On fractional GJMS operators

TL;DR

This paper offers a new geometric interpretation of fractional GJMS operators as generalized Dirichlet-to-Neumann maps, providing insights into their energies, positivity, and maximum principles on curved spaces.

Contribution

It introduces a novel geometric perspective on fractional GJMS operators via weighted GJMS operators, extending the understanding of their properties and applications.

Findings

Energy identities for fractional GJMS operators in Poincaré–Einstein manifolds

Nonnegativity of $P_{2 extgamma}$ under certain curvature conditions

Strong maximum principle for $P_{2 extgamma}$ when $Q_{2 extgamma}$ is non-zero

Abstract

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli--Silvestre extension for $(-\Delta)^\gamma$ when $\gamma\in(0,1)$, and both a geometric interpretation and a curved analogue of the higher order extension found by R. Yang for $(-\Delta)^\gamma$ when $\gamma>1$. We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincar\'e--Einstein manifold, including an interpretation as a renormalized energy. Second, for $\gamma\in(1,2)$, we show that if the scalar curvature and the fractional $Q$-curvature $Q_{2\gamma}$ of the boundary are nonnegative, then the fractional GJMS operator $P_{2\gamma}$ is nonnegative. Third, by assuming additionally that $Q_{2\gamma}$ is not identically zero, we show that $P_{2\gamma}$ satisfies a strong maximum principle.