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

78 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
c
• • •
discrete
essential spectrum
Figure 6.
of D 2 is c.
Contour of integration if bottom of the essential spectrum
estimate in the Schatten p–norm
‖Ae −tD2 dim M+ε
−a/2−
‖ p ≤ C(t 0 , ε) t
2p
, 0 < t ≤ t 0 . (B.13)
Note that C(t 0 , ε) is independent of p. The same estimate holds for ‖e −tD2 A‖ p .
2. Denote by π 1 , π 2 : M × M → M the projection onto the first resp. second factor
and assume that π 2 (supp A) ∩ π 1 (supp B) = ∅. Then
‖Ae −tD2 B‖ 1 = O(t ∞ ), 0 < t < t 0 , (B.14)
with N arbitrarily large.
3. Let ϕ ∈ C ∞ (M) be a smooth function such that supp dϕ is compact. Moreover
suppose that supp ϕ∩π 1 (supp A) = ∅. Then the estimate (B.14) also holds for ϕe −tD2 A.
Proof. 1. From Proposition B.2 and the contour integral (B.12) we infer the inequality
(B.13) for p = 1. For p = ∞ it follows from the Spectral Theorem. The Hölder
inequality implies the following interpolation inequality for Schatten norms
‖T ‖ p = Tr(|T | p ) 1/p ≤ ‖T ‖ 1−1/p
∞ ‖T ‖ 1/p
1 , 1 ≤ p ≤ ∞. (B.15)
From this we infer (B.13).
The remaining claims follow immediately from the contour integral (B.12) and the
corresponding resolvent estimates.
□
For the next result we assume additionally that D is a Fredholm operator and we
denote by H the orthogonal projection onto Ker D. H is a finite rank smoothing
operator. Put c := inf spec ess D 2 . Then e −tD2 (I − H) = e −tD2 − H can again be
expressed in terms of a contour integral as in (B.12) where the contour is now depicted
in Figure 6.
This allows to make large time estimates. The result is as follows:
Proposition B.6. Assume that D is Fredholm and let A ∈ Ψ a (M, W ) be a pseudodifferential
operator with compact support. 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 ≤ ∞
‖Ae −tD2 dim M+ε
−a/2−
(I − H)‖ p ≤ C(δ, ε) t
2p
e −tδ , 0 < t < ∞. (B.16)