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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.

##### MSC:
 58J05 Elliptic equations on manifolds, general theory
##### Keywords:
vector bundle; elliptic operator; spectral resolution
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##### References:
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