Wave extension problem for the fractional Laplacian

TL;DR

This paper presents a novel interpretation of the fractional Laplacian as a Dirichlet-to-Neumann map via a degenerate wave equation, providing new solution representations and asymptotic analysis methods.

Contribution

It introduces a wave extension framework for the fractional Laplacian, connecting it to hyperbolic problems and oscillatory integral analysis, with alternative Bessel function methods.

Findings

Fractional Laplacian viewed as Dirichlet-to-Neumann map for a degenerate wave equation

Solution representations derived from Schrödinger group and oscillatory subordination

Asymptotic analysis identifies domains for initial data in extension problems

Abstract

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schr\"odinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.