Essentials of Nonlocal Operators

This preliminary chapter aims at providing some basic knowledge on nonlocal operators. Notions which are necessary to know about the fractional Laplacian and about more general nonlocal operators will be addressed. The expert users may skip this introduction.The goal of this preliminary chapter is to bring the non-expert reader closer to the beautiful world of nonlocal operators. By no means exhaustive, this introduction gives a glance at some basic definitions, notations and well known results related to a few aspects of some nonlocal operators. With these premises, we take a look at fractional Sobolev spaces, at the fractional Laplacian and at a more general class of nonlocal operators (of which the fractional p-Laplacian is the typical representative).