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

72 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
with respect to the product metric on R × ∂M. Then the following conditions
hold true:
(i) Let y denote local coordinates of ∂M, (y, ξ) the corresponding local coordinates
of T ∗ ∂M, and τ the cotangent variable of the cylinder variable
t ∈ R. Then the symbol ã(t, τ, y, ξ) can be (uniquely) extended to an entire
function in τ ∈ C such that uniformly in t, uniformly in a strip | Im τ| ≤ R
with R > 0 and locally uniformly in y
∥
∥∂t k ∂τ∂ l y α ∂ β ξ ã
∥ ≤ C k,l,α,β (1 + |τ| + ‖ξ‖) m−l−|β|
for l ∈ N, β ∈ N dim M−1 .
(ii) There exist symbols ã k (τ, y, ξ) ∈ S m( C × T ∗ ∂M; π ∗ Hom(E, F ) ) , k ∈ N,
and r n (t, τ, y, ξ) ∈ S m( R × C × T ∗ ∂M); π ∗ Hom(E, F ) ) , n ∈ N, which all
are entire in τ and fulfill growth conditions as in (i) such that for every
n ∈ N the following asymptotic expansion holds:
n∑
ã(t, τ, y, ξ) = e kt ã k (τ, y, ξ) + e (n+1)t r n (t, τ, y, ξ).
k=0
(iii) The Schwartz kernel K eB of the operator ˜B := Ã−Op(ã) with Op(ã) defined
by Eq. (A.11) can be represented in the form
∫
K eB (t, p, t ′ , p ′ ) = e i(t−t′ )τ ˜b(t, τ, p, p ′ ) dτ
with a symbol
R
˜b(t, τ, p, p ′ ) ∈ S −∞( T ∗ R × ∂M × ∂M; π ∗ Hom(E, F ) )
which is entire in τ and which for every ˜m ∈ N, k, l ∈ N and every pair of
differential operators D p and D p ′ on ∂M (acting on the variable p resp. ′ p′ )
satisfies the following estimate uniformly in t, p, p ′ and uniformly in a strip
| Im τ| ≤ R with R > 0
∥
∥∂t k ∂τD l p D p ′ ˜b
∥ ≤ C ′ ˜m,k,l,D,D ′ (1 + |τ|) ˜m .
(iv) There exist symbols
˜bk (τ, p, p ′ ) ∈ S −∞( C × ∂M × ∂M; π ∗ Hom(E, F ) ) ,
for k ∈ N and symbols
r n (t, τ, p, p ′ ) ∈ S m( R × C × ∂M × ∂M; π ∗ Hom(E, F ) ) ,
for n ∈ N which all are entire in τ and fulfill growth conditions as in (iii)
such that for every n ∈ N the following asymptotic expansion holds:
n∑
˜b(t, τ, p, p ′ ) = e kt ˜bk (τ, p, p ′ ) + e (n+1)t r n (t, τ, p, p ′ ).
k=0