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 (-) when (0,1), and both a geometric interpretation and a curved analogue of the higher-order extension found by R. Yang for (-) when >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 Poincare-Einstein manifold, including an interpretation as a renormalized energy. Second, for (1,2), we show that if the scalar curvature and the fractional Q-curvature Q(2) of the boundary are nonnegative, then the fractional GJMS operator P-2 is nonnegative. Third, by assuming additionally that Q(2) is not identically zero, we show that P-2 satisfies a strong maximum principle.(c) 2016 Wiley Periodicals, Inc.