Fractional Laplacian in conformal geometry

Gonzalez, Maria del Mar; Chang, Sun-Yung Alice

doi:10.1016/j.aim.2010.07.016

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Fractional Laplacian in conformal geometry

Author(s): Sun-Yung Alice Chang , María del Mar González

Publication date Created: January 2011

Publication date (Print): January 2011

Journal: Advances in Mathematics

Publisher: Elsevier BV

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An extension problem related to the fractional Laplacian

Luis A. Caffarelli, Luís Silvestre (2006)

The operator square root of the Laplacian \((-\lap)^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.