Stein–Weiss Operators and Ellipticity
Stein–Weiss Operators and Ellipticity
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Stein–Weiss 算子和椭圆度
DOI：
10.1006/jfan.1997.3162
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发表时间：
1997
影响因子：
1.7
通讯作者：
T. Branson
中科院分区：
文献类型：
--
作者：
T. Branson
摘要 Stein 和 Weiss (1968, Am. J. Math. 90 , 163–196) 引入了广义梯度的概念：具有结构群 SO( n ) 或 Spin( n ) 的不可约向量丛之间的等变一阶微分算子 G 。除其他外，他们证明了某些系统的椭圆性，类似于柯西-黎曼方程以及（黎曼签名）麦克斯韦和狄拉克方程，由这些广义梯度构建。在本文中，我们对所有此类椭圆系统进行分类；答案对于所有黎曼或（如果自旋结构进入）黎曼自旋流形都有效。特别是，我们发现椭圆率可以通过组装极少的广义梯度来获得。所采用的方法产生了一个附带好处：对于每个广义梯度 G ，标准球 S n 上的 G * G 的光谱分辨率。这种光谱分辨率以前仅适用于“小束”上的算子，例如微分形式算子 δd 和 dδ 以及狄拉克算子的平方。
Abstract Stein and Weiss (1968, Am. J. Math. 90 , 163–196) introduced the notion of generalized gradients : equivariant first order differential operators G between irreducible vector bundles with structure group SO( n ) or Spin( n ). Among other things, they proved ellipticity for certain systems, analogous to the Cauchy–Riemann equations, and to the (Riemannian signature) Maxwell and Dirac equations, built from these generalized gradients. In this paper, we classify all systems of this type which are elliptic; the answer is valid for all Riemannian or (if spin structure enters) Riemannian spin manifolds. In particular, we find that ellipticity may be attained by assembling surprisingly few generalized gradients. The method employed yields a side benefit: the spectral resolution of G * G on the standard sphere S n , for each generalized gradient G . This spectral resolution was previously understood only for operators on “small bundles”—for example, the differential form operators δd , and dδ , and the square of the Dirac operator.