Bergman and Calderón Projectors for Dirac Operators

Abstract

Abstract. For a Dirac operator Dḡ over a spin compact Riemannian manifold with boundary (X, g), we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on X, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of Dḡ and the analysis of the complete conformal metric g = g/ρ 2 where ρ is a smooth function on X which is equal to the distance to the boundary near ∂X. We then show that 1 2(Id + e S (0)) is the orthogonal Calderón projector, where e S(λ) is the holomorphic family in { (λ) ≥ 0} of normalized scattering operators constructed in [24], which are classical pseudo-differential of order 2λ. Finally we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.

Bernstein-Sato identities and conformal symmetry breaking operators

“…Hyperbolic metric: The eigen-equation associated to the square of the Dirac operator [GMP12] for the hyperbolic metric g hyp = x −2 n (dx 2 1 + • • • + dx 2 n ) also induces the operator / P (λ). In particular, the eigen-equation / D g hyp − iλ = 0 on Γ(Σ…”

“…In recent years there appeared several approaches to a classification scheme for conformally covariant differential operators P 2N , / D 2N +1 and L (p) 2N (acting on functions, spinors and differential p-forms) on semi-Riemannian (spin-)manifolds, cf. [GJMS92, GMP12,BG05]. Furthermore, the operators P 2N and / D 2N +1 were extended to a theory of 1parameter families of conformally covariant differential operators [J09,FS14], nowdays known and termed as the residue families.…”

On conformal powers of the Dirac operator on Einstein manifolds

“…for all smooth sections ψ ∈ Γ S(M, h) , and · denotes the evaluation with respect to h. Conformal odd powers of the Dirac operator were constructed in [HS01, GMP12,Fis13], and are denoted by D 2N +1 = / D 2N +1 + LOT, for N ∈ N 0 (N < n 2 for even n). Here, LOT stands for "lower order terms."…”

“…[GJMS92, GZ03,GP03], and this is analogous for the Dirac operator, cf. [HS01, GMP12,Fis13]. Their original construction is based on the ambient metric, or, equivalently, on the associated Poincaré-Einstein metric introduced by Fefferman and Graham [FG85,FG11].…”

A Family of Riesz Distributions for Differential Forms on Euclidian Space

“…Remark 5.5 (Structure of Branson-Gover operators and conformal powers of the Dirac operator) The Branson-Gover operators [BG05], again conjectural, and conformal powers of the Dirac operator [GMP12] are also residues of certain scattering operators. Less is known about their structure.…”