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

74 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
which we denote by the same symbol like the original operator.
(2) The b-Sobolev-space b H 1 (M, E) is the natural domain of any elliptic first order
b-pseudodifferential operator acting on sections of E.
(3) If A has order l = 0, then A is bounded.
A.4. Indicial family. Assume A ∈ b Ψ m (M; E, F ). Denote by à and ã the induced
operator and its complete symbol on the cylinder R × ∂M as above in condition ( b Ψ4).
Consider the zeroth order term ã 0 in the asymptotic expansion ( b Ψ4)(ii) and put for
τ ∈ C, u ∈ Γ ∞ (∂M; E) and p ∈ ∂M
I(A)(τ)u(p) := Op ( ã 0 (τ, −) ) u(p) =
∫
1
=
(2π)
dim M−1
T p∂M×T ∗ p ∂M
α 0 (p, exp v) e −i〈v,ξ〉 ã 0 (τ, p, ξ) τ E( u(exp v), p ) dv dξ,
(A.19)
where, as explained in Section A.2, α 0 : M × M → [0, 1] is a cut-off function vanishing
outside the injectivity radius and τ E is a connection induced parallel transport on E.
One thus obtains an entire family I(A) of pseudodifferential operators on the boundary
∂M which is called the indicial family of A. The indicial family plays a crucial role in
deriving the Atiyah–Patodi–Singer index formula within the b-calculus (cf. [Mel93]).
Appendix B. Basic resolvent and heat kernel estimates
For the convenience of the reader we are going to summarize some basic estimates for
the resolvent and the heat operator associated to an elliptic operator. These estimates
are an essential tool for the investigation of the asymptotics of the Chern character in
Section 3.
During the whole section M will be a Riemannian manifold without boundary and
D 0 : Γ ∞ (M; W ) −→ Γ ∞ (M; W ) will denote a first order formally self–adjoint elliptic
differential operator acting between sections of the Hermitian vector bundle W . We
assume that there exists a self–adjoint extension, D, of D 0 . E.g. if M is complete and
D 0 is of Dirac type then D 0 is essentially self–adjoint; if M is the interior of a compact
manifold with boundary then D can be obtained by imposing an appropriate boundary
condition. For the following considerations it is irrelevant which self–adjoint extension
is chosen. We just fix one.
B.1. Resolvent estimates. We fix an open sector Λ := { z ∈ C \ {0} ∣ 0 < ε ≤
arg z ≤ 2π − ε } ⊂ C \ R + in the complex plane.
We introduce the following notation: for a function f : Λ → C we write f(λ) =
O(|λ| α+0 ), λ → ∞, λ ∈ Λ if for every δ > 0, λ 0 ∈ Λ, there is a constant C δ,λ0 such that
|f(λ)| ≤ C δ |λ| α+δ for λ ∈ Λ, |λ| ≥ |λ 0 |.
We write f(λ) = O(|λ| −∞ ), λ → ∞, λ ∈ Λ if f(λ) = O(|λ| −N ) for every N; the
O–constant may depend on N.
L 2 s(M; W ) denotes the Hilbert space of sections of W which are of Sobolev class
s. The Sobolev norm of an element f ∈ L 2 s(M; W ) is denoted by ‖f‖ s . For a linear
operator T : L 2 s(M; W ) → L 2 t (M; W ) its operator norm is denoted by ‖T ‖ s,t .