On Fractional GJMS Operators

By Jeffrey S. Case, Sun‐Yung Alice Chang. 2014

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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.© 2016 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c9d8cb16613d20bd4cbddba0685f2eba111ed562

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Citations in the corpus

The Nirenberg problem and its generalizations: a unified approach (2014, background): “Some new interpretation and properties of those fractional operators and their associated fractional Q-curvatures were later given by Chang and González [13] and Case and Chang [12], which also provide geometric interpretation of the extension formulations for fractional Laplacian (− ) established b”
Existence theorems of the fractional Yamabe problem (2016, background): “In [10] (see also [9]), Chang and González observed that the fractional conformal Laplacian, defined through the scattering theory (see e.”
Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities (2019, methodology): “It relies on the localization formula for fractional Laplacian operators developed by Caffarelli-Silvestre [7], Chang-González [9], Yang [47], Case-Chang [8], as well as some technical analysis of Poisson integrals.”