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

80 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
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
(B.21)
In particular, a j ≤ 1, j = 0, ..., k, then
‖(A 0 (I − H), ..., A k (I − H)) √ tD ‖
= O(t −a/2−(dim M)/2−0 e −tδ ), for all 0 < t < ∞.
(B.22)
Proof. We first reduce the problem to the case that all A j are compactly supported. To
this end choose ϕ j0 −1, ϕ j0 ∈ C ∞ 0 (M) such that supp ϕ j0 ∩ supp(1 − ϕ j0 −1) = ∅ and such
that ϕ j0 A j0 = A j0 ϕ j0 = A j0 . Decompose A j0 −1 = A j0 −1ϕ j0 −1 + A j0 −1(1 − ϕ j0 −1).
First we show that the estimates (B.19), (B.21) hold if we replace A j0 −1 by A j0 −1(1−
ϕ j0 −1):
Case 1. Proposition B.5.3 gives
‖A j0 −1(1 − ϕ j0 −1)e −σ j 0 −1tD 2 ϕ j0 ‖ 1 ≤ C t0 σ N j 0 −1t N , for σ j0 −1t ≤ t 0 . (B.23)
The operator norm of the other factors can be estimated using the Spectral Theorem,
taking into account the D a j
–boundedness of A j :
Hence by the Hölder inequality
‖A j e −σ jtD 2 ‖ ≤ C t0 (σ j t) −a j/2 , for σ j t ≤ t 0 . (B.24)
‖A 0 e −σ 0tD 2 A 1 · . . . · A j0 −1(1 − ϕ j0 −1)e −σ j 0 −1tD 2 · A k e −σ ktD 2 ‖ 1
( k∏
≤ C t0 σ −a j/2
j
)t − P k a j +N
j=0 , 0 < t ≤ t0 ,
j=0
(B.25)
which is even better than (B.19).
Case 2 (D Fredholm). From (B.23), Proposition B.6 and the fact that H is a finite
rank operator with e −ξD2 H = H we infer
‖A j0 −1(1 − ϕ j0 −1)e −σ j 0 −1tD 2 (I − H)ϕ j0 ‖ 1
≤ C δ e −σ j 0 −1tδ , for all 0 < t < ∞.
(B.26)
To the other factors we apply Proposition B.6 with p = ∞:
‖A j e −σ jtD 2 (I − H)‖ ≤ C δ (σ j t) −a j/2 e −σ jtδ , 0 < t < ∞. (B.27)
The Hölder inequality combined with (B.26),(B.27) gives (B.21).
Altogether we are left to consider A 0 , ..., A j0 −1ϕ j0 −1, A j0 , ...A k where now A j0 −1ϕ j0 −1
and A j0 are compactly supported. Continuing this way, also to the right of j 0 , it remains
to treat the case where each A j has compact support.