Multiparameter resolvent trace expansion for elliptic boundary problems

0.21 MB·English

by  Boris Vertman

MULTIPARAMETER RESOLVENT TRACE EXPANSION
FOR ELLIPTIC BOUNDARY PROBLEMS
BORIS VERTMAN
Abstract. We establish multiparameter resolvent trace expansions for elliptic
boundary value problems, polyhomogeneous both in the resolvent and the auxil-
iary parameter. The present analysis is rooted in the joint project with Matthias
Lesch on multiparameter resolvent trace expansions on revolution surfaces with
3 applications to regularized sums of zeta-determinants.
1
0
2

n
a 1. Introduction and formulation of the main result
J

0 Consider a compact manifold M of dimension m with boundary ∂M, equipped
3
with a Hermitian vector bundle (E,hE) of rank p. We recall the notion of an elliptic

] boundary problem from Seeley [See69]. Let A ∈ Diffq(M,E) denote a differential
P
operator on M with values in E of q-th order with pq ∈ 2N . The operator A is
S 0
. elliptic if its principal symbol σ(A)(p,ζ) is invertible for (p,ζ) ∈ T∗M\{0}. Assume,
h
t A satisfies the Agmon condition in a fixed cone Γ′ = {z ∈ C | arg(z) ∈ (θ ,θ )} of
a 1 2
m the complex plane, i.e. (σ(A) +zq) is invertible for zq ∈ Γ′. Consider a system of
differential operators B = (B ,..,B ) on ∂M, such that (A,B) defines an elliptic
[ 1 pq/2
boundary problem satisfying the Agmon condition on Γ′ in the sense of [See69, Def.
1
v 1,2].
3
9 Under this setup, (A,B) defines a closed unbounded operator A on L2(M,E),
B
2
obtained as the graph closure of A acting on u ∈ C∞(M,E) satisfying the boundary
7
. conditions Bu = 0. Moreover, (A + zq) is invertible to zq ∈ Γ′ sufficiently large
1 B
0 and the seminal work of Seeley [See69] establishes an expansion of the resolvent
3 (A +zq)−1 as |z| → ∞. In the present paper we fix W ∈ C∞(M,E) and a finite
B
1
collection of scalar smooth potentials V ,..,V ∈ C∞(M), constant along ∂M.
: 1 n
v
Xi Assume for simplicity that each V (0) = V ↾ ∂M > 0. Set Γ := {z ∈ C | zq ∈ Γ′}
k k
r and write Γn for its n-th Cartesian product. Consider λ = (λ1,...,λn) ∈ Γn and the
a
corresponding the multiparameter family
n
A (λ) := A + λqV +W.
B B k k
k=1
X
For qN > m the N-th power of the resolvent (A (λ)+zq)−N is trace class and our
B
mainresultestablishesamultiparameter expansionoftheresolvent traceTr(A (λ)+
B
zq)−N, polyhomogeneous in (z,λ) ∈ Γn+1.
Date: This document was compiled on: January 31, 2013.
The author was supported by the Hausdorff Center for Mathematics.
1
2 BORISVERTMAN
Theorem 1.1. Consider any multiindex α ∈ Nn and β ∈ N . Fix N ∈ N such that
0 0
qN > m. Then there exist e ∈ C∞(M ×(Γn+1 ∩Sn)), such that
i
∞
(λ,z)
∂α∂β Tr(A (λ)+zq)−N ∼ e |(λ,z)|−1−qN−j−|α|−β+m.
λ z B j |(λ,z)|
j=0 (cid:18) (cid:19)
X
The interest in the multiparameter resolvent trace expansions arose in view
their application for the computation of regularized sums of zeta-determinants in a
joint project with Matthias Lesch. In particular, consider a manifold with fibered
boundary Sd ×M, with an elliptic boundary problem (A,B) represented after the
eigenspace decomposition on Sd by an infinite sum of elliptic boundary problems
(A ,B ),n ∈ N on M. Assume, the boundary problems admit well defined zeta-
n n 0
determinants. Then the methods elaborated in [LeVe13] together with our main
theorem above equate the zeta-determinant of A to the regularized sum of zeta de-
B
terminants for A , up to a locally computable error term. We refer to [LeVe13]
n,Bn
for further reference.
This paper is organized as follows. In §2 we recast the symbolic expansion of the
resolvent (A +zq)−1 in terms of polyhomogeneity properties of its Schwartz kernel
B
lifted to an appropriate blowup of R+×M2. In §3 we establish a composition result
for the polyhomogeneous conormal distributions on the blowup of R+ ×M2. In §4
we employ this microlocal characterization of the resolvent kernel to establish the
multiparameter resolvent trace expansion.
Contents
1. Introduction and formulation of the main result 1
2. Resolvent kernel as a polyhomogeneous conormal distribution 2
3. Composition of polyhomogeneous Schwartz kernels 6
4. Multiparameter resolvent trace expansion 10
Acknowledgements 14
References 14
2. Resolvent kernel as a polyhomogeneous conormal distribution
Seeley [See69] provides a careful construction of the resolvent for the elliptic
boundary problem A . More precisely, in view of [See69, Theorem 1, Lemma 2,
B
(25), (32), (49))] we may state the following
Theorem 2.1. Let Γ := {µ ∈ Γ | |µ| ≥ R}. Consider a local coordinate
R
neighborhood U ⊂ ∂M × [0,1) in the collar of the boundary, with local co-
x
ordinates {x,y = (y ,..,y )}. Then for any j ∈ N there exist d ∈
1 m−1 0 −q−j
C∞(U × Rm × Γ,Hom(E ↾ U )), homogeneous of oder (−q − j) in (x−1,ζ,γ,µ),
where (x,y,ζ,γ,µ) ∈ U × Rm−1 ×R ×Γ , such that for R > 0 sufficiently large
ζ γ µ
MULTIPARAMETER RESOLVENT TRACE EXPANSIONS 3
and µ ∈ Γ the Schwartz kernel
R
(A +µq)−1(x,y,x,y)−(A+µq)−1(x,y,x,y)
B
N−1
− (2π)−m e e eihy−ye,ζie−ixeγd (ex,ey,ζ,γ,µ)dγdζ
−q−j
Rm−1 R
j=0 Z Z
X
N−1
=: K (x,y,x,y;µ)−K (x,y,x,y;µ)− Op(d ),
AB A −q−j
j=0
X
is uniformly O(|µ|m−q−N)ease|µ| → ∞. Heree,ethe first term K denotes the re-
AB
solvent kernel of A near the boundary, whereas the second term K refers to the
B A
interior resolvent parametrix of A, defined without taking into account the boundary
conditions.
Assume for simplicity that µ varies along a ray within Γ, which we identify with
R+. Then the Schwartz kernel K of the resolvent (A + µq)−1 is a distribution
AB B
on R+ × M2. Choose local coordinates (x,y) and (x,y) on the two copies of M
1/µ
in a collar neighborhood of the boundary, where x and x are the boundary defining
functions, with non-uniform behaviour at e e
D := {µ = ∞,(x,y) = (x,y)},e
(2.1)
C := {µ = ∞,x = x = 0,y = y}.
e e
This non-uniform behaviour is resolved by considering an appropriate blowup M2
b
of R+ × M2 at C and D, a procedure introeduced by eMelrose, see [Mel93], such
thatthekernels K ,K lifttopolyhomogeneous distributionsonthemanifoldwith
AB A
corners M2 in the sense of the following definition.
b
Definition 2.2. Let X be a manifold with corners, with all boundary faces embed-
ded, and {(H ,ρ )}N anenumeration of its boundaries andthe corresponding defin-
i i i=1
ing functions. For any multi-index b = (b ,...,b ) ∈ CN we write ρb = ρb1...ρbN.
1 N 1 N
Denote by V (X) the space of smooth vector fields on X which lie tangent to all
b
boundary faces. All distributions on X are locally restrictions of distributions de-
fined across the boundaries of X. A distribution ω on X is said to be conormal
if ω ∈ ρbC∞(X) for some b ∈ CN, and V ...V ω ∈ ρbC∞(X), for all V ∈ V (X)
1 ℓ j b
and for every ℓ ≥ 0. An index set E = {(γ,p)} ⊂ C × N satisfies the following
i
hypotheses:
(1) Re(γ) accumulates only at plus infinity,
(2) For each γ there is P ∈ N , such that (γ,p) ∈ E iff p ≤ P ,
γ 0 i γ
(3) If (γ,p) ∈ E , then (γ +j,p′) ∈ E for all j ∈ N and 0 ≤ p′ ≤ p.
i i
An index family E = (E ,...,E ) is an N-tuple of index sets. Finally, we say that
1 N
a conormal distribution w is polyhomogeneous on X with index family E, we write
ω ∈ AE (X), if ω is conormal and if in addition, near each H ,
phg i
ω ∼ a ργ(logρ )p, as ρ → 0, (2.2)
γ,p i i i
(γX,p)∈Ei
withcoefficientsa conormalonH , polyhomogeneouswithindexE atanyH ∩H .
γ,p i j i j
We also need to consider polyhomogeneous distributions on a manifold with cor-
nersX, conormal toanembedded submanifoldY ⊂ X. Thebasic space Im(Rn,{0})
4 BORISVERTMAN
consists of compactly supported distributions with the Fourier transform given by
a symbol of order (m−n/4). Im(Rn,{0}) is invariant under local diffeomorphisms
and thus makes sense on any manifold around an isolated point.
For an embedded k-submanifold S ⊂ X, any point in S admits an open neigh-
borhood V in X which can be locally decomposed as a product V = X′ ×X′′ so
that V ∩S = X′×{p},p ∈ X′′. The space Im(X,S) is defined (locally) as the space
of smooth functions on X′ with values Im+dimX′/4(X′′,{p}). The normalization is
chosen to give pseudo-differential operators their expected orders. All distributions
in Im(X,S) are locally restrictions of distributions on an ambient space, which are
conormal to any smooth extension of S across ∂X.
Choosing now index sets E for each boundary face of X as in Definition 2.2, we
define a space AE (X,Y) as the space of distributions conormal to Y, with poly-
phg
homogeneous expansions as in Eq. (2.2) at all boundary faces and with coefficients
conormal to the intersection of Y with each boundary face.
We now continue with the definition of a blowup M2, so that the Schwartz kernels
b
K and Op(d ) lift to polyhomogeneous distributions, possibly conormal to an
A −q−j
embedded submanifold. Blowing up R+ × M2 at C and D amounts in principle
to introducing polar coordinates in R+ × R+ at C and D together with a unique
2
minimal differential structure with respect to which these coordinates are smooth.
We first perform a blowup of C. The resulting space [R+ × M2,C] is defined
as the union of R+ × M2 \ C with the interior spherical normal bundle of C in
R+×M2. The blowup [R+×M2,C] is endowed with the unique minimal differential
structure with respect to which smooth functions in the interior of R+ × M2 and
polar coordinates on R+×M2 around C are smooth. This blowup introduces a new
boundary hypersurface, which we refer to as the front face ff. The other boundary
faces are as follows. The right face rf is the lift of {x = 0}, the left face lf is the lift
of {x = 0}, and the temporal face tf is the lift of {µ = ∞}.
The actual blowup space M is obtained by a blowup of [R+ ×M2,C] along the
b
lift oef the diagonal D. The resulting blowup space M is defined as before by cutting
b
out the submanifold and replacing it with its spherical normal bundle. This second
blowup introduces an additional boundary hypersurface td, the temporal diagonal.
M is a manifold with boundaries and corners, illustrated below.
b
ρ
rf
ξ
lf ff τ
x
tf tf
td e
Figure 1. The blowup M2 = [[R+ ×M2,C],D].
b
Denote by Y := {(µ,p,p) ∈ R+ × M2 | p = p} the diagonal hypersurface. We
denotebyAl,p(M2,β∗Y)thespaceofSchwartzkernelsthatlifttopolyhomogeneous
phg b
e e
MULTIPARAMETER RESOLVENT TRACE EXPANSIONS 5
conormal distributions on the blowup space M2, with leading order (−m + l) at
b
the front face ff, leading order (−m + p) at the temporal diagonal td, index sets
(N ,N ) at the left and right boundary faces, conormal at the interior singularity
0 0
β∗Y. The space of such Schwartz kernels without a conormal singularity is denoted
by Al,p(M2).
phg b
Projective coordinates on M are given as follows. Near the top corner of ff away
b
from tf the projective coordinates are given by
1 x x y −y
ρ = , ξ = , ξ = , w = , y, (2.3)
µ ρ ρ ρ
e e
where in these coordinates ρ,ξ,ξ are thee defining functions of the faces ff, rf and lf
respectively. For the bottom corner of ff near rf the projective coordinates are given
by e
x y −y
τ = (µx)−1, s = , u = , x, y, (2.4)
x x
where in these coordinates τ,s,x are the defining fuenctions of tf, rf and ff respec-
e e
tively. For the bottom corner of ff near lf the projective coordinates are obtained
e e
by interchanging the roles of x and x. The projective coordinates on M near the
e b
top of td away from tf are given by
e s−1 u
η = τ, S = , U = , x, (2.5)
η η
In these coordinates tf is the face in the limit |(S,U)| → ∞, ff and td are defined
e
by x,η, respectively. The blowup M is related to the original space R+ ×M2 via
b
the obvious ‘blow-down map’
e β : M → R+ ×M2,
b
which is in local coordinates simply the coordinate change back to
(1/µ,(x,y),(x,y)). The blowup M2 is similar to the blowup space construc-
b
tion for incomplete conical singularities by Mooers [Moo99] with the difference
that here the blowup is not parabolic in µ−1−direction.
e e
Theorem2.3. Considerthe elliptic boundaryvalue problem(A,B) with a differential
operator A of order q on a compact manifold M of dimension m with boundary ∂M.
Then the resolvent kernel K and the interior resolventkernel K are both elements
AB A
of Aq,q(M2,β∗Y), where Y := {(µ,p,p) ∈ R+ ×M2 | p = p} denotes the diagonal
phg b
hypersurface. Moreover, their difference K −K =: K ∈ Aq,∞(M2).
AB A phg b
e e
Proof. For R > 0 sufficiently large and µ ∈ Γ , we may write according to [See69,
R
(26), (28)]
Op(d ) = (2π)−m eihy−ye,ζid (x,y,ζ,x,µ)dζ, (2.6)
−q−j −q−j
Rm−1
Z
whered ishomogeneous ofdegree(−q−1) ine(x−1,x−1,ζ,µ). Moreover, [See69,
−q−j e
(29)] asserts the following estimate
e
xixk∂α∂β∂γ∂δd (x,y,ζ,x,µ) ≤ Cexp(−ec(x+x)(|ζ|+µ))
x xe ζ µ −q−j
(2.7)
(cid:12)(cid:12) (cid:12)(cid:12)×(|ζ|+µ)1−q−j−k−i+α+β−|γ|−δ,
e
(cid:12) e e (cid:12) e
6 BORISVERTMAN
with constants c,C > 0. This estimate is stable under differentiation in y ∈ Rm−1.
We may now study the asymptotics of the lift β∗Op(d ) in the various projective
−q−j
coordinates near the front face of M2. For instance, in coordinates (2.4) we find
b
β∗Op(d ) = (2π)−mx−m+q+j eihu,νid (s,y,ν,1,τ−1)dν. (2.8)
−q−j −q−j
Rm−1
Z
Hence the lift β∗Op(d ) is of order (−m+q+je) at the front face ff (x → 0), and
−q−j e
smooth at rf (s → 0). In view of the estimate (2.7) the expression is also vanishing
to infinite order at the temporal face tf (τ → 0). Similarly, in coordinates (2.5)
e
β∗Op(d ) = (2π)−m(ηx)−m+q+j eihU,νid (S +η−1,y,ν,η−1,1)dν. (2.9)
−q−j −q−j
Rm−1
Z
Hence the lift β∗Op(d ) is of order (−m + qe+ j) at the front face ff (x → 0).
−q−je
In view of the estimate (2.7) the expression is also vanishing to infinite order at the
temporal face tf (|(S,U)| → ∞) as well as temporal diagonal td (η → 0).
e
Summarizing we have shown β∗Op(d ) ∈ Aq,∞(M2). Similar arguments ap-
−q−j phg b
plied to the classical symbol expansion of the interior parametrix K = (A+µq)−1
A
explain K ∈ Aq,q(M2,β∗Y). Since β∗µ = ρ ρ ρ , we infer from Theorem 2.1
A phg b ff td tf
N−1
β∗K = β∗K + β∗Op(d )+O(ρ ρ ρ )−m+q+N, (2.10)
AB A −q−j ff td tf
j=0
X
as (ρ ,ρ ,ρ ) → 0. Taking the limit N → ∞ proves at the statement. (cid:3)
ff td tf
3. Composition of polyhomogeneous Schwartz kernels
Let X and X′ be two compact manifolds with corners, and let f : X → X′ be
a smooth map. Let {H } and {H′} be enumerations of the codimension one
i i∈I j j∈J
boundary faces of X and X′, respectively, and let ρ , ρ′ be global defining functions
i j
for H , resp. H′. We say that the map f is a b-map if
i j
f∗ρ′ = A ρe(i,j), A > 0, e(i,j) ∈ N∪{0}.
j ij j ij
i∈I
Y
The map f is called a b-submersion if f induces a surjective map between the b-
∗
tangent bundles of X and X′. The notion of b-tangent bundles has been introduced
in [Mel93]. Assume moreover that for each j there is at most one i such that
e(i,j) 6= 0. In other words no hypersurface in X gets mapped to a corner in X′.
Under this condition the b-submersion f is called a b-fibration.
Suppose that ν is a density on X which is smooth up to all boundary faces and
0
everywhere nonvanishing. A smooth b-density ν is, by definition, any density of the
b
form ν = ν (Πρ )−1. Let us fix smooth b-densities ν on X and ν′ on X′.
b 0 i b b
Proposition 3.1. [Mel92, The Pushforward Theorem] Let f : X → X′ be a b–
b
fibration. Let u be a polyhomogeneous function on X with index sets E the faces H
i i
of X. Suppose that each (z,p) ∈ E has Rez > 0 if e(i,j) = 0 for all j ∈ J. Then
i
the pushforward f (uν ) is well-defined and equals hν′ where h is polyhomogeneous
∗ b b
on X′ and has an index family f (E) given by an explicit formula in terms of the
b
index family E for X.
MULTIPARAMETER RESOLVENT TRACE EXPANSIONS 7
Rather than giving the formula for the image index set in general, we provide the
index image set in a specific setup, enough for the present situation. If H and H
i1 i2
are both mapped to a face H′, and if H ∩ H = ∅, then they contribute to the
j i1 i2
index set E +E to H′. If they do intersect, however, then the contribution is the
i1 i2 j
extended union E ∪E
i1 i2
E ∪E := E ∪E ∪{((z,p+q +1) : ∃(z,p) ∈ E , and (z,q) ∈ E }.
i1 i2 i1 i2 i1 i2
We now employ the Pushforward theorem to establish the following fundamental
composition result. Let Y := {(µ,p,p) ∈ R+ × M2 | p = p} denote the diagonal
hypersurface. Consider any K ∈ Aℓ,k(M2,β∗Y) and K ∈ Aℓ′,∞(M2). Their
a phg b b phg b
composition is defined by e e
K (p,p;µ) = K (p,p′;µ)K (p′,p;µ)dvol (p′). (3.1)
c a b M
ZM
Proposition 3.2.
e e
Aℓ,k(M2,β∗Y)◦Aℓ′,∞(M2) ⊂ Aℓ+ℓ′,∞(M2). (3.2)
phg b phg b phg b
Proof. Consider K ∈ Al,k(M2,β∗Y) and K ∈ Al′,∞(M2). Their composition
a phg b b phg b
K = K ◦ K is defined in (3.1). This expression can be rephrased in microlocal
c a b
terms. Consider the space R+ ×M3 , and the three projections
1/µ (p,p′,pe)
π : R+ ×M3 → R+ ×M2 ,
c 1/µ (p,p′,pe) 1/µ (p,pe)
πa : R+1/µ ×M(3p,p′,pe) → R+1/µ ×M(2p,p′), (3.3)
π : R+ ×M3 → R+ ×M2 .
b 1/µ (p,p′,pe) 1/µ (p′,pe)
We reinterpret K ,K and K as ‘right densities’
a b c
K ≡ K (p,p′;µ)dvol (p′),
a a M
K ≡ K (p′,p;µ)dvol (p),
b b M
K ≡ K (p,p;µ)dvol (p).
c c M
e e
Then we can rewrite (3.1) as
e e
K = (π ) (π∗K ·π∗K ).
c c ∗ a a b c
The basic idea in the proof of polyhomogeneity of K is a construction of a triple-
c
space M3 which is a blowup of R+ × M3 obtained by a sequence of blowups,
b 1/µ
designed such that there are maps
Π ,Π ,Π : M3 −→ [R+ ×M2,C] =: M2
a c b b 1/µ rb
which ‘cover’ the three projections defined above. The construction is reminiscent of
the triple space construction for the heat space calculus for conical singularities, see
[Moo99], but differs from the latter since there is no convolution in the parameter
µ−1 variableand the blowups are not parabolicin theµ−1 direction. Oneach copy of
M we use the local coordinates p = (x,y),p′ = (x′,y′),p = (x,y) ∈ R+×Rm−1 near
the boundary ∂M with (x,x′,x) being the three copies of the boundary defining
function. First we blow up the submanifold
e e e
F = {(x,y,x′,y′,x,ye,µ) | µ = ∞,x = x′ = x = 0,y = y′ = y}
= π−1C ∩π−1C ∩π−1C,
a b c
e e e e
8 BORISVERTMAN
which is the intersection of all highest codimension corners C introduced in (2.1),
pulled back under the three projections to R+×M3. Then we blow up the resulting
space [R+ ×M3,F] at the lifts of each of the three submanifolds
F = π−1C = {(x,y,x′,y′,x,y,µ) | µ = ∞,x = x = 0,y = y},
c c
F = π−1C = {(x,y,x′,y′,x,y,µ) | µ = ∞,x = x′ = 0,y = y′}, (3.4)
a a
e e e e
F = π−1C = {(x,y,x′,y′,x,y,µ) | µ = ∞,x′ = x = 0,y′ = y}.
b b
e e
Identifying notationallyeachF withtheirliftsto[R+×M3,F], wemayaltogether
a,b,c
define the triple space e e e e
M3 := R+ ×M3,F F ,F ,F .
b a b c
If we ignore the µ−1−direction, the spacial part of M3 can be visualized as below.
(cid:2)(cid:2) (cid:3) (cid:3)
b
101
001 100
111
011 010 110
Figure 2. The spacial component of the triple space M3.
b
Here, (101),(011)and (110) label the boundary faces created by blowing up F ,F
c b
and F , respectively. The face (111) is the front face introduced by blowing up F.
a
We denote the defining function for the face (ijk) by ρ . The triple space comes
ijk
with a natural blowdown map β(3) : M3 → R+ ×M3, which as in the discussion of
b
M2 amounts in local coordinates to a coordinate change back to (x,y,x′,y′,x,y,µ).
b
Now consider the projections π , π and π introduced in (3.3). Theseeineduce
c a b
projections Π , Π and Π from M3 to the reduced blowup space M2. It is not
c a b b rb
hard to check that the choice of submanifolds F,F that have been blown up
a,b,c
ensures that these projections are in fact b-fibrations.
Denote the defining functions for the right, front and left faces of each copy of
M2 by {ρ ,ρ ,ρ }, respectively. These lift via the projections according to the
rb 10 11 01
following rules
Π∗(ρ ) = ρ ρ ,
c ij i0j i1j
Π∗(ρ ) = ρ ρ , (3.5)
a ij ij0 ij1
Π∗(ρ ) = ρ ρ .
b ij 0ij 1ij
Now consider the behaviour in the parameter µ−1-direction. Let τ be the defining
function for the boundary face in M3 which is mapped onto {µ = ∞} by the
b
blowdown map. Since each F,F is a submanifold of {µ = ∞}, we find
a,b,c
β∗ µ−1 = τρ ρ ρ ρ . (3.6)
(3) 111 110 101 011
MULTIPARAMETER RESOLVENT TRACE EXPANSIONS 9
Let β : M2 → R+ ×M2 be the blowdown map for the reduced blowup space.
(2) rb
Then β∗ µ−1 = Tρ , where T is the defining function for the temporal face tf in
(2) 11
M2. Note that β ◦Π = π ◦β and hence acting on functions on R+×M2
rb (2) a,b,c a,b,c (3)
we have
Π∗ ◦β∗ = β∗ ◦π∗ .
a,b,c (2) (3) a,b,c
Consequently, in view of Eq. (3.5) and Eq. (3.6), we conclude
Π∗(T) = τρ ρ ,
c 110 011
Π∗(T) = τρ ρ , (3.7)
a 101 011
Π∗(T) = τρ ρ .
b 101 110
Using these data, we may now prove the anticipated composition formula. Con-
sider the ‘right densities’, K (x,y,x′,y′;µ)dx′dy′ and K (x′,y′,x,y;µ)dxdy. Their
a b
product is given by
K (x,y,x′,y′;µ)·K (x′,y′,x,y;µ)dx′dy′dxdye e e e
a b
Its integral over dx′dy′ gives K (x,y,x,y;µ)dxdy. To put this into the same form
c
required in the pushforward theorem, write t =e µe−1 and multeiplyethis expression by
dtdxdy.
e e e e
Blowing up a submanifold of codimension n amounts in local coordinates to intro-
ducing polar coordinates, so that the coordinate transformation of a density leads
to (n−1)st power of the radial function, which is the defining function of the cor-
responding front face. Hence we compute the lift
β∗ (dtdxdydx′dy′dxdy)
(3)
= ρ3+2(m−1)ρ2+(m−1)ρ2+(m−1)ρ2+(m−1)ν(3) (3.8)
111 101 110 011
e e
3+2(m−1) 2+(m−1) 2+(m−1) 2+(m−1) (3)
= ρ ρ ρ ρ τ (Πρ )ν ,
111 101 110 011 ijk b
where ν(3) is a density on M3, smooth up to all boundary faces and everywhere
b
nonvanishing; ν(3) is a b-density, obtained from ν(3) by dividing by a product of all
b
defining functions on M3; and (Πρ ) is a product over all (ijk) ∈ {0,1}3. Set
b ijk
κ = β∗ K and κ = β∗ K . Since κ is vanishing to infinite order as T → 0, its
a (2) a b (2) b b
lift Π∗κ vanishes to infinite order in τρ ρ by Eq. (3.7).
b b 110 101
Since κ is not polyhomogeneous on the reduced blowup space M2, the lift Π∗κ
a rb a a
is not polyhomogeneous in τ. However, due to infinite order vanishing of Π∗κ in
b b
τ, their product Π∗κ ·Π∗κ is polyhomogeneous and vanishing to infinite order in
a a b b
τρ ρ ρ . We obtain
110 101 011
Π∗κ ·Π∗κ β∗ (dtdxdydx′dy′dxdy) = ρℓ+ℓ′+1(Πρ )Gν(3),
a a b b (3) 111 ijk b
where G is a bounded polyhomogeneous function on M3, vanishing to infinite order
b
in (τρ ρ ρ ), with index sets N at theefacees (001), (100) and (010). Applying
110 101 011 0
the Pushforward Theorem now gives
(Π ) Π∗κ ·Π∗κ β∗ (dtdxdydx′dy′dxdy)
c ∗ a a b b (3)
(3.9)
= β∗ (cid:0)(K dtdxdydxdy) = ρ2+ℓ+ℓ′G′ν(2), (cid:1)
(2) c 11 b
e e
where ν(2) is a b-density on M2 and G′ is a bounded polyhomogeneous function on
M2, whbich vanishes to infiniterborder ineT,eand has the index set N at the left and
rb 0
10 BORISVERTMAN
right boundary faces. By [EMM91, Proposition B7.20] the pushforward is smooth
across β∗Y.
Note also that the pushforward by Π does not introduce logarithmic terms in the
c
front face expansion of κ , since the kernel on M2 is vanishing to infinite order at
c rb
(101). Hence, for κ and κ with integer exponents in their front face expansions,
A B
same holds for their composition.
By an argument similar to Eq. (3.8), we compute
β∗ (dtdxdydxdy) = ρm+1(ρ ρ ρ T)ν(2). (3.10)
(2) 11 10 11 01 b
Consequently, combining Eq. (3.9) and Eq. (3.10), we deduce that β∗ K vanishes
(2) c
to infinite order in T, is of leadingeoreder (−m+ℓ+ℓ′) at the front face and has the
index sets N at the left and right boundary faces. This proves the statement. (cid:3)
0
4. Multiparameter resolvent trace expansion
We continue in the notation introduced in §1. We abbreviate
n n
µq := λqV (0)+zq, λ(V,W) := λq(V −V (0))+W,
k k k k k
k=1 k=1
X X
and expand A (λ)−1 in Neumann series, for the moment only formally
B
(A (λ)+zq)−1 =: I +(A +µq)−1λ(V,W) (A +µq)−1
B B B
∞
(cid:0) (cid:1)
= (−1)j (K λ(V,W))jK −(K λ(V,W))j K
AB AB A A
Xj=0 (cid:16) (cid:17) (4.1)
∞ ∞
+ (−1)j(K λ(V,W))jK =: (−1)jRj +R .
A A 0 1
j=0 j=0
X X
=: R +R .
0 1
The Neumann series converges in the operator norm for z ∈ Γ with |z| ≫ 0 suffi-
ciently large. Trace norm estimates of Rj depend on the following basic result.
0
Proposition 4.1. Consider R ∈ Aℓ,∞(M2). Then R defines a Hilbert Schmidt
phg b
operator with the Hilbert Schmidt norm
kR(·,µ)k = O(µ−ℓ+m/2), µ → ∞.
HS
Proof. We explify the argument in projective coordinates (2.4). R lifts to a polyho-
mogeneous function on M2 and
rb
β∗(R2dxdydxdy) = x−m+2ℓGdsdxdudy
= µ−2ℓ+mτm−2ℓGdsdxdudy,
e e e e
where G is bounded in (x,s,u,y) and vanishing to infinite order as τ → ∞. Similar
estimates hold at the other two corners of thee front face in the reduced blowup space
M2. Consequently we obtain as µ → ∞
rb e
kR(·,µ)k2 = R(p,p;µ)2dvol (p)dvol (p) = O(µ−2ℓ+m).
HS M M
ZM ZM
(cid:3)
e e