Operatori pseudodifferenziali su varietà lisce

In this thesis we present various aspects of the theory of pseudodifferential operators on smooth manifolds. The first chapter is devoted to an overview of the basic theory of function spaces, Fourier transform and smooth manifolds, which we use extensively. In the second chapter we recall the fundamental ideas of Schwartz space and tempered distributions, and the definition of pseudodifferential operators in the Euclidean setting, that is, operators which generalize differential operators by means of Fourier analysis. A great deal of attention is given to elliptic operators, which turn out to be especially important in the theory of regularity for (pseudo)differential problems. We then use these operators to define Sobolev spaces of real order, modelled on the spaces of p-summable functions. Finally, we state several important continuity results for pseudodifferential operators, and employ them to describe the theory of elliptic regularity in more detail. In the third chapter we turn our attention to smooth manifolds, explaining how the previous notions can be extended to define pseudodifferential operators and Sobolev spaces within this more general setting. In particular, we illustrate the extension of the continuity results, and some elements of the theory of operators between vector bundles. In the fourth and final chapter we give an overview of index theory for Fredholm operators and its applications to elliptic pseudodifferential operators on closed manifolds. We conclude by presenting a few formulae and techniques to calculate the index of such operators, focusing on the case of pseudodifferential operators on the unit circle.