On fractional GJMS operators

الأصل · EN

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.