Fractional Laplacian, homogeneous Sobolev spaces and their realizations

TL;DR

This paper compares two classical definitions of the fractional Laplacian and homogeneous Sobolev spaces on R^d, establishing their equivalence and exploring properties of the fractional Laplacian.

Contribution

It provides an explicit correspondence between two classical definitions of fractional Laplacian and homogeneous Sobolev spaces, demonstrating their equivalence and shared representations.

Findings

Two definitions of fractional Laplacian are shown to be equivalent.

Explicit correspondence between the two Sobolev space definitions is established.

Properties of the fractional Laplacian are proved.

Abstract

We study the fractional Laplacian and the homogeneous Sobolev spaces on R^d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way we also prove some properties of the fractional Laplacian.