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.