A pseudodifferential approach to the fractional Laplacian and magnitude

Abstract

In this thesis we consider two problems related to the fractional Laplacian. One of these is the study of the singular behaviour of solutions of boundary value problems for the fractional Laplacian in smooth and polygonal domains. We study these problems in model geometries using a harmonic extension of the solution to the upper half-space. We then discuss how these problems fit in the theory of edge pseudodifferential calculus and outline the additional steps necessary to address the general case. On the other hand, we study magnitude, a metric invariant of compact metric spaces, for Euclidean domains or compact manifolds with boundary. We provide a framework for its analysis, relating it to a boundary problem for a non-local integral operator, which upon localisation near the diagonal is a negative order parameter-elliptic pseudodifferential operator coinciding with a negative order fractional Laplacian up to lower order terms. The inverse of this operator, when acting between suitable Sobolev spaces, is asymptotically computed using Wiener-Hopf factorizations and methods from semiclassical analysis. As a main result, we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton by explicit computation.