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Another approach to Juhl’s conformally covariant differential operators from \(S^n\) to \(S^{n-1}\). (English) Zbl 1365.58020
Summary: A family \(({\mathbf D}_\lambda)_{\lambda\in \mathbb C}\) of differential operators on the sphere \(S^n\) is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of \(S^n\) which preserve the smaller sphere \(S^{n-1}\subset S^n\). The family of conformally covariant differential operators from \(S^n\) to \(S^{n-1}\) introduced by A. Juhl is obtained by composing these operators on \(S^n\) and taking restrictions to \(S^{n-1}\).

58J70 Invariance and symmetry properties for PDEs on manifolds
43A85 Harmonic analysis on homogeneous spaces
Full Text: DOI arXiv
[1] Beckmann, Ralf and Clerc, Jean-Louis, Singular invariant trilinear forms and covariant (bi-)differential operators under the conformal group, Journal of Functional Analysis, 262, 10, 4341-4376, (2012) · Zbl 1244.43004
[2] Clerc, Jean-Louis, Covariant bi-differential operators on matrix space, Universit\'e de Grenoble. Annales de l’Institut Fourier, (None) · Zbl 1365.58020
[3] Eelbode, David and Sou\vcek, Vladim\'\ir, Conformally invariant powers of the Dirac operator in Clifford analysis, Mathematical Methods in the Applied Sciences, 33, 13, 1558-1570, (2010) · Zbl 1201.30065
[4] Fischmann, M. and Juhl, A. and Somberg, P., Conformal symmetry breaking differential operators on differential forms, (None)
[5] Gel’fand, I. M. and Shilov, G. E., Generalized functions. Vol. I: Properties and operations, xviii+423, (1964), Academic Press, New York – London
[6] Juhl, Andreas, Families of conformally covariant differential operators, \(Q\)-curvature and holography, Progress in Mathematics, 275, xiv+488, (2009), Birkh\"auser Verlag, Basel · Zbl 1177.53001
[7] Knapp, Anthony W., Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, 36, xviii+774, (1986), Princeton University Press, Princeton, NJ · Zbl 0604.22001
[8] Kobayashi, Toshiyuki and Speh, Birgit, Symmetry breaking for representations of rank one orthogonal groups, Memoirs of the American Mathematical Society, 238, 1126, v+110 pages, (2015) · Zbl 1334.22015
[9] Kobayashi, Toshiyuki and Pevzner, Michael, Differential symmetry breaking operators: I. General theory and F-method, Selecta Mathematica. New Series, 22, 2, 801-845, (2016) · Zbl 1338.22006
[10] Kobayashi, Toshiyuki and Pevzner, Michael, Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Mathematica. New Series, 22, 2, 847-911, (2016) · Zbl 1342.22029
[11] Kobayashi, Toshiyuki and Kubo, Toshihisa and Pevzner, Michael, Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Math., 2170, ix+192, (2016), Springer, Singapore · Zbl 1353.53002
[12] Olver, Peter J., Classical invariant theory, London Mathematical Society Student Texts, 44, xxii+280, (1999), Cambridge University Press, Cambridge · Zbl 0971.13004
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