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
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