Introduction to pseudo-differential operators

The present notes give introduction to the theory of pseudo-differential operators on Euclidean spaces. The first part is devoted to the necessary analysis of functions, such as basics of the Fourier analysis and the theory of distributions and Sobolev spaces. The second part is devoted to pseudo-differential operators and their applications to partial differential equations. We refer to the monograph [1] by Ruzhansky and Turunen for further details on this theory on the Euclidean space, torus, and more general compact Lie groups and homogeneous spaces. This book will be the main source of examples and further details to complement these notes. The course will contain the material from these notes and from [1], but not everything, with the rest of the material provided here for the reader’s convenience. The reader interested in the relations of this topic to the harmonic analysis may consult the monograph [3] by Stein, and those interested in the relations to the spectral theory can consult the monograph [2] by Shubin, to mention only very few related sources of material.