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On Fractional GJMS Operators

Author(s): Case, Jeffrey S; Chang, Sun-Yung A.

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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 (-) when (0,1), and both a geometric interpretation and a curved analogue of the higher-order extension found by R. Yang for (-) when >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 Poincare-Einstein manifold, including an interpretation as a renormalized energy. Second, for (1,2), we show that if the scalar curvature and the fractional Q-curvature Q(2) of the boundary are nonnegative, then the fractional GJMS operator P-2 is nonnegative. Third, by assuming additionally that Q(2) is not identically zero, we show that P-2 satisfies a strong maximum principle.(c) 2016 Wiley Periodicals, Inc.
Publication Date: Jun-2016
Electronic Publication Date: 9-Feb-2015
Citation: Case, Jeffrey S, Chang, Sun-Yung Alice. (2016). On Fractional GJMS Operators. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 69 (1017 - 1061. doi:10.1002/cpa.21564
DOI: doi:10.1002/cpa.21564
ISSN: 0010-3640
EISSN: 1097-0312
Pages: 1017 - 1061
Type of Material: Journal Article
Version: Author's manuscript

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