Connes-Chern Character for Manifolds with Boundary and ETA ...

70 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
Next let us define the symbol spaces S m( T ∗ M 0 ; πT ∗ ∗ M E) . For fixed m ∈ R this
space consists of all smooth sections a : T ∗ M 0 → πT ∗ ∗ M 0
E such that in local coordinates
x : U → R dim M 0
of over U ⊂ M 0 open and vector bundle coordinates
(x, η) : E |U → R dim M 0+dim R E the following estimate holds true for each compact K ⊂ U
and appropriate C K > 0 depending on K:
∥ ∂
α
x ∂ β ξ (η ◦ a)(ξ)∥ ∥ ≤ CK (1 + ‖T ∗ x(ξ)‖) m−|β| for all ξ ∈ T|KM ∗
0 . (A.10)
Given a symbol a ∈ S m( T ∗ M 0 ; πT ∗ ∗ M Hom(E, F )) one defines now a pseudodifferential
operator Op(a) ∈ Ψ m( M 0 ; E, F ) by
( )
Op(a) u (p) :=
∫
1
:=
α
(2π) dim M
0 (p, exp v) e −i〈v,ξ〉 a(p, ξ)τ E (u(exp v), p) dv dξ, (A.11)
T pM 0 ×T ∗ p M 0
where u ∈ Γ ∞ cpt(E) and p ∈ M 0 . Moreover, there is a quasi-inverse, the symbol map
σ : Ψ m( M 0 ; E, F ) → S m( T ∗ M 0 ; πT ∗ ∗ M Hom(E, F )) which is defined by
(
) (π(ξ) )
σ(A)(ξ)(e) := A α 0 (p, −) τ E (e, −) e i〈ξ,Φ(p,−)〉 , (A.12)
where p ∈ M 0 , ξ ∈ Tp ∗ M 0 , e ∈ E p . It is a well-known result from global symbol calculus
(cf. [Wid80, FuKe88, Pfl98]) that the map Op maps S −∞( T ∗ M 0 ; πT ∗ ∗ M Hom(E, F ))
onto Ψ −∞( M 0 ; E, F ) and that up to these spaces, Op and σ are inverse to each other.
A.3. Classical b-pseudodifferential operators. Let us explain in the following the
basics of the (small) calculus of b-pseudodifferential operators on a manifold with
boundary M. In our presentation, we lean on the approach [Loy05]. For more details
on the original approach confer [Mel93].
In this section, we assume that M carries a b-metric denoted by g b . Furthermore,
let π E : E → M and π F : F → M be two smooth Hermitian vector bundles over M,
and fix metric connections ∇ E and ∇ F . Then observe that by the Schwartz Kernel
Theorem there is an isomorphism between bounded linear maps
A : J ∞( ∂M, M; E ) → J ∞( ∂M, M; F ′) ′
and the strong dual J ∞( ∂(M × M), M × M; E ⊠ F ′) ′ ( , where J
∞
∂M, M; E ) :=
J ∞( ∂M, M ) · C ∞ (M; E). This isomorphism is given by
(
A ↦→ K A := J ∞( ∂M, M; E ) ˆ⊗J ∞( ∂M, M; F ′) )
∋ (u ⊗ v) ↦→ 〈Au, v〉 , (A.13)
where we have used that
J ∞( ∂(M × M), M × M; E ⊠ F ′) = J ∞( ∂M, M; E ) ˆ⊗J ∞( ∂M, M; F ′)