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. The knowledge is supposed to be
somehow exhaustive for a mathematician.
He/she will be able to solve exercises and to prove autonomously
basic properties in the topics introcuced in the course.
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 advanced scientific books on the discipline.

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

a) AC and BV functions. Nowhere differentiable functions.

b) Differentiation of measures. Radon Nikodim Theorem. Hardy-Littlewood maximal function.

c) Distributions. Temperate distributions.
Fourier transform of functions and temperate distributions.

d) Sobolev spaces in 1 D. Sobolev embeddings,
Rellich theorem. Poincaré inequality. Boundary value problems in 1D.

e) Sobolev spaces in N D. Sobolev-GagliardoNirenberg theorem. Morrey theorem. Rellich theorem. Poincaré inequality
in N D. Boundary value problems in N D.

- 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

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.

Lectures, exercises and discussions in class. The didactical matherial will be
put on Moodle. Homeworks will be proposed to the students and a
part of them will be discussed in class.

The course will possibly host the lessons of a professor in the framework
of an Erasmus exchange.

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

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs)