Abstract
We provide an elementary formula for the non-integer powers of the Laplace operator in the Euclidean spaces. Such formulas are helpful in establishing the elliptic nature of some pseudo-differential operators arising from the study of energies of knotted loops in space, and possibly of embedded submanifolds in a Riemannian manifold.
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Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton Univ Press, 1970
He Z-X. The Euler-Lagrange equation and negative gradient flow for the Möbius energy of loops. preprint, January, 1998. Postscript file available at http://math.ucsd.edu/∼zhe/
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He, Z. A formula for the non-integer powers of the laplacian. Acta Math Sinica 15, 21–24 (1999). https://doi.org/10.1007/s10114-999-0058-4
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DOI: https://doi.org/10.1007/s10114-999-0058-4