On fractional GJMS operators

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.