Residue Family Operators on Spinors and Spectral Theory of Dirac operator on Poincar\'e-Einstein Spaces

We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincar\'e-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;\lambda)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;\lambda)$ including conformal covariance, factorization properties by conformally covariant operators for both flat and curved semi-Riemannian $Spin$-manifolds, and Poisson transformation.