Aspects of Dirac operators in analysis (Q931047)

This is an expository paper, devoted to the analysis behind several examples of Dirac-type operators. The classical Dirac operator in \(\mathbb R^ n\) is \(\sum _ {j=1}^ {n} e_ {j} \frac{\partial }{\partial x_ {j}}\) ( \(\mathbb R^ n\) being considered as embedded in a real, \(2 n\) dimensional Clifford algebra). The application of Clifford analysis to classical harmonic analysis proved useful to the study of boundary value problems, namely by providing a new proof of the Coifman-McIntosh-Meyer theorem on the \(L^2\) boundedness of the double layer potential operator on a Lipschitz graph lying in \(\mathbb R^ n\). Other Dirac-type operators analized here include generalization to Riemannian manifolds (Hodge-de Rham operator) and Atiyah-Singer-Dirac operators on spin manifolds. The square root of the hyperbolic Laplacian on upper half euclidean space is also a new type of Dirac operator.