposed and discussed in a second time in class.

Extended Programme

1. AC and BV functions. Nowhere differentiable functions. Lebesgue theorem on differentiability a.e. of monotone functions. Functions with bounded variation (BV). Absolutely continuous functions (AC), fundamental theorem of calculus in Lebesgue framework.

2. Differentiation of measures. Hanh decomposition theorem, Radon-Nikodim Theorem. Hardy-Littlewood maximal function. Symmetric derivative of a measure, Lebesgue points theorem for L^1 functions.

3. Distributions. Test functions. Distributions of finite and infinite order. Derivatives in the sense of distributions. The "Théoème de structure". Distributon with compact support. Convolution of distributions. Fourier transform of L^1 functions. Schwartz space, Temperate distributions. Fourier transform of temperate distributions. Plancherel theorem. Fourier-Laplace transform of a distribution with compact support. Paley-Wiener theorem.

4. Sobolev spaces in 1 D. Definition and characterization of Sobolev Spaces in 1 D. Results of extension and density. Sobolev embeddings, Rellich theorem. Poincaré inequality. Boundary value problems in 1D. Maximum principle in 1D.

5. Sobolev spaces in N D. Friedrichs lemma. BV functions in N D. Extension of Sobolev functions defined in open sets. Sobolev-Gagliardo-Nirenberg theorem, Morrey theorem. Rellich theorem. Poincaré inequality in N D. Boundary value problems in n D.

End-of-course test

Final oral examination on the whole content of the course. Also some easy exercise will be possibly part of the examination.

Other information

The course will possibly host the lessons of a professor in the framework of an Erasmus exchange.
Any possible change to what described here, which will become necessary to ensure the application of the safety protocols related to the COVID19 emergency, will be communicated on the Department, Study Program and teaching website.

Texts/Books

- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer.

Units Libray position
https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0345510

- E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer.

Units Libray position
https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0608477

- L. Hörmander, Linear Partial Differential Operators, Spinger.

Units Libray position
https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0714367

- A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover.

Units Libray position
https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0549953

- W. Rudin, Real and Complex Analysis, McGraw-Hill.

Units Libray position
https://www.biblioest.it:443/SebinaOpac/.do?idopac=TSA0238135

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