On higher order extensions for the fractional Laplacian

TL;DR

This paper extends the Caffarelli-Silvestre extension technique to general positive, non-integer orders of the fractional Laplacian, establishing a higher-order characterization linking boundary norms to interior seminorms.

Contribution

It generalizes the fractional Laplacian extension method to higher orders, providing a new equivalence between boundary and interior function norms for non-integer orders.

Findings

Extension technique applies to all positive, non-integer orders

Establishes equivalence between boundary H^s norm and interior seminorm

Enables analysis of higher-order fractional Laplacians

Abstract

The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold for general positive, non-integer orders of the fractional Laplace operator, by showing an equivalence between the H^s norm on the boundary and a suitable higher-order seminorm of U.