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

More documents

Recommendations

Info

74 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM which we denote by the same symbol like the original operator. (2) The b-Sobolev-space b H 1 (M, E) is the natural domain of any elliptic first order b-pseudodifferential operator acting on sections of E. (3) If A has order l = 0, then A is bounded. A.4. Indicial family. Assume A ∈ b Ψ m (M; E, F ). Denote by Ã and ã the induced operator and its complete symbol on the cylinder R × ∂M as above in condition ( b Ψ4). Consider the zeroth order term ã 0 in the asymptotic expansion ( b Ψ4)(ii) and put for τ ∈ C, u ∈ Γ ∞ (∂M; E) and p ∈ ∂M I(A)(τ)u(p) := Op ( ã 0 (τ, −) ) u(p) = ∫ 1 = (2π) dim M−1 T p∂M×T ∗ p ∂M α 0 (p, exp v) e −i〈v,ξ〉 ã 0 (τ, p, ξ) τ E( u(exp v), p ) dv dξ, (A.19) where, as explained in Section A.2, α 0 : M × M → [0, 1] is a cut-off function vanishing outside the injectivity radius and τ E is a connection induced parallel transport on E. One thus obtains an entire family I(A) of pseudodifferential operators on the boundary ∂M which is called the indicial family of A. The indicial family plays a crucial role in deriving the Atiyah–Patodi–Singer index formula within the b-calculus (cf. [Mel93]). Appendix B. Basic resolvent and heat kernel estimates For the convenience of the reader we are going to summarize some basic estimates for the resolvent and the heat operator associated to an elliptic operator. These estimates are an essential tool for the investigation of the asymptotics of the Chern character in Section 3. During the whole section M will be a Riemannian manifold without boundary and D 0 : Γ ∞ (M; W ) −→ Γ ∞ (M; W ) will denote a first order formally self–adjoint elliptic differential operator acting between sections of the Hermitian vector bundle W . We assume that there exists a self–adjoint extension, D, of D 0 . E.g. if M is complete and D 0 is of Dirac type then D 0 is essentially self–adjoint; if M is the interior of a compact manifold with boundary then D can be obtained by imposing an appropriate boundary condition. For the following considerations it is irrelevant which self–adjoint extension is chosen. We just fix one. B.1. Resolvent estimates. We fix an open sector Λ := { z ∈ C \ {0} ∣ 0 < ε ≤ arg z ≤ 2π − ε } ⊂ C \ R + in the complex plane. We introduce the following notation: for a function f : Λ → C we write f(λ) = O(|λ| α+0 ), λ → ∞, λ ∈ Λ if for every δ > 0, λ 0 ∈ Λ, there is a constant C δ,λ0 such that |f(λ)| ≤ C δ |λ| α+δ for λ ∈ Λ, |λ| ≥ |λ 0 |. We write f(λ) = O(|λ| −∞ ), λ → ∞, λ ∈ Λ if f(λ) = O(|λ| −N ) for every N; the O–constant may depend on N. L 2 s(M; W ) denotes the Hilbert space of sections of W which are of Sobolev class s. The Sobolev norm of an element f ∈ L 2 s(M; W ) is denoted by ‖f‖ s . For a linear operator T : L 2 s(M; W ) → L 2 t (M; W ) its operator norm is denoted by ‖T ‖ s,t .
CONNES-CHERN CHARACTER AND ETA COCHAINS 75 For an operator T in a Hilbert space H we denote by ‖T ‖ p the p–th Schatten norm. To avoid confusions the letter p will not be used for Sobolev orders. Note that the operator norm of T in H coincides with ‖T ‖ ∞ . Proposition B.1. Let A, B ∈ Ψ • (M, W ) be pseudodifferential operators of order a, b with compact support. 1 1. If k > (dim M)/4 + a/2 then A(D 2 − λ) −k , (D 2 − λ) −k A are Hilbert–Schmidt operators for λ ∉ spec D 2 and we have ‖A(D 2 − λ) −k ‖ 2 = O(|λ| a/2+(dim M)/4−k+0 ), as λ → ∞ in Λ. (B.1) The same estimate holds for ‖(D 2 − λ) −k A‖ 2 . 2. If k > (dim M + a + b)/2 then A(D 2 − λ) −k B is of trace class for λ ∉ spec D 2 and ‖A(D 2 − λ) −k B‖ 1 = O(|λ| (dim M+a+b)/2−k+0 ), as λ → ∞ in Λ. (B.2) 3. 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 A(D 2 − λ) −k B is a trace class operator for any k ≥ 1 and ‖A(D 2 − λ) −k B‖ 1 = O(|λ| −∞ ), as λ → ∞ in Λ. (B.3) Proof. 1. Sobolev embedding and elliptic regularity implies that for f ∈ L 2 (M; W ) the section A(D 2 − λ) −k f is continuous. Moreover, for r > dim M/2, |λ| ≥ |λ 0 |, and x in the compact set supp A =: K ‖ ( A(D 2 − λ) −k f ) (x)‖ ≤ C‖(D 2 − λ) −k ‖ a+r,K ≤ C‖(D 2 + I) (a+r)/2 (D 2 − λ) −k f‖ 0 ≤ C|λ| −k+(a+r)/2 ‖f‖. For the Schwartz–kernel this implies the estimate ∫ ‖A(D 2 − λ) −k (x, y)‖ 2 d vol(y) ≤ C|λ| −2k+a+r , sup x∈supp A M and since A has compact support, integration over x yields ‖A(D 2 − λ) −k ‖ 2 2 ∫ ∫ ≤ ‖A(D 2 − λ) −k (x, y)‖ 2 d vol(x)d vol(y) supp A M ≤ C|λ| −2k+a+r , (B.4) (B.5) (B.6) proving the estimate (B.1). The estimate for (D 2 −λ) −k A follows by taking the adjoint. 2. The second claim follows from the first one using the Hölder inequality. 3. To prove the third claim we choose cut–off functions ϕ, ψ ∈ C ∞ 0 (M) with ϕ = 1 on π 2 (supp A), ψ = 1 on π 1 (supp B) and supp ϕ ∩ supp ψ = ∅. 1 This means that their Schwartz kernels are compactly supported in M × M.
Page 1 and 2:
CONNES-CHERN CHARACTER FOR MANIFOLD
Page 3 and 4:
CONNES-CHERN CHARACTER AND ETA COCH
Page 5 and 6:
Page 7 and 8:
∫ b M CONNES-CHERN CHARACTER AND
Page 9 and 10:
Page 11 and 12:
where CONNES-CHERN CHARACTER AND ET
Page 13 and 14:
where the map ι(h) is defined by C
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
∫ This means that b M CONNES-CHER
Page 23 and 24: CONNES-CHERN CHARACTER AND ETA COCH
Page 53 and 54: Applying (5.9) to A 1 we get e −
Page 57 and 58: lim t→0+ CONNES-CHERN CHARACTER A
Page 73: CONNES-CHERN CHARACTER AND ETA COCH
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?