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{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 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.
Original language | English (US) |
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Pages (from-to) | 1017-1061 |
Number of pages | 45 |
Journal | Communications on Pure and Applied Mathematics |
Volume | 69 |
Issue number | 6 |
DOIs | |
State | Published - Jun 1 2016 |
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics