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

CONNES-CHERN CHARACTER AND ETA COCHAINS 79
Proof. For t → 0+ the estimate follows from Proposition B.5.1.
For t → ∞ and p = 1 the estimate follows from Proposition B.1 and (B.12) by
taking the contour as in Figure 6. For p = ∞ the estimate is a simple consequence
of the Spectral Theorem. The general case then follows again from the interpolation
inequality (B.15).
□
Finally we state the analogue of Proposition B.4 for the heat kernel.
Proposition B.7. Let A ∈ Diff a (M, W ) be a differential operator such that [D 2 , A] has
compact support (in the sense of Definition B.3) and is of order ≤ a + 1.
Then for t > 0 the operator [A, e −tD2 ] is of trace class. For t 0 , ε > 0 there is a
constant C(t 0 , ε) such that for all 1 ≤ p ≤ ∞ we have the following estimate in the
Schatten p–norm
‖[A, e −tD2 dim M−1+ε
−a/2−
]‖ p ≤ C(t 0 , ε) t
2p
, 0 < t ≤ t 0 ; (B.17)
C(t 0 , ε) is independent of p.
If D is a Fredholm operator then for any 0 < δ < inf spec ess D 2 and any ε > 0 there
is a constant C(δ, ε) such that for 1 ≤ p ≤ ∞
‖[A, e −tD2 dim M−1+ε
−a/2−
(I − H)]‖ p ≤ C(δ, ε) t
2p
e −tδ , 0 < t < ∞. (B.18)
Proof. For p = 1 this follows from Proposition B.4 and the contour integral representation
(B.12) by taking the contours as in Figure 5 for t → 0+ and as in Figure 6 in
the Fredholm case as t → ∞. For p = ∞ the estimates are a simple consequence of
the Spectral Theorem. The general case then follows from the interpolation inequality
(B.15).
□
B.3. Estimates for the JLO integrand. Recall that we denote the standard k–
simplex by ∆ k := { (σ 0 , ..., σ k ) ∈ R k+1 ∣ ∣ σ j ≥ 0, σ 0 + ... + σ k = 1 } . Furthermore, recall
the notation (2.3).
Proposition B.8. Let A j ∈ Diff a j
(M; W ), j = 0, ..., k, be D a j
–bounded differential
∑
operators on M, a := k a j . Furthermore, assume that supp A j0 is compact for at least
j=0
one index j 0 .
1. For t 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
(B.19)
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 + . (B.20)
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)