On Fractional GJMS Operators

Jeffrey S. Case, Sun Yung Alice Chang

Research output: Contribution to journalArticlepeer-review

59 Scopus citations


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{small element of}(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{small element of}(1,2), we show that if the scalar curvature and the fractional Q-curvature Q of the boundary are nonnegative, then the fractional GJMS operator P is nonnegative. Third, by assuming additionally that Q is not identically zero, we show that P satisfies a strong maximum principle.

Original languageEnglish (US)
Pages (from-to)1017-1061
Number of pages45
JournalCommunications on Pure and Applied Mathematics
Issue number6
StatePublished - Jun 1 2016

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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