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

42 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
p = 2. We estimate the L 2 -norm of the kernel on R × R by:
∫ ∫
1
(x−y)2
−2α|x|−
e 2t +2β|y| dxdy
4πt R R
≤ 1 ∫ ∫
e −2α|x| e − z2
2t +2β|z|+2β|x| dzdx
4πt R
R
≤ √ 1 ∫
e −2(α−β)|x| e 2β2t dx
2πt
= √ 1 1
2πt α − β e2β2t ,
proving the result for l = 0 and p = 2. Again, the case l = 1 is similar.
R
p = 1. Put c = (α + β)/2. Then the semigroup property of the heat kernel gives
‖e −α|X| e −t∆ R
e β|X| ‖ 1
≤ ‖e −α|X| e −t/2∆ R
e c|X| ‖ 2 ‖e −c|X| e −t/2∆ R
e β|X| ‖ 2 ,
and using the proved case p = 2 gives the result for p = 1.
(3.25)
(3.26)
The previous Proposition and standard estimates for the heat kernel on closed manifolds
(cf. the Appendix B) immediately give the following result for the heat kernel of
the model Dirac operator on the cylinder.
Proposition 3.7. Let N be a compact closed manifold and D = Γ ( d
+ A) a
dx
Dirac operator on the cylinder M = R × N (cf. Remark 1.10). Furthermore, let
Q ∈ b Diff q cpt((−∞, 0) × N; W ) a b-differential operator of order q with support in
some cylindrical end (−∞, c) × N. Then for α > β > 0, t > 0 the integral operator
e −α|X| Qe −tD2 e β|X| with kernel
1
√ e −α|x| Q x e −(x−y)2 /4t+β|y| e −tA2 (3.27)
4πt
is p-summable for 1 ≤ p ≤ ∞. Furthermore, for ε > 0, t 0 > 0, there is a constant
C(ε, t 0 ), such that for 1 ≤ p ≤ ∞, 0 < t < t 0 , 0 < β < α
‖e −α|X| Qe −tD2 e β|X| ‖ p ≤ C(ε, t 0 )(α − β) −1/p dim M+ε
− −
t q 2p 2 . (3.28)
If in addition the operator A is invertible then for 0 < δ < inf spec A 2 and ε > 0 there
are constants C j (δ, ε), j = 1, 2 such that for 1 ≤ p ≤ ∞, 0 < t < ∞, 0 < β < α we
have the estimate
□
‖e −α|X| Qe −tD2 e β|X| ‖ p
≤ ( C 1 (δ, ε)t − q 2 + C2 (δ, ε) ) (α − β) −1/p dim M+ε
−
t
2p
e (α2 +β 2−δ)t . (3.29)
For the definition of b Diff q cpt, b Diff q see Proposition A.1 and Eq. (A.16).
Proof. By Proposition A.1 we may write Q as a sum of operators of the form
( d
) l
f(x, p)P
(3.30)
dx
with