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 (−Δ)γ when γ ∊ (0,1), and both a geometric interpretation and a curved analogue of the higher‐order extension found by R. Yang for (−Δ)γ when γ > 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 γ ∊ (1,2), we show that if the scalar curvature and the fractional Q‐curvature Q2γ of the boundary are nonnegative, then the fractional GJMS operator P2γ is nonnegative. Third, by assuming additionally that Q2γ is not identically zero, we show that P2γ satisfies a strong maximum principle.© 2016 Wiley Periodicals, Inc.