An Introduction to the Fractional Laplacian

Abstract

We introduce here some preliminary notions on the fractional Laplacian and on fractional Sobolev spaces. The definition and equivalent representations for the fractional Laplacian are introduced and the constant that appears in this definition is explicitly computed. Fractional Sobolev spaces enjoy quite a number of important functional inequalities. We will present here two important inequalities which have a simple and nice proof, namely the fractional Sobolev Inequality and the Generalized Coarea Formula. Moreover, we present an explicit example of an s-harmonic function on the positive half-line, i.e. \((-\varDelta )^{s}(x_{+})^{s} = 0\) on \(\mathbb{R}_{+}\) and an example of a function with constant Laplacian on the ball. We also discuss some maximum principles and a Harnack inequality, and present a quite surprising local density property of s-harmonic functions into the space of smooth functions.