zbMATH — the first resource for mathematics

Stein-Weiss operators and ellipticity. (English) Zbl 0904.58054
Stein and Weiss introduced the notion of generalized gradients: equivariant first order differential operators \(G\) between irreducible vector bundles with structure group \(SO(n)\) or \(\text{Spin} (n)\). Among other things, they proved ellipticity for certain systems, analogous to the Cauchy-Riemann equations.
In the paper under review, the author classifies all systems of this type which are elliptic. He obtains also the spectral resolution of \(G^*G\) on the standard sphere \(S^n\) for each generalized gradient, which was previously understood only for operators on “small bundles”, e.g., for \(\delta d\), \(d\delta\) or the square of the Dirac operator.

58J05 Elliptic equations on manifolds, general theory
Full Text: DOI
[1] Ahlfors, L, Conditions for quasiconformal deformation in several variables, Contributions to analysis. A collection of papers dedicated to L. bers, (1974), Academic Press New York, p. 19-25
[2] Boerner, H., Darstellungen von gruppen, (1955), Springer-Verlag Berlin · Zbl 0068.01603
[3] Branson, T., Conformally covariant equations on differential forms, Comm. partial differential equations, 7, 393-431, (1982) · Zbl 0532.53021
[4] Branson, T., Group representations arising from Lorentz conformal geometry, J. funct. anal, 74, 199-291, (1987) · Zbl 0643.58036
[5] Branson, T., Harmonic analysis in vector bundles associated to the rotation and spin groups, J. funct. anal., 106, 314-328, (1992) · Zbl 0778.58066
[6] Branson, T., Sharp inequalities, the functional determinant, and the complementary series, Trans. amer. math. soc., 347, 3671-3742, (1995) · Zbl 0848.58047
[7] Branson, T., Nonlinear phenomena in the spectral theory of geometric linear differential operators, Proc. sympos. pure math., 59, 27-65, (1996) · Zbl 0857.58042
[8] Branson, T.; Ólafsson, G.; Ørsted, B., Spectrum generating operators, and intertwining operators for representations induced from a maximal parabolic subgroup, J. funct. anal., 135, 163-205, (1996) · Zbl 0841.22011
[9] Fegan, H., Conformally invariant first order differential operators, Quart. J. math. Oxford, 27, 371-378, (1976) · Zbl 0334.58016
[10] Folland, G., Harmonic analysis of the Rham complex on the sphere, J. reine angew. math., 398, 130-143, (1989) · Zbl 0671.58036
[11] Gallot, S.; Meyer, D., Opérateur de courbure et Laplacian des formes différentielles d’une variété riemannienne, J. math. pures appl., 54, 259-284, (1975) · Zbl 0316.53036
[12] Humphreys, J., Introduction to Lie algebras and representation theory, Graduate texts in mathematics, 9, (1980), Springer-Verlag Berlin
[13] Ikeda, A.; Taniguchi, Y., Spectra and eigenforms of the Laplacian onS^nandpn(\(C\)), Osaka J. math., 15, 515-546, (1978) · Zbl 0392.53033
[14] Kalina, J.; Pierzchalski, A.; Walczak, P., Only one of generalized gradients can be elliptic, Annales polonic. mathematic, LXVII.2, 111-120, (1997) · Zbl 0901.53017
[15] Kalina, J.; Ørsted, B.; Pierzchalski, A.; Walczak, P.; Zhang, G., Elliptic gradients and highest weights, Bull. acad. polon. sci. ser. math., 44, 511-519, (1996)
[16] Knapp, A.; Stein, E., Intertwining operators for semisimple groups, Ann. of math., 93, 489-578, (1971) · Zbl 0257.22015
[17] Kosmann, Y., Dérivées de Lie des spineurs, Ann. mat. pura appl., 91, 317-395, (1972) · Zbl 0231.53065
[18] Kostant, B., A formula for the multiplicity of a weight, Trans. amer. math. soc., 93, 53-73, (1959) · Zbl 0131.27201
[19] Ørsted, B., The conformal invariance of Huygens’ principle, J. differential geometry, 16, 1-9, (1981) · Zbl 0447.35059
[20] Palais, R., Foundations of global non-linear analysis, (1968), Benjamin New York · Zbl 0164.11102
[21] Sahi, S., Jordan algebras and degenerate principal series, J. reine angew. math., 462, 1-18, (1995) · Zbl 0822.22006
[22] Stein, E.; Weiss, G., Generalization of the cauchy – riemann equations and representations of the rotation group, Amer. J. math., 90, 163-196, (1968) · Zbl 0157.18303
[23] Strichartz, R., Linear algebra of curvature tensors and their covariant derivatives, Canad. J. math., 40, 1105-1143, (1988) · Zbl 0652.53012
[24] Wang, M., Preserving parallel spinors under metric deformations, Indiana univ. math. J., 40, 815-844, (1991) · Zbl 0724.53031
[25] Wünsch, V., On conformally invariant differential operators, Math. nachr., 129, 269-281, (1986) · Zbl 0619.53008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.