On fractional GJMS operators

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 Poincaré--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.

38 pages. Final version, to appear in Communications on Pure and Applied Mathematics