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

CONNES-CHERN CHARACTER AND ETA COCHAINS 43
• f ∈ b Γ ∞ cpt((−∞, c) × N; End W ),
• P ∈ Diff b−l (N; W |N ) a differential operator of order b − l on N which is constant
in x-direction.
Note that e −α|X| commutes with f. Furthermore, f is uniformly bounded. Thus
∥ ∥ ∥
∥
∥e −α|X| fP ∂xe l −tD2 e β|X| ∥∥p ∥∥e
≤ ‖f‖ −α|X| ∞ P ∂xe l −tD2 e β|X| ∥∥p
. (3.31)
Inside the p-norm is now a tensor product of operators
e −α|X| P ∂ l xe −tD2 e β|X| = ( e −α|X| ∂ l xe −t∆ R
e β|X|) ⊗ ( P e −tA2 )
. (3.32)
Since the p-norm of a tensor product is the product of the p-norms the claim follows from
Proposition 3.5 and standard elliptic estimates for the closed manifold N (Propositions
B.5, B.6).
□
3.3. Trace class estimates for the JLO integrand on manifolds with cylindrical
ends. The heat kernel estimate for the Dirac operator on the model cylinder
Proposition 3.7 together with the comparison result in Section 3.1 will now be used
to obtain trace estimates for b-differential operators similar to the one in Proposition
B.8 if the indicial operator of at least one of the operators A 0 , ..., A k vanishes. Let us
mention here that in the following we will use the notation introduced in Section 2, in
particular Eq. (2.3).
Proposition 3.8. Let M = (−∞, 0] × N ∪ N X be a complete manifold with cylindrical
ends and let D be a Dirac operator on M. Let A 0 , ..., A k ∈ b Diff(M; W ) be b-differential
operators of order a 0 , ..., a k . Assume that for at least one index l ∈ {0, ..., k} the indicial
family of A l vanishes. Then for t > 0, σ ∈ ∆ k the operator
A 0 e −σ 0tD 2 A 1 · . . . · A k e −σ ktD 2 (3.33)
is trace class. Furthermore, there are the following estimates:
1. For t 0 > 0, ε > 0 there is a constant C(t 0 , ε) such that for all σ = (σ 0 , ..., σ k ) ∈ ∆ k ,
σ j > 0,
‖A 0 e −σ 0tD 2 A 1 · . . . · A k e −σ ktD 2 ‖ 1
( k∏
)
≤ C(t 0 , ε) σ −a j/2
j t −a/2−(dim M)/2−ε , 0 < t ≤ t 0 .
j=0
(3.34)
In particular, if a j ≤ 1, j = 0, ..., k, then
‖(A 0 , ..., A k ) √ tD ‖ = O(t−a/2−(dim M)/2−0 ), t → 0 + . (3.35)
2. Assume additionally that D is Fredholm and denote by H the orthogonal projection
onto Ker D. Then for ε > 0 and any 0 < δ < inf spec ess D 2 there is a constant C(δ, ε)
such that for all σ ∈ ∆ k , σ j > 0
‖A 0 e −σ 0tD 2 (I − H)A 1 · . . . · A k e −σ ktD 2 (I − H)‖ 1
( k∏
)
≤ C(δ, ε) σ −a j/2
j t −a/2−(dim M)/2−ε e −tδ , for all 0 < t < ∞.
j=0
(3.36)