The course deals with some important topics in Mathematical Analysis and is divided into two parts.
The first half is devoted to the Theory of Distributions. In the second part, we will deal with the theory of Sobolev Spaces.

Frontal lectures

The final examination will be oral

Main properties of Banach and L^p spaces.

First Part: Introduction to the Theory of Distributions. Tempered distributions. Measures. The notion of convolution in the framework of distributions. Fourier Transforms for tempered distributions. Applications to Partial Differential Equations, with particular emphasis on fundamental solutions for the most important Partial Differential Equations of Second Order.

Second Part: starting from the Theory of Distributions, we will introduce the definition of Sobolev Spaces, their main properties, Sobolev and Poincare' inequalities, trace theorems, Sobolev spaces with real indices, and dual Sobolev spaces.



First Part

Definition of distribution; main operations on distributions; notion of support of a distribution; compactly supported distributions; division of distributions; notion of convolution; tempered distributions; Fourier transform of a tempered distribution; link between degree of vanishing of a distribution and regularity of its transform; Paley-Wiener theorem; fundamental solution of Laplace equation in R^N; fundamental solution of the heat equation in R^N; Cauchy problem for the heat equation in R^N; fundamental solution for the wave equation in R^2 and in R^3.



Second Part

Sobolev spaces; Gagliardo-Nirenberg theorem; embedding of W^{1,p} into L^q; Rellich theorem; Poincaré inequality; Sobolev inequality; Sobolev spaces with non integer indices; applications to the regularity of solutions to elliptic equations; traces theorem in L^p and in W^{s,q}.

S. Kesavan: "Topics in functional analysis and applications". John Wiley & Sons, New York, 1989.



E. DiBenedetto: "Real Analysis". Birkhauser, Boston, 2002.



F.G. Friedlander: "Introduction to the theory of distributions". Cambridge University Press, Cambridge, 1998.



P.W. Ziemer: "Weakly differentiable functions. Sobolev spaces and functions of bounded variation". Springer-Verlag, New York, 1989.



H. Brezis, Analisi Funzionale, Liguori Editore, 1986.



W. Rudin, Functional Analysis, McGraw-Hill Series in

Higher Mathematics, McGraw-Hill, 1973.