The kernel bundle of a holomorphic Fredholm family

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by  Thomas Krainer

THE KERNEL BUNDLE OF A HOLOMORPHIC FREDHOLM
FAMILY
3
1 THOMASKRAINERANDGERARDOA.MENDOZA
0
2 Abstract. Let Y be a smooth connected manifold, Σ ⊂ C an open set and
n (σ,y) → Py(σ) a family of unbounded Fredholm operators D ⊂ H1 → H2
a of index 0 depending smoothly on (y,σ) ∈ Y ×Σ and holomorphically on
J σ. We show how to associate to P, under mildhypotheses, a smooth vector
4 bundle K → Y whose fiber over a given y ∈ Y consists of classes, modulo
2 holomorphicelements,ofmeromorphicelementsφwithPyφholomorphic. As
applications wegivetwoexamples relevantinthegeneraltheoryofboundary
] valueproblemsforellipticwedgeoperators.
P
A
.
h
1. Introduction
t
a
m It has long been recognized that, as with uniformly elliptic linear operators on
smooth bounded domains, also for other classes of linear elliptic operators A it is
[
the case that boundary conditions should be expressed as conditions on (some of)

1 the coefficients of the asymptotic expansion at the boundary of formal solutions
v
of Au = f. Such asymptotic expansions are proved to exist in many instances,
1
for example for elliptic b-operators or operators of Fuchs type (Kondrat′ev [6],
1
8 Melrose [9], Melrose and Mendoza [10, 11], Rempel and Schulze [12], Schulze [15],
5 in an analytic context Igari [5], etc.), where such expansions, well understood,
1. form a finite dimensional space (Lesch [7]). More generally, such expansions also
0 existundersomeconditionsforelliptic e-operators(Mazzeo[8],andinasomewhat
3 differentcontext,CostabelandDauge[1,2]andSchmutzler[13,14]). Theseclasses,
1
which include regular elliptic differential operators, come up in certain geometric
:
v problems on noncompact manifolds. Slightly modified, they are central in the
i analysisofellipticproblemsoncompactmanifoldswithsingularities(conicalpoints,
X
edges, and corners, for instance).
r
a Because of their importance andpotential applicability there is greatinterestin
developing the tools to handle boundary value problems for these classes. To this
endwepresentherethedefinitionofasmoothvectorbundlewhosesectionsarethe
principal parts of the analogues of the traces (in the sense of boundary values) in
thecaseofclassicalproblems. Boundaryconditionsaretobeimposedasdifferential
orpseudodifferential conditions onthese sections. We refer to the vectorbundle as
the trace bundle. It depends, in general, on the differential operator itself.
Togivesomeconcretecontext,considerfirstthecasewhereAisaregularelliptic
linear differential operator of order m > 0 on a manifold with boundary. After
2010 Mathematics Subject Classification. Primary: 58J32;Secondary: 58J05,35J48,35J58.
Key words and phrases. Manifoldswith edge singularities, ellipticoperators, boundary value
problems.
WorkpartiallysupportedbytheNationalScienceFoundation,GrantsDMS-0901202andDMS-
0901173.
1
2 THOMASKRAINERANDGERARDOA.MENDOZA
localization and flattening of the boundary,
A= a (x,y)DkDα
k,α x y
X
k+|α|≤m
in a neighborhood of 0 in R ×Rn. The coefficients are smooth, the boundary is
+
x = 0 and the interior of the manifold contains the region x > 0. The operator
P = xmA is an example of an edge operator, and x−mP (so A itself) an example
of a wedge operator. The operator P is equal to
a (x,y)xm−k−|α|p (xD +i|α|)(xD )α
k,α k x y
X
k+|α|≤m
where p (σ)=(σ+i(k−1))(σ+i(k−2))···σ. The Ansatz that a formalsolution
k
ofPu=0 is a formalseriesin (possibly complex)powersof xatx=0leads to the
conclusion that in fact u has a classical Taylor expansion,
∞
u∼ u (y)xk.
k
Xk=0
Theprimaryreasonforthiscanbeseenthroughthesameanalysisaswithordinary
differential operators with regular singular points: the indicial equation for the
operatoraboveisa (0,y)p (σ)=0,anequationwhoserootsarethe elements of
m,0 m
I ={0,−i,−2i,...,−i(m−1)}.
These are the only roots because a (0,y) is invertible due to the assumed ellip-
m,0
ticity of A. The coefficients of xiσ with σ ∈I in the formal Taylor expansion of a
solution of Au = 0 are the objects on which conditions are placed. The upshot of
this brief analysis is that if A acts on sections of a vector bundle E, then the trace
bundle of A is the direct sum of m copies of the restriction of E to the boundary;
thismaybeviewedsimplyasthekernelofxmDm. Thisleadstothetrivialcomplex
x
vector bundle of rank m if A is a scalar operator.
This analysis generalizes, with a different conclusion, to operators of the form
A=x−mP where P is an edge operator,
P = a (x,y,z)(xD )k(xD )αDβ,
k,α,β x y z
X
k+|α|+|β|≤m
(see [8]) where again a is smooth and, in the simplest case, z ranges in a
k,α,β
compactmanifoldZ withoutboundary; Z isapointinthecaseofaregularelliptic
operator. Looselyspeaking,thevariabley liesinanopensetsomeEuclideanspace,
more generally in manifold Y, while x ≥ 0. The case Y = {pt.} models conical
singularities. Edge-ellipticity of P is the property that its edge symbol,
a (x,y,z)ξkηαζβ,
k,α,β
X
k+|α|+|β|=m
is invertible if (ξ,η,ζ)6=0. If P is e-elliptic, its indicial family,
P (σ)= a (0,y,z)σkDβ,
y k,0,β z
X
k+|α|≤m
is elliptic as a family of differential operators on Z (typically acting on sections of
a vector bundle). This family depends smoothly on (y,σ) and holomorphically on
σ ∈ C. For any given y the elements σ for which P (σ) is not invertible forms a
y
THE KERNEL BUNDLE OF A HOLOMORPHIC FREDHOLM FAMILY 3
closeddiscrete set spec (P )⊂C called the boundary spectrum of A at y (see [9]);
b y
by a well known trick, this set is the spectrum of an operator. Quite evidently,
spec (P ) typically depends on y. If Z is a manifold with boundary, then the
b y
indicial family is accompanied by homogeneous boundary conditions coming from
the original setup—again making the b-spectrum discrete for each y. The possible
dependence on y may lead, because of variable multiplicity, to the effects referred
tointheliteratureasbranchingasymptoticsmakingtheanalysisofboundaryvalue
problemsforA=x−mP considerablymoredifficultincomparisonwiththeclassical
case.
It is this branching that we address here. What concerns us is the global def-
inition of the trace bundle, specifically, its C∞ structure: in order to develop a
generaltheoryofboundaryvalueproblemsforellipticoperatorssuchasA,aglobal
problem by its very nature (consider for instance the APS boundary condition),
one needs to be able to refer to traces as global objects.
The rest of the paper is organized as follows.
We collect in Section 2 all the assumptions underlying our construction of a
vector bundle and definition of its C∞ structure, of a vector bundle closely related
tothetracebundle. Thisvectorbundle,whichwecallthekernelbundle,isdefined,
as a set, in Section 3; see Theorem 3.2. Our approach—defining the kernel bundle
rather than a trace bundle directly—allows for enough generality to treat at the
same time trace bundles both when Z closed and when ∂Z =6 ∅ (the latter case
withsomeboundarycondition). Allowingforsuchgeneralityopensthe doortouse
an iterative approach to handle the more complicated situations that arise in the
presence of a stratification of the boundary.
We constructthe putative smooth frames in Section 4. An equivalent versionof
theselocalframeswasalreadydefinedbySchmutzler[13,14]usingKeldyshchains,
however,these papers do not address the regularity of the transition functions.
Our construction ofthe specialframes lends itself to be readily used to show,in
Section 5, that two frames of the same kind are related by smooth transition func-
tions,inotherwords,thatthe kernelbundle hasa smoothvectorbundle structure,
see Theorem 5.1. This structure is natural in that the property that a section is
smoothcanbecheckedintrinsically. TheproofofTheorem5.1reliesheavilyonthe
smoothness and nondegeneracy of a pairing between the kernel bundle (associated
to a family of operators)andthat ofits dual. This resultis statedas Theorem5.3;
its proof relies heavily on ideas used in [4]. Section 5 is at the heart of this work.
The last two sections provide examples. In Section 6, the first of these two
sections,weconsiderthecaseofageneralellipticwedgeoperatorx−mP,P ∈Diffm,
e
where the boundary fibration has compact fibers. In the second, Section 7, we
illustrate with a toy example (motivated by what would be codimension 1 cracks
in linear elasticity) the use of the kernel bundle when the fibers of the boundary
fibration (the Z) are nonclosed.
2. Set up
Let Y be a connected manifold and
℘ :H →Y, ℘ :H →Y
1 1 2 2
smoothHilbertspacebundles. FurtherletD →Y beanothersmoothHilbertspace
bundlecontinuouslyembeddedinH withfiberwisedenseimageandsuchthatthe
1
4 THOMASKRAINERANDGERARDOA.MENDOZA
trivializations of H (smooth and unitary) restrict to smooth trivializations of D.
1
We write H , H and D for the model spaces.
1 2
LetΣanopenconnectedsubsetofC. Withπ :Y×Σ→Y denotingthecanonical
projection, let
P :π∗D ⊂π∗H →π∗H (2.1)
1 2
be a smooth bundle homomorphism covering the identity consisting of fiberwise
closed Fredholm operators depending holomorphically on σ. Suppose further that
for each y ∈Y there is σ ∈Σ such that
P (σ):D ⊂H →H (2.2)
y y 1,y 2,y
is invertible. Passing to trivializations over an open set U ⊂ Y, (2.1) becomes a
smooth family
P (σ):D⊂H →H , (y,σ)∈U ×Σ,
y 1 2
holomorphic in σ.
Since (2.2) is Fredholm for all σ ∈Σ and invertible for some such σ,
sing (P )={σ ∈Σ:P (σ) is not invertible}
b y y
is a closed discrete subset of Σ and
sing (P)={(y,σ)∈Y ×Σ:σ ∈sing (P )}.
e b y
isaclosedsubsetofY×Σ. Thenotationismotivatedbythecorrespondingobjects
in the context of b- and e-operators,spec and spec (see [9], [8], also [3]). We will
b e
assume the stronger condition that
sing (P )isafinite setforeachy ∈Y andsing (P)isclosedinY×C. (2.3)
b y e
The condition that sing (P) is closed in Y×Σ means that for every y ∈Y there
e 0
is δ >0 and a neighborhood U of y such that dist(sing (P ),C\Σ)>δ if y ∈U.
0 b y
Write P (σ)∗ for the Hilbert space adjoint of (2.2). Assume that the domains
y
of the P (σ)∗ join to give another smooth Hilbert space bundle D∗ continuously
y
embedded in H and such that the smooth unitary trivializations of H restrict
2 2
to trivializations of D∗. Defining P∗(σ) = P (σ)∗ we obtain another smooth
y y
homomorphism
P∗ :π∗D∗ ⊂H →π∗H
2 1
depending holomorphicallyon σ, where now π is the projection Y×Σ→Y, which
satisfies the same Fredholm, analyticity, and invertibility properties as P.
3. The kernel bundle
If K is a Hilbert space and V ⊂ C is open, we write M(V,K) for the space of
meromorphicK-valuedfunctionsonV andH(V,K)forthesubspaceofholomorphic
elements. Thus f ∈ M(V,K) if there is, for each σ ∈ V, a number µ ∈ N such
0 0 0
that σ 7→(σ−σ )µ0f(σ) is holomorphic near σ . Suppose Ω⋐V is open and has
0 0
smooth (or rectifiable) boundary. If f ∈M(V,K) has finitely many poles in Ω and
no poles on ∂Ω, then the sum of the singular parts of f at each pole in Ω is given
by
i f(ζ)
s (f)(σ)= dζ, |σ|≫1
V
2π I ζ−σ
∂Ω
with the positive orientationfor ∂Ω. Replacing Ω by a disjoint union of open discs
with small radii, each containing at most a single pole of f and contained in Ω we
see that the formula determines an element of M(C,K).
THE KERNEL BUNDLE OF A HOLOMORPHIC FREDHOLM FAMILY 5
Let V be an open subset of Σ. Since σ 7→P (σ) is holomorphic, it gives maps
y
P :M(V,D )→M(V,H ), P :H(V,D )→H(V,H )
y y 2,y y y 2,y
so there is an induced map
[P ] :M(V,D )/H(V,D )→M(V,H )/H(V,H ). (3.1)
y V y y 2,y 2,y
Any element [φ] ∈ ker[P ] is represented uniquely by the sum of the singular
y V
parts of any given representative φ at the various poles in V. Define
K ={s (φ):φ∈M(Σ,D ), P φ∈H(Σ,H )}.
y Σ y y 2,y
Thus K is canonicallyisomorphicto the kernelof [P ] . It is a vectorspace over
y y Σ
C,finite-dimensionaldueto(2.3)andthevariousotherhypothesismadeonP. An
element of K is in particular a D -valued meromorphic function on C with poles
y y
contained in sing (P ); and if the element is regular, then it is the zero function.
b y
It is also convenient to define, if σ ∈Σ
0
K ={s (φ):φ∈M(D,D ):P φ∈H(D,H )}
y,σ0 D y y 2,y
where D is a disc in Σ centered at σ with
0
sing (P )∩D\{σ }=∅.
b y 0
Thus
K = K .
y y,σ0
σM0∈Σ
Theorem 3.2. Define
K = K , π :K →Y the canonical map,
y
yG∈Y
and let B∞(Y;K ) be the space of right inverses of π which viewed as D-valued
functions on (Y ×C)\sing (P) by way of the trivializations of H are smooth in
e 1
thecomplement of sing (P)andholomorphic in σ. If (2.3)holds, then π :K →Y
e
has a smooth vector bundle structurewith respect to which its space of C∞ sections
is B∞(Y;K ).
We may replace Y by an open subset U ⊂ Y in all of the above, in which case
we naturally write, with only slight abuse of the notation, B∞(U;K ).
Definition 3.3. The vector bundle K → Y is the (meromorphic) kernel bundle
of P.
Since P(σ) commutes with multiplication by functions that depend only on y,
B∞(Y;K ) is certainly a module over C∞(Y).
TheproofofTheorem3.2willoccupythenexttwosections. Inthefirstofthese,
Section4, wewillfind foreveryy ∈Y, a neighborhoodU ofy overwhichD, H ,
0 0 1
and H are trivial and elements
2
φs ∈B∞(U;K ), s=1,...,S, j =1,...,J , ℓ=0,...,L −1 (3.4)
j,ℓ s s,j
giving a pointwise basis of K for each y ∈ U. Then we will show, in Section 5,
y
that for any φ∈B∞(U;K ), the functions fj,ℓ such that
s
φ= fj,ℓφs
s j,ℓ
sX,j,ℓ
6 THOMASKRAINERANDGERARDOA.MENDOZA
are smooth. Consequently, declaring the φ to be a frame over U gives the
s,j,ℓ
desired smooth vector bundle structure. The somewhat peculiar indexing of the
components of the frame reflects the nature of the problem, as will be clear as we
develop the proof.
4. Frames
Fix y ∈Y. In the rest of this section we will workin a neighborhoodof y over
0 0
which D, H and H are trivial. We write P (σ) : D ⊂ H → H for the family
1 2 y 1 2
viewedonthetrivializationandwritesing (P )inplaceofsing (P )andsing (P)
b y b y e
for the part of sing (P) over the given neighborhood of y . In this section we aim
e 0
at finding a neighborhoodU of y and elements (3.4) giving a basis of K for each
0 y
y ∈U.
Denotebyσ ,s=1,...,S ,thepointsinsing (P ). LetK ⊂DandR ⊂H
s y0 b y0 s s 2
be, respectively, the kernel and image of P (σ ). The latter space is a closed
y0 s
subspaceofH . The spacesK andR⊥ arefinite-dimensionalsubspacesofH and
2 s s 1
H , respectively, of the same dimension because P (σ) has index 0. Write P (σ)
2 y0 y
in the form
K R⊥
p1 p1 s s
s,1 s,2 : ⊕ → ⊕ ; (4.1)
(cid:20)p2 p2 (cid:21)
s,1 s,2 K⊥ R
s s
the pi are smooth in (y,σ) and holomorphic in σ. The space K⊥ is the subspace
s,j s
ofD orthogonaltoK withrespecttothe innerproductofH . Likewise,thespace
s 1
R⊥ is the orthogonal of R in H . All entries except p2 vanish at (y ,σ ), and
s s 2 s,2 0 s
p2 (y ,σ ) is invertible.
s,2 0 s
There is ε>0 such that
(1) the family of discs
D ={σ :|σ−σ |<2ε}, s=1,...,S
s,2ε s y0
is pairwise disjoint with each D(σ ,2ε) contained in Σ;
s
(2) for each s, p2 (y ,σ) is invertible if σ ∈D .
s,2 0 s,2ε
By continuity, (2) implies there is a connected neighborhood U of y such that
0
(3) for each s, p2 (y,σ) is invertible if (y,σ)∈U ×D .
s,2 s,ε
Define P (y,σ):K →R⊥ for (y,σ)∈U ×D by
s s s s,ε
P =p1 −p1 (p2 )−1p2 . (4.2)
s s,1 s,2 s,2 s,1
Inviewof(3),theinvertibilityofP (σ)ifσ ∈D isequivalenttothatofP (y,σ).
y s,ε s
In particular, by (1), P (y ,σ) is invertible if σ ∈D \{σ }.
s 0 s,ε s
The spaces K and R⊥ have the same dimension so, after fixing a basis for each
s s
of these spaces, it makes sense to talk about the determinant of a map K →R⊥.
s s
Let then q =detP be computed with respectto some such pair of bases. This is
s s
a smooth function on U ×D , holomorphic in σ ∈D and q (y ,σ) is nonzero
σs,ε s,ε s 0
on D \{σ } by (1). Note that
s,ε s
sing (P )∩D ={σ ∈D :q (y,σ)=0}.
b y s,ε s,ε s
Shrinking U further we may thus also assume that
THE KERNEL BUNDLE OF A HOLOMORPHIC FREDHOLM FAMILY 7
(4) for each s, q is nowhere zero in U ×{σ :ε/2≤|σ−σ |<ε}, equivalently,
s s
Sy0
y ∈U =⇒ sing (P )⊂ D .
b y s,ε/2
s[=1
It follows from (4) that the number of zeros, counting multiplicity, of the function
D ∋σ 7→q (y,σ)∈C
s,ε s
is independent of y ∈ U; we assume throughout that U is connected. Letting
d be that number, we have in particular that the function q (y ,σ) factors as
s s 0
(σ−σs)dshs(σ) where hs, defined in Ds,ε, is holomorphic and vanishes nowhere in
its domain.
For each y ∈U let
K ={s (φ):φ∈M(D ,K ), P (y,·)φ∈H(D ,R⊥)}.
s,y Ds,ε s,ε s s s,ε s
This space is canonically isomorphic to the kernel of the operator
M(D ,K )/H(D ,K )→M(D ,R⊥)/H(D ,R⊥) (4.3)
s,ε s s,ε s s,ε s s,ε s
inducedbyP (y,·). The elements ofK areK -valuedmeromorphicfunctions on
s s,y s
CwithpolesinD ∩sing (P ). Indeed, P (y,σ)−1 ismeromorphicinD with
s,ε/2 b y s s,ε
poles in D ∩sing (P ).
s,ε/2 b y
By [4, Lemmas 5.2 and 5.5] there are elements φKs(y ,σ)∈K , j =1,...,J ,
j,0 0 s,y0 s
with pole only at σ of some order L such that
s s,j
(a) the elements
φKs(y ,σ)=s (σ−σ )ℓφKs(y ,σ) , j =1,...,J , ℓ=0,...,L −1 (4.4)
j,ℓ 0 Ds,ε s j,0 0 s s,j
(cid:0) (cid:1)
form a basis of K ,
s,y0
(b) the values of the (σ−σs)Ls,jφKj,0s(y0,σ) at σs from a basis of Ks,
(c) the R⊥-valued functions
s
β (σ)=P (y ,σ)(φKs(y ,σ)) (4.5)
s,j s 0 j,0 0
are holomorphic in D and their values at σ form a basis of R⊥.
s,ε s s
Define
φKs =s (σ−σ )ℓP (y,σ)−1 β (σ) , ℓ=0,...,L −1, j =1,...,J .
j,ℓ Ds,ε s s s,j s,j s
(cid:0) (cid:0) (cid:1)(cid:1)
Remark 4.6. Condition (4) above implies that the elements φKs(y,σ) are holo-
j,ℓ
morphic in the complement of
(U ×D )∩sing (A)
s,ε/2 e
in U ×C.
Lemma 4.7. There is a neighborhood U′ ⊂U of y such that for each y′ ∈U′ the
0
restrictions to {y =y′} of the functions φKj,ℓs give a basis of Ks,y′.
Proof. We first argue that dimK = d for all y ∈ U. By [4, Lemma 5.5] the
s,y s
dimension of K is d = L . This is also the number of zeros of q (y ,·) in
s,y0 s j s,j s 0
Ds,ε counting multiplicity P(i.e., the order of vanishing of qs(y0,·) at σs, its single
zero). To see this, note that the elements (σ −σs)LjφKj,0s(y0,σ) are holomorphic
at σ = σ and that their values there, therefore also nearby, form a basis of K .
s s
We noted that the β (σ) in (4.5) give a basis of R⊥ when σ = σ , hence also for
s,j s s
8 THOMASKRAINERANDGERARDOA.MENDOZA
σ near σ . With respect to these bases, the matrix of P (y ,σ) is diagonal with
s s 0
entries (σ−σs)Lj, therefore detPs(y0,σ)= j(σ−σs)Lj modulo a nonvanishing
factor. If y ∈ U is arbitrary and σ′ ∈ Ds,ε ∩Qsingb(Py), then the same argument
gives that
dim{φ∈K :pole(φ)={σ′}}
s,y
is equal to the order of vanishing of q (y,·) at σ′. Since K is the direct sum
s s,y
of these spaces, the dimension of K is the number of zeros, d again, of q (y,·)
s,y s s
counting multiplicity. Thus dimK =d .
s,y s
We now show that there is a neighborhood U′ of y in U such that the d
0 s
functions φKs(y,·) are linearly independent for every y ∈ U′. If not, there is a
j,ℓ
sequence {y }∞ ⊂ U converging to y and for each k, numbers aj,ℓ not all zero,
k k=1 0 k
such that
aj,ℓφKs(y ,·)=0.
k j,ℓ k
Xj,ℓ
We may assume |aj,ℓ|2 = 1 and then, passing to a subsequence, that the
j,k k
sequence {aj,ℓ}∞Pconverges, say lim aj,ℓ = aj,ℓ. The functions φKs are in
k k=1 k→∞ k j,ℓ
particular defined and continuous when ε/2<|σ−σ |<ε, so
s
0= lim aj,ℓφKs(y ,·)= aj,ℓφKs(y ,·)
k j,ℓ k j,ℓ 0
k→∞Xj,ℓ Xj,ℓ
if |σ−σ |>ε/2(see Remark 4.6). Since the φKs(y ,σ) are meromorphicin C, the
s j,ℓ 0
equality holds everywhere. Since not all a are zero, we reachthe conclusionthat
j,ℓ
the elements (4.4) are linearly dependent, a contradiction. Thus it must be that
the φKs(y,·) are linearly independent for y near y . This completes the proof of
j,ℓ 0
the lemma. (cid:3)
Replacing U by U′ allows us to assume:
(5) the elements φKs form a basis of K for each y ∈U and s=1,...,S.
j,ℓ s,y
The elements φKs form a basis of K for each y ∈U and s=1,...,S. (4.8)
j,ℓ s,y
Suppose now that φ∈B∞(U;K ). Passing to trivializations, there is a smooth
function
u:(U ×Σ)\sing (P)→D
e
such that v =P u is holomorphic in Σ for each y ∈U and
y y y
φ=s (u)
Σ
Thefunctionu ismeromorphicinΣwithpolesinsing (P ). Wewillomity andσ
y b y
fromthenotationinthefollowingfewlines. DecomposingupointwiseoverU×D
s,ε
as
u=uKs +uKs⊥
according to K ⊕K⊥ =D, similarly
s s
v =vR⊥s +vRs,
we obtain using (4.1) that
uKs =P−1(vR⊥s −p1 (p2 )−1vRs), uKs⊥ =(p2 )−1(vR⊥s −p2 uKs).
s s,2 s,2 s,2 s,1
THE KERNEL BUNDLE OF A HOLOMORPHIC FREDHOLM FAMILY 9
Since(p2s,2)−1(vR⊥s )is smoothonU×Ds,ε andholomorphicinthe secondvariable,
u≡uKs −(p2 )−1p2 uKs
s,2 s,1
modulo a smooth function on U ×D depending holomorphically on σ. Since
s,ε
sDs,εuKs isanelementofKs,y foreachy ∈U,Lemma4.7(i.e. Condition(5))gives,
for each such y, unique numbers fj,ℓ(y) such that
s
s (uKs)= fj,ℓφKs,
Ds,ε s j,ℓ
Xj,ℓ
that is,
uKs − fj,ℓφKs
s j,ℓ
Xj,ℓ
is holomorphic in σ for each y ∈U when σ ∈D . So the same is true of
s,ε
u− fj,ℓφKs −(p2 )−1p2 fj,ℓφKs ,
s j,ℓ s,2 s,1 s j,ℓ
(cid:0)Xj,ℓ Xj,ℓ (cid:1)
hence passing to singular parts and adding we also have, with
φ= fj,ℓ φKs −s (p2 )−1p2 φKs , (4.9)
s j,ℓ Ds,ε s,2 s,1 j,ℓ
sX,j,ℓ (cid:0) (cid:1)
that u −φ is holomorphic in D for each y ∈ U and s = 1,...,S. Using
y y s s,ε
that, as a consequence of (2.3), sSpecb(Py) ⊂ sDs,ε, we conclude that uy −φy is
holomorphic in Σ. It follows that the elementSs
φs =φKs −s (p2 )−1p2 φKs (4.10)
j,ℓ j,ℓ Ds,ε s,2 s,1 j,ℓ
(cid:0) (cid:1)
with s = 1,...,S, j = 1,...,J , ℓ = 0,...,L −1, form a basis of K for each
s s,j y
y ∈ U. By means of the trivialization of H over U we view φs as an element of
1 j,ℓ
B∞(U;K ) with singularities within U ×D .
s,ε/2
5. Smoothness of transition functions
The proof of the following theorem will complete our proof of Theorem 3.2.
Theorem 5.1. The φs in (4.10) are elements of B∞(U;K ) giving a basis of K
j,ℓ y
for each y ∈U. If φ∈B∞(U;K ) then
φ= fj,ℓφs for y ∈U (5.2)
s j,ℓ
sX,j,ℓ
with smooth functions fj,ℓ :U →C.
s
Proof. Wehavealreadyseenthat,asaconsequenceofCondition(5)intheprevious
section and (4.9), if φ ∈ B∞(U;K ) then there are unique functions fj,ℓ : U → C
s
such that (5.2) holds. The task now is to show that the fj,ℓ are smooth.
s
Define K ∗ tobe the meromorphickernelbundle ofP∗;this willnotnecessarily
be the dual bundle of K but the notation is convenient. The following theorem is
the key result.
10 THOMASKRAINERANDGERARDOA.MENDOZA
Theorem 5.3. Let Ω ⋐ Σ be open with smooth positively oriented boundary and
sing (P )⊂Ω. The sesquilinear pairing
b y
K ×K ∗ →C,
y y
1 (5.4)
[φ,ψ]♭ = φ(σ),P⋆(σ)ψ(σ) dσ
y 2π I∂Ω(cid:0) y (cid:1)H1,y
is nondegenerate.
The theorem is proved below. Assuming its validity, we proceed as follows.
Revertingtotrivializations,writeP∗(σ)(withσ ∈Σ)forthelocalversionofP∗(σ)
overU. Possibly after shrinking U about y we can carry out the constructions we
0
did for P(σ) in obtaining the elements φs to obtain elements
j,ℓ
ψj,ℓ ∈B∞(U;K ∗)
s
giving a pointwise basis of K ∗ over U. In doing this we note that sing (P∗) is
b y
the conjugate set, sing (P ), of sing (P ) and take advantage notationally of the
b y b y
fact that by [4, Lemma 6.2] we can use the same indices s,j,ℓ as for the φs . We
j,ℓ
also arrange that the singularities of ψj,ℓ lie within U ×D , the “conjugate” of
s s,ε/2
U ×D .
s,ε/2
If φ∈B∞(U,K ) and ψ ∈B∞(U,K ∗), then
1
U ∋y 7→[φ(y),ψ(y)]♭ = φ(y,σ),P⋆(σ)ψ(y,σ) dσ ∈C
y 2π I y H1
Xs ∂Ds,ε(cid:0) (cid:1)
is smooth, simply because the integrand is smooth in the complement in U ×Σ of
sing (P)∩(U ×Σ). Consequently, the functions defined by
e
as,j′,ℓ′(y)=[φs (y),ψj′,ℓ′(y)]♭
s′,j,ℓ j,ℓ s′ y
are smooth, as are the functions
bj′,ℓ′(y)=[φ(y),ψj′,ℓ′(y)]♭.
s′ s′ y
Sincetheφs (y)andψj′,ℓ′(y)formbasesof,respectivelyK andK ∗,thenondegen-
j,ℓ s′ y y
eracyof (5.4)impliesthatthematrixa=[as,j′,ℓ′]isnonsingular. Ifφ∈B∞(U;K )
s′,j,ℓ
then (5.2) gives
bj′,ℓ′ =[φ,ψj′,ℓ′]♭ = fj,ℓ[φs ,ψj′,ℓ′]♭ = fj,ℓas,j′,ℓ′.
s′ s′ s j,ℓ s′ s s′,j,ℓ
sX,j,ℓ sX,j,ℓ
Since ais nonsingularandits entriesandthoseofb=[bj′,ℓ′]aresmooth,soarethe
s′
fj,ℓ. This completes the proof of Theorem 5.1. (cid:3)
s
Proof of Theorem 5.3. The assertion is a pointwise statement, so we may assume
y = y throughout the proof to take advantage of the notation introduced so far;
0
we also work with the trivializations near y of the various Hilbert space bundles.
0
The proof we give here collects arguments spread throughout Sections 5, 6, and 7
of [4], used there to prove an analogous statement. Our notation here is slightly
different from the one used there.
If φ ∈ K , then φ = φs with φs = s φ. The element φs has a pole only
y0 s Ds,ε
at σs and is of the form P
φs =φKs −s (p2 )−1p2 φKs
Ds,ε s,2 s,1
(cid:0) (cid:1)