HODGE COHOMOLOGY
OF
NEGATIVELY CURVED MANIFOLDS
by
RAFE ROYS MAZZEO
10
S.B.,
Massachusetts Institute of Technology
(1982)
SUBMITTED TO THE DEPARTMENT OF
MATHEMATICS IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE,
1986
@ Rafe Roys Mazzeo
The author hereby grants to M.I.T. permission to reproduce
and to distribute copies of this thesis document in whole
or in part.
Signature of Author .....
9ajtment of M!4hematics
May 2. 1986
Certified by .....
Professor Ric ard B. Melrose
sis Supervisor
Ap &T
#i
Accepted by .......
Professor Nesmith C. Ankeny
Graduate Committee
Departmental
Chairman,
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
AUG 0 4 1986
LIBRAnIES
x&?bl
HODGE COHOMOLOGY
OF
NEGATIVELY CURVED MANIFOLDS
by
RAFE ROYS MAZZEO
Submitted to the Department of Mathematics on May 2, 1986
in partial fulfillment of the requirements for the Degree
of Doctor of Philosophy in Mathematics.
ABSTRACT
The Hodge Laplacian acting on differential forms is
examined for a certain class of complete Riemannian
manifolds of dimension
n.
This class consists of the
interiors of compact manifolds with boundary, each endowed
Such a metric, by
with a 'conformally compact' metric.
definition, is of the form
g = p- 2h
smooth nonsingular metric and
p
where
h
is any
is a defining function
These manifolds are all negatively
for the boundary.
curved near infinity; examples include the hyperbolic space
Hn
and those of its quotients which have no cusps.
A parametrix is constructed for the Laplacian acting
on the space of
(n11).
L
2
k-forms for all degrees
k
1
n
Thus, in these degrees the Laplacian is Fredholm,
and in particular its null space is finite dimensional.
The space of
L2
harmonic
k-forms is
then identified with
the relative and absolute deRham cohomology of the manifold
when
k < L(n-1)
and
k > 1L(n+1),
respectively.
It is
also shown that the range of the Laplacian is closed when
k = n/2,
although its null space is of infinite dimension.
The parametrix construction is microlocal:
the
Laplacian should be thought of here as a degenerate elliptic operator on a compact manifold, and a space of
pseudodifferential operators large enough to contain its
Green operator is defined and studied.
calculus, including
L
A fairly complete
continuity properties, is
developed along the way.
Thesis Supervisor:
Title:
Richard B. Melrose
Professor of Mathematics
Acknowledgements
Among the many people who have bettered my life in
some way during the past several years I wish especially
to thank my family, Chet Roys, John Cohn, Diane Mariano,
Joanne Murray, Dr. Michael Smith, Jimmy Page, Rick
Kreminski, Chris Wendel and John Ostlund.
For
the mathema-
tics they have taught me I am grateful to Woody Lichtenstein, Ted Shifrin, Victor Guillemin, Gunther Uhlmann and
particularly David Jerison.
Thanks also to Viola Wiley for
a magnificent typing job.
Finally, well deserving a paragraph of his own is
Richard Melrose.
suffice for all
No amount of gratitude on my part would
that he has done on my behalf.
5
"Don't brood too much on the superiority of
the unseen to the seen.
It's true, but to
brood on it is mediaeval."
-
from Howards End
by E. M. Forster
Table of Contents
Introduction
History of the Problem
Statement of Results
7
7
16
1.
Conformally Compact Metrics
Definitions
A.
Geodesics at Infinity
B.
Curvature Asymptotics
C.
Boundary Regularity
D.
Laplace's Operator
E.
22
22
24
30
33
39
2.
The Calculus of Vo Pseudodifferential
Operators
Vo Vector Fields
A.
B. The Stretched Product Construction
C. Vo Kernels
Indicial and Normal Operators
D.
E. Continuity Properties
43
43
49
63
81
89
3.
Analysis of the Normal Operator
Identification of the Model Operators
A.
The Inverse of the Hyperbolic Laplacian
B.
Structure of the Kernel
C.
Refined Mapping Properties
D.
99
99
110
138
152
4.
Hodge Cohomology
A. The Parametrix Construction
Proof of the Main Theorem
B.
164
164
170
Bibliography
181
Introduction
The many recent advances in the analysis of partial
differential equations notwithstanding, the study of
general
linear elliptic operators on unbounded domains
remains as yet inchoate.
The lack of a systematic theory
is due no doubt to the fact that the geometry of the domain
and the asymptotic degeneracies of the equation each can
radically affect the analytic nature of the operator.
A
natural class of problems in which the geometry and analysis are rather intimately intertwined is in the study of
Laplace's operator on a complete Riemannian manifold, acting either on functions or differential forms.
Even this
case is far from well understood.
A typical question here is whether, on a complete noncompact Riemannian manifold
M,
there exist functions, or
k-forms, which are both harmonic and square-summable (or
bounded).
As was widely suspected from the classical sepa-
ration of Riemann surfaces into those which do or do not
admit bounded harmonic functions, the operative feature is
the curvature.
(Of course, the
L2
class is not well
defined in the conformal geometry of Riemann surfaces,
whereas boundedness is an invariant concept, so long as
attention is restricted to functions.)
theorem of this
The first general
type in higher dimensions is due to Yau
[28], and asserts the nonexistence of nonconstant bounded
harmonic functions on complete manifolds of nonnegative
Ricci curvature.
Negative curvature produces the opposite effect, as
may be discerned from the harmonic analysis of the hypern
1H.
bolic space
functions,
L
Although this space has no
2
harmonic
it admits many bounded harmonic functions--in
fact, by a Poisson-type representation theorem, there is
one corresponding to each continuous function on
aBH n
Sn-1_
(and others with less regular boundary values).
Much attention over the past several years has focused
on the Laplacian of complete negatively curved manifolds,
and some quite general results have been obtained.
Let us
discuss some of these, first for the Laplacian on funcHere the concern is only with bounded harmonic
tions.
functions since there are no nontrivial
n dom
u E L2
fact, if
A
ones.
L2
In
is harmonic, an integration by
parts-which relies on the completeness of
the lack of a boundary term-implies that
M
to ensure
du = 0.
is constant, and must vanish if the volume of
M
Thus
u
is
infinite.
atures
is simply connected and has all sectional curv-
M
If
asserts that
each
p C M.
infinity,
theorem of Cartan-Hadamard
negative, then the
KM
S.,9
exp
: T M -pp
M
is a diffeomorphism for
It is possible to define a sphere at
for such manifolds, cf. Eberlein-O'Neill
[15].
Each point
x E S.
to one another, and these are said
geodesics all asymptotic
to converge to
x.
S.
is identified with a set of
itself is the aggregate of such
mutually asymptotic classes of geodesics.
natural
topology which makes
M = M U S.
There is a
into a compact
topological manifold with boundary (in fact, a disc).
for any
asymptotic Dirichlet problem may then be posed:
given continuous function
on
M
f
on
S.,
The
u
find a function
which is harmonic and assumes the boundary values
asymptotically.
f
It should be remarked that it is insuffi-
cient merely to require
lim u(-Y(t))
= f(x)
t -W
for any geodesic
T(t)
converging to
x E So,;
the convergence must occur in the topology of
stronger than this
instead,
M,
which is
'radial' convergence.
This asymptotic Dirichlet problem was posed by Choi in
[7] and partially solved.
For various technical reasons
one must assume that the sectional curvatures are bounded
away from zero:
KM
-a2 < 0.
Choi proved the result with
this hypothesis, but also assuming a convexity condition at
infinity-namely that for
two distinct points
x, y E SW
there are disjoint geodesically convex neighbourhoods
U ,
This extra condition is
y E U .
with x E U ,
y
x
y
Mike Anderson finished
unnecessary when M is a surface.
U
the proof [1] by showing that if there is a two-sided bound
on curvature,
-b2 < KM (
convexity condition.
result
a2
<
0,
M
then
satisfies the
Sullivan independently proved the
[27] using probabilistic methods.
M
This positive resolution has the consequence that
carries many bounded harmonic functions.
In fact,
the
asymptotic maximum principle of Yau asserts that, in this
situation, the infimum and supremum of
So,
hence coincide with those of
f.
u
are attained on
At this juncture it
is reasonable to seek a Poisson-type representation
formula.
This was accomplished in Anderson-Schoen [3].
Their paper also addresses a quite interesting and
relevant question:
namely, to what extent may
sidered a smooth manifold?
M
be con-
Inasmuch as there is a
S M in the tangent
p
one may form the composite map
homeomorphism between the unit sphere
S0.,
space at any point and
T
pq
: S
p
and study its regularity.
-M
S0
S M
q
Anderson and Schoen prove that
a,
this map is Hlder continuous with exponent
a/b
depending only on the curvature bounds.
with
a =
Little else
is known one way or the other about this question, though
it seems likely that
ever,
T
is rarely
C .
We note, how-
the result of Fefferman [16] concerning the Bergman
metric in the interior of a smoothly bounded strictly
pseudoconvex domain.
This metric has asymptotically
constant holomorphic sectional curvatures, and he proves
in
(which is only defined locally) is C ;
pq
the map from S M to the boundary of the domain is
p
T
that each
fact,
C.
Finally, on a spectral note, it is also known that the
spectrum of the Laplacian (on functions) is contained in
the interval
M
(n-1)
2 2
a /4,co),
cf. McKean [21].
In addi-
(is simply connected and) has sectional curva-
tion,
if
tures
tending uniformly to
its Laplacian has pure
-o,
point spectrum, cf. Donnelly-Li [13].
Much less is known about the Laplacian acting on
differential forms.
Hodge
Of course, when
M
is compact the
theorem provides topological meaning to the dimension
of the space of harmonic forms in each degree.
When
M
is
complete but noncompact, it is natural to allow the
Laplacian to act on
L 2fQk = f
the space of
L2
Ek
= F(AkM) :
k-forms over
1w1 2 < ,
The subspace of harmon-
M.
ic forms is denoted
k ={
Recall
that
EL2nk
wE dom A
n dom
means
A
Aw = 0}.
dw E L2 Qk+1,
EL
2
k-1
and
2 k
d6w E L 2k.
6dw,
The (by now) classical
theorem of
Andreotti-Vesentini [4] allows one to integrate by parts,
as on a compact manifold, to deduce that
Ik =
provided
dom A
9k
M
if
L2Qk
(wE
is complete.
w E L 2k
: dw = 6(
= 0}
Notice that this implies
Aw = 0
and
pointwise.
the Hodge cohomology space of degree
w E
We shall call
k.
Another similar, but perhaps more widely studied,
space is the
L2Hk =
cohomology space of degree
{w E L2Qk : d
Since each
Ik
L2
(o E 9k
> L2 Hk
= 0}/d{7 E L 2k-1
k:
: dr
is closed, there is a natural map
for each
k.
However, unlike the compact
setting, the two spaces are rarely isomorphic.
their
E L2 k
A gauge of
relationship is afforded by Kodaira's weak Hodge
decomposition [9]:
L 2k = d{L 2k-1
n dom
d} @ 6{L 2Qk
which is valid quite generally.
wise orthogonal.
Wk
From this
~ {w E L 2 Qk
n
dom 6}
D Wk
The summands here are pair-
it is easy to see that
: dw = 0}/d{L2Qk-1
n
dom d}
Ik
whence the map
L2 k
-
is always injective.
Furthermore, the two spaces coincide precisely when the
range of
d
L2
The
is closed.
cohomology will rarely be mentioned again.
The bulk of this dissertation is devoted to the computation
of
Ak
pected for quite a while that if
is simply connected,
M
and its sectional curvatures satisfy
then most of the harmonic spaces
It
It has been sus-
for a certain class of manifolds.
*k
a2 < 0,
-b2 < KM
should be trivial.
is straightforward to do the necessary calculations
n
1n,
for the hyperbolic space
or in fact for any complete
rotationally symmetric manifold (which need not satisfy the
curvature constraints),
following conclusion:
of
+
IR,
such a manifold has
0 CS
to arrive at the
see Dodziuk [10],
if
in polar coordinates the metric
the form
ds
2
= dr
2
+ f(r)
2
2
dG2,
r
n-1
,then
1
if
k = 0, n
f(r)n-1 dr < 0
and
%0
if k = n/2 and
dim Ik =
0
<O
otherwise.
The first line simply asserts that constant functions are
in
L2
iff the volume of
M
is finite.
The second is
E
more interesting;
it reflects the fact that both the kernel
n/2.
invariants on
that
M
norm are conformal
L2
of the coboundary operator and the
the integral means
The finiteness of
is conformally equivalent to a Euclidean disc of
introduce
finite radius:
radial variable.
R(r) = exp
Jrr
dt
ft)
as the new
Then one only need note that there cer-
tainly is an infinite dimensional family of smooth harmonic
forms on such a disc.
It is striking that in this proposition, the important
geometric feature is the growth of the metric rather than
This is unexpected since the spaces
the curvature.
:k
implicitly depend on the derivatives of the metric, by way
of the coefficients of the Laplacian.
equivalent to g,
c1 g
g'
c 2 g,
domain different than that for
C2
close to
g.
g
If
g'
is uniformly
then its Laplacian has
unless
g'
is uniformly
In contradistinction, the spaces
are patently stable under
C0
such a
L2 Hk
perturbation of the
metric.
Further support for the conjectured vanishing of Hodge
cohomology outside of the middle degree(s) was given by the
discovery of Donnelly-Xavier [14]
that, so long as
vature is (negative and) tightly enough pinched,
e(n,k),
1/2.
then
Ak = 0
for all
k
such that
the curIa/b-1|
Ik-n/21 >
Donnelly-Fefferman [12] extended the method to show
that for
the Bergman metric of a strictly pseudoconvex
<
domain in
Cn,
k
where
splits as
O
Cqp+q=k
then
',
p + q # n
dim Wpfq
p + q = n
-
Recall that the Bergman metric has holomorphic sectional
curvatures asymptotically constant at the boundary of the
domain.
It was thus quite surprising when Mike Anderson [2]
constructed counterexamples to the conjecture;
simply connected manifolds with
which
Xk
thermore,
KM
-b2
there exist
-a2
is infinite dimensional for any given
<
0
k.
for
Fur-
it is seen that the curvature pinching ratio of
Donnelly-Xavier is sharp.
Many questions are thereby raised dealing with how one
might recognize the negatively curved manifolds with suitable vanishing of the harmonic spaces.
It appears that the
key to this problem lies in the (intrinsic) regularity of
the sphere at infinity.
ifolds for which
S.,
The characterization of those man-
is smoothly attached is very likely
quite difficult, so a first step might be to a priori
assume some regularity at infinity.
Among the many forms
this assumption could take, we pursue the course of considering manifolds with metric differing from that of the
interior of a compact Riemannian manifold with boundary by
a singular conformal factor;
these we call conformally
compact.
boundary,
Consider the metric
aM
dp IBM s 0).
(so that
8M
g = p-2h.
has the effect of pushing
to see that,
am,
a metric nondegenerate and smooth up to
h
a function defining
p
and
be a compact manifold with
M
More explicitly, let
The singular factor
p-2
It is not hard
to infinity.
like the Poincar6 metric on the unit ball
after which it is patterned, this metric is complete and,
There are
aM.
more importantly, negatively curved near
other ways to produce metrics with 'inverse-square asymptotics' and the correct geometry.
In particular, the
Bergman metric is obtained as the complex Hessian of
c log(S p n-1 +
with
*i., *2
smooth,
tance to the boundary.
02 = 0.)
h
1
2 log p)
$1
(For the unit ball,
p
But, were it to be written as
then partially degenerates at
in the directions of the
Our principal result
Theorem (4.8):
p
nonvanishing, and
CR
aM
-in
the dis= 1
the metric
h,
fact, precisely
TaM.
bundle in
is the
For the metric
g = p
2
h
and
on
M,
described above, there are natural isomorphisms
as
k <n-i
kM,6M)
Ik
k
Ak ~ Hk
n+l
w E Ik
obtained by sending a (necessarily closed) form
By virtue of conformal
its de Rham class.
is infinite dimensional when
k = n/2.
A ,
g
L2 (dg).
I -
and its adjoint
Q
where
=
Q,
Q
A
E
g
such that
E
a pseudodifferential operator
A E
Ik
invariance,
This is proved by constructing a parametrix for
i.e.
to
=
I
-
Q
are compact and smoothing on
From a fairly explicit knowledge of the Schwartz
Q
kernels of
and
Q
,
it is possible to deduce the
boundary asymptotics of elements of
k
Ik.
This, in turn,
A
allows the isomorphisms of the theorem to be proved.
closely related 'Poisson' operator solves the asymptotic
Dirichlet problem, but due to constraints of time and space
this will not be developed here.
E
The main difficulty in the construction of
from the fact that the principal symbol of
quadratically at
therefore have a
corner of
M
x
M.
8M.
fairly complicated
degenerates
A
The Schwartz kernel of
arises
E
must
singularity at the
The systematic introduction of singular
coordinates at this corner allows one to adequately
'resolve' this singularity.
In effect, the degeneracies
are transferred from the Schwartz kernel of the operator in
question to the geometry of the new 'blown up' manifold
which replaces
M x M.
This type of construction was first
developed by Melrose [22] to deal with differential operators degenerating somewhat less
considered here.
thoroughly than the ones
Melrose-Mendoza [23] studies
that class
of operators further, in particular proving Fredholm properties for the elliptic elements.
Both that work and the
present one are part of a more general construction, to be
contained in the, as yet mythical, Melrose-Mendoza [24]-from which inspiration for the following pages comes.
The interest in these methods as presented here lies
perhaps in the ease of their applicability to a natural
geometric question (albeit in a somewhat artificial setting).
It seems likely that similar techniques will prove
effective in other related geometric problems.
For in-
stance, Mazzeo-Melrose [20] employs essentially the same
construction as here to deduce the meromorphic extension of
the resolvent and Eisenstein series for certain quotients
of
Hn
This paper is organized as follows.
The first chapter
examines conformally compact manifolds from a differential
geometric point of view.
Their curvature is seen to have
fairly simple behaviour, and in particular is negative at
Asymptotics of
infinity.
the geodesics are also studied,
and it is shown that in a special case the ideal boundary
has an intrinsic regularity structure.
ter,
operator calculus is developed, including
V
the
In the second chapL2
continuity of its elements, and the parametrix construction
In Chapter 3, a certain model for
is outlined.
analyzed.
In these circumstances this model
A
is
is nothing but
the (constant curvature) hyperbolic Laplacian; however we
require some rather unusual mapping properties of it.
Finally, the hard work now complete, the actual construction and proof of the Main Theorem are contained in the
last chapter.
brief
In conclusion we recall an old conjecture of Heinz
Hopf's,
for it has stimulated much of the work directed
toward the computation of
folds.
Xk on negatively curved mani-
One formulation of the classical uniformization
theorem classifies surfaces by which constant curvature
metric they admit.
Motivated both by this and the higher-
dimensional Gauss-Bonnet Theorem, Hopf proposed the
generalization:
Conjecture:
If
M2 m
is compact and admits a metric with
strictly negative sectional curvatures, then
(-1)mz
M)
> 0.
The corresponding query for compact flat manifolds is
trivial, and the one for compact manifolds of positive
curvature is likewise unresolved.
The proposed methods of
proof are quite different for the positive and negative
cases;
in the positive case much work has been devoted
toward showing that the Gauss-Bonnet integrand-the
Pfaffian-is pointwise positive.
hopeless task by Geroch [17]
This was shown to be a
(who gives a local counter-
example) and Bourgouignon-Karcher
[6] (who give a global
one).
Following a similar pattern of history, Anderson's
counterexample [2] likewise dashed hopes that the negative
case could be resolved by settling the hopefully less
complicated issue of determining the
L2
harmonic spaces
on complete noncompact negatively curved manifolds.
In
fact, Singer had suggested that combining Atiyah's index
theorem for covers [5] with suitable vanishing of the Hodge
cohomology would settle Hopf's conjecture.
This
index
theorem equates the alternating sum of the Betti numbers of
M,
a compact manifold
which of course is just the Euler
characteristic, with the alternating sum of certain
Betti numbers on the universal cover
Y(M) =
F = w 1 (M).
These
(-1)kbk(M) =
b
L2
M:
(-1)( k(
are nothing but the (normalized)
dimension of the
negative
k
Wk = 0
M
for
is simply connected, if one could show that
k A m =
pullback metric on
x(M ) =
The
that
1
M,
dim M
for the negatively curved
it follows that
F
(-1)kb(M) = (-1)mb
are always nonnegative, and it is not hard to show
bk
b
Thus, since if the curvature is
Xk.
m
is strictly positive, so the conjecture is
proved.
This method of proof does work when the curvature is
close to being constantand in particular when
hyperbolic-by virtue of Donnelly-Xavier's work.
M
is
We remark
also that Anderson's counterexamples are not universal
covers of compact manifolds since their isometry groups
have no cocompact subgroups.
true
Thus it may well still be
that such periodic covering manifolds have vanishing
Hodge cohomology outside the middle degree, but to prove
this would likely be quite difficult.
question that warrants attention.
It is,
I think, a
Chapter
A.
1.
Conformally Compact Metrics
Definitions
The interior of a compact manifold with boundary
M
may be endowed with a complete Riemannian metric, and thus
becomes a complete open manifold.
that
The general effect is
is placed at infinity; geodesics take an infinite
8M
time to reach it.
The seminal example, and model,
n
Bn.
is the Poincar6 metric on the unit ball
for us
In local
coordinates near the boundary the components of the metric
become arbitrarily large.
The rate of this blow-up is
essentially fixed by balancing the requirements
that the
metric be complete and that its sectional curvatures be
negative and bounded away from zero near
aM.
Explicitly,
suppose
h
is a nondegenerate smooth metric on the
(closed) manifold
(1.1)
M.
p
is a defining function for
P~
(0) = BM,
and
dp
OM:
p
0,
is nonvanishing on N"BM\O
and set
g = p -h,
a > 0.
Then, from computations in sections B and C of this chapter
23
it will follow that
g
is complete for
a > 2,
whereas
its sectional curvatures are bounded above by a negative
constant only when
a < 2.
to negative infinity if
a
In fact the curvature diverges
is strictly less than
2.
This thesis focuses on a study of the analytic properties
of
g = p
(1.2)
2
h
8M,
which is both complete and negatively curved near
in particular of its Laplacian.
that
and
Let us however mention
there are other examples of metrics of this general
form which are both complete and negatively curved, but
which are not "conformally compact"-i.e. as in (1.2) with
p
and
ple,
h
satisfying the hypotheses of (1.1).
the Bergman metric on the unit ball in
correct geometry (and there is a vanishing
Cn
For examhas the
theorem for its
Hodge cohomology), but in Cartesian coordinates it blows up
as
in directions converging to the complex
(-Iz12)-1
subbundle of
TB n
(1|z|
2
)-2
only in the other
It would be quite interesting to under-
two directions.
stand the ways
and as
p
and
h
could degenerate so that
p2 h
still is complete and of bounded geometry.
This chapter is devoted to a discussion of the geometry of
(M,g),
g
as in (1.2).
In particular, we examine
the behaviour of diverging geodesics fairly thoroughly, as
well as asymptotic properties of the curvature tensor.
computations use
addition,
C (M).
must
B.
p
and
g = (p)-
2 2h
h
routinely, but note that,
E
g
therefore be invariant under such a transformation.
Geodesics at Infinity
1
zn = 0
such that
inward pointing
z
8/azn
is
1
n-i
)
(z ,-,z
=
p E
near a point
on the boundary and
h-unit normal.
the
are
It is more convenient to work
BM.
then coordinates on
n
(z ,---,zn)
Introduce coordinates
OM
in
*
for any strictly positive
formulae for the geometric quantities of
All
All
with the co-geodesic flow, the equations for which, in
terms of
-i
z
(1.3)
-
=
-P
_g
1
p
_p
are
2
ijE
f
=p h
2 Bh pq
1
pq
h
f
p q
iz
Here all
f ,---,n
and the dual coordinates
z
1
-
-
2
h_
2
Bz
1
'
f
p q
i
1 ~ 2
indices vary between
p
and
n,
the summation
convention on repeated indices is used, and attention is
restricted to the energy surface
p h
(1.4)
1.
=
Notice that (1.4) implies
Ci,
|f
C1 /p
(1.5)
C2 /p
_
2
2
1I
+---+
E2
n*
depend only on the largest and largest eigenvalues
C2
h'3 .
of the matrix
Now consider a maximally extended geodesic
(z i(t),---,zn t))
and suppose
p(r(O)) < e,
where
e
r(t)
zn(O) < 0
-r is, of course, the
is sufficiently small.
projection of a Hamiltonian curv ,e
which solves (1.3).
1
n
(z ,---,z , y,---,fn)
By virtue c f the choice of coordi-
nates,
Op/8z
for some
C
i =
= ai(z)p(z),
functions
is smooth up to
8M.
I.,***,n-1,
and in particular
ai,
Using (1.5)
i
t. 1 = 0(1)
=
i
n
p1 Op/Oz
in (1.3) one concludes
1,---,n-1
(1.6)
-k1/p
ap/azn > 0
-k2
26
k1 , k 2
where
If
(1.7) Lemma:
and
e > 0
depend only on
We first prove
e.
is a geodesic as described above,
r(t)
then
is sufficiently small,
-n(t) < 0
for all
t > 0.
By (1.3)
Proof:
a
the sum in
then
hnn n;
zn
n
1
is from
F(0) < 0
na
By (1.5) and (1.3),
nn
2 na
n)
a + h
p (h
2 ni
p h
Set
n-1.
to
where
F(t) = h na
+
by assumption, and we compute
+hnnn
a
z
=
h a
i
+
Pz
O(p)
so
i
a + Bhnn
z
+ 8
i
n
-na
z fa,
z
n
are
0(1).
Thus
F'(t) = h nnn + 0(1)
where
C
< h nt
+ C
again depends only on the metric but not on
By applying the bound (1.6) and assuming
e
is small
enough we may ensure that
n(t)
where
K
< -K
is chosen so that
zn (t)
if
-
< zn(O)
-K + C < -1.
2
Now set
-r.
27
t0 = inf{t
If
0 < t < t0
Finally, if
> 0 :
n (t)
zn(t) < zn(O),
then
hnn > 1/2
in
{z
> 0}.
so
En(t) < -K.
p(z) < e},
we conclude
that
+ C < -1
hnn n + C ( -!K
2
F'(t)
for all
zn(t0)
(1.8)
0 < t < t0 .
< 0,
g
Proposition:
< 0,
F(t 0 ) < F(O)
< 0,
i.e.
a contradiction.
particular, if
z (0)
Hence
and
is a complete metric.
is a geodesic with
7(t)
p(-Y(O))
Y
is small enough, then
e
In
< e,
can be
extended to infinite length.
Proof:
First observe that the function
zn
of our
coordinate system may be defined on a collar neighbourhood
of the full boundary.
On this neighbourhood there is a
product metric
k = (zn -2 h lm + (d(nh2i
Clearly for some
c,
g > ck.
This
implies
that
clzn|/zn = -cZn/zn
|r'(t)|1
1 =
tt
4 t =
|t'(r)|
(-c) Z (T) dT
dT
Z(T)
0
0
which by Lemma (1.7) reduces
to
n
()dn
t > c
=
c log(zn(o),z (t)).
z n (t)
0,
zn (t)
tends to infinity as
t
In other words,
has infinite length.
i
and consequently
Notice that completeness still obtains if
n -a
(zn)-,
replaced by
tends to
in
a > 2,
proof is modified accordingly.
k
(zn)- 2
is
and the rest of the
Now, as might be hoped, it
is also true that geodesics approach definite points of the
boundary.
More is true:
boundary as
t
tends to a definite point of the
-(t)
(1.9) Proposition:
o.
-
Furthermore,
taking
Proof:
Since
/
8 zn
n
-
at the boundary.
-n,
we may use
nonsingular parameter, and then
as
the new
tangent to the
variable, the reparametrized geodesic is
h-unit normal
zn
in
as a
29
Oz
/8f
n
p2 h
=
f./(-2
= p 3 h 'j
i =
1,---,n-1,
=
pn
p
3
/(-2pn
p/Oz n
-(t)
+ O(p)) = O(p2 ),
From (1.5) it follows that
C -2
|Bzj/afn I
Hence
+ 0(1))
n-1
1
(zo ,---zW
,0),
tends to
i
Z0 +
z
zi(t) t
0
z (0) +
=
where
(Blz/afn Oan
En (0)
is well-defined since the integral converges.
Next,
(1.5)
and (1.6)
En
-C/p,
* Ofn/t
Recalling that
integrates to
< 0,
imply
n
kn
,
2-k2
k > 0.
this differential inequality
30
< -Ce
a = O(1),
On the other hand,
|fa(t)|
a = 1,---,n-1,
and thus
< Ct.
Finally,
dz i/dzn
n
=
h
estimates above,
dz i/dz
C.
Ei./Phnj
+ hin
hna(Ea /f n)
a n
+ hnn
a 0,
fa
2h
n)
hia
By assumption
_
hnn
0.
n
---
1,
and
by the
We conclude that
> 0.
Curvature Asymptotics
The conformally compact metric
g
of (1.2)
is
patently modelled on the Poincar6 metric on the unit ball
in
IRn
The obvious desire is that much of the well
understood hyperbolic geometry (and analysis) will be
reflected in this more general setting.
Indeed, this is
the case with geodesic behaviour at infinity, as we have
demonstrated in the last section.
sectional curvatures of
g
It is also true that the
are negative when
p
is small;
there is a well defined limiting curvature tensor
in fact
is isotropic.
OM
for which
-Y(t)
tion at
The sectional curvatures in any direc-
p E OM.
approaches
tend to
(8p/8zn) (p)
-
> o.
t
as
h-unit normal to
is the
i
8 zn
Here, as usual,
be a geodesic which
'-(t)
Let
(1.10) Proposition:
BM
at
p.
this proposition implies
BM,
on
0
pn
Since
Remark:
that the sectional curvatures are bounded between two negative constants in some collar neighbourhood of the boundAlso, recall from the end of section A that we may
ary.
+
function
for any strictly positive
by 02 h,
h
Op,
by
p
replace
g.
without altering
The quantity
invariant under this transformation at
new
'2h-unit normal is
-1
0
a
(Op)
nn
z
Proof:
nates
a/azn
-1
9ao/az p = pn
n
+ 0
is
Indeed, the
and
at
p = 0.
For the duration of the proof only, we use coordin
1
(z ,.--,z)
for which
,n-1,
which are normal with respect to
corresponds
p
orthogonal to
1,-
= p
0
p = 0.
ap/azn
BM
at
p.
ap/zn (p) > 0.
to
0,
Thus
and
and such that
ap/az i(p) = 0,
a/azn
i =
h,
is
32
h..
13
Set
= log p,
u
= 6..
Notice
that
13
of
g
u
with subscripts on
Tk
derivatives, and denote by
h. ..
+ O(|zI).
13
referring to
z
the Christoffel symbols of
F ii = O(Iz|).
The Christoffel symbols
are
k
Ti
(1.11)
=
~k
k
u
u6
-
k
ekh
h
+
Insert these into the formula exp ressing the components of
the curvature tensor in terms of the
A straightforward but me ssy calculation shows that
atives.
R
+
Here
-2
..
= p-{R i..
131l
iji~j
Ivu| 2 (6.13.-1)
-
1
11
2u.u 6
1j
2u
ij
6.
ij
+ u
22
ij
ja .
+ u. + O(|x|
)}.
and assume
i
p
Substitute
h.
ui
=
2
+ u.
+ uii -
are components of the curvature tensor for
R, R
respectively
R. ..
and their deriv-
F's
R
=2PP
u=pp
= p /p,
j
..
p
2 2
-pu/p
to get
+ p
{
-
Ivp| 2
+ p 2 0(|z1 2 )}.
g,
33
Therefore, the sectional curvature in the
vpI
R-2
t
n]2
-
LznJ
(h..h. .-h . )
311
ja3
p
as
2
plane is
i-j
C.
-+
It is easy to follow through this proof using the
metric
k
F..
for any
p ah,
a > 0,
Indeed, the new
instead.
are obtained from those for
a = 2
by replacing
u
13
au.
with
The sectional curvature in the
i-j
plane now
becomes
2
aP7
-
-1
a-2 2
a2a
2 1p
(p
p +p ) + 2p
(-2
vp 2 + F({1)p
+p
+ O(pU).
This substantiates the claim of section A that only when
is the curvature bounded between two negative con-
a = 2
stants near
D.
BM.
Boundary Regularity
By their very definition, conformally compact mani-
folds have
Co
compactifications, but it is quite unclear
to what extent this regularity is
g.
intrinsic to the metric
In the present section we partially resolve this issue.
Geometric regularity of an 'ideal' boundary is natu-
rally gauged by the asymptotic properties of diverging geodesics.
In this particular setting, let
be a geodesic ray emanating from
p
p E M
and
i(t)
and tending toward the
34
boundary.
Define
: w
V
lim exp
--
This maps a neighbourhood of
sphere in
to
T pM)
8M.
r'(0)
Since
(tw).
p
t-+)
p
g
in
(the unit
S M
p
has negative curvature
bounded away from zero in the cone around
r(t),
sults of Eberlein-O'Neill [15] imply that
vP
homeomorphism.
on
the re-
is a local
To impose an intrinsic smoothness structure
it is sufficient to show that the composition maps
8M
> S M all possess a certain fixed regularS M
'ir
q
p
q p
Anderson-Schoen [3] prove quite generally that each
ity.
V ~
such map is
Ca,
where the H~lder exponent
only on the upper and lower curvature bounds.
a
depends
Here, how-
ever, we are able to strengthen this considerably, but only
with the strong proviso that the limiting sectional curvature function on
bourhood of
8M
-
r (-'(0)) =
(p/zn )2
r..
is constant in a neigh-
This bears marked resemblance
to Fefferman's analogous result [16] for the Bergman metric
of a strictly pseudoconvex domain, in which case the curvature is asymptotically constant along the whole boundary.
I do not yet understand how essential this hypothesis is
for conformally compact metrics.
(1.12) Proposition:
that
-
(ap/azn )2
Suppose, for the metric
g = p 2h,
is constant on a neighbourhood of
35
Tr (-,'(O))
= 7.
7'(0) E S M
p
in
8M.
Proof:
aM.
in
Then
maps a neighbourhood of
r
diffeomorphically to a neighbourhood of
The technique, inspired by [16],
rize the geodesic equations (1.3)
is to reparamet-
so as to obtain a nonsing-
ular system on a finite interval.
It is convenient to
hnn = 1, han = 0.
choose coordinates so that
the following, we assume all Greek indices
between
1,---,n.
1
and
i,j,----
while
n,
(1.13)
2va
za
n
1
v ,---,v
-n
Here and in
a,Iv,-r--
vary
take the values
p(-r(O))
By renormalizing we also assume
Introduce new functions
-r,
=
1.
by
n
so that
(1.14)
va = hap
n =
n'
From the relationship (1.4) we have
p 2h
n 2 =1.
+ (v)
v
v af3
By differentiating (1.14) and using this last equation, a
bit of algebra shows that the
v
satisfy
36
-a
v
Oz
~13
h Bz
Yn
83haP
n
2
6 hap v -r6
v + phP
= p h P5OIf
(1.15)
happ
-n
v
h
p
2
a n
2
= pp v v
h
zn
a
T
'zP
C
a (
P pnhap V V
af
ah
1 3
2 p h pah~ aCZn
Now introduce
hap ahp
C vr
v v
The resulting
as the new parameter.
equations of motion are
dza/dz n
(1.16)
a
dv /d z
-
n
hap
a
n
h ap
=ph P6
ph
h
hap
pp
2
v
2
1
-
n
_
(2 vn
dvn/dzn = p va -
Y 6
h
yt y /vn +
z
p
hap ah"
vCv /Vn
/vn
_h
h
a
ahhap3
P3r aIz n
AV
h
a P
n
a vp azn
These equations are nearly regular,
the obstructions
being only the
vn
factors in the denominators and the
third summand in the equation for
va.
However, from
(1.14) and the proof of (1.9) we have the a priori bound
0 > -c1
where
c , c2
) vn
-c2
2
-
are independent of the particular geodesic.
This resolves the former difficulty;
write
p(z)
function
g,
a(z')
The coefficient
is naturally associated with the metric
aplazn
being simply
when
b = O((zn )2).
= a(z')zn + b,
a(z')
as for the other one
0
p
. Hence
= O(p 2)
is locally constant, and in this case
precisely
p /p2
is smooth.
Since we already know that solutions of (1.16) exist
zn = 0,
down to
the standard theory for ordinary differ-
ential equations-in particular,
smooth dependence
on ini-
tial conditions-implies that the map
(z'(0),v(0))
--
from the initial values at
t -
W
Iv(0)|
= 1
is a
C
-
t = 0
(z
,v
)
to limiting values as
diffeomorphism from the energy surface
Of course we need to show that
to its image.
Vw-(0)
:
v(0)
-
z
38
is a diffeomorphism.
It has already been noted that
7
is known to be a homeomorphism, so it suffices to prove
that its differential is nonzero.
If
To this end a Jacobi field argument is natural.
M
w E T
is orthogonal to
then essentially by
-r'(0),
definition
dir
()
(w)
=
lim
J
te
)
Here
is the Jacobi field along
J w(t)
J,(0) = 0,
(t).
w
w
such that
It satisfies the equation
J'(0) = w.
J"
-'(t)
+ R(J ,),
=
w
0.
By calculations very similar to those in section C, the
components of
tor
R0
and of the corresponding curvature opera-
for the metric of constant curvature
blow up as
O(p 2),
R
p -2.
-a2
Furthermore, using strongly that
both
p
=
it is also the case that
|R-RO| = 0(1).
Hence, by the
clude that
J
technique of asymptotic integration we congrows at
the same rate as
constant curvature Jacobi field
basis
1/ -
n
a/az ,*'8/z ,this
J
0
.
w
the corresponding
With respect to the
latter field
39
tends
to a
finite nonzero
limit as
t
-
O.
Thus
does also, and the proof is complete.
E.
Laplace's Operator
This chapter concludes with a calculation to express
the action of the Laplacian for the metric
ential
OM,
and
on
differ-
We use, now and for the rest of this paper,
forms.
coordinates
g
n
1
(z ,.--,Z )
a/azn
the
with
z
n
vanishing simply at
h-unit normal there.
With the hindsight of experience, the metric on
(1.17)
< , >g = p2k<
A kM
h
is best regarded as a nondegenerate metric on singular
k-forms.
Introduce the space of sections
(dzi
p
over
T * M.
dn
p
In the next chapter we shall define a new vec-
tor bundle for which these are nonsingular sections.
any rate,
k-fold wedge products of these
basis for
AkM
1-forms give a
over any point of the interior.
combinations of these basis forms may be written
~01
At
Smooth
~%
M.
is a genuinely smooth form on
w
where
If
T1 = p
-k
T,
then
7
gW =
<W(q>,
A W it is
g
more convenient to study the induced operation on w:
w
Instead of studying the operation
(1.18)
A (p k)
g
= p
-
k
-k
P = pA p.
g
PW,
Much of the 'hard' analysis later will focus on determining the mapping properties of a simpler operator which
models
P.
Now, since there is a factorization
k -k+1
)(
P = (p dp
it
6
g
k-1
6 g p -k
)
+
(p
k6
gp
-k-1
)(p
k+1
is easier to first calculate the conjugates of
First, then,
. j-1 dpAw)
p j+1 (p -j dw-jp
-j
j+1
p pa=
(1.19)
=
As for
pdw -
jdp A w.
the adjoint, use duality:
dp
-k
d
)
and
>hp
j-ndh
<d,
<dTa, p do)> dg =
=f
=f
h
<,
j-nW)>hdh
<7, 6 h( j-n W)> p2-2j+ndg
(7, p-j+1
j-form
for any
W
and
j-1-form
=
pPj -1 S
6 p -j
p
j-nW]>
dg
In other words
Ti.
n-j+16
=
n-j+16hP
j-n
Sh
Furthermore, for any metric
6(f-a) = f~a =
Combining
p
these
1
P
t
vf a.
last two formulae yields
= pn-j+1 (Pj-n h-(j-n)p
j-n-1 Lv
(1.20)
=
Here
vp
Finally,
p6hw
+
(n-j)uy(.
is the gradient of
(1.19)
and (1.20)
computation, show that
p
with respect
together, along with a bit of
42
Pw = P 2 Ah& + (2-k)pdp
(1.21)
L
vp
= dc
vp
+
L
A Shw - kp6h(dpAw)
+
(n-k)pL
w -
2pt
+
(n-2k)dp
A t
e
VP
d
dw
-
k(n-k-1)yp (dp)o
is the Lie derivative.
The Calculus of V 0 Pseudodifferential Operators
Chapter 2.
A.
Vo
Vector Fields
The only distinguished submanifold of a manifold with
boundary
M
Related to this is the ob-
is its boundary.
servation that the ring of differential operators
has
two natural subrings of geometric origin:
Diffb(M)
The former is the space of operators which
Diff 0 (M).
and
Diff(M)
are sums of products of vector fields, unrestricted in the
interior but required to lie tangent
to the boundary-the
so-called totally characteristic vector fields,
analogously using the
at
8M.
Both
Vb
The latter space is defined
V
of which is denoted
and
the class
Vo-vector
fields, the ones vanishing
are Lie algebras under the
V
usual bracket operation for vector fields.
Diffb(M)
the related space of totally
*b(M),
and
characteristic pseudodifferential operators, were introduced and studied by R. Melrose in [22];
[23] contains further developments.
Melrose-Mendoza
It is the purpose of
this chapter to define and examine the ring
pseudodifferential operators generalizing
% 0 (M)
of
Diff 0 (M).
Much
of the material here derives from information gleaned during conversations with Richard Melrose.
If
usual
z = (y,x) E
type, with
is generated as a
zn
n-1
x [+
= x)
are coordinates of the
vanishing on
C (M)-module by
OM,
then
V0
44
zn a ,---,zn
zl
(2.1)
a
Bzn
Thus a typical element of
Diffm(M)
(2.2)
aa(z)(zn aa
P =
has an expression
|aIsm
The (modified) Laplacian of a conformally compact metric
(1.21)
lies in
2
Diff 0 (M).
Associated to
V0
diffeomorphisms of
M
is the group composed of those
which fix
nential of a vector field in
V0
aM
pointwise.
The expo-
belongs to this group.
Any diffeomorphism induces a linear action on the tangent
space of one of its fixed points.
aM
and
Mp
In particular, if
is the inward pointing half of
a member
T pM,
T M
p
of this class induces a linear transformation on
which preserves
(2.3)
where now
M
(x,y) --
(x,y)
p
.
Such a map is of the form
(sx,y+xu)
s E R
+
,
u
are linear coordinates on
order reversed from the previous paragraph).
note this
R+
with
p E
C R
M
n-1
p
Let
81M
p
(with the
Gp
de-
linear group which is the semidirect product of
Rn-1.
Its composition and inverse laws are
45
=
(s,u)-(t,v)
(st, tu+v)
(2.4)
(s,u)
-1
-1
-1
(s ,-s u)
=
(2.3) should be recognized as the left action of
itself.
G
on
p
In addition, the vector fields in (2.1) are the
For future
infinitesimal generators for this action.
reference let us record the right action, along with its
invariant vector fields
(x,y)
-
(s
(x,y).(s,u)
=
- x
---
-1
x,s
-1
(y-u))
(2.5)
It
-
y
'
G = ANK
S = AN
Furthermore, the
hyperbolic.
Gp
in the Iwasawa decompo-
of the group of hyperbolic isometries:
G = SOO(1,n),
fields
n-1J
is interesting and significant to identify
with the solvable subgroup
sition
'
A
S
=
+
IR
,
N
=
n-1
IR
=
,K
invariant metric on
Thus, the introduction of the
G/K
V0
SO(n).
~ Mp
is
vector
leads rather inexorably to constant negative curva-
ture geometry.
46
In order to view
convenient to define the
bundle,
vector
tangent bundle
V0
0TM
as that
sections of which are precisely the
C
the
somewhat more globally it is
V0
V0
fields
V0
r(
=
TM).
This way of constructing a bundle will be used several
later on, so we pause to elaborate briefly in this
times
0T
The individual fibre
case.
alence relation on
V, V'
is obtained by an equiv-
M
V
V
E V0;
0
Vf = V'f
p
p
V' +=*
p
V f E C (M), p C 6M; d(V-V')f
= 0
V f E CM(M), p E aM.
(2.1)
tions;
0 TM.
exhibits a spanning set of sections with no rela-
The dual bundle
bundle structure on
C
these determine a unique
is also quite useful.
0T*M
A basis
of sections here, dual to the vector fields of (2.1),
(2.6)
dz
-
z
dzn
-,
n
z
n
=
d dx
-
xx
is
has a well defined symbol,
Diffo(M)
An operator in
which is a homogeneous polynomial of degree
If
0T*M.
fibres of
P
is written as
0 am(P)(z,C)
(2.7)
m
on the
in (2.2),
then
aa(z)(a
=
|a|=m
zn a/az
i.e. each
is replaced by
terms are discarded.
and lower order
C
Invariance follows by continuity from
the interior.
The operator
P
is elliptic in this calculus if
oam(P)(z, C)
The Laplacian (1.21)
Our goal
= 0 4*
C = 0.
is elliptic in this extended sense.
in this paper is to prove Fredholm properties for
this particular operator acting on certain weighted
spaces.
tic
V
L2
The general theory applies of course to any ellipdifferential operator.
In the study of
Diff 0 (M),
as in the usual theory of
standard differential operators, it is important to understand simpler models of operators in this class.
interior of
M
In the
these are obtained by freezing the coeffi-
cients so as to obtain constant coefficient operators on
each tangent space, also discarding all but the top order
terms, exactly as usual.
able since
aa
z
This last reduction is justifi-
is 'stronger' when
lal
= m
than when
48
|ai <
Gp
modelled more effectively by a certain
ator on
M p,
which we call
will be
P
on the boundary,
p
At a point
m.
invariant oper-
the normal operator
N p(P).
zn = 0
Its definition is suggested by the fact that near
az
zi Bz
are all equally strong.
k >
|ai
,
zn
[zn 0
n 6
1
[zn 6_ a
and each
Oz
azi
However, any
(zn)k[_ a
with
is dominated by one of these operators, hence
weaker.
Motivated by these observations we describe a procedure to define
(2.43)
is differential.
P
when
N p(P)
further on contains a more general definition for pseudoWe require first the notion of a
differential operators.
normal fibration at
a neighbourhood
T M
p
T
p E aM.
of
This is a diffeomorphism from
of
'
to a neighbourhood
p
such that
f(p) = 0,
f(T) C M
,
f(6qI) C am4,
f*
0
=
Id.
Such a map is readily constructed using for example the
geodesic flow of a metric for which
geodesic.
(2.8)
aM
is totally
Then
N (P)u = lim (Rr) f
r-+0
** -1 **
) (R1/r) u
P(f
0 E
where
R
z
moment
by
z
is dilation by the factor
r
on
M,
induced
P
then from (2.8) one may easily check
'.
n 86za
-
Np(a(z)
(2.10)
N p(P-Q) = N p(P)-N p(Q),
zn
These formulae show that
B.
Letting, for the
denote the linear coordinates on
(2.9)
Diffo(M)
r.
to the
G
= a(0) zn
Np
M
aa
_
,
p = {z = 0}
P, Q E Diff 0 (M)
is a homomorphism from
operators on
(left)-invariant
p
M .
p
The Stretched Product Construction
Any attempt to construct a pseudodifferential inverse
for an elliptic
P E Diff 0 (M)
must reconcile itself with
the rather complete degeneracy of
P
at
aM.
In partic-
ular, the Schwartz kernel of such an inverse must possess a
singularity somewhat more severe than usual at the submanifold where the diagonal of
am x am.
M x M
intersects the corner
In order the more thoroughly to display this
singularity we shall define the appropriate class of
kernels on a slightly larger manifold
stretched product of
abode;
M
with itself.
M x0 M,
the
V0
It is their natural
the additional singular behaviour is transferred to
the geometry of
possible here.
M x0 M
and these kernels are as smooth as
50
M
We first examine the product
M
x
has rather more natural structure than
more closely.
alone.
M
It
Thus
there are two natural hypersurfaces, the left and right
boundaries
ae(MxM)
= aM
x
M,
8r(MxM) = M
M
These are mapped surjectively to
left projections
M
x
M
irr
and
ir.,
x
aM.
by the right and
respectively, which send
onto the right and left factors.
They intersect in
the corner
a,(mxm)
n
ar(MxM)
= am
x
aM.
Finally, the fixed point set of the involution
interchanges the two factors of
This,
,?At .
I
is the diagonal
M
which
AL.
in turn, is bounded by the diagonal of the corner
&I
VO-pseudodifferen-
As noted earlier, the kernels of
tial operators should exhibit a singularity at
aAL
greater than their customary conormal one along the whole
diagonal.
to the
V
This new behaviour is best exhibited by passing
stretched product
have a certain extension property.
fined by taking the real blow-up of
or
where these kernels
M x0 M
This manifold is deBAL,
around
M x M
to put it more familiarly, by introducing polar coordi-
nates around this submanifold.
formed from the disjoint union of
is
M x0 M
Abstractly
(MxM)\OAL
and
SN++8AL,
the closed inward pointing sector of the spherical normal
bundle
to
unique
C
8At.
[25] contains a proof that
M x0 M
has a
structure such that the blow-down map
Mx
b : M x0 M -
(2.11)
M
defined as the identity away from the new face, and as the
bundle projection on
SN++BAc,
is smooth and of rank
along this new face.
(= dim 8AL)
n-1
We shall content our-
selves by relying on coordinate descriptions.
Using the coordinates
M
(x,y)
on the first factor of
in the product, and an identical pair
(xy) on the
second, then
AL =
{x = x, y = y},
8AC = {x = x = 0, y = y}
52
Let
Y = y - y,
and define the singular coordinate system
(R,w,y)
(2.12)
R = [x 2 +
Y2 + x 2] /2
R
O ,
n
E S++
R~
=
(j
{W E
-
n+1
wo0
so
(,Y,X)
Wn
(
n)
=
> 0}
that
x
= RwO,
x
= Rw
(2.13)
y = y + Rw'.
This system lifts to be
sion one boundary, where
The original
C
on
R = 0,
M
x0
M.
The new codimen-
is called the front face.
two codimension one boundaries lift to
and bottom faces:
the top
F
=
{R = 0},
T
=
= 0} = b~
{(0
({x = 0})\F
(2.14)
B =
= 0}
= b
({x
= 0})\F
Figur e 2
It
M
is often easier to picture
x
M
instead as
Figure 3
The front face is the
loca tion where the analysis of
the next chaper takes place.
one might first suspect.
quarter-sphere bundle over
I t has more structure than
First and foremost it is a
aAt
with fibre
F
p
~ Sn
++
over
54
(p,p),
This follows either from (2.12) or its
p E OM.
intrinsic description as
completely natural origin
0 ,
p
also has a
Fp
Each
SN++AtL.
F
which is where
inter-
p
sects the lifted diagonal
At0
(2.15)
0 = F n At 0 =M
The bundle of origins
rated from
R, y
={w = (1/-s2,0,1/%/),
In fact
6F.
is clearly sepa-
0
8(Mx 0 M)
intersects
At0
arbitrary}.
transversally, and only at the codimension one boundary.
This remark indicates the fundamental gain incurred by
passing to the stretched product, as we discuss in the next
section.
The fibre
Fp
G
admits two separate transitive
Gp
actions in its interior, where
is the group of
(2.4).
The groups
x
Id,
act on the quarter space
M
G
p
= G
p
p
Gr =Id
p
x M
p
C T
x
G
(p,p)
p
(MxM),
BAt,
they also fix T
fixes OM
(p,p)
p
p
these actions descend first to the normal bundle
Since
G
N
B At
= T
p E aM.
hence
(MxM) +/T BAL
thence to its projectivization
SN++aAt
1=
F .
Because of
0 ,
p
the natural origin
either by way of
F
the action of
le
Gp
or that of
corres-
Fp
I.
intertwined by the (lift of the) involution
(2.16)
Gp
on itself, and are
G
pond to the left or right action of
the map on
p
r
Through this identification these actions on
ally,
G
may be identified with
p
Specific-
Mp x Mp
( 2 .1 6)
*((x 1 ,y 1 ),(x2'y 2 )) = (xyl)*(2'2
- 1
1
y1
2]
2' x12
is invariant both under dilations and translations by elements of
T(BAc).
Hence it diffeomorphically identifies
F
p
with
G
p
Then
.
G
p
acts as
(tv)-((x1 ,y1 ),(x 2 ,y 2 )) =
,
whereas for
(Mp xM p)/T BAL =
interior of)
the projectivization of (the
y -Y2
Lx2 ' x2
x
((tx 1 'y1 +x 1 d,
(x2'y 2 ))
1
1
2
2 2
172
Gr
p
(t,v)-((x 1 ,yl) (x2 ,y2 )) =
[1
.- - 1 -1
t x 2' t
proving the claim.
lY2
x2
-- v]
t
((xly
=
1
1
),(tx
2
'y2 +x 2v))
1Y~2](tv) -1
-,(tv
x 2' x2
56
Next we describe the infinitesimal
action, which
p
leads to an important perspective on the nature of
F .
p
(2.17) Lemma:
a
The vector fields
M
the left factor of
trictions of these
M,
x
xg-,
thence to
lifts to
F
G
a
ay
xJ
M
x0
M
on
M.
lift
to
The res-
lie tangent to each
Fp;
C -invariant vector fields.
p
with respect to (i.e.
These restricted lifts are in V 0
there they span the space of
vanish on)
tangent to)
be
F
and in
fl T
F
p
space of
F
'mixed V 0
Bn
fl B
-
with respect to (i.e. are
Finally, the quarter-sphere
fl B.
regarded as a ball
boundary, with
Vb
Vb'
F
p
may
blown up around a point on its
as the created face.
The full
vector fields, generated by the
Geinvariant ones described above, is then the lift of
p
under this blow-up.
VO(Bn)
Figure 4
Remark:
All assertions concerning tangency properties of
various lifts in this statement are special cases of a
if a
general property of the blow-up construction:
is a
V
manifold is blown-up around a submanifold, and
lifts on
then it
vector field tangent to this submanifold,
to the new boundary,
the blow-up to a vector field tangent
which is the pre-image of the submanifold.
as in (2.12), on
(R,w,y),
to
M
Next,
(x,y,x,y)
Use the coordinates
Proof:
x0
Then
M.
x8
and
x
M
,
xa
lift
to vector fields with the same expression.
M
x
M
M
on
b
if
is the map of (2.11), then a brief computation
shows that
n
bM(W 2 RBR +
-0(0
jB
=O
0
}) =xa
(2.18)
n
bX(W 0 WiRaR +
oO
~ Wi
j=0
8j }
i1
Since
b
-
=x
'
-
is a diffeomorphism on the interior, and all
these vector fields are smooth, the vector fields in parentheses here are the unique smooth lifts of
Their restrictions
to
F
are
x x.
xa
.
n
X
=
(
W0f aw
j= 0
j.8
.j=0
JO
(2.19)
Y. = w
1
0
{a
-
0.
i
9.
=},0
0.
=
1,---,n-1.
j=0O
w8() ,
These are patently orthogonal to
to
F.
Furthermore, since they contain no
F P.
are also tangent to each
their vanishing at
Fp
n
T
The factor
where
w0 = 0.
Y -
= 0
hence tangent
terms, they
6~
y
W0 ensures
On the other
hand,
X-6 (0
at
F
p
fl B,
= 0
o
where
=
i
1
(0
is the normal,
8
and
so
n(0
they are tangent here, as claimed.
It is by no means obvious that
are
It
G Rinvariant.
convenient
is
of (2.19)
Y
and
X
to
therefore
use
the
p
projective coordinates of (2.16)
(2.20)
(x,y),(xy) E MP .
u=
s = x/x,
x
These are nonsingular on the interior of
neighbourhoods of
Notice
that
F
p
fl T
F
bounded away from
and even on
,
fl T
F
p
n B.
59
s=-,
0
u
'
-
n
n
Another short calculation now shows that
(2.21)
X = sas'
i = S
invariance is obvious.
p
As for the last assertions of (2.17), first observe
from which
G
q
that the blow-up of a ball around a point
in its bound-
ary is smoothly equivalent to a quarter-sphere.
Denote the
blow-down map by
p:nF
B
Fp =S++
It carries
F
sends
VO(Bn)
F
p
P
F \F
p p
fl B
to
q.
fl B
p
n
diffeomorphically to
Bn\{q},
and
We need to verify that the lifts of
F
vector fields vanish at
fl T
and lie tangent to
fl B.
The vanishing is obvious from the bijectivity of
near
F p fl T;
the tangency is proved by essentially the
same computation as goes into (2.18), which we leave to the
reader to check.
(2.22) Definition:
full space of
at
F
p
fl T
Cm
This completes
E
the proof.
is the vector bundle over
Fp
whose
sections are the vector fields vanishing
and tangent to
F
p
fl B
described above.
We
60
have shown that
P* (T("T(Bn))) = F(E).
Our description of the basic natural b aggage carried
by
concludes with a discussion of certain vector
M x0 M
bundles we shall be using.
density
V0
k-form bundles.
V
bundles and the
These are the
Let us commence with the density bundl es, as they are
density
fine then
of (1.2),
dg
Fo(M)
M
To(M)
line bundle over
M,
such that
near
p-n
which grows as
OM.
De-
densities
C
to be the space of all
on the interior of
the closure.
the Riemannian
Recall
somewhat simpler to understand.
v
ext ends smoothly ti
pn *v
is the space of sectio ns of a
C
and a typical element may be written
in coordinates as
v = h(xy)Ix-ndx
h E CW(M).
dy
This generalizes naturally to a manifold
sion one) boundary components
fining functions
(2.23)
pl,**-,pN,
X1 ,---X N,
p --- pNv
v
with (codimen-
which have de-
respectively.
F 0 (X) = (smooth densities
that
X
on
Let
X
extends smoothly
such
to
X}.
61
is
V 0 -geometry
The connection between these densities and
illustrated by
(2.24) Lemma:
TO(Mx0M)
lift to densities in
FO(MxM)
Densities in
under the usual blow-down map.
In fact,
the line
bundle for which the latter space is the space of sections
b
is the pullback under
of the
line bundle corresponding
to the former space.
F
Away from
Proof:
b
since
dinates
and
there is nothing to prove
OAt
At
is a ,diffeomorphism there.
(R,w,y)
use the coor-
F
of (2.12):
dx dy dx dy
n n
The left term spans
_
RndR dw dy
~
~(RwO)n (Rn n
TO(MxM)
since
dy
dR d
I
x
n nn
and
x
'
are defin-
ing functions for the left and right boundary components,
whereas the term on the right generates
T,
B
are defined by
R, w0' Wn,
To(Mx0 M)
respectively.
F,
as
This
proves the lemma.
One may also consider the powers of either of these
they too correspond under pullback. The
1/2
(MxM)
F
Both
half-densities are the ones we use later.
density bundles;
and
1/2 (Mx0 M)
will commonly be denoted simply
the base space should be clear from the context.
F
2
62
k-form bundles; consider first
Let us turn now to the
the
left factor of
"A 1
bundle
By pulling it back to the
0T*M.
cotangent bundle
V
M x0 M,
and then to
M x M,
one obtains a
The subscript
over the stretched product.
signifies the involvement of the left factor of
A
).
oA
A .
Call this new bundle
of sections are denoted
b
The blow-down
(p,p) E aAc.
Oe,
0
Their spaces
r'
Fp
carries
oA ,
Hence
Using the right
is also defined.
oAk
r
o k
o k
factor of the product,
By
Ak (oTM) as above coincides
functoriality the pullback of
with
M x M.
e
to the single point
for example, restricts to a
F .
p
canonically trivial bundle over
To identify this res-
triction with a more familiar object, compute the lifts of
x
-
M x0
to
M
on
'x0
with
(s,u,x,y),
as in (2.20), which is nonsingular
(s,u)
away from the bottom face
x
d~
ds
dx
_
(2.26) Lemma:
phism
B,
dyi
1%
-
2dx
x
du
dx
u,
^.
s
s-x
-
x
F
'formal pullbacks' to
We define their
(2.25)
dy 1
dx0
x
s
Using the coordinate system
M.
d
dy.
du.
s
x
s
p
-
The correspondence (2.25) induces an isomor-
63
OA F
E
where
~ Ak (E*)
is the dual of the bundle
E
described in
(2.22).
s
Proof:
ds,
s~
du.
are dual to
sa s,
su
which by
,
1
(2.17) generate the bundle
there that
In addition, we showed
F
is a smooth bundle over the closed face
also extends smoothly to the closure.
E
hence
E
E.
p
The rest
of the proof is obvious.
Remark:
(3.3).
We shall prove a slight extension of this Lemma in
Also, the rules (2.25) apply to the
V0
density
bundles since densities are simply 'absolute values' of
So analogously
n-forms.
To(MxoM)I F
C.
Vo
FO(F).
Kernels
The temperate distributions on a manifold with boundary
M
are those with specific extension properties across
this boundary.
If
M
is an open region containing
its interior, then distributions on
constitute
M
M
in
M
with support in
the space of distributions dual
to
C (M).
On
the other hand, restrictions of arbitrary distributions on
64
M
to the interior of
M
lie in the space dual
C (M)-functions vanishing to infinite order on
resulting spaces on
M,
to
6M.
The
of supported and extendible
distributions, respectively, are quite similar but have
slightly different extension properties nonetheless.
On a manifold-with-corner, for example
on
M x M
there
For instance, an ordinary
are various other possibilities.
pseudodifferential operator on
M x M,
M
has a Schwartz kernel
which has a conormal singularity along the diag-
onal and is
elsewhere.
Co
In fact
k
conormal along the 'longer' diagonal of
in some sense fixes its singularity at
then extends to be
M x M,
8AL.
and this
There are,
however, other quite reasonable extension properties which
k
could be required to have.
recall
To formulate one of these,
that a manifold with boundary can be doubled across
its boundary.
[Mx0M]2
k
In particular, we may form the double
of the stretched product across its front face.
NM . M I
Figure 5
65
The lifted diagonal joins with its double, resulting in a
boundary-less submanifold contained in the interior of
[Mx 0 M] 2
(2.27) Definition:
1/2
K 0 (M;F1/)
is the space of distribu-
tional sections of
TO12 (Mx0 M)
which extend to
extended diagonal
extension of
C
(tensored with
1/2
)
TO
(see [19]) along the
m
as distributions conormal of order
[Mx 0 M] 2
sections of a smooth
and vanishing to all orders on
8([Mx 0 M]2 )
1/2
m
%P(M;T1/), the space of
VO
pseudodifferential
operators, is the collection of operators with Schwartz
kernels on
Remark:
A E
belongs to
K(A)
IC
If
M x M
m1/2
lifting to elements of
then we shall say that its kernel
0m
0
Km
Notice also that the pushforward of
00
1%K is well defined:
the blow-down map
F,
diffeomorphism away from
F
with
KO(M;TO
is transversal
b
is a
and the intersection of
BAL
so by wave front set considerations
this operation makes sense.
Let us examine how
A E %
m
0
1/2
)
(M;F 0
acts.
Set
dx dy 1/2
x
(2.28)
ds du dx dy
s x
1/2
dx dy ds duj1
s x
2
66
where
s
and
(s,ux,y)
u
are defined in
are regular except along
quently be used.
B,
system
and shall
fre-
x(A) = k(s,u,x,y)-r
k
On the hal f density
(2.30)
s -
= s
-1
~
x
ity is
represented by
x,
-
+
u = 0},
=
|ul
,0
and
0.
-
we have
f -p,
y = y -
since
(s
on
s
0,
(Af)(x,y) =
~0
m
is conormal of order
decreases rapidly as
(2.31)
The full
Then
(2.29)
where
(2.20).
k(s,u,
f
s
-1
xu.
i(Id) = 5(s-1)6(u)-
s
,y
s
)f(
s
,y-
u)!
s5
In particular,
0
E K0 (M;/O
d
-
the ident-
1/2
We record another representation of this action, now using
the coordinates
the half density
(x,y,t,v),
r
t = x/x,
equals
dx dy dt dv 1/2
ntn
and
v = (y-y)/x.
Here
(2.32)
if
(Af)(xy) =
}
k'(x,y,t,v)f(xt,y+xv)--dv -p
tn
K(A) = k'(x,y,t,v)
in these coordinates.
Although
(2.32) appears both more natural and neater than (2.30),
the first expression is better for our purposes.
reason is
that we are frequently interested in the behavAf
iour of
The
x
as
the top face
T,
0,
-
hence
but
are singular along
(t,v)
less convenient.
It is straightforward to extend these definitions to
include operators acting on the
section.
k-form bundles of the last
Thus
(2.33)
A E *m(M;OAk
O1/2
0F
<=
x(A) E K 0 (M;T 1/2) @ Hom(A , A ).
Unfortunately the class of operators we have defined
is not large enough to contain inverses of elliptic
differential operators.
Diff
For if
E
is
to invert
V0
P E
it must somehow incorporate information about the
,
asymptotic behaviour of (formal)
the boundary.
solutions to
Pu = 0
near
The kernels we have already introduced
vanish to infinite order at the relevant boundary components
T
and
B.
68
We remedy this shortcoming as follows.
Let
T
pT
Suppose
and
Vb
pB
B
[Mx 0 M]2
denote also the extended top and bottom faces in
with
and
their corresponding defining functions.
is the space of vector fields on this double
which are tangent to both boundary components;
set
a,b
{u E
V
5'([Mx0M]
E Vb'
2
V 1 ... V u
) :
E
for all
i = 1,--e,j,
N CO
ab
PTPB(log
j
B)
T log
and some
L
N
independent of
j}.
This has an invariantly defined subspace of polyhomogeneous
elements
CO
dab _
phgi
N.
a+i
a,b :u ~
i=0
j
u=0 P
j =0
(log
near T,
(2.34)
N'
Ni
u ~
u
i=0 j=0
=0log
)
near
B}.
69
u
Here
and
u'.
are
Co
and depend only on the
'tangential' variables at the appropriate boundary.
(2.35) Definition:
K' ma~b
0phg
i)
ah b
mX 0
1/2
i ma,b
)
(MFI
K'
ii)
A E
iii)
0M
C (Mx0M;T 1/ 2
'0O
+
m
1/2
KMMF
OMI
C
1/2
m,.a,b 4* i(A) E Kma,b
00
1/2
(Mx 0 M)
TO
The kernels in i) are just sections of
are
w,ab
K
which
in the interior and up to the front face, and
T
conormal with classical expansions at
and
B.
The extension of these last definitions to operators
acting on differential forms is slightly arcane since we
need to consider forms with tangential and normal components vanishing with different rates at the boundary.
additional
Some
structure is needed so that these components are
well-defined;
for the present problem we have the conformal
class of the metric
h
with which to work.
the ordinary form bundle splits at
k
A (M)
lM
k
= At
Using this,
aM:
k
n
Similarly, from the formal pullback (2.25) we have
70
0kM
|
= oAk
ok
n
(t
Next, by virtue of the metric
h
again, we may actually
assume that such a splitting is defined in a neighbourhood
of the boundary.
point is
This is
less natural, but the important
that in any two such splittings the summands agree
to first order at the boundary.
is to say that
Another way to phrase this
the conformal class distinguishes the col(y,x)
lection of coordinate systems
the boundary and
a/ax
for which
is orthogonal to
aM.
x = 0
on
Henceforth
only such coordinates will be used, and we assume that a
splitting of the
V
form bundle is given.
The real purpose of these machinations is so that we
may attach an invariant sense to the requirement that the
components of
w = wt + wn
vanish at different rates at
am:
S=
If
Wt'
(y,x)
on
O(x)
a
b
Wn = O(x ).
are computed with respect to the coordinates
and are seen to vanish at these rates, and then
recomputed with respect to new coordinates
rates might be intermingled.
(yx),
these
If, however, both coordinate
systems are of the special type above, then from the fact
that
71
Ox
ai
O(X),=
0(x)
B9y
ax
it is easy to see that the new components
vanish
Wt
respectively.
min(a+1,b),
min(a,b+1),
at the rates
Wt'
Hence, these rates are well defined provided we assume
Similarly, it is likewise invariant
b+1.
b-1 ( a
to re-
quire that
CL
la-bi
if
E
t
a
b
phg
n
pgh'
The spaces here are simply those of (2.34)
1.
when only one boundary is present, and of course are
extended to contain vector-valued elements.
Now, using all of these technical hypotheses, we
oA k
assume that both
F
of
in
bundles.
M x0 M
Thus any
split in a neighbourhood
oA r
and
10
into their tangential and normal subG E Hom( A ,A r)
may be associated to
a matrix
G
(2.36)
Gnt]
tt
=
Gtn
Gnn
where
G
tt
Ak
r
t
-
(oAk
e
t
G
nt
:(Ak
r n
oAk
(A
t
G
tn
:
Thus, for
The
G..,
(0Ak )
r
(A
-
t
w =
e
+ W n,
t
=
G
(Go)t
=
(Gw)
= G
i, j = t,
G
: (oAk
nn
r n
n
ttt
(GW)t +
oAk
R n
(GG)
where
+ G nt n
t
+ Gn
.
are also called the components of
n,
These byzantine preliminaries allow us finally to define kernels generalizing those of (2.35).
when expressed as a matrix, con-
G.,
simply to let each
ab
A
phg
tain entries in
The idea is
More precisely, suppose
a, T
are
two by two real matrices
a
tt nt
ttant
=T=
.tn
a nn,
tn
Tnn-
(2.37) Definition:
i)
K
G
G.
EK
ii
ii)
iii)
1/2
okT(M;'Ak
(MT1,2
0
K
Km,a,r
0
A E *
00
m + K
0
#
0)
i,
j
= t,
n,
H m
1 2 ) @ Hom(( 0A ).,(A )}
aiTi(M'r
=
(G
=G
0
(A) E Km '.
As before,
these spaces are well defined only when
Imin(att'ant) - min(atn' a nn)
|min(Ttt'nt)
V
isomorp hism.
1.
pseudodifferential
VO
their composition properties
operators, we discuss
with
min(Ttn'Inn
treatment of
To conclude this
cially
-
(espe-
differenti al operators) and the symbol
The following results will only be stated and
proved for scalar operators
but obviously still hold for
operato rs on forms, by, for example, referring t o a basis.
*,a,b
0
(2.38) Proposition:
Diff0 0
m,a,b
0
Proof:
m,a,b C
0
- Dif
It suffices firs t of
inclusion,
for the other
m+p,a,b
0
all
only to prove the first
by taking adjoints (note
%m,ab
0
Furthermore it also suffices to take
iterate.
module
f 0 c *m+pa,b
0
fol lows
that adjoints of elements in
Diff
is a two-sided
lie
in
%m,b,a
~0
p = 1
Consider then the vector fields
and then
xd ,
xa
1
.
74
(2. 17)
these lift to
M
M
x0
to vector fields tangent to
all boundaries, whence they preserve the conormal spaces
wa b.
we
Furthermore, using the coordinates
(s,u)
of (2.20)
showed there that
x ,
bM(s s
Thus
if
A E
x(A) =
and
m,a,b
b(sOu )=x 0
1/2
ds du dx dy
k(s,u,x,y)Insnd
s x
= k-r
=lk -
-A)
ic(xO
then
as
k-
2 J
(2.39)
K(xO
y
-A)
=
sk - -r.
iu
K m+1,a,b ,
0
These new kernels each belong to
so we are
done.
Diff 0 (M)
A symbol map on
was introduced in (2.7);
there is a natural generalization of this map to
Recall first
,0
that
Im
0
This implies that if
-o,a,b
-0
0
0
A = A' + A"
is a decomposition as in
0 am(A')
am (A) =
then it makes good sense to let
(2.35) ii),
and only define the symbol on
% .
Now,
the map
here, apart from simplifications in the density factors, is
just the usual symbol on the space of distributions conorAt 0 :
mal along
am
Sm (N*(AL
: K (M; F 1/2
0
);
Fo(M) o T(fibre))
The space on the right contains symbolic densities on the
At0;
fibres of the conormal bundle of
T(fibre)
-- densi-
ties on the fibres of this bundle-arises through the use
ro(M)
results
of the invariant Fourier
transform.
Also
from the restriction of
1/2
(Mx0 M)
TO
to the diagonal, for
the half densities on each factor combine along
At0
to
give densities of weight one.
The
ing the map
tors.
N*(At 0 )
steps.
0 am
symbol map
V
a
is obtained by first apply-
above, and then removing the density fac-
It suffices to construct a natural density on
by which to 'divide'.
This proceeds in three
First note the isomorphism
TO(M) ® F(fibre) = TO(N
(At0
the latter being the space of densities along
singular like
R-n
natural isomorphism
at
F.
N*(AL 0 )
Next, observe that there is a
6 : N*(At 0 )
>
This is dual to the isomorphism
of
TM
-
which is
away from
near this boundary it comes from the identification
xO x
lifts
0 TM,
N(At 0 )
N(AL)
the usual natural isomorphism
AL0;
'T*M.
xa
,
y.
which span
,
sas, sa u,
0
TM
near
N(At 0 )
which span
BM,
with their
near
CAt
0
'
1
Finally, by way of the natural map
0T M
T M --
(which is not an isomorphism at
density on
T*M
the symplectic
corresponds to a density on
density is singular at
0T*M.
This
yet a nonsingular section of
cM,
The resulting naturally defined element of
TO(OT*M).
may thusly be removed so as to obtain the
FO(M) @ F(fibre)
V0
OM)
symbol map
0
(2.40)
a
m
Here we have also used
N (AC0 )
(2.41)
m E ER.
to
q
0(T)
6
m 0T*M).
to transfer the symbols from
OT*M.
Theorem:
90
For any such
filtered by the symbol maps
0
am
we have a short exact sequence
77
0
m
0r-i
m
m 0 *M)/Sm-1
=a
(A)am
1r
0
oT*M
and the product formula
0
Proof:
+m(A
A
am1 +m 2l'
2)
(A2.
2i'm
The exactness of the sequence is a consequence of
the same property of
normal to
the symbol map on distributions co-
The product formula follows from the usual
At o
one on standard pseudodifferential operators, provided we
operator.
VO pseudodifferential
is in fact a
A -A2
ascertain that
For, both the symbol map and the composition
rule agree with their ordinary analogues away from
8M,
hence the product rule must extend by continuity, provided
the left hand side makes sense.
The first step is to derive an expression for the
Use the representation (2.30),
kernel of a composition.
near
k2
and
k
since we may assume that
as functions of
This is
x, y, s, u).
(rather than
x, y, s, u
A , A2
of
ki, k2
kernels
but write the
are supported quite
With
At0.
s
=
x
x
y-y'
x
-V
,U
=
x
,
s2
=
x
,
legitimate
u2
x
,
and recalling the half densities of
f
= f(x','
|
I-
)-p
)
f
->
fk
2
(x,y,s 2 ,u )f s
2
x
ds
-NI
_d2
d2
2
'y--s-u2
2
2
kl(x,y,sl,ul)k2 s ,YX Us1
s1 1u 2')
As1
ds2 du2
ds 1 du"
f(
the composition
kernels,
becomes, on the level
A1A f
(2.28),
s1s2
,y-
s
s2
( s 2u 1+u 2)
- II.
s1s2
Now, introduce
x
x
s 1s 2,
x
= s2 u 1 + u 2 '
Hence
A1 A2 f = f k(x,y,a,v)f(l,
where, by the substitution
k(x,y,a,v)
y-xv/a)-
s 1= a/s 2'
u
dv
=
(v-u
2
)/s
2
=
ds 2du2
kl(x, y ,a/s2, (v-u2)/s2)k2(ax/s2, y-x(v-u2)/a, s2, u2I
n-1
2
79
The
k
are conormal along
so we must show that
k,
(s i
=
1, u
= 0},
i = 1,2,
as given above, is conormal
along, and supported near,
{a = 1, v = 0).
The support
condition is obvious from the similar assumption on
k2 .
As for the conormality, consider first the special
= y = 0:
x
case
(2.42) Lemma:
Gp
k ,
If
k , k2
are distributions on the group
of (2.4) which are conormal at, and supported near, the
(1,0).
identity
Then
k(a,v) = f k 1 ((av)-(su)~ )k2 (su)s-n ds du
enjoys the same properties.
Proof:
We reduce this to the corresponding assertion for
the group
IRn
where this result is well known.
The main
point is that the space of distributions conormal to some
submanifold (e.g.
(1,0))
is coordinate invariant, and
furthermore that, under a submersion, conormal distributions pull back to conormal distributions.
(a,v,a-s,v-u)
a/s = (1 -
a
-
-
v-u
s
Thus,
(a/s,(v-u)/s)
_ -u
a
_ a-s -1
a
since
80
kl(a/s, (VU)
which is conormal at
kj(a,v,a-sv-u)
a-s = 0,
v-u = 0,
and
C0
else-
The pairing
where.
J
k
defining
proves
defined by
k
is a surjection, there is a distribution
kj(a,v,a-sv-u)k2 (s u)s-n ds du
is therefore of the required regularity, which
the lemma.
To understand the group convolution when the parameters
x, y
necessary.
are present, only a slight modification is
Thus, as before, there is a distribution
kY(x,y,a,v,s 2 ,u 2 ) = k 2 (x(l
also conormal along
a, V
0
is conormal along
-
ay
-(v-u 2 ),s 2 ,u 2 )
and for which
As in the lemma
a = 1,
u = 0.
the
x,
y,
kernel
The proof of
is complete.
Remarks:
Diffm,
2 -
s2 = 1, u 2 = 0
are smooth parameters.
k(x,y,av)
(2.41)
-a
Though we did not verify it,
that
0 am(P)
P E
as defined here agrees with the earlier definition
(2.7)
is quite easy to check.
In addition, the product
rule
oa
m+L
A E %
(P-A) E Oa (P)oa (A),
m
0
is simpler to prove, since we may operate directly on the
kernel of
ing
A
with the lifted operators on
M x0 M
cover-
P.
the
The residual space for this symbol calculus is
%~ ;
kernels of the operators in this space are
on the
stretched product
M x 0 M,
but not on
though smoothing in the interior of
pact on any reasonable space.
M,
M x M.
C
Thus, al-
they are not com-
This is discussed at length
in the next two sections.
D.
Indicial and Normal Operators
The first step in constructing a parametrix for
Diff0
P E
is to iteratively 'remove' the conormal singulari-
ties of its
kernel along
At 0.
As in the ordinary con-
struction, this is reduced to algebraic manipulation by way
of
the symbol calculus.
But, as noted at the end of the
last section, the resulting error after
compact.
this step is not
This necessitates the use of two additional
steps, both iterative in nature, and both employing models
for the various operators involved.
These we describe now.
If
E0
is the operator obtained in the first step:
PE 0 = I
x0 ,
then
the
all boundaries.
at
B;
and
T
trivial
-
kernel of
F,
(which causes
QO0
QO,
is
x0
In fact,
at
Q0
0
C
on
M x0 M up to
vanishes to infinite order
however, its Taylor series is nonQO
to be compact).
not
Thus the
second step of the construction is the removal of this
Taylor series.
PE 1
at
the
xi,
and
F.
kernel of
-
vanishes to infinite order
lying
in
1
9-o,a,b
I0
for
Consequently
a,
Q R o*-00
QE
9a'b,
0
where the space of
b
b
ic
IC
Rw
-
E R KO
a,b
kernels on the right consists of those
KO 'a,b
elements of
and
E1
(We are restricting discussion to scalar oper-
a, b.
a
1
Unfortunately, in order to accomplish this, we must
ators for simplicity.)
If
Q
Q ,
settle for a correction term
some
such that
E
We seek an
which vanish to infinite order on
are sufficiently
large then
Q
F.
is actually
Usually, though,
compact and the construction is complete.
the re-
it is necessary to carry through the third step:
of
moval
E
Here we seek an
face.
x1
the conormal singularity of
2
P2
Q2
The operator
~
1
such that
E R q'-w ab
0
0
42 E
~2'
top
the
along
L2
is compact on appropriate weighted
spaces and
P-(E 0 +E 1 +E
so that
E = E0 + E
+ E2
2
) =
I
-2
is the desired parametrix.
From this somewhat vague discussion it should be clear
E
that
cally,
at
F
and
E1
E2
P
should only depend on the operator
and the relevant Taylor series of
KO,
must rely only on the infinitesimal action of
and the conormal expansion of
Consider first the
K
frozen
whereas
near
P
E2
T
there.
former of these steps.
stated goal, define the normal operator of
(2.43)
Specifi-
may be 'locally' constructed.
Toward the
A E %0
a,b
Np (A) = x(A) IF
By the remark following (2.26), and recalling that
is actually a section of the half density bundle,
ic(A)
N p(A)
84
is a section of
T1/2
0
p) .
a,b
aF
phg F p )
i.e. smooth in the interior, an d with the appropriate conormal singularity at
n Fp
T
and
B
n
F
p
As
.
it
stands,
this seems to bear no relations]hip to the previous definition (2.8) for
we must
Fp
P E Dif:f 0*To make this comparison
N p(P),
interpret
as an operator.
N p(A)
Remember that
is naturally identified (in its interior) with the
identification,
p
is to be thought of as a left convolution operators
G
p
group
N p(A)
on half densities over
(s,ux,y)
of (2.20),
G
Using the
in two ways.
F
Thiis, in the coordinates
.
p
p = (0,0
N p(A) = k(s,u,0,0)-
I
~1/2
ds du ds
~ du
n'n
s s
(2.44)
Np(A)f
=
k((su)-(s,u)
f = f(s,u) ds du/
where
Now, if
(2.21)
P E Diff 0
its lift to
ds du 1/2
ds du
,0,0)f
s,u)
~n
s
'
s
n
s 1/2.
is expressed as in (2.2),
M x0 M
is
then by
-~
'I
aa(sxy+xu)(sau)
a)
a
a
(SOS)
n
a = (a'
,
,an
|a|(m
Hence, applied to the
=I a(P)
kernel of the identity (2.31),
a nb(s-1)6(u)_-y
aa(sx,y+xu)(sa) a(ssO
|a|(m
so that,
b)
according to (2.43)
N (P) =
a
aa(0,0)(sau) a (sOs
u s 2
a
n 6 (s.-1)
6
d
d
d1/2
(u) ds du ds du
ss
|alsm
However, according to (2.8)
c)
Np(P)
a(0,0)(s
=
)a
(sOS)
n
|a|(m
Finally then it is not hard to see that the expression c),
applied to a half density
f(s,u)Ids du/snl/
2,
coincides
with the result of setting expression b) into (2.44).
We
have shown that (2.43) is a proper generalization of (2.8).
(2.45) Proposition:
There is an exact sequence
86
N
O-0 >
W
0
a,b
-00
a, b
0
da,b (F)
p
phg
If
: 0.
then
A E 9-I,a,b
0
P E Diff m,
0
@ T 1 /2 (F )
p
0
N p(P-A) = N p(P)-N p(A).
Remark:
For
operators whose
Proof:
k
R k0
k = 0,1,2,---,
-m
a b
'
the space of
is
kernels vanish to order
F.
at
k
The exactness of the sequence is obvious, save per-
haps the surjectivity of
N
p
But that too is easy since
.
one may readily construct in local coordinates a
with fixed restriction to
F .
p
As for the product formula,
either by applying the operator a) above to
setting
x = y = 0,
kernel
and then
K(A)
or by applying the operator c) to
in (2.44) one arrives at the same section of
N (A)
p
for
x and y are merely parameters in a).
The conjugate
k
R N R
shall not use this fact.
Z+,
the
residual
-k
acts on
p
k 0
R k0%
Nonetheless, as
space
R %-o,a,b
0
a~b
'
k
,
1/2
'
T
though we
varies in
87
is the outer limit of the effectiveness of
N
elements in this space, when pushed down to
conormal
aM
x
at
M,
Kernels of
.
M
x
M,
lie in
1/2 (Mx1M).
a,b (MxM)
consists of those distributions which are
da,b
0
The space
p
(with expansions) along the boundary components
M
x
M
of
aM
x
and which vanish to all orders
M,
BAt.
The final step utilizes the indicial operator, which
It
kernels in the last residual space.
once again models
is defined by the exact sequence
Thus, for
A E R
I
a+1,b -
o
a,b
1/2
0
0
a+1,b 01/2
0- ,a,b
I(A)
R
(2.46)
kernel of
the expansion of the
-,a,b
-
0
0
0.
is simply the top term of
A
OM
on
x
M.
It too may
be defined as an operator, but we have no need of this.
also has an indicial
P E Diffm(M)
On the other hand,
operator, which we do want to regard as an operator.
that
A
E
R*
-,
0
a,b
*
P-A
E
R0%-w,a,b
0
Note
88
I(P)
is now defined to be that operator acting
or equivalently on
OM'.
'infinitesimally at
N8M,
for
which
I(P-A) =
(2.47)
I(P)I(A).
Strictly speaking there are many operators satisfying
(2.47).
But since
X4
:a,
x
0
a+l,b
b
a+1 ,b
a,b
0
xva
P >
0
V
0
I(P).
there is a natural choice for
aa
k+ IaIm
> 1
k
) (x8 )
y)(xa xy
(x.
k~a
a
then set
I(P)
(2.48)
=
ak,0
(o, y) (xa x) k
kfm
This is well defined, provided only coordinate systems of
the type discussed in section C are used, and obviously
satisfies (2.47).
induced by
I
is
The residual
space of the filtration
Rw%-w
the
is
E.
, w,
b
kernels of which are compact on
L2
(so long as
b
large enough), as we prove in the next section.
Continuity Properties
We now study the continuity properties of
the
L2
pseudo-
Following the usual scheme of such
differential operators.
proofs, first
VO
boundedness of residual operators is
demonstrated-this is the most arduous part of the proofthen the symbol calculus immediately implies boundedness
Finally a few more refined
for operators of order zero.
statements, including criteria for compactness, are proved.
Other less general results are discussed in the last secAs before, we limit all
tion of the next chapter.
consideration to scalar operators;
the analogues for opera-
tors on the form bundles are either obvious or will be
commented on when required.
(2.49) Proposition:
Any
induces a bounded
A E * 0,a,b
map
dy)
A : xrL29-ndx
c
-
xr' L2
loc
(X-ndx dy)
90
a + b
a - r >
r' ( r,
provided
n-i
b + r' > -,
n-i
2 ,
and
> n-1.
The slightly bizarre method we employ is to reduce
Proof:
L2 bounded-
to the familiar one concerning
the statement
ness of ordinary pseudodifferential operators of order
B
where it is conormal.
It is
< 1.
T
except at
M x 0 M,
when lifted to
C
of course
lyl, |y|
0 < x, x < 1,
is supported in
A
of
k
kernel
First note that we may assume that the
zero.
and
It also suffices to demonstrate
boundedness of
-r '-(n-)/
2
Agr+(n-)/
2
dy) .
2 -dx
L(x
on
kernel of this last operator is
The
k c=
x
r-r' rx]-r-(n-1)/2 k = x~r-r'*]r'+(n-l)/2
k.
x
2x
and the other in
x
2x.
the first of these divided by
divided by
k
and
k2
.
x
k2
Write
and
x K
into a sum of two kernels, one supported in
kc
Split
since
k
xr-r
k 1 + k2
dominates
as a function of
k1
and
>
let
0,
k1
be
the second
k2
We need only demonstrate boundedness of
kc
x, y, s, u
x, y, t,
v
is supported in
0
as a function of
by assumption
r - r'
As
as
in (2.30)
as in (2.32).
x < 1,
Iy|
Then
1,
o ( s
in
k2
and
K 2,
|y|
0 ( x < 1,
( 1,
0 ( t < 2;
both are supported within the domain of regularity of their
Furthermore
coordinate systems.
a)
for
k 1 E Co
sa-r-(n-1)/2
with an expansion of lowest power
and a symbol
s = 0
is conormal at
s A 0,
-
of order
u
in
a-b
(2.50)
b)
for
k 2 E Co
tb+r'+(n-1)/2
with an expansion of lowest power
symbol
and a
,
[x,y,
k
alone.
k
Consider first
x
jf(,
-
of order
u.
in
a-b
t = 0
at
is conormal
t A 0,
By the Plancherel formula
dx dy=
)nx
x
e
x
dx d77
f(x,ri)x
x
since
to
y
the Fourier transform of
is
,1
x
with respect
Next,
introduce into
--
k
x1.
kl(x,y,x/x,-xl)e-
-t
this
last
integral
the new variables
e
=
x,
e
-t
obtain
^
f kl(e
-t
-
,y,e
t-t)
,e
-t
TI)e
- iy
- T^
f(e
- t
,~
~
td
-
=x
to
92
t > 0,
t > 0,
where the integrand is supported in
Notice that by (2.50) a) the first factor in-
t-t > -1.
in which case
creases exponentially unless a-r > (n-l)/2,
it decreases rapidly.
-~
=
Cj(t,y,T,T)
Finally set
r kl(e-t
-a
,ye
-t
,e
r)e
-iy-a -iarT
e
da
so that the former integral becomes
ei~-)~~-)a
Kt,y,T,,q)f(e~
at
e t Q = 0,
k
that
and so represents an
dT1.
of order
In
L2 (dt dy)
L2 (dt dy) = L2 (- 1dx dy),
Since
operator.
T,
except for a mild conormal
(1,0)),
(and type
is a symbol in
K
We shall show that
,y)dt dy dT
0
singularity
bounded
this shows
has the required boundedness.
To prove the claim we need only examine the effect of
differentiating the integral which defines
notice
that by (2.50) a),
kl(x,y,s,u)
in
u
Ky
Since
imply that its Fourier transform is
a+b > (n-1),
k
also enjoys these properties.
(n-l)/2
as assumed,
First
the symbolic properties of
rapidly decreasing in the dual variable
0.
K 1.
K1
'
and conormal at
is bounded there.
Hence
In addition, if
a-r >
is rapidly decreasing in
T.
or
T
Thus differentiating with respect
to either
y
93
preserves this rapid decrease.
is a smooth parameter,
x
As
fourth.
either affects the first slot of
t
respect to
Differentiating with
the former preserves
In the latter case a factor
regularity.
or the
k
e
-t
ri
is pro-
duced in front:
{
=
t
(e t k1 + e
-t
^
r
k~e
-iy--irT
d
x
is of the same regularity as
in particular bounded.
S1,
Finally,
i(
.
1
1
-
Ck
is also uniformly bounded.
k ^ )e -iy-7-iUTada
i+iny
By iteration it
da
is readily seen
that
Bae
t1 Y
1 (t yrtr)
C
t
and, as mentioned before ,
singularity at
Ti = 0,
By the standard theory
tion on
L2
+
ITI +
I1i)'
has a bounded conormal
uniform in all other parameters.
kg1induces a bounded transforma-
-1 dx dy).
The proof for
k2
requires the hypothesis
proof of
V.1
Jean (1
(2.49).
is quite similar, only here one
n-1
b + r' >
-.
This completes the
94
Before proving continuity of operators of order zero,
let us note that
%Pm
0
is invariant under the operation of
taking adjoints, for any
kernel of
A
m.
A E *
Thus if
is
1/2
dx dy dx dy
n'n
x x
K *(x,y,x,y)
A
-
0
_1/2
dx dy dx dy
n'n
x x
A(x3Ytx'YJ
which certainly has the same regularity as
In addition, it
(2.51) Theorem:
A E 9O,a,b
0
M x0 M.
is bounded from
rL 2F1/2
0
0
a+b
> n-1,
Write
A = A
a-r
>
n-1
-,
2
calculus of (2.41) to find
sufficiently large
b +
+ A2 E %0 +
0
1 2
between these spaces by (2.49).
bol
on
ua(A).
rL2 1/2
Proof:
KA
is easy to see that
o mA ) =
provided
then the
M E IR
r'
,O'
0
As for
B E %
0
n-i
> -,
2'
'ab A
A
2
r > r'.
is bounded
use the sym-
such that for some
95
A A
1
+ B*B = M-I
+ R,
R E 4
.
f = f(x,y)|x-n dx dy1/2
Then, for
| Ajf| 2 +
| 2 +
|jBf| 2 = M If|
which implies, using the boundedness of
f
1A
f|2
M'
Rf-f
that
R,
|f12
as desired.
The next step is to consider the action on Sobolevtype spaces.
Since the interest here is with parametrices
of differential operators,
developed.
(2.52)
For
Hb(MF
the full
m = 0,1,2,---
set
) = (u E L2 r1/2
i + lal
(2.53) Corollary:
A E %-m,a,b
0
r L2 1/2
0
for
r, r',
a, b
as in (2.51).
theory shall not be
(xO
i(x
8
Y)au E L 2
< m}
is bounded from
xr' Hm
b
1/2
)
96
If
Proof:
k
then recalling (2.39)
A,
is the kernel of
(xa ) (xa )ak C KO,a,b
for
i +
|al
m.
The corollary follows.
The space of conormal half densities
daA(M; 1/2
FO
1/2.
FO
f = f(x~y)Idxndy
x
) af(xy) E pa(log pN L
(xO ) (x
for all
has,
as
(2.54)
and some fixed
i, a,
in (2.34), the subspace of polyhomogeneous elements
1/2
dhM
Aphg (M;FO0
iO
{f C da .
the natural range of
fi
f(y)xa+i (log X)}
V0
tors applied to smooth functions.
special case:
N
_
i=0
This is
N}
0
pseudodifferential operaAgain we only prove a
97
(2.55) Proposition:
A E
b, m E R,
For any
m,a,b
p0
maps
a
600
C ~
~phg
(where the space on the left denotes the smooth half densities vanishing to infinite order on
Proof:
If
k E Km"ab
is the
aM).
kernel of
A,
1/2
~
then the
product
dx dy dx dv
k(x,y,x,y)
x n n
x x
f E C
,
infinite order on
vanishes to
~ 12
f (XY) dx dy
xn
x
{x = 0}
and has an
ordinary conormal singularity along the diagonal.
Its
integral then is that of an ordinary pseudodifferential
kernel against
f.
Furthermore, if
k ~
L
k. E
1
k ,
1
a+i,b
phg
then each
^0
'%
dx dy dx dy
k (x,y,x,y)f(x,y)
which gives the result.
0~.0
n
x
x
n
1/2
C
Aa+i
98
Our final result concerns compactness of operators of
negative order with appropriate boundary conditions.
(2.56)
Proposition:
-m,a,b
A E R'qO
is a compact map
between
r ' L2 1/2
r L2 1/2
0
0
for
a+b > n-1,
m > 0,
a-r >
Proof:
x~FA = B E *-m,a-e,b
small,
B
-1,
b+r
and if
,
>
The compactness of
uniform smallness of
21
is sufficiently
6
r ' Hm
xFB
b
A = x B
is furnished by the
:x >
6,I
for each
BM
in a neighbourhood of
the uniform equicontinuity of functions in
{(x,y)
n-i
is bounded between
xr L2
also.
n-1
6l
> 0,
cf.
Hm
b
[29].
on
and
99
Analysis of the Normal Operator
Chapter 3
A.
Identification of the Model Operators
The novel
step in constructing a parametrix for
Ag ,
as indicated in section D of the previous chapter, is the
use of simpler models for this operator at the boundary,
where it degenerates.
These models, the normal and indi-
cial operators, are employed in iterative schemes analogous
to the initial symbolic step of the construction to obtain
It is
a parametrix for which the remainder is compact.
important then to reach a thorough understanding of their
mapping properties;
ter.
this is the goal of the present chap-
In this section we identify these operators explicit-
ly and analyze the indicial operator.
Since it
is an
this is quite
ordinary differential operator of Euler type,
The normal operator, however, is a partial differen-
easy.
tial operator, and the derivation of those of its properties we need later will require more of an effort.
We have given two different, yet equivalent, definitions of the normal operator of
one,
P E Diffo(M).
The first
(2.8), is an operator on the half tangent space
p E aM,
M ,
while the second, (2.43), is a convolution kernel
of the front face. This latter
p
interpretation is the one we ultimately use, but the former
on the fibre
F
is easier to manipulate, hence the one of interest in this
chapter.
given by
Recall now that the action of
A
on
Ak (M)
is
100
P
the operator
N (A )
(1.21),
is an operator on
A k(M p);
so that
k
-k
P = p A p
.
Then
acting on sections of
Mp
the induced operator is
N p(A )
that
of
N p(P).
It turns out
is actually the Laplacian of a constant curv-
ature metric, whence it has greater invariance properties
G p-invariance.
than merely the expected
N p(A )
(3.1) Proposition:
stant curvature metric
is the Laplacian of the con-
(dp )
p
-2
thought of as a linear function on
Ak (Mp)
on
Proof:
on
M
is given by
Choose coordinates
near
such that
p,
and
z
|/BzIn
x
=
coordinates on
Then
The induced action
.
N p(P).
-n-1
-1
z = (z ,---,z
8M,
-n
,zn) = (y'x)
-1
-n-i
)
y = (z ,--,z
z = (y,x)
Let
M
p
.
are geodesic
centered at
p.
the corresponding linear
be
Finally, write
= a(y)x + b(yx),
Idp |2 = a(0) 2 . Now, for
p
p
is to be
p
is a defining function for the boundary and
h = 1.
p(j,2)
M
restricted to
h
coordinates for
dp
here
h ;
p
b = O(2
w E 0 (M)
101
PW
+
= p2 Ah
(2-k)p
-
6 h(dpAw)
kp
(3)
(n-k)pLpw
-
du
2pt
(5)
(4)
+
Sh w
(2)
(1)
+
A
dp
(n-2k)dp A
-
cvw
(dp)w
k(n-k-1)t
(7)
(6)
is
dz-I ,
Suppose
(
ik
We shall
(
=
W
I
=
(i
, ---
(1)
p2Ah
~
p
k
-1
ik
consider each term (1)
always using the product rule
(PW)
Ag (k
p
defined by
the operator of (1.21),
-
(7)
separately,
(2.10).
has normal operator
n
I 2I
I-a(0)2
(3z
(2)
dx
)2
p dp A 5h ' =
I
-n+-n
-n
(az +b)(adz +z da+db)A(-6.i
d-n
d
z
=a2-n
5vWI
d
(-1s)
dz
az s
which becomes
s I\i
s +---
102
- 2z
-az
Cl
-
dz'
if
i
=
n
azn
r
a
2
)
k+s-1
z
n
I\i
s
cz
.
. n.
.
Adz
1±i
az s
s
(3)
I
__
-I
A w Idz )
pbh(dpAw) = (az +b)6h( (adz +--)
k+1 I n
( k 1,I\i
=(az'+b)
S
S
,n
v n(a1
(an+b)(-1)k+1,I
*S
)dz
-I\i
(aw)dz
+-
which reduces to
2 n aw I dz
azn
s
k+sn
(-1)az
s
if
if
i k
< n
ik = n
sA dzn
A dzn
103
(4)
=
pL v(*
.
S
.i
--s-1i i s+1
8p s dz_11
azj
- n +b)(vp(w )dz- I +w
(az
1
2-n
+
0
8(I)
2 n
a z
which remains as
I
-z
n dz
for any
dz
I
azn
(5)
-nb
3
p vp dw = (az +b) c
1.z 1
= (azn+b)pi
Adz )
azi
(dzjAdz)
+
0 0
81/82n
if
= O(p)
recalling
A dz]
(dz
= an OW
ai z
dz
i 8a
i
< n
yielding
2zn
az
___
i.
< n
dz i A dzI\n
if
dz
if
de
a2
kzn
ozJ
k = n
k)
104
(6)
dp
A
tLO
=
+---)
(adz
a
=
and all
2
-n
dz
A
. gdz
I
f)
_
-I
dz
that remains
2
a w
dz
O
(7)
-I
i
A
Finally,
dp(vp)o
leaves
a
2
I
gdz
=
2
ao
i
ik
dz
k = n
< n.
-I
+00
for any
Heuristically, the procedure throughout is to expand the
coefficients of
P
which the power of
in a power series, discard any term in
z
exceeds the number of derivatives,
and then replace all coordinates
logues
-i
z
by their linear ana-
i
z .
Collect all terms above with the appropriate coefficients to obtain
105
2
a(0)2
- (zn)2
i2+
i=1
(8z )
nW I
(n-2)z
k(n-k-1)w1
-
-
azn
I
dz
(3.2)
+
2
I
n
z
(- 1 )k
(-1)
s-dz
sI\i
A dzn
C
-z
s
ik
if
< n
N p(P)(w) =s
2
n
a(0)2 1-(zn)2
2 +
(8z )
(n-2)zn
az
nI
-
I
(k-1)(n-k 1(.
d
1dz
n-1
n 1
+2(-1)k+1
j= 1
dzi A dzI\n
(3z
if
ik = n.
Furthermore, the product formula (2.10) shows that
N p(A ) = (zn)-k N p(P)(zn )k
][
so
by
that
N (A )
p g
(A )=
(z n) k
9
Ak (M )
on
p
N (P)W
i.e. the action induced
(znyk
as claimed.
is given by N (P),
p
106
Finally, to see that
metric
is the Laplacian of
Np(A )
(a(0)zn)-2 dz 2,
simply observe that
the
this a special
case of all of the calculations above, and preceding
(1.21),
with
p = a(0)zn,
h = dz 2
P
The corresponding
is already dilation invariant
and so no terms are lost in passing to the
limit in (2.8).
The assertion now follows, for (3.2) depends only on the
dimension
n,
the degree
Idp 12n
and
k,
p n
In the next section we construct an inverse for the
action of
N (A )
Ak(M P),
on
for
i.e.
N p(P).
However,
it is really the normal operator, as defined by (2.43),
1/2w
ok
that we need to
F
1O
acting on sections of oAk
0
p
invert.
It is true, though not immediately obvious, that
this bundle and
phic.
are naturally isomor-
Ak (M ) ® r1/2 (M)
p
p
O
Thus'it suffices to study the operator over
then transfer the results back to
F
.
p
the details of this bundle isomorphism.
identification only for the
Mp,
Let us now examine
We prove the
k-form bundle; a similar argu-
ment applies equally well to the half-densities.
In fact,
the density bundles will be systematically neglected
through most of this chapter.
Their
later inclusion is
by
effected by 'conjugating the whole discussion
xn/2
,
107
oAk (M )
Both
(3.3) Lemma:
and
OA kF
are trivial,
There is a canonical equivalence
though not naturally so.
between these two bundles over the interiors of their
Furthermore, each is also naturally equivalent to
bases.
again only over the interiors of
0Ak Bn) = A (oT*Bn),
their bases.
M x M
Ak (M ) lifts to the left factor of
p
p
p
the resulting bundle we denote
thence to M x 0 M ;
p Op
oAk(M x M
It then pulls back to the fibre F
over
0
R p o p
Examination of the rules (2.25) for
x OM .
(0,0) C OM
p
p
formal pullback show that this restriction is isomorphic to
Proof:
oAk(M P).
Now, the linear model
order with
ducts
and
Mp x 0
Op M
lent near
F0
oAk (M)(MX0M)
M x O0 M
F P,
and
and
is equivalent to first
This means that the stretched pro-
p.
at
M
MP
are also first order equiva-
respectively.
OA (Mx0 M)
agree to order zero at
these fibres, so their restrictions to
cide in a natural manner.
Hence
F
least over the interiors of the bases.
conformal map
0Ak(M P
n
]R+
--
o Ak (Bn)
B
F
coin-
Finally, Lemma (2.19) asserts
the equivalence of any of these bundles with
isomorphism
and
Ak (Bn),
at
Notice that the
is not induced by the usual
n
As we have indicated earlier, the indicial operator of
P
is an ordinary differential operator of Euler type;
it
108
Using the coordinates of
is actually an uncoupled system
(3.1) with
p
0,
corresponding to
from (3.2) we derive
WI
a(O)2 -(zn )2
-1('zn'2
(n-2)
k(n-k-l) WIdz
(3.4)
I P(P)(w, dz
2
- zn)2
WI
z
I P(P)(W)
cial
roots.
s
The numbers
dy
s1
ik
= n.
solutions of
and
=
x
s2
2 dyi A dx,
III = k,
are called
the indi-
or
n-k-1
unless
k =
or
n-k
unless
k =
(3.5)
s2 = k-l
I
az n
= 0
W
,
if
A short calculation shows that
= k
n
2
the
= (y,x),
I
si
w = x
(n-2)
dz
(k-1)(n-k)I
Reverting now to the notation
+
(azn)
I
IJI = k-i.
ik
)I=
a(O)
are of the form
if
az n
n+1 -g
109
The solutions in the exceptional cases
k = (n+l)/2
involve additional logarithmic factors, but we are not concerned with these here.
The basic reason the parametrix construction breaks
down in these exceptional cases is that only for these
rates at
do the indicial roots not vanish at different
k
values of
x = 0.
For the other values of
only one of
k,
the solutions corresponding to each pair of indicial roots
space which is natural for the problem.
L2
lies in the
Thus:
Both
xn-k
Both
x
Both
xk
x n-k-i
2 ( -n
E
C Lloc
dx)
x
if f
k < n-1
2
if f
k
(3.6)
2 (X -n dx)
k-1 E Lloc
w
Slightly restated, if
(-n
L2
loc
dx),
solves
IP(P)w = 0
and
> n+1
2
w E
then (with summation intended)
+ c xn-kdyZ A dx
J
1
W= cx n-k-1 dy
I
if
k < n-i
if
k
(3.7)
(j
=
x dy
+ c x k-dy
A dx
I1
We shall
PW
there.
w E L2 (dg),
then it is polyhomogeneous conormal at
= 0,
(3.7)
prove in the next chapter that if
is
the
> n+i
2
aM,
leading term in its asymptotic expansion
and
110
We should remark, as a general note, that the anaylsis
elliptic problem is greatly simpli-
of this particular V
fied by the fact that the indicial roots (3.5) are independent of the base point
p.
In the general problem they may
well vary, and complications arise from repeated roots,
etc.
B.
The Inverse of the Hyperbolic Laplacian
Let
a(0)2
denote the operator of (3.2), neglecting the
N
may be interpreted
N
From Lemma (3.3),
factor.
as the operator induced by the constant curvature Laplacian
or
Ak (IRn)
either on
have both of
Ak (Bn).
It is important that we
is constructed using the Fourier
ables in the upper half space.
N
The inverse for
these models available.
transform on the
y
vari-
This is quite natural, for
these hypersurfaces are horospheres, hence have natural
Euclidean structures.
Then, however, the additional rota-
tional symmetry of the ball
behaviour of
So,
N
this
inverse near
00 E
n
R+'
has a partial symbol given by
N (xa ,x y) f e' '(n)dn
where
is useful in understanding the
-q = T 1 dy
+---+
=
nn-1dy n-
eiY
is
-N(xax,ix7)w'(T)d1
dual
to
y,
and
111
={
(1)
w(y)dy
e
the usual Fourier transform on the components of
w.
easy to see from (3.2) that
2^^
Np(x8
2 6mBOw
+ (n-2)xg
-x 23x
I
,ixI)COdy =
+ (x2
1
2-k(n-k-1))W ]dy
+ 2(-1)
k
L(ixq)w Idy
I
A dx
(3.8)
Np(xa
,ixn)wj dy
A dx =
2
[-x2
+
C1x 2C
(n-2)x 0
+ (x2 212- k-1)(n-k))wi dy" A dx
k+l
+ 2(-1)
The
r
ix
A wjdy
J
in the first of these formulae is the
- y= +-+
77 n-1
-l
By
and is to be contracted with
The operator of
only mildly coupled.
dy
By
h-dual of
n-1
A dx.
(3.8), which we also call
N ,
This allows all solutions to
is
Nw =0
112
to be found, whence the inverse may be constructed and
the
ri now as a fixed parameter,
Thus, regarding
analyzed.
coordinates may be rotated so that (3.8) assumes a
y
particularly simple form.
1
Iril dy
In fact, rotate
y
so that
7 =
.Then
7 A dy
j0 > 1
d O 4
There are
three cases to consider:
(3.9)
i)
dy I
o = o
=
i
A dx s 0 t*
L(n)dy
1.
> 1.
2^
N
2
= 1-x
W
ii)
=
e
W
nI
W
+
(n-2)xg--
A dx,
dy
j
+ (x
=
||-k(n-k-1))w I dy
1.
2^
^2
Nax=
P
a
C3
(n-2)x=
WJ
2'+
-
+ (x2
-4
iii)
(ii,---,i
=
^
c = W I dy
k) =
I
2-(k-1)(n-k))oj dy
^-
+ O
'1
J
dy
A dx,
'''
k-1)
A dx
113
No
^
=
2^^
2
I
2 aI WIknk1
2 + (n-2)x -- + (x
-x
dx
|n|
2
-k(n-k-1))I
+ 2(-1) k+ 1ixl
+
-x2
2
dx
2 -(k-1)(n-k))w
+ (x 2
+ (n-2 )x
Jdy
+ 2(-1)kIl
dyJ A dx.
The first two of these operators are quite easy to invert.
The third one is slightly less so, since not all of its
solutions are known explicitly.
Nonetheless, it may be
regarded as a compact perturbation of the diagonal system
formed by i) and ii).
The program now is as follows.
Ultimately we seek to
prove that the partial differential operator
N
of (3.2),
acting on
L2
+
n-1
-n dx dy)
has a bounded inverse.
n
L2
+
-n dx;L2
n-1,dy))
Yet more refined regularity proper-
ties of this inverse are proved in the next section.
Upon
conjugation by the Fourier transform it suffices to show
that the ordinary differential operator
acting on
N
of (3.8),
114
L2
+
-n dx;L2
n-1dp))
also has a bounded inverse.
There exists a unique kernel
(3.10) Proposition:
G(x,x,n)
which is bounded on
L2 R+
-n dx;L2
n-1dl))
and such that
Np(xax,iX71)G(Xx~T1)
(The right side of
= X 6(x-x).
this equation is the Schwartz kernel of
the identity on the space above.)
The proof is quite long and involves several lemmas
Consider first the two operators (3.9)
along the way.
ii).
Set
P
2
=
- -2
d2
+
(n-2)xdx
(x2 1j 2 -k(n-k-1))
-2
d2
dx
+
d
(n-2)x -
(x 1_
dx
These are Fuchsian operators
as expected, are
k
<
i),
n-1
(the case
k,
n-k-1
k > n+1
2 -(k-1)(n-k)).
the indicial roots of which,
and
is
k-1,
n-k.
In fact, for
treated similarly, and
115
shall be mentioned only rarely) we find, cf.
n-1
n-1
P 1u
= 0 * u (x)
= C1 X
2
[19],
,i~2
I
(XIT)
+
K 1(xlIi),
1
c2 x
1
n-2k-1
2
V1
(3.11)
n-1
n-1
P22 = 0 * u2(x) = clx
Il 2 (xIIq|)
IV
amd
Kv
l)
n-2k+1
V2
The
K V2
+ c2
2
'
are Bessel functions-specifically, the
modified Bessel function of the first kind and Macdonald's
Their asymptotics
v > 0.
function, respectively, of order
are well-known [19]
I
K (x)
-xV/2VT(1+v),
-
1)
~1 T(
2
0+
x -
(3.12)
I(x)
-
e /vIr,
KV(x)
-
e~X7r/2x
x
Thus, near
x = 0,
n-1
2
1
(xiii)
1
_ xn-k-
liiv
-
0o
116
n-k
-v
xkhi'
(xlIl)
x2 K
1
(3.13)
n-1
(XIl)
2 1
n-k 1
1
"2
2
n-1
2 K(XII)
(up to constant factors).
L o (x-n dx)
Recalling (3.7),
c2 = 0,
only if
k-n
i = 1,2.
"2
1
u. E
On the other
hand, these solutions are exponentially increasing at
The other pair, with
infinity.
at
decrease suitably
c 1= 0,
infinity, but do not vanish to sufficiently high order
at
x = 0.
I
and
V
K
k = (n+1)/2,
(For the exceptional degrees
V
differ only by a logarithmic factor at
x =
0.)
Now we construct kernels inverting
fact
only
P1
and
P 2.
these two operators are treated so similarly that
the work for
is displayed.
P
n-1
n-1
u(x)
= x
With
Io (xr|), v(x)
1
we know the kernel is of the form
= x 2 K
1
the
117
G1 (x,x,1) = a(x)u(x)H(x-x) + P(x)v(x)H(x-x)
and satisfies
n~
P 1 G1 (x,x,n) = x 6(x-x),
which is the kernel of the identity on
form is chosen so that
x X x,
x)
L (x-n dx).
This
is a 'free' solution for
G
satisfying the boundary conditions (with respect to
of sufficiently rapid decay at
Now, as
shows
a(x)xu(x)v'(x)-u'(x)v(x)].
dx-x)[-22-nd
xn nd
2-n d ) +
-xn
P
A bit of
c.
a(x) = Y(x)v(x), P(x) = Y(x)u(x),
calculation, with
P1GS-
x = 0,
(x2
-k(n-k-1)),
the
expression
x 2-n(u(x)v'(x)-u'(x)v(x))
is simply the Sturm-Liouville Wronskian of
independent of
series as
ing
7(x)
x
x.
> 0,
P1 ,
hence
By examining, for example, its power
it is also independent of
to be the appropriate constant
p.
Choos-
118
G 1 (x,x,7) = Ck,nX
(3.14)
n-i n-1
2 ~ 2
+ I
with
placing
V
2
I)H(x-x)
(xI1)KV(xl7f|)H(x-x)].
for
Analogously, the kernel
(x 171
1 (xln|)K
is obtained by re-
P,
.
It is convenient to introduce the space
W
(3.15)
= H ([0,2];x
where
p E Z+, r E IR, HP
order
p
=
dx) + xrHp([1,w);dx)
is the usual Sobolev space of
and
{f
2
L ([O,2] ;x
f
dxa
x
-n dx)
for
p > i > j
0}
so as to formulate the
For
(3.16) Proposition:
p
is
i
an isomorphism,
k < n-1
2
W(0)
r+2
:12(2)
r
k >
or
i
=
2
1,2
with inverse represented by the kernel
Gi(x,x,r7).
Proof:
Obviously
and
it suffices to show that
P2
map
W(2)
into
Xr+2
so
119
G.
i
W(2)
r
i = 1,2
By construction it must then actually invert
is bounded.
P..
:1W(0)
r+ 2
Uniqueness is also immediate since
solutions in
has no
r.
for any
W(2)
r
P u = 0
The basic tool to prove boundedness of a kernel is
Schur's Test:
G(x,x)
The positive kernel
map on
induces a bounded
if there exist positive
L2 (dji)
measurable functions
f G(xx)q(x)dl(x)
q
and
p
such that
J G(x,x)p(x)dp(x)
ap(x);
Now, given the decomposition of
W(O)
r+2
as a sum in (3.14)
it suffices to prove boundedness on each summand.
f E W(O)
r+2'1
f2
f = f
write
xr+2 L2 ([1,);dx).
+ f
2
2
'
Bq(x)
Thus if
1 E L2 ([O,];x-n dx),
1
Accordingly
1
%1
lif1|2r,(0) _
0
Case I:
(x-r-2f
f1 (x) 2 x-n dx +
f E L ([O,1],x-n
2
( )
%1
dx) = x n/2L2 ([0,1],dx).
dx.
120
It is convenient to include
Thus let
in the kernel.
J(x,x) =
Then
x
-n/2
-n
L2
: ([0,1],x-n dx)
G
the various weight factors
~
~ ~-n/2
.
G1 (x,x)x
2 [,,
-n dx)
> H([0,1],x
is
bounded i f and only if
L 2 (dx)
f
3
I--
x
n/ 2 f
fj--
G1 (x,x)x
n/2
J(x,x)f(x)dx C H2([0,1];dx)
=
Furthermore, it even suffices to prove
is also bounded.
L2 (dx)
J
on
are similar to
J
in their asymptotics as
0.
derivatives
(These
as a factor, but never
coefficient;
and
x
6'(x-x),
x
x
0
-
contain terms with
lie in
6(x-x)
[0,1]
on
L
2
.
we may replace
by
Ja(x,x) = x
-n/2
[x
~-n/2
k'n-k-1
n-k-l1k
H(x-x)]x
x H(x-x) + x x
-k-lk-n
k-- -k-1
2""2
2
x
H(x-x) + x
x
_
2
=
x
and
multiplied by a bounded
these are obviously bounded
Now, since
J
since
x2 d 2J/dx2
xdJ/dx,
boundedness of
-
~
dx
G (x,x)x -n2f(x)dx
f x-/
x
~ ~-n
f(x)x
H(x-x).
121
For
k < 2 1
-k-
k-n
x
Ja(xx)dx =
10
p = q = 1:
test with
apply Schur's
--k-i~k-a
1
x
dx
x
2
x
2
dx
0
k-n1
n
+ -k-1
a-k
x
(1-k)
xk- 2
x
V
(k-n1 ) IxI
=-x
22(x21-k
/
1-k))
n_-k-1
nor
k-2
since neither
k--+ 1
_2
a-k- 1
2
-1,
equal
--k-1
< C
and
> 0.
2
1
J(x,x)dx <
Symmetrically
0
For
(-log
use
(the only excluded case),
k
x)/V'X.
1
1
e"1/2
Ja(xx)(-log x)/x
0
I-1
~
%0
-log
x
1
3/x
3/2
1/2
x
d
~"
dx
p = q
122
=
x
-1 (-2x
"1/2
log gx + 4x~1/2 ) I
+ (2xv- 1/2
-1/2
~
l og x + 4x
)I
and likewise for
( C(-log x)/Fx
21
Ja (x,x)(-log
On the other hand, as
dx.
x)/&
C- the only nonvanishing
x
term in
dx
G 1 (x,x)f(x)x
is,
by (3.14)
1
dx
f (x)xn
v(x)u(x)
since
v(x)u(x)f (x)x -n
=
~
dx
0
supp f C [0,1].
This decreases exponentially
provided the integral makes sense.
But
n-k-1
I so
that
Ii
~n-k-1
~ ~-n
dx
f(x)x
x
0
['J
f2 -n d x
~n-2k-2
1/2
x
%O
1/2
dx]
<
C.
123
f E x r+2L 2([1,O);dx).
Case II:
Again we combine weight factors into the kernel.
Thus
we need to prove boundedness of the composition
3
L 2(1i,o);dx)
k--
jf
x
= x
J(xx)
prove that
J
-r
J(x
(3.17) Lemma:
~~' ~r+2-n
is bounded on
If
2
,x)f(x)dx C H ([1,c],dx)
As before it suffices to
G (x,x)x
for its derivatives are
L
We require the
handled simil arly.
p E R
x e
then as
dxI x
eX
x
m,
--
-.1
1
x
x
Proof:
x) dx
~
~r+2-n
-rn~
G(x,x)x
f(x)dx
=
where
Gj(x, x)~xr+2-n f
f--
xr+2f
x--
f
eX
dxx
Apply L'Hopital's rule.
e
-x-e--
1.
124
J
Now by (3.12) we may replace
-r-1 r+1 -n
2 [e x-x H (X-X) + e XxH(x-x)].
x
x
Ja(XX)
p = q =
Apply Schur's test with
-r-1
"%A
%0
x
Ja(x,x)dx =
f1
x2
+
by
e
x
2e x
dx ( C
X
2
ex
r+1-
-x
dx
by (3.17).
%x
By symmetry,
< C.
Ja(x,x)dx
1
Finally,
then as
supp f C [1,),
G(0
-n
dx =
G (x,x)f(x)x ndx
x
> 0
u(x)v(x)f(x)x
-n
dx
%x
= u(x)
v(x)f(x)x
-n
~^. 2
dx E H ([0,1];x
-n
since the integral exists and is independent of
u(x)
dx)
x,
lies in this space.
This concludes the proof of Proposition (3.16).
and
125
We must now construct an inverse for the operator of
Set
(3.9) iii).
02
L
0
[0
L 1= L 0 + A.
Then that operator is
Lt
+
= L
1 1
0
-2(-1)kixial
P 2-
By (3.16),
2(-1) k+1j
1.
More generally, let
tA,
0 < t < 1.
(2)
r
1(0)
r+2
for any
L
is invertible.
:
0
(Strictly speaking, these spaces are
defined to contain only scalar functions.
Here of course
we extend the definition to allow vector functions by
requiring each component to lie in the space.)
each
L
maps
t
(3.18) Lemma:
Proof:
1(2)
r
A : X(2)
r
Certainly
is bounded.
into
X(0)
r+2'
*(O)
r+2
is compact.
A : xrH2 ([1,o),dx)
The inclusion
r+1 H2
Clearly
X r+2L 2
->
xrH2
([1, o),dx)
126
Theorem.
version of the Arzela-Ascoli
L2
is compact by the
On the other hand, between
^2-n
H ([0,2],x
is
A
dx)
2
L ([0,
->
2
],xn dx)
the strong limit of the same multiplication operator
acting between
^2-n
H ([e,2],x
dx) ~
2
T2
([6,2],x
e > 0,
which, for each
-n
dx)
-
-n
2
L ([e,2],x
is compact as before.
dx)
Such strong
limits are compact provided a 'uniform smallness' condition
at the boundary holds, cf.
f
varies over a bounded set in
functions
Af
Here this states that if
-n
"" 2
dx),
H ([0,2],x
the
have uniformly small norm in
L 2([0,6],x-n dx).
A.
[29].
This follows from the factor of
x
in
The proof is complete.
By the Lemma,
L
t
:
(2)
r
(0)
r+2
is an analytic family of Fredholm operators.
t:
Thus,
for any
127
index L t
(3.19)
W( 2 )
4 1(0)
r
r+2
= index L
L
To prove that
u =
Note
some
u E
1
2
r
r *
for some
0.
(3.21) Corollary:
each
is an isomorphism we need only show
L 1u = 0,
(3.20) Proposition:
r+2 = 0.
(2)
rr2
0
L
'(2)
ecr
>
()
r+2
is invertible for
r.
that from (3.16), a solution to
W(2)
r
must actually lie in every
L u = 0
*(2),
r
lying in
-o
<
r
<
o.
For
u E 1(2),
(L+A)u = 0 * L u = -Au C X(0)
r+1
r00
> u C W(2)
r-1
and the assertion follows inductively.
In essence, (3.20) is nothing but the statement that
IHn
has no
L2
harmonic forms outside the middle degrees.
However, rather than developing this connection directly,
which requires a fair bit of effort, it seems better to
give a proof more nearly consonant with our overall
niques.
There is actually an easy proof that
L
techis
128
strictly positive based on the inequality
-x2
+
ddx 2d
(n-1)
(n-2)x -
-- which is equivalent to the well-known fact that
trum of the hyperbolic Laplacian on functions
below by
degrees
(n-1) 2/4.
k =
the spec-
is bounded
Unfortunately this fails in the two
2 -+ 1.
The method which covers all cases involves an integration by parts to replace the second order system
NP
= 0
by the larger first order system-which may be solved-
=
NdW
N
0,
6
=
0.
Obviously this is just the normal operator analogue of the
fact that
L2
harmonic forms on
11-n
are both closed and
coclosed.
Recall first that the conjugated hyperbolic Laplacian
may be factored
N
(xk d
Here
6H
-k+l
k-i
H-k
= xxkAH-k
+ (xk
H-k-1i
is the coboundary operator on
by the product formula (2.10),
n
IH.
k+l d x-k
Furthermore,
129
N
= N(xk dx k+1)N(xk-1
H X-k)
+ N(xk 6H x-k-1 )N(xk+1 d x-k
(neglecting basepoints).
Now, from (1.19) and (1.20) a
calculation similar to that in Proposition (3.1)
the (conjugate by the Fourier transform of)
shows that
the normal
operators are
k
Nd
( xaiI
kw 1 )dy
A dx + ixn A (Wsdy +WJ dy Adx)
(3.22)
N W = (-1)k
_.
-
(n-k)wJ )dy
In
Using again the reduction
Idy'
-
ixL(n)(wIdy +W dyAdx).
=
7,
into three cases, just as in (3.9).
T
(I
then,
Ld
for
Only the last of these
I
interest here, and arises when
possibilities is of
(1,J):
this system splits
w =
k
oydy
d
_I
+ W dy
-k
- kw,)
+ ixa
A dx
1
l-1
I
dy
A dx
(3.23)
Ldw =
(-1 )k
and the relationship
- (n-k)j )
ix ~nleI dy
,
-
130
L1 = LdL
+ L Ld
k
(The degree
is still valid.
Ld
in
L
and
has been
suppressed, so in this last equation it is different in the
two
Ld's,
L 's.)
and also in the two
-~
w = f Idy
Let
(3.24) Lemma:
aKdyK + P dyL A dx
be
+ g dy J+-A dx
I
i =
(k-1)-form valued func-
k-form and
tions, respectively, on
and
Then using the pointwise
IR.
Euclidean scalar product on forms we have
^
^
-n
<Ndjp>x
^^
(w,N6 pAx
dx=
0
-n
dx
%0
{g(x
I:
if and only if for each
1ff~)dx
=
0.
0%
<Ndo>x
Proof:
F
n dx
df
-n
%0
<(-)kx-
-J
kk df
(-)(x
%0
kf1 )dy
--
-d kf,)P,
A dx + ix
A w4p>x
^ i
^
+ <(J,-ixt(77)p> Ix
-n
dx
dx
131
77T)
(x
(-1)k
I 1 ~, x-n)
dx 1-ngy)-kb
(-1)(- 1I d
1
4>x- n
£0
+
(-1)
k d
1-n
f iP3)dx
d-(x
0
<j,
10
^ -n
dx
N p4>x
N,
Hence
and
*1O
k d
0
(-1)
1-nf
dx(x
)
f iPI)dx.
are adjoints iff this boundary
NA
term vanishes.
Proof of Proposition (3.20):
A dx
g dy
suppose
solves
= 0
L1
I
(here
and
F
2
W EL (x
0
-n
J
dx)
(LW, Wx
%0
w E :(2)
r
and
indicate
and decreases rapidly at inf ini ty.
^( ^
=
I = (1,J)),
are fixed,
The comments following (3.21)
r.
for some
w = f dy
-n
^
^.
(L dL 6 w + LSLdoo)x
dx
0
-n
that
Then
dx
132
(IL6w|2 + ILd^I
2 )-n
dx
%0
provided
a)
d:
1-nx1n
(xd& - (n-k)g + (-1)k+1ixinf)g dx = 0
%0
and
b)
Now
L
{
[xl-nxif - kf + (-1)kixInig)?Idx = 0.
is a Fuchsian operator, the indicial roots of
which coincide with those of the indicial operator of
From (3.6) and a bit of calculation,
g
if
k < n,
f
P.
and
have convergent expansions
f = c 0 xn-k-1
+---+
log x-(c6xn-k+1
+-
--
)
(3.25)
g = d0 n-k +---+
log x
0(d
xn-k +---).
The logarithmic factors arise since the indicial roots
differ by a positive integer.
That these factors actually
occur follows from an explicit
computation of the first few
terms of the Frobenius series.
For later reference we
133
record that when
k >
2
= c0
g
= d 0 xk-1
k
log x-(c6xk +---)
+.--+
f
(3.26)
+---+
log x-(d6xk+1
+
0..)
To prove that the boundary contributions vanish, use
these expansions near
x
-
o.
x = 0
Then, for example
and the rapid decrease as
in a),
when
k
<
n-i
2
the
integrand is
d-(a n-2k+1
dx 0
while in b) it is
n-2k-1
d-(b
dx 0
By the second fundamental
terms vanish as desired;
theorem of calculus the boundary
similar reasoning holds when
k >
n+1
22
We have so far shown that an
L
solution of
N pW = 0
also satisfies
LdW = 0,
L6
= 0.
In terms of the coordinate components
f
and
g
these
134
equations are
df
(n-k)g +
-
x
kf + (-1)kixIn7g
-
= 0
(-1) k+1ixI7If
= 0.
These
equations may actually be uncoupled and solved.
Thus,
with a bit of work one sees that
x2
d2 f
d2
dx
%dx
+ (k(n-k+1)-x2 1
0
2)f
22
x2
dx
Iri2)g
= 0.
are Bessel functions, [19]:
g
and
f
Once again
+ ((k+1)(n-k) x 2
nx
2dx 2
n+1
n+l
f = cx 2I
(xln|) + c2x
(xlI|)
v2
n-2k+1
2
Ko 1 (xlii)
1
n-2k-1
2
K
(3.27)
n+1
n+1
g
=
d
x
2
Io (xlii|)
1
In order that
and
d2
f, g
+ dX2
d2x
lie in
L2 (x-n dx)
must vanish, as in (3.13).
near zero,
c2
But the solutions thus
obtained increase exponentially, which is a contradiction.
This proves Proposition
(3.20).
'
135
Notice that the solutions
d2 = 0
with
above are the factors of the logarithmic
Proof of (3.10):
G(xx,i)
terms in
By virtue of (3.16) and (3.21) there is a
for which
Np(xaxix1)G(x,x,n) =
and such that, for each
T,
x
n
6(x-x)
the corresponding operator
L 2 (-n
G : L2 (X-n dx) -
is bounded.
c 2=
Finally we may complete the
(3.25) and (3.26).
kernel
g
and
f
dx)
As of yet we may not assert this uniformly in
Of course this uniformity is
-q, which is what we require.
obvious for the
'uncoupled part' of
G
(3.14);
for the
'coupled part' we know little save that it is an analytic
solution of
for
x > x.
invariance of
Thus let
1.
of the form (3.25) for
Ng,
x
<
x
and (3.26)
For this extra information we use the dilation
N .
G(x,x,i)
be the solution above when
It is obviously smooth in
N (xO ,ixlnriQ)G
obtained by replacing
x, x
=
by
n.
The solution
I
x1 6(x-x)
xI-fl, x-rjl
G
must be
|I|
to
=
136
where
rI/|77|.
=
T1
|)
|,1 /1
G(x,x,i) = |,|1-nG(xl |,
(3.28)
The known part of the kernel (3.14) does
indeed satisfy this.
The
boundedness of
L2
as in (3.10) is now quite
C
Define the transformation
easy.
D
:f(x)
=
f (x)
-4
a
1-n
2
a E
f(ax),
This correspondence is readily seen to be an
isometry.
(3.29)
f
where
C
Set
|
I
.
-n dx)
into the inequality
fa
"~F
~'~-n
G(x,x,)f(x)x
dx
x
-nn2
dx
C
replacing
fa
J If 2xn
PO n-2
ri E Sn.
may be assumed independent of
right hand side, with
a.
L2
+
dx
The new
is independent of
f,
On the other side, a change of variables
x
I--
ax
inner integral on the left side
results in a new
n-1
G(x,x,r)f (x)x
-n
~2
dx =
Applying the dilation
a
n-1
G(a
D _1
-1
x,a
a
G(x,a
-n
1~
xn)f(x)x
transforms this
~ ~-n
-1~ ^
dx
x,n)f(x)x
to
dx.
137
L2 (x-n dx)
which retains the same
|9|~
with
a =
the definition (3.28) implies
1,
(3.30)
f
C
where
Thus,
norm.
G(x,x,)f(x)x
is independent of
allowed to depend on
sides of
p
dx
dx < C
x
n E Rn-1\{}.
f(x)| x
If
dx
is now
f
also, then by integrating both
(3.30) with respect to
Ti
we obtain the required
estimate of (3.10).
The kernel inverting the partial differential operator
is
G(x,y,x,y) = cn
(3.31)
It
is
bounded on
e
L (x-n dx dy),
(x,x,n)d
hence is the unique Green
funct ion for the constant curvature Laplacian.
some
.
It also has
isometry invariance, namely that of translation and
di lat ion:
G(x,y+v,x,y+v) = G(xy,x,y)
G(az,az) = G(zz),
z = (x,y),
z = (x,y)
the latter following from a change of variables in (3.31).
138
C.
Structure of the Kernel
of
G
We now discuss the structure of the kernel
(3.31), which is of the type discussed in the last chapter.
-2,a
G(z,z)-r E KO0
(3.32) Theorem:
T
n
1/2
n
0
T(If+
O +
0
1
@
0 k
Ak)
where
n-k-1
T=
Ln-k
ar =
k
k
n-k+1
n-kJ
n-1
2
k]
k,
k-1
t
T = a
tn-i
,
for
> -,
k
and
-r
2
k+l
is the half-density of (2.28).
First we prove
As always, the proof is rather involved.
uniform symbol estimates in the regions
x, x < x.
x
This implies the requisite regularity in the interior and,
together with properties of the indicial operator, yields
the correct estimates at the boundary.
We commence by examining the 'uncoupled' part of the
kernel:
(3.33) Lemma:
The
kernels
G1
furnish inverses for the operators
the estimates,
for
|a7l
1,
G2
and
P
of (3.14), which
and
P
satisfy
139
x,
|B x B X B.a'G.(x,
1
7)|
-
.
.
C
,1,j
a
uniformly in
6
> 0,
( x < x <
6
i = 1,2.
-1
~~
-1
,
6
6
for any
( x < x < 6
In particular, neglecting the
dependence, they are symbols of order
7
a classical expansion as
uniformly in
-1,
G.
has
of lowest homogeneity
0
-
(x,x)
Finally, each
(xx) plane.
these regions of the
-x|N|Q
i+ji e-
1-|a|+
(1+|T?|)
|T |0.
Proof:
Consider only
in the region
G1
x
x;
the other
Then
cases are quite similar.
n-1
G)
G 1 (x,x,ri) = (xx)
where
assume.
v = (n-2k-1)/2
Both
I 1v
for
K
and
V
2
x
IV(xI1I)K
k < (n-1)/2,
V(xIr1I)
which we also
have asymptotic expansions for
large values of their arguments, the first terms of which
All derivatives of
are given in (3.12).
I , K
also have
expansions which must equal the ones obtained by
I , K
differentiating those for
.
All these expansions
are of step size one in descending powers of
xjri|,
hence it suffices
Thus replace
G1
n-2
(xx)
or
to argue only for the first terms.
by
~) 2
xIqI|
~
e
-(x-x)|n|
| Tl
-1
140
IH
as
O,
-
The estimates of the Lemma are
e < x < x.
the only point to be noted is that
quite easy to verify;
aae-(xx)I1
-(x-x)I|l
-
I3lIla|
1(31
where each
p3
1#|
degree
|
||
-{a term homogeneous in
(x-x)
}.
of
n
Since
-(Xx) I
I-|aI(-x) I
C
(1+|
|)
|a|
this too has the correct growth.
As
for
the behaviour as
71
0,
-
both
I
and
K
V
have convergent Frobenius series (the one for
K
contain-
ing logarithms) so that from (3.13)
n-k-1~k
GJ(x,x,n) = c 0x
(with an additional
x
|n |
+
|n|(cl x
log
171
n-kk
x +c ix
n-k-1~k+1
) +--x
term when
k = 1 -
1).
This proves all claims.
To extend this to the 'coupled' part of the kernel
G
c
,
which inverts
better.
L ,
1'
we must understand
its structure
This necessitates a closer examination of the
solutions to
L 1 u = 0.
Altogether, this nullspace is four
dimensional and two independent elements are identified in
141
these, both components are exponentially
In one of
(3.27).
v ;
increasing-call this solution
the other solution
has both components exponentially decreasing.
record that when
|aI=
Let us
1
n/2 x
n-k+1
u1 =
u
u1
[bxn-k +,,,
Kxn/2 x
e
x
b~x
+--
b
c~x
---
c x
0
X
+---
(3.34)
k
d0xk+1 +000'
1
n/2 e -x +--
-
0
dxn/2 e-x+..i
"1
Other solutions are not known explicitly, but fortunately their asymptotics may be derived as follows.
A
Frobenius series calculation posits two additional solu-
|7|
tions of the form (also when
= 1)
-
n-k-1
u - - n-k
x
Lk
(3.35)
c0
v2
In (3.34),
bO, b,
+Au
1 log
dO
x
++ Bvy1
k-1
x
(3.35) none of the coefficients
bO, B, c 0 , c.
c 0 , d O,
log x
+-
d6,
Eo
a 0 , a0 , a 0 , A,
vanish.
142
On the other hand, the singularity of
One may find formal
is not regular.
L
at
x = O
solutions which are
x
(possibly divergent) series in descending powers of
Thus a straightforward
multiplied by an exponential.
computation produces series
x
2
+
a 1x+a
-_1
a_
x
-1
+
---
x
eI
x 1x
+
:0 + ---
n
2
x
x
-1-
+ a-2 x
1+
+00
---
+---
x
+---.
x
X
+---W
are different of course in each of these solu-
a., 11P.
The
x
a_ 1x
2
#x
e
n-_
a 1x +---
-_1
a x
x
Once
tions, and none of the leading coefficients vanish.
these formal solutions have been found, the theory
developed in Chapter 5 of [8]
the coefficients of
L
(for which the analyticity of
is essential) implies the exis-
tence of actual solutions obeying each of these asymptotics
as
x
-p
.
Indeed both
and
u
v1
of (3.34) fit this
pattern-their series being the ones commencing in
Thus for some choice of
a.,
b,
c.,
d
in (3.35)
x
1
143
ax2
u2
x +---
1-2
ex +--b 6x 2
,
a6,
b6
,
c',
d6
0
(3.36)
2 -x
c 6x 2e
v2
n_ 2
.dx2
The kernel
-x+
o
0
-x +
C (x,x,i),
||
=
has a form similar
1,
to (3.14):
Gc(x,x,
U, V
where
) = U(x)A(x)H (x-x) + V(x)B(x)H(x-x)
are two by two matrices, the columns of which
are linear combinations of
and
A, B
u , u 2,
respectively
vi,
v2'
are matrices chosen so that
~
L LOn
c = x 6(x-x)-I.
But
L
is self-adjoint with respect to
formally and by (3.24) as an operator),
too:
G (x,x) = G (x,x)
Hence
A(x) = V(x)
,
B(x) = U(x)*
and
xn
dx
so that
(both
Gc
is
144
Gc = U(x)V(x) H(x-x) + V(x)U(x)H(x-x).
(3.37)
The only obstruction to completing our task is the explicit
identification of
u
,
v
.
the columns of
in terms of the
U, V
L Gc
However, the condition that
n6
=
entails
that
U'(x)V(x)
1 n-2
V'(x)U(x)
-
x
-
I.
Of necessity then, from (3.34) and (3.36)
(3.38)
U(x) =
for some
a, b, c, d,
bu]
au 2
,
V(x) =
[cv 1
dv 2 ]
none of which may vanish.
Since
G (x,x, n) = |
a verbatim
repeat of
|8 i8 3BaG (x , x.
x -n c
x
uniformly in
the proof of (3.33) shows that
more, each term of
x
x
G_
6
,
6
(
nI
> 1,
l)1-|aI +i+j e-x-x||nl
Sei-,j a(l+
7)X
6
the estimates, for
satisfies
G
(3.39) Lemma:
n nG,(xIn ,xIn|In/|In|)
x
<
x
<
F6
Further-
has a classical expansion (in
fact
145
a convergent series) as
> 0,
-
uniform in
x, x < C
I|0
and of lowest homogeneity
We now have sufficient information to study the kernel
G(z,z)
of (3.31).
when
C
is
G(z,z)
(3.40) Lemma:
z X z
and
x, x > 0.
The singularity along the diagonal is classical conormal,
with an expansion involving (quasi-) homogeneous terms of
z
increasing degree in
the first such term, up to a
z;
~ -n
.
1z-z|
constant factor, is
Proof:
-
By virtue of the uniform symbol estimates of the
two previous lemmas, the usual arguments imply that
is
CO
G(z,z)
The singularity along the
in the stated region.
diagonal is the normal one for the inverse of an elliptic
From our vantage point,
differential operator.
symbolic behaviour of
that when
x = x
powers of
y
homogeneity
behaves
-
away from
G(x,x,n)
G(z,z)
77 = 0
shows
has an appropriate expansion in
the most singular of which has degree of
y,
-(n-1)
-
(-1)
= 2-n.
That
thusly in all directions along
invoking the rotation invariance of
G.
might consider the Fourier transform of
with respect to
though, the
t
identically one near
(where
t = 0
p
the singularity
{z = z}
follows by
Alternately one
G(x,x+t,ri)p(t)
is a cutoff function
and compactly supported in,
146
(-
say,
,
(r
-2
of degree
(T,77)
is symbolic in
this
It is straightforward to show that
)).
being dual to
t)
and polyhomogeneous.
(3.41) Proposition:
x > 0},
{x = 0, x
>
{x = x = 0, y s y}.
and
0}
{x = 0,
is conormal on
G(z,z)
Its
components each have expansions on either of these codimension one faces, with leading terms determined by the two-bytwo matrices
a
{x = 0}
for
and
{x = 0}
for
r
of
(3.32).
Proof:
We prove all assertions only for the inverse
transform of the kernel
k
< n-.
kernel,
of (3.14), and assuming
G
The methods apply equally well
the full
to
though the notation becomes more intricate.
Consider first the region near
x = 0, x > e.
From
(3.33) the integral
(xa) aaap
x
f ei(Yy)-G (xxI)
dr
x Y y
n-1
e ( -Y . (xax)e jA
=
#
_
.,i ) a
) 2
VII
|
I|)K
V(xln|)dn
x
converges absolutely for any
1
~%0 -1
lyl,
e,
IyI
a, P,
and uniformly in
indeed it decreases
147
Furthermore, it obviously has the
exponentially there.
same regularity as
{x = 0, x > 0}
independently of these
This yields the conormality along
indices.
various
> 0
x
Similar arguments
was arbitrary.
e
since
{x = 0, x > 0}.
apply to
For the corner, where both
x
x,
|y-y|
convenient to observe that we may assume
the dilation invariance
G(az,az) = G(z,z)
x
Now, assuming only that
ply both sides of the equation above by
place
,~
x
by
(
since
{x = x =
x,
(y-y)
we may multiand re-
to get
xO~
x
e i(Y-Y)
> e,
allows for all
computations to be performed well away from
y - y = 0}.
is
it
0,
-
18
)
n-1
(x
2 1 (x1n|)K (x(|I |)
p
) ((in)a
x
as
the new right hand side.
rj
is permissible since
(The integration by parts in
the whole expression is to be
considered as an oscillatory integral.)
7
so that
-1
integral over
over
I|
1
+
e
+ j +
|lii >
1
d
|a|
+
1|
-
Now, by choosing
|7| <
-n,
the
is convergent, whereas the integral
can only produce a
C
contribution.
Furthermore, the regularity of this expression is stable as
148
x,
x
> 0.
-
y - y X 0,
Since
the conormality along the
corner is now established.
As for the various expansions, notice that when
> 0,
x
x >
6,
in
the first factor
n-1
S(x
(y-y)-n d
)K(xII)(xx) 2
may be replaced by its Frobenius series .
x
< 1e,
2'
each term of
this
(convergent)
grand decreases exponentially with
77,
Again, provided
sum in the intehence we have a
series
0
n-1+j~
c.(x,y-y)
x
j =0
furnishing the asymptotic behaviour as
obtain this expansion uniformly down to
x
To
0.
-
x = 0,
but still
of course, we may instead replace
assuming
x < x
IV(xJ77|)
by the first
terms of its Frobenius series
N
plus an exponentially increasing remainder.
Each term in
the resulting integrand exhibits symbolic behaviour, and
the first
N
summands still decrease exponentially in
These provide the first
sion;
N
TI.
terms of the asymptotic expan-
the remainder term is indeed lower order as may be
seen by multiplying it by
(y-y)
for
|-r
sufficiently
149
large and integrating by parts so that the resulting
integral converges.
Applying the arguments of the last paragraph to the
full kernel yields the correct expansions in all components, for these all are derived from (3.34), (3.35).
Proof of Theorem (3.32):
The lift of
has the correct behaviour along
n
n
G(z,z)
to
IR+
R+
and
B,
as
T,
At
Indeed the only point
demonstrated in (3.40) and (3.41).
not yet discussed is the behaviour near the front face
However
F.
G
this is trivial since the dilation invariance of
is equivalent to the independence of the lifted kernel on
R.
Hence not only is
but the expansions at
G
smooth on
AtO, T, B
F,
except at
all continue uniformly
F.
down to
Let us now untangle
the various definitions and iden-
tify both the operator induced by the Laplacian on
T1/2
and its inverse.
G
for which
First recall
then is
tion
G
that the operator
P,
-k
P =_kx kAx-
the kernel induced by the standard Green func-
k k
E F Hom(A ,A ).
In fact,
since
A
k
and
canonically identified over the interior we may let
act on
0Ak 0
is an inverse, satisfies
A~-k9_
(PG) <=
k ) = x_-kp
A(x-
G
BAt0'
0Ak:
o k
A
GA
are
150
~k ~ ~-k
(i(x,y_)
k
x
GA(x,y,x,y)x
f
= x
-k
~
~
~ ~ ~ n
dx dy
W(xy)x
~
G(x,y,x,y) (x,y)x
-n
~
dx dy
i.e.
GA(xyx.y) = x
-k
~ ~ ~k
G(x,y,x,y)x
(3.42)
AGA = xn 6(x-x)6(y-y).I.
To incorporate half-densities, use
(2.28) and define
A(x
-k
W-
) = x
-k
P
(PW)-
4
The corresponding kernel
-which
1/2
F0
P = x
k+n/2
(3.43)
G
Ax
-k-n/2
A
r Hom(A
E T
n/ 2 Axn/2 '
GA)
(x2
-() =
is the kernel of
From
-
A
on
k)
n/2
x
n/2
Px
o k
-n/2
1/2
®1T0
must satisfy
= x 6(x-x)6(y-y)-I
the identity with respect to
(3.42) one may check that
GA = x
-r of
and
by
It is of course the operator induced by
A-(G
p
~-n/2
GA x
=x -k+n/2 Gx~k-n/2
151
The kernel
transfers to
G
Bn
under all hyperbolic isometries there.
and is invariant
The preceding dis-
cussion may be carried through on the ball,
x
difference being that
is
the only
must be replaced by
the Euclidean coordinate on
B n).
1 -
|w1 2
The appropriate
kernel inverting the Laplacian acting on, 0Ak Bn)
is
1/2 n
(B )i
F0
(3.44)
GA = (1-Iw|
Suppose now that
~ 2 -n/2
~
2 n/2
.
)G,(w,w)(1-Iw
)
is any two-by-two matrix-the compo-
a
nents of which regulate decay rates of the tangential and
normal components of a section of
-and
Hom(Ak ,oAk)
define the new matrix
a + q = (a. .i+q),
i, j =
t,
n,
q E
R.
13]
In summary, we have proved
(3.45) Corollary:
-2,a-k+n/2,r+k-n/2 n o k 1/2
GA(ww)*t E KO
(B
A O
a, r
as in (3.32).
A(G A-)
Furthermore
= (1-|wi 2 )n6(w-w)-r*I.
(w
152
D.
Refined Mapping Properties
In this section we further refine the mapping properTwo results are
ties of the Laplacian and its inverse.
proved;
the first, a straightforward extension of Proposi-
tion (2.55), is valid more generally, but its statement and
Let us
proof are simpler in the present circumstances.
define a form valued version of the space (2.54).
(M,oAk),
SE Aa
a =
phg
(3.46)
+ Wtn
=
a.
k
1,
oAk _ oAkt
The decomposition
near
0012k
E o k
1
C A phg~(M)
(a t ,an )
0An
i
=
t,
near
aM
n.
is defined
aM
following the discussion in Chapter 2, Section C, and the
spaces above are invariant under coordinate changes only
when
|at-anI
1.
(3.47) Proposition:
(n-k-ln-k)
and
if
f E dg
k < (n-l)/2,
a + q = (a +q),
is a unique
Let
a
w E A
phg
q E Z+,
nok
)
n
Ao =
(Bn oAk),
a = (k,k-1)
where
if
in either case.
such that
f.
a =
k > (n+l)/2
Then there
153
Proof:
and
e
Let
p
Bn,
be a defining function for
f
and normal components of
0,
Now, the tangential
n-1.
3Bn
the variable in
p
have expansions
i
=
t,
n
j=0
a +q+j
(log
tj
=0
Inasmuch as
near the boundary.
A -
for any
a,
p)
104
a+1
: Aa
IA
w.
we proceed by first choosing
,
i =
t,n
so that
=
I U.
A 10
f.
10
a. +q
o.
10
=
c:
e
1
()p
(log
p)
e=0
Then, assuming
so that
w im
have been chosen,
m < j,
pick
j
154
I
A
.
).
=f.
ij
ij
N..
a.+q+j
(J)jc =
UR(O)p
(log
1
p)
e=0
are quasihomogeneous terms of the correct degree
f:.
The
which depend only on
m
<
j.
All
methods.
f
and
(A-IA)Wtm' (A-A)"nm'
these equations are solvable by finite series
Note only that uniqueness holds at each step
since each exponent
a
+ q + j
is greater than the
largest of the indicial roots of
duced by
I
ok
Ak.
on
I
--the operator in-
For the same reason the highest
power of the logarithm at each stage is never increased.
L co-~-Ethen
0.. If
If
Aw0
-
f ~ g EC (B n ) @ o Ak .
(2.55) and (3.45) together guarantee the existence of a
w
such that
AW 1
Then
A(o 0 -
1
) = f,
=
g.
as desired.
The other solvability result is of the same type as
(3.47), but with somewhat more singular right hand sides.
Recall from (2.17) that it is useful to think of the
quarter-sphere
point
w0
Sn
as the ball
in its boundary.
Bn
blown up around a
Analogous to the spaces (3.46)
155
are
oAk
a(Bn
A
B
OAk
n 5AAphg
d
Aphg (S++
(3.48)
(348
OAk
An element is a section of
n).
= P
a = (at an)'
which is conormal, with classical expansion, at the top and
bottom edges of the quarter-sphere.
a, 3
The two-vectors
determine the most singular terms in these expansions;
in (3.46) we require
|a t
analogous spaces over
Bn
aBn
-
ing function
|w-w
r(w) =
1.
or
The
are those with expansions at
In particular, using the defin-
{w0 }.
and at
{w0 }
an ' IIt PnI = 0
as
0
|,
the tangential component of
an element has an expansion
SN.
(3.49)
0ier t
ft
0
r
(log r)
i=0 e=0
and similarly for the normal component.
functions of
space to
Bn
(3.50) Theorem:
w0 .
*
here are
the sphere in the half tangent
8 E Sn-1,
at
The
We may now state the
For any
a+q,p
f E dphg
there is always a solution
w E dajh(Bn oAk
(B
n o k
, A )
q E Z
+
156
Aw = f,
to
a
are integers and
3
(3.47).
2-vector of
the
is
so long as the entries of
The typically arduous proof involves removing the
singularity at
additional
This step is essentially local,
to (3.47).
Bn
R +0
n
by
A
k
and regard
N
than
rather
corresponding to
w0
with
by
A
replace
so as to reduce the problem
w0
ok
Ak.
and
w
so we replace
0.
In addition,
f
as sections of
The principle tool is the Mellin
+
r E IR
has polar coordinates
n-1
E S+
,
.
z E R+
where
r,
transform, taken in the radial variable
This
the
is
transform
u(r,8)
r C-1 u(r,O) dr
uM(co) =
-
(3.51)
u(r,8)
In general,
space
uM
Re C < a,
= (27r)-l
r uM(C,6) dC.
is only defined (and analytic) in a halfand so the integral defining the inverse
transform is taken over a line
Re C = c,
c < a
fixed.
(For all facts quoted here regarding the transform, see
a
The constant
closely
related to the decay of
0
for large values of
at
r =0,
then
|Im
C| -
c,
uM
Re C
uM
which limits the domain of
[23].)
r
u
as
r
> 0.
If
is
u =
and vanishes at a definite rate
is rapidly decreasing as
fixed.
157
The most important property of
uM
for us is that if
then
large,
continues meromorphically to the whole
uM
+ Z7+
C E P
C-plane, with poles only at
Furthermore, the
Conversely, if
+ 1.
Re C = constant, then
rapidly on each line
u(r,0) =
fRe
rcv([,8)
C=c
dC
has an expansion as in (3.49) with the exponent
s
rs (log
r)
occur for which
is the
with real poles and decreases
C
is meromorphic in
v(C,O)
N
is
PO + i
order of the pole at
0
(if
most singular exponent in the expansion).
term
r
has an expansion as in (3.49), and vanishes for
u(r,O)
the location of a po le of
v.
in
s
Only those
and the order of the pole at
Re s > c,
is one greater than the highest power of the
= s
rithm multiplying
each
loga-
s
r .
L
Define the operator
(N
u)M
by
=
L
uM
Clearly
LC
Since
N
=
|z~ INgIz|
.
is invariant under the homothety
this conjugate is also.
z
) az,
Therefore it is differential only
158
in the
8
variable;
in
C
it is purely algebraic.
Sn-i
Furthermore, as an operator on
|9|
= 1,
boundary
9
2
Diff 0
it is in
n > O}
= 0
='...')
--with respect to the
A short calculation shows
of course.
that its indicial operator is one we know:
I(L )
(3.52)
=
IP.
L ,
The present goal is to invert
equation
L
M
M.
C.
every value of
Naturally,
I ts
inverse
is not invertible for
L
GC
so as to solve the
is meromorphic, and
its poles contribute to the expansion of
I l
C (IR) equal one for
formally, the inverse for
1/2,
dilation by
t
G(r p(r)f(@))}
the inverse of
oAk,
U1 (Sn-1
and define, at least
LC
G f = lim M {r
t4O
f E
N,
M
and
Substituting the integral for
G
formula we get
~% ~% ~
(3.53)
G C f(e) =
n
Sn-
where
p(r) E
Let
wM.
HC (9,)f(9)(0)n)
(n-1~
l
d8
the
in this
159
(3.54)
Hf (0,0) =
C
For any value
G( ,To)T
C dr
~
TO
O
.
n
at which this last integral con-
then serves as an actual inverse for
G
verges,
J
fact, but for an insignificant difference,
a
VO-pseudodifferential operator.
values of
C
as
7
-
0
and
T
all other components exhibit
n-i
then
Suppose k < 2 ,
of decay.
H
-(n-k-1)
(n-k-1) < Re C < n-k-1.
-
extends meromorphically to the whole
+
C-plane with poles at
± C E k-i + Z+if
order two;
=
0
-
Hence (3.54) converges when
(3.55) Lemma:
G(8,TG)
G,
'more favourable' rates
(,O
Gtt tn-k-1,
ev
and
the extreme case is
However,
G tt
-G
8 A 0
decays at various rates
G
W.
-
is actually
G
Fix
where (3.54) converges.
The components of
0,8).
In
Let us now examine the
notice that by the dilation invariance of
G(r
L .
t
E n-k-i + Z
are kernels of finite rank.
k
<
n-i
n2 1
and
These poles are of at most
k > n.
the residues at
if
(
=
N
of
H
and
(C-N)HC
160
Proof :
Choose a partition of unity
1 = 0 1 (T)
*3 ( T)
where
is supported in
,
/2,
$ 3 (t)
1
(T- 1 ).
'0
be defined by the int egral
H
1
-H
H
Write
in (1/4,4).
*2
and
by le tting
=
+
0 2 (r)
+ H
(3.54)
2
+
3
+ H
with
inserted into the integrand.
di (T)
H
2
discu ss
?
entire in
is
which
8 X 0,
for
singularity we
By a change of variab les
later.
H (0,0)
C d
$*(T)G(8
=
n
=
H (0,0)
J
-0
1 (T)G(T9,8)T
n
A = +
r+,
by
r
Replace
of
r+
The family
sions.
each of these expres-
in
,
homogeneous distributions is
to
well known to extend meromorphically
Now,
sions as
T
r Gm1 ,
rm log
0.
-
These expansions
where the
8
or
Gme
8
by their expan-
G(T0,8)
and
G(8,T9)
replace
is
X = -N
the residue at
X E -Z+;
poles at
with simple
C
involve terms
are kernels in
for
in
respectively.
The expansions commence with
,
any
T k-,
or
X,
term of
m,
and
k >
TX
n.
Inasmuch
log T = dX(r),
as
the
T X
which
(0,0)
or
G(r8,O)
are polynomial
rm Gmo
G(O,r8),
n-k-i
nk,
r=
+m
k <
for
integrals of each
the expansions should be thought of as a pairing of
161
r+M
or
T
+M log r
a kernel of finite rank since it
Finally, since
0.
for example,
will have a pole resulting
3
H ,
1
H .
and not in
Since
L G
(
=
n-k-l+i
log T
H
is
occurs only
The proof is complete.
n-1
ni) 6(0-)
=
and
G(6,TO),
Similarly, the pole at
regular.
n-k-l+i
n
(0,0),
in the expansion for
)
(9,
G
C =
Specifically, at
n-k+i
rn -ki
only from the two terms
in
H
or
only one or two
i > 0,
terms will actually be singular.
-(n-k-l+i),
9
is polynomial in
(0) = 0,
*1
the coefficient here is
(T);
with
in the region where
must continue to hold in the extended
(3.54) converges, it
domain, away from the poles, by the uniqueness of analytic
The kernel
continuation.
K-2,09,-1
strate.
First of all,
a few extra
(0,0).
=
S n-
over
r
essentially lies in
H
S n-,
X
as we shall now demon-
is the pushforward of
H
factors) under the submersion
{9 = 9, r =
is conormal along
along
(with
(9,0,)
-
The restriction of this map to the submanifold
is still a submersion onto its image, and
0}
G
{0 = 9}.
Next,
1},
has
H
so
H
{0
G(6,rO)
is conormal
the proper singularities
along the top and bottom faces, as may be ascertained by
setting the expansion for
one way in which
H
G
into (3.54).
differs from the
V
Finally, the
kernels we have
already studied is that it has a logarithmic singularity at
the
front face.
The easiest way to check this is to
162
GA
observe that the kernel
(3.42) is actually a
of
function of the hyperbolic distance between
z
and
z.
This simplifies the calculations necessary to display this
singular behaviour, which are then straightforward and much
the same as those in [20], hence will be omitted.
Despite this one difference,
H
still enjoys all the
continuity properties discussed in Chapter 2.
From this,
and using (3.52), the proof of (3.47) may be repeated to
show that
L
has a unique solution
meromorphically on
f E
=
(
n-1 ,Ak
+
a+q
phg
k
n-i
a
,A )
w E dphg S+
depending
Its poles are precisely those of
C.
Hr.
Proof of (3.50):
a+q,f no0 k
)
f E sdq (BA n
phg
Take
to the upper half-space, with
by
xk
so as
components
w0 = 0.
meromorphic;
r.
its poles are at
The new form,
and it
wM(c e) = G fM(c,9).
It
0.
By
are integers,
WM
0.
Let
too is meromorphic in
C,
and lies in
virtue of the hypothesis
is
fM(c,e),
C E min(Pt ,Pn ) + Z+,
d phgC in
in
Now, express its
in polar coordinates, and take their Mellin
transform with respect to
is still
Also, multiply it
k-form.
to get an ordinary
and transfer it
that the
P3
Aaphg
in
163
+
has poles at
(
E n-k-1 + Z+
(or
k-1
+ Z+)
and
min(13 t'1n) + Z4.
Def ine
f
Wi(rO) = (2 r)i
where
the
integral
is over
No
(3.47) to find
w2
r = 0,
o-
92is
1/2.
Then
A
and upon being
(Bn,Ak).
such that
N
-
- f =g
lies in
Bn,
dC
Re C = min(Pt' n)
vanishes to infinite order at
transferred back to
r&)M(,9O)
1
W=
gs
the desired solution:
Ng(w 1 -& 2 )=f
Now apply
164
Chapter 4:
A.
Hodge Cohomology
The Parametrix Construction
We now have all the analytic tools at our disposal
necessary to construct a suitable parametrix for the
k-forms for a conformally compact metric
Laplacian on
(1.2).
As described in section D of Chapter 2, the conEach of these involve
struction proceeds in three stages.
the normal
--the symbol,
A
inverting model operators for
operator, and the indicial operator at the diagonal,
front face, and the top face of
M x0 M,
the
respectively.
The first step uses the symbol calculus of Theorem
(2.41), but is otherwise identical to the usual microlocal
procedure.
Thus we seek an operator
E
EP
2
(M;oAk
1/2
the kernel of which is supported quite near
M
M,
x0
and of course such that
(4.1)
First choose
1,
in
AL
E,0 E
-00
Q0 '
Ag E 0
-2
0
Q0
satisfying
which is possible by the
VO
0
o
o
a 2 (Ag) 0-
ellipticity of
2
(EO00 )
A .
165
Then
A E
g
0
= -Q
-I
,0
E q1.
0
0,1
Similarly, by induction, we may find
EO'j
0
such
that
A E
j = 1,2,---.
of each
-
= -Q
Q
E
0
Obviously it may be assumed that the kernel
E 0 'j,
and thus
QO,j,
is supported near
At0'
Now take
E0~
EOj
and the first step is complete.
Next we seek an operator
(4.2)
A E1
where the kernel of
front face
F.
Q
Although
E1
= QO -
for which
Q1
vanishes to infinite order on the
x(Q 0 )
is supported well away
from all other boundaries, it is not possible to choose
and
(4.3)
Q
with this property.
E E *O
In fact, we will find that
1 E R 00
E
166
where
a' = a-k+n/2, T'
of (3.32).
T+k-n/2,
=
and
a
r
the matrices
We note that it is precisely at this step where
the proof breaks down in the middle degrees
k = n/2,
(n 1)/2.
process.
is obtained by an iterative
E
As in the first step,
is chosen to satisfy
E1,0
The first term
N p(A )Np(E1,0)
for each
p E OM.
In fact, by (3.47) there is a function
N (E l0) E d
p
= NP (QO)
10
phg
T'
(F ) @ T1/2
0
solving this equation, and then, by the exactness of
sequence in (2.45) there is an
A E
g
It is crucial
1
,0
with this function as
E1,0
Thus
its normal operator.
-
= -Q'
Q
Q
0
Q
E R- 0
to observe now that
Q
slightly better:
Q'
1,1
,''T
0
,1
the
E R 0-"a'+1,r'
0
is actually
167
Qj
=
*i
Q'', 1
xQ' 1 1,
E q1cI
0
,T
The reason for this gain is that the normal operator of
A
g
annihilates the top order term of the asymptotic expansion
E1,0
for
Np(E
1
Indeed, this is so because
at the top face.
is given by applying the Green function for this
,0 )
(which is supported in the
Np(QO)
normal operator to
interior), and now (3.47) implies that
E1,0
has the
correct top order term.
terms in the series for
To get further
assume that,
instead of a Taylor series in
let us
El,
R,
we let
so that
g
Define
Q
1, j
=
kQ"
x
x 1, j
A
.;
E
g 1,j
1
L'-
g
1,j*
then the inductive step is
-
Q
1,j
=-xQ"
1,3+1
where
1, j E
-Q
0 , a'+1,
0
'-j
,
Q1,j
00
',
T'-j+
to solve
168
j = 0.
This has already been accomplished for
values of
N
p
(A
)N
g' p
j,
(
For greater
we solve this equation by first solving
N (Q
= N(Q
p
1,j
1,j
1/2
k
a'.+1,T'-j
E d-j)
C
T'
(F p ; AkTr 0
0
).
By Theorem (3.50) an appropriate solution may indeed be
found;
its extension into the interior of
M
x0
M
then
satisfies
A E
. - Q 1,j.= -RQ ,+.
g 1,j
0
1,j+l
However, by reasoning as above-and now the point to note
is that by hypothesis the top order term in the expansion
for
Q
.
1,j
is of type
a'
+ 1
and so doesn't interfere
with the argument-we actually have
A
E
l,j+l
as desired.
=-xQ
.-
g 1,1,
Finally, since
xE 1,j.E
1,+
0
E
9-0,a
0
C
E
,T
,T
j
169
and
L X'j
1
This
,1 j
0 0,
'
T
solves (4.2).
E
The final step of this construction is to find
R
-
a
,r
such
that
A E 2 ~
(4.3)
This is quite easy:
of
(3.47).
1
Q2
.
R
one merely solves for each term in the
using the indicial operator, cf.
T
expansion at
E2 E
the proof
This completes the construction.
Set
E = E1 + E2 + E3'
(4.4) Theorem:
for
E E %k02',T'
)1/2is a parametrix
(M;oA
A
g
A E =I -Q,
g
(4.5) Corollary:
compact, and thus
O - 000T'
Q ER~ 00
Q : L2 (
TAk )
o
k
1/2
(0M0A
L
(
12
O )
is
170
A
B.
gOO
>1L2 ( A k1/2)
L2 ( 0A k1/)
is Fredholm.
Proof of the Main Theorem
It
follows from Corollary (4.5) that the harmonic
space
Ak =
2k dg)
{w E L
: A o = 0}
g
It remains for us to identify it in
is finite dimensional.
terms of the topological cohomology of
First observe
M.
that the adjoint of the remainder
Q
ER'
k
,o"(M; (A®
oU
0
x(Q )
To double check the order of
1/2
0
.
on the top face,
notice that
E
A
Ik
*
g
=
I
-
implies
(4.6)
Thus
that
W E
the order
a'
Q*(Weii)
= W-P.
is the only one possible which ensures
171
I(A
= 0
)
modulo terms of one order higher.
Remove the density factor and regard
of the ordinary form bundle.
2
as a section
w
From (2.55) and (4.6)
2
phg
,Ak
(4.7)
k < n-i
(n-2k-l,n-2k)
a =
k > n+1
2
(0-1)
L2
These explicit asymptotics of
harmonic forms, together
with various consequences of the existence of the parametrix, allow us to identify
spaces of
Xk
with the deRham cohomology
M.
Let us first briefly review the deRham theory on a
compact manifold with boundary.
There are, of course, two
flavours of cohomology to consider, the relative and the
absolute, and these are dual.
M
Inasmuch as
and its
interior share the same topological cohomology, it is
reasonable to let the absolute cohomology be defined by
Hk (M) = {WE
2k (M)
: dw = 0}/{da :r
E
)I.
Mk-
172
In this equality we must specify what sort of regularityand boundary values-these forms should possess.
it
suffices either to let
just distributional
constraints.
W and
T1
(currents), cf.
be
Co
in
In fact,
M
or
[9], with no boundary
However, when proving the Hodge theorem for a
nondegenerate metric it is more convenient to use smooth
M
forms on
cf.
which satisfy
'absolute' boundary conditions,
[26], which we shall not define here since we do not
need them.
On the other hand, the relative cohomology not
surprisingly always requires constraints at
Hence,
it is well known that we may compute
the boundary.
Hk (M,M)
using the complex of forms satisfying relative boundary
conditions
H k (M,c3M) =
i :M
Here
-
= 0, dw = O}
i*
(M)
{wE C
co k-i
(M), i W = O}
{dT : q E C0
M
We could equally
is the inclusion.
well use forms supported in the interior of
M,
as shown
by an argument using the chain homotopy operator in the
proof of our next theorem.
We need to define the relative groups using somewhat
less regular forms.
on
M,
There are two spaces of distributions
briefly mentioned already in Chapter 2, but see
[22] for more details.
Let
C
denote the smooth func-
tions vanishing to infinite order at
BM
and let
173
(M) = (C (M))',
C
C
(M) = (C (M))'.
These are the 'extendible' and 'supported' distributions,
respectively.
M
If
is an open extension of
A
former space contains the restrictions to
distributions on
M,
distributions on
M
M,
then the
of arbitrary
while elements of the latter are
actually supported in
M.
The
sequence
0 ---
+ C~w(M,aM) ----
+
O~(M)
---
+ C~M(M)
0
is exact, the initial space being of distributions on
supported in
8M.
M
One of the few differences between these
spaces arises by comparing how differential operators act
on them.
As may be checked from duality, an extendible
distribution is to be differentiated in
M,
the result
extending to the boundary by continuity; a supported
distribution is differentiated as a distribution on
M,
so
that boundary layers may well occur.
Having stated these facts, we now come to
the discussion.
the point of
the case (which unfortunately is not
It is
to be found in the literature) that the absolute and relative cohomologies of
of
M
forms with values in
mention all
may be computed from the complexes
C
and
C
,
respectively.
We
this to justify a minor technicality concerning
regularity in the proof of our
174
(4.8) Main Theorem:
Ak 3
There are natural isomorphisms
--
k <
[w] E H k(M,OM)
n+1
k
k
22
harmonic forms are closed and coclosed since
k
n+1
k
W
k >
Hence when
is complete.
g
the metric
2
n-i
represents an absolute cohomology class.
If k (2
Proof:
L2
-,
.
may then consider it as lying in
vanish at
w
then from (4.7) both components of
-00
C
(M),
8M.
We
and it is still
Thus
closed in the sense of supported distributions.
w
It will suffice to show that
represents a relative class.
below the middle degree the map is injective, while above
the middle degree it is surjective.
n-i
Suppose
then that
for
W = d,
k
< n2 1,
i*
o E
k
f
,
= 0.
Choose finitely many coordinate patches near
h
an
= 0, h
ally near
nn
= 1,
and the coordinate
the boundary.
we have
Then, since
zn
8M
such that
is defined glob-
6 ce = 0,
we compute
175
IoII 2 = <dn,w> -
<I,6 9 w> =
g
=
2k-n
h
lim
{z n=6}
F-+0
By assumption on the coordinates, only
(-*h
t
1t
= O(pn2k
and
wn
We know that
enter into this last integral.
W
but
A * W)
d(
JMg
log p)
it is less clear at what rate
Tt
vanishes.
We study this issue by using the chain homotopy operator of [26].
Thus if
dy
A dx E Ok
then we define
Rw = (Rw)
w = W
and
w 1 (y,O) = 0,
dy
+ W
J
dyi
by
%1
(Rw)
=
(-1) k-1
w
(y,tx)x dt.
This is actually coordinate independent, and satisfies
dRw +
so long as
i w = 0.
Rdw = w
Note also that
i*Rw = 0.
176
If
w = dRw
is closed then
w
is defined.
Rw
of the boundary where
in the neighbourhood
If
P
is a func-
tion supported in this neighbourhood which is identically
one near
then
aM,
w
-
d(
PRw)
represents the same relative cohomology class and vanishes
near the boundary.
trivial,
j3 for which
so there exists a form
i*
which is
We are assuming that this class is
C
= 0,
up to
w
-
d(NPRw)
= dP
This shows that we may take
aM.
n = P + %pRw
and so
lit
=
O(p
log
p).
The limit above must now vanish, hence
w = 0
and the map
into relative cohomology is injective.
n+1
On the other hand, when k > -+- take a smooth
representative
class.
a
Obviously
for an arbitrary absolute cohomology
a
is square-integrable with respect to
177
since
p2k-n
|a
is bounded.
2
has
A
We know that
closed range with a finite dimensional complement, so we
may write
+ w
13P
a = A
for some
# E L 2k
the interior of
w E * .
and
M,
[a] = [o]
Hence
da = 0
and since
1
a = d6
Obviously
+
1
is
C
in
it follows that
o.
and the theorem is proved.
As the last part of this proof shows, we have actually
proved something about the
L2
cohomology spaces discussed
in the introduction.
(4.9) Proposition:
For
k
<
n-i
2
2k
L 2H k
L2Hk
or
k >
n+1
2+1
we have
k
is finite dimensional.
and so,
in these degrees,
Proof:
We have established a strong Hodge decomposition
for these values of
k:
178
a = d6
L2k (M,dg)
k
W E I.
so
f3 + 6 df3 + w
is closed iff the second summand vanishes, and
a
[a] = [w]
2 k
L2H
in
The existence of a parametrix away from the middle
degrees
implies that the spectrum of
A
is
L 2k
on
bounded below by a positive constant for these values of
k.
k = (n+l)/2
However, when
-so
that the dimension
is odd-then judging from the special case
cf.
IH,
n
[11],
the continuous spectrum should in general extend down to
zero, and
A
About
would not have closed range.
final case, when
n
is even and
k = n/2,
the
we can reach a
definite conclusion.
(4.10) Proposition:
the orthocomplement
positive lower bound.
L2 n/2
n/2
Proof:
Since
n/2
(n/
2
)
the spectrum of
A
has a
g
Hence the range of the Laplacian on
is closed, a strong Hodge decomposition holds, and
2 n/2
on
is of infinite dimension, but on
n/2
L 2n/2
(the space
class of the metric),
(M,dg) = L2n/2 (M,dh)
n/2
and
6
= p 2h
depends only on the conformal
it is necessary only to solve the
elliptic boundary problem
179
= 0,
A
io
=
pi
for the nondegenerate metric
hence
in
Wn/2
is
L2 (dg),
n/ 2 (aM),
E C0
h;
i *6 h
=
each solution is
and an infinite dimensional
0
C
subspace of
thereby obtained.
The argument for the next part follows [14] closely.
Choose
so that
e > 0
[0,e]
does not
intersect the spec-
n/2+1
(the superscript indicating the degree of
g
Now suppose
is supposed to act).
form on which A
trum of
A
=((A n/2 )a # 0.
g
By the assumption on
dw = dK(A
6g
and so
e,
n/2
)a =
g
n/2+1
)da = 0
,/(A
g
= 6 x(An/ 2 )a = x(An/ 2
g
w E i(n/2.
g
g
This proves
spec(A n/2)
g
g
)6 a = 0
g
that
[0,6] = {}
as desired.
The last assertions are now immediate, and we are
done.
180
It seems quite likely that a fairly complete spectral
picture of the Hodge Laplacian for a conformally compact
metric in any degree could be obtained from refinements of
the analysis in this dissertation.
be addressed later
Perhaps this issue will
if it seems warranted.
181
Bibliography
1.
Anderson, M.: The Dirichlet Problem at Infinity for
Manifolds ofNegative Curvature, J. Diff. Geom.,
18, no. 4 (1983), 701-721.
2.
L 2 Harmonic Forms and a Conjecture of
Anderson, M.:
Dodziuk-Singer, Bulletin of the A.M.S. 13, no.
(1985), 163-165.
3.
Positive Harmonic FuncAnderson, M. and R. Schoen:
tions on Complete Manifolds of Negative Curvature; Preprint.
4.
Carleman Estimates
Andreotti, A. and E. Vesentini:
for the Laplace-Beltrami Equation on Complex
Manifolds, Publications Mathematiques IHES, no.
25 (1965).
5.
Atiyah, M. F. : Elliptic Operators, Discrete Groups
and von Neumann Algebras, Asterisque, 32-33
43-72.
(1976),
6.
Curvature
Bourguignon, J.-P. and H. Karcher:
Pinching Estimates and Geometric
Operators:
Examples, Ann. Scient. Ec. Norm. Sup., 11
1978), 71-92.
7.
Thesis, University of California,
Choi, H. I.:
Berkeley, 1982.
8.
Theory of Ordinary
Coddington, E. and N. Levinson:
Differential Equations, McGraw Hill, New York
(1955).
9.
Differentiable Manifolds, SpringerdeRham, G.:
Verlag ( 1984).
10.
L 2 Harmonic Forms on Rotationally
Dodziuk, J.:
Symmetri c Riemannian Manifolds, Proc. Am. Math.
Soc. 77 (1979), 395-400.
11.
The Differential Form Spectrum of
Donnelly, H.:
Hyperbolic Space, Manuscripta Math. 33, (1981),
365-385.
12.
L 2 -Cohomology and
Donnelly, H. and C. Fefferman:
Index Theor em for the Bergman Metric, Ann. of
Math., 118 (1983), 593-618.
182
13.
Pure Point Spectrum and
Donnelly, H. and P. Li:
Negative Curvature for Noncompact Manifolds, Duke
Math. J., 46, no. 3 (1979), 497-503.
14.
On the Differential Form
Donnelly, H. and F. Xavier:
Spectrum of Neg atively Curv ed Riemannian Mani106 (1984), 169-185.
folds, Amer. J. Math.,
15.
Visibility Manifolds,
Eberlein, P. and B. O'Neill:
1 (1973), 45-109.
no.
46,
Math.,
J.
Pacific
16.
The Bergman Kernel and Biholomorphic
Fefferman, C .:
Mapping s of Pseudoconvex Domains, Invent. Math.
26 (197 4), 1-65.
17.
Positive Sectional Curvature does not
Geroch, R.:
Imply Positive Gauss-Bonnet Integrand, Proc.
A.M.S. Vol. 54 (1976), 267-270.
18.
The Analysis of Linear
Hdrmander, L.:
III,
Differ ential Operators, Vol.
Verlag (1985).
19.
Special Functions and their ApplicaLebedev, N. N.:
tions, Dover (1972).
20.
Meromorphic Extension
Mazzeo, R. and R. B. Melrose:
of the Resolvent on Complete Spaces with
Asymptotically Constant Negative Curvature, Preprint.
21.
A on
An Upper Bound to the Spectrum of
McKean, H.P.:
a Manifold of Negative Curvature, J. Diff. Geom.,
4 (1970), 359-366.
22.
Melrose, R. B.: Transformation of Boundary Problems
Acta Math., 147 (1981), 149-236.
23.
Elliptic Operators of
Melrose, R. B. and G. Mendoza:
Totally Characteristic Type, to Appear in J.
Diff. Eq.
24.
Degenerate Elliptic
Melrose, R. B. and G. Mendoza:
Boundary Problems, In Preparation.
25.
Interaction of
Melrose, R. B. and N. Ritter:
Progressing Waves for Semilinear Wave Equations
II, to Appear in Ark. Math.
26.
R-Torsion and the Laplacian on
Ray, D. and I. Singer:
Riemannian Manifolds, Adv. in Math., 7 (1971),
145-210.
Partial
Springer-
183
27.
The Dirichlet Problem at Infinity for a
Sullivan, D.:
Negatively Curved Manifold, J. Diff. Geom., 18,
no. 4 (1983), 723-732.
28.
Harmonic Functions on Complete Riemannian
Yau, S. T.:
Manifolds, Comm. Pure Appl. Math., 28 (1975),
201-228.
29.
Adams, R.A.:
(1975).
Sobolev Spaces, Academic Press, New York