On fractional GJMS operators (Q2811169)

In the interesting paper under review, the authors describe a new interpretation of the fractional Graham-Jenne-Mason-Sparling (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 [\textit{L. Caffarelli} and \textit{L. Silvestre}, Commun. Partial Differ. Equations 32, No. 8, 1245--1260 (2007; Zbl 1143.26002)] for \((-\Delta)^\gamma\) when \(\gamma\in(0,1),\) and both a geometric interpretation and a curved analogue of the higher-order extension due to R.~Yang for \((-\Delta)^\gamma\) for \(\gamma>1.\)NEWLINENEWLINEThree applications of this correspondence are given. First, the authors 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 \(\gamma\in(1,2),\) it is shown 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, the authors show that \(P_{2\gamma}\) satisfies a strong maximum principle.