00020 - Advanced Analysis

Academic Year 2022/2023

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)

Learning outcomes

 Working knowledge of modern distribution theory and Fourier transform as the basic tools for the modern theory of Partial Differential Equations.

Course contents

 The method of characteristics for the solution of partial differential equations of order one. Theory of distributions and of their Fourier transform.


  1. L. Hörmander: Linear Partial Differential Operators, Springer Edizione del 1969, Chapter 1.
  2. L. Grafakos: Classical Fourier analysis, 3d edition (2014), Chapter 2, sections 2.2; 2.3; 2.4; Chapter 5, sections sezione 5.1.1.
  3. L. Hörmander: Linear Partial Differential Operators I, Springer 2nd edition 1990, chapters 2, 3, 4, 6, 7.
  4. F.G. Friedlander. Introduction to the theory of distributions. Second edition. With additional material by M. Joshi. Cambridge University Press, Cambridge, 1998.
  5. C. Parenti e A. Parmeggiani, Algebra Lineare ed Equazioni Differenziali Ordinarie, 2nd edition, Springer Unitext 117, capitolo 5.
  6. C. Zuily: Eléments de distributions et d'équations aux dérivées partielles. Dunod.

Teaching methods

In-person lectures consisting of theory, examples, exercises and applications, also aimed to the applied math curriculum.

Assessment methods

Course evaluation is based on a final, oral examination consisting of an exposition of a topic chosen by the student on the material covered in class, to be followed by questions pertaining to theorems proved in class, and/or the solution of exercises assigned during the course.


This course is on of the two-modules integrated course on Advanced Analysis and Differential Geometry: the course grade is the average of the grades earned in the two modules. A grade of 29 in the analysis module does not preclude from earning 30 Cum Laude for the integrated course

Office hours

See the website of Loredana Lanzani