E. Vesentini ( E d.)
Differential Operators on Manifolds Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, August 24 - September 2, 1975
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11113-6 e-ISBN: 978-3-642-11114-3 DOI:10.1007/978-3-642-11114-3 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 st Reprint of the 1 ed. C.I.M.E., Ed. Cremonese, Roma 1975 With kind permission of C.I.M.E.
Printed on acid-free paper
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CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
I11 Ciclo - Varenna dal 25 Agos,to a1 2 Settembre 1975
DIFFERENTIAL OPERATORS ON MANIFOLDS Coordinatore: Prof. E. VESENTTNI
M. F. ATIYAH
:
Classical groups and classical differential operators on manifolds
R. BOTT
:
Some aspects of invariant thery in differential geometry
n
49
Singular integral operators and nilpotent groups
)>
147
E. M. STEIN
:
Pa&
5
P. MALLIAVIN
:
Diffusion et g6ometrie differentielle globale
S. HELGASON
:
Solvability of invariant differential operators on homonogeneous manifolds >> 281
B
207
C.I.M.E.
CLASSICAL GROUPS AND CLASSICAL DIFFERENTJAL OPERATORS ON MANIFOLDS
M. F. ATIYAH
Corso tenuto a Varenna dal -25 Agosto a1 2 Settembre 1975
M.F. Atiyah Lecture I Introduction The aim of these first two lectures is to review some basic facts about the representation theory of the unitary group U(n) orthogonal group
SO(n).
and the special
What is involved is just
linear algebra but it is very important to have a good understanding of it for the applications to differential geometry. 51.
The Exterior Algebra If V
is a complex vector space of dimension n
we can form the exterior powers :A~(v). vector spaces of dimension ("1
and
9
an
naturally on them.
Taking V =
representations Aq
of GL(n,C)
If v
E
V
GL(V) acts
this gives
and hence of U(n).
then exterior multiplication by
defines a linear transformation Tv : If v
0
(1.1)
o
is exact.
They are
the sequence T + AO(V) Y A~(v)
v
+
...
nq(v)
v +
Aqtl(v)
T
l-V
hn(v)
This may be proved as follows.
+
o
Let
Jr
T denote the adjoint of Tv with respect to the v * natural inner product of A (V) given by an inner product on V.
where
Then
I is the identity.
This is easily checked
M. F. Atiyah by using an orthonormal base of v = Xel.
for which
Formula (1.2) immediately implies the
exactness of (1.1) because
=
V
Tv(x) = 0 gives
%( 2 3 j-2 @ e1e2 = 1 8 e j s 2 j Since the elements JI(e.1 satisfy the relations (1.1) 3
it follows that
JI
:
JI extends to an algebra homomorphism
.. cn+2
-
+
cn @
-
C2
This takes a basis into a basis and hence is an isomorphism.
-
It remains to check that C1 z C 8 C (which is -,
clear) and that C2 is the 2x2 matrix algebra:
an explicit
isomorphism is given by sending el,e2 into the matrices
For any n the' mapping 1~~
Cn+l given by
+
.-
.
ei * eien+l extends to an isomorphism Cn c:I1 ,.. Thus, for n even, (2.2) implies that ciV is a direct sum of two full matrix algebras. If we give R~~ the structure of a complex vector space V then we get a natural isomorphism
Cwhere End denotes the algebra of all linear maps (
rh
*
A V 3 A V) given as foalows.
To every v
E
V we
M. F. Atiyah associate Av = Tv where
Tv
- Tv*
E
*
End (A V)
is exterior multiplication by v as in
Lecture I. Formula (1.21, together with the identities 2 = 0, shows that T = 0, (7':l2 2 Av = - 1v112 1
.
Hence v* Av
extends to an algebra homomorphism
-C2k
*
.
End (A V) .. Since this is non-zero and C2k is a full matrix
A
:
+
algebra, A must be injective. that A is also surjective.
Dimensions then show
Note that, restriction
to the even part of (2.3) gives, tev End ( A ~ ~ v@) End (fIoddv) (2.4) 2k A full matrix algebra End E does not determine
.
'
the original vector space E canonically.
It does
however determine the projective space P(E):
points
of P(E) correspond to minimal right ideals of the algebra End E.
Any automorphism of End E
therefore induces a natural automorphism of P(E). Applying this with E = A
* cn
and using (2.3) we
see that every element of SO(2k) (and even of O(2k)) induces an automorphism of P(A *C k3 i.e. we have a * k projective representation of SO(2k) on P(A C 1.
.
Moreover since SO(2k) preserves the grading of C2k it follows from (2.4) that the action of SO(2k) on
':2 k P(A C )
leaves invariant the two subspaces p(hevck)
and P ( A ~ ~ ~ c ~ ) These . are called the half-spin representations while the action of the whole space * k P(A @ is called the Spin representation. Thus the even and odd parts of A*Ck, which are (decomposable) representations of U(k), projective representations of S0(2k),
extend as and these turn
out to be indecomposable: the Z-grading by individual dimensions is not invariant under SO(2k). In general a projective representation of a connected Lie group G is the same as a linear
.. G
representation of a (connected) covering group
-
To find SU(N)
+
G
G
+
SU(N)
G
+
PU(N)
we have to pull back the cyclic covering
PU(N),
of order N, to G and pick the identity
..
component.
Thus the order of G
In our case
N
+
G
divides N. % .
is a power of 2 but in fact G
is of order precisely 2. returning to the
G
This can be seen by
real Clifford ..
+
algebra C2k.
Since
its complexification C2k is a matrix algebra over C, it follows that Cgk is a matrix algebra over R or M (the quaternions).
Thus our Spin representation of
O(2k) maps into the real or symplectic projective
M. F. Atiyah group and the corresponding coverings SO(2k)
+
P0(2k),
Sp(2k)
-*
PSp(2k) are of order 2.
The covering group of SO(2k) is called Spin (2k). Note that the case k = 1 is degemrate and we make the convention that Spin(2) is the double cover of SO(2) and that the (linear) Spin representation is induced by the isomorphism SOi2) The character
2
U(1).
a of the (linear) Spin
representation follows by combining (1.4) and (2.3). We get k
o
(2.5)
=
n (tif i=l
+ tif)
Writing o = at t a- for the decomposition into the half-spin representations we get
Multiplying (2.5) and (2.6) and comparing with (1.6) we see that (a'
(2.7)
+ a-)(a+
- 0-1
= P+k
-
k P-
On the other hand we have
4
12 we identify the exterior algebra A ( R ~ ~ )
with the Clifford algebrq CgkY as representations of S0(2k), given by
then the involution
T
defined in 01 is
M.F. Atiyah
where w = ele2
.:.
e2k
E
Cpk
.
Here the ei
denote aq orthonormal base of lR2k and all products are taken in the Clifford algebra. Note that w ev As a general rule it is is in the centre of C2k
.
much simpler to use Clifford multiplication by than the standard
*,
w
since the unpleasant signs tend
to disappear. The Spin representation of Spin(2k) gives a representation of a double covering of U(k) which is projectively equivalent to A k(C 1. However the character formula (2.5) shows that it differs from it by the factor ( A k ) 4 :
This formula also shows that the representation
C X of ~ U(k) extenqs to a representation of the group
where, on the right, we divide by the element (-1,-1) The groups Spin (2k) and spinc(2k) may be explicitly described as subgroups of the invertible "
elements in Cgk and C2k respectively. fufther details see [ 7 ] .
For this and
.
M. F. Atiyah
Lecture I11 Introduction The exterior algebra and the Clifford algebra lead, by Fourier transforms, to certain standard differential operators on vector spaces.
These can
be globalized to manifolds by introducing, where necessary, a Riemannian metric.
We shall describe
these operators and relate them via the Hodge-de Rham theory to homology. 1
The Hodge
-
de Rham theory
We review first the de Rham theory. be a cW manifold of dimension n and let
K
Let
nq(x)
denote
the space of cW exterior differential forms of Then we have a natural differential
degree q on X. operator d :
nq(x)
nqtl(x)
+
differential of a function.
o
+
nO(x)
+
nltx)
+
which extends the The sequence
.. . + nntx)
+
o
is a complex (i.e. d2 = 0) and the main theorem of de Rham asserts that the cohomology of this complex is isomorphic to the cohomology of X (with real coefficients).
In more concrete terms if yl
...yN
is a basis for the homology in dimension q then there exists a q form w periods
I
yi, and
with dw = 0 having prescribed w
is unique modulo the
addition of forms dB. If X = il?" then
flq(x)
can be identified with
M. F. Atiyah
*
cm maps IR" + A~(R" ) and d becomes a constant coefficient operator whose Fourier transform is exterior multiplication by it, 5 Suppose now that
X
E
(R")
is compact oriented and that
we have a Riemannian metric on
X.
We then define
nq(x)
a pesitive definite inner product on
*
where
:
nq(x)
* .
+ Q"-~(X)
by
is the isomorphism given
by the metric Band orientation) as in Lecture I. Because of the identity
~(u,*v) = ~u,*v we get 0 =
t
(-I)~u,,~*v,
SX
duA*v t (-llq
lX
u
u,d*v
nq, v
E
E
nq"
.
Hence (du,v) Thus the
* adjoint d
of d on
d* = (-l)4+:*-ld*
where
E
.
= (-l)qtl(u,*-ld*v) =
E
nq
is given by
*d*
= 21 depending on n,q.
The Hodge-Laplace operator on
* A=dd
* t d d
nq
is defined as
and the hanmonic forms gq are the solutions of Au = 0.
The main theorem of Hodge theory is that #q
is isomorphic to the q-th de Rham group and hence to the q-th cohomology group of X.
M. F. Atiyah The Euler characteristic and signature
52.
It is convenient to consider the operator d acting on
* (XI
= @ 9
nq(xI.
we have A = (dtd*)
Since d2 = 0, ($12
8
fi2q
d
= 0,
and the harmonic forms are also
*
the solutions of (d+d )u = 0.
nev
as an operator
J(
t
-+
If we consider d+d
*
its null space is
s-todd
while the null space of its adjoint is
Thus its index is the Euler characteristic of X. We recall that for any elliptic differential operator P on
X
we put index P = dim M(P)- dim
where
#(PI
Y(P* I
is the null-space of P.
For any differential operator P
the highest
order terms have an invariant meaning on the cotangent bundle:
this is called the symbol of P.
For the
exterior derivative d its symbol is exterior multiplication by ic.
* dtd
For A
it is
- 11 5 112
and for
it is iA where A is the operator defined in 5 5 Lecture 11, namely Clifford multiplication by 5 . Suppose now we consider the case n = 2 R use the involution
T
and
of Lecture I, which as we saw
in Lecture I1 is given by
T
= iR w
where
w
denotes
Clifford multiplication by the volume form *l.
*
I claim that dtd' and hand it is clear that
T
anti-commute.
* (d+d IT
and
On the one
* ~(d+d
are both
M. F. Atiyah of the form +*d +d*, so it is sufficient to check the symbols. cw
= -wE
But in the Clifford algebra we have
for 5
eigenspaces of
T
E
*.
Thus if
* T , d+d
maps
SIt
a*
denote the +I-
into !,IThis .
restricted operator d+d
*
:
a,
+
a-
we call the signature operator and denote by A. We shall now compute its index when n = 4k, i.e. R = 2k.
Clearly this index is equal to dim
where p(* If q
R
fl+
-
dim
8- .
are the spaces of harmonic forms in 0,
, the
space S t q
is stable under
which just interchanges the two factors.
T
Thus we
get a zero contribution to our index from such a For q = R = 2k we have ~ ( a )= iL(R-l)tR*a = *a
spece.
4
E
SIR 1.
Hence
index A = dim&+II whereti*R
-
a dim)l-
are the 21-eigenspaces of
Since the inrlar product in
fiR
*
on
is given by
see that
R
$i
s
.
uA*v we
index A = Sign X where Sign X is the Signature of the quadratic form on H ~ ~ ( x ; given R) by the cup-product. explains our name for the operator A.
This
It plays a
very important role in the general index theory.
M. F. Atiyah
Lecture IV Introduction We shall now describe the Dirac operator and its relation to the 8-complex.
This will use the
material in Lecture 11. 51.
The Dirac operator We begin by considering Euclidean space R 2k
.
-,
The Clifford algebra C2k is a full matrix algebra acting on a space S (the Spin space) of dimension gk.
Moreover S = S+ 8 S- is the decomposition
... 5 k : .. . E2k be the
given by the eigenspaces of w = ele2 precisely u = ?ik on sf.
Let El,
linear transformatio~on S representing el,... e2k. Then the differential operator
satisfies D ' = S.
-
A - I , where
I
is the identity of
We call D the Dirac operator.
anti-commute with S+ and S-.
w
Since the ei
the Dirac operator interchanges
Restricting to
st
we therefore get an
operator
.
B : c ~ ( R ~ ~ , s-++ )c~(IR~~,s-) Note that, in the standard metric, the Ei are unitary and hence skew-adjoint (since
~f
= -1).
Since a is also formally skew-adjoint it follows axi
M. F. Atiyah that D is formally self-adjoint which means that the formal adjoint of B is the restriction of D to
s-. We now pass to the global situation of a compact
oriented manifold X of dimension 2k.
First we must
assume that there is a Spin structure on X.
This
means that the principal SO(2k) bundle P of X (consisting of oriented orthonormal frames) lifts to a principal Spin (2k)-bundle Q, i.e. Q
+
P is a
double covering inducing the standard covering Spin(2k)
+
SO(2k) on each fibre.
If 'S
are the
two half-spin representations of Spin(2k) we consider
s*.
= Q
the associated vector' bundles on X: 'E
Sections of these are called Spinor fields on X. The total Dirac operator on X will be a differential operator acting on E = E+
Q
E- and switching
factors
To define D we must use the Riemannian
as above.
(Levi-Civita) connection on P (which lifts to one on
Q).
This enables us to define the covariant derivative
V : cW(x,E)
-+
cW(x,E 8 T*)
and we define-D as the composition pf C=(X,E €3 T*)
-+
c~(x,E)
V
with the map
*
induced by Clifford multiplication (note T s T the metric).
by
In terms of an orthonormal base ei
of T at any point we can write
h4. F. Atiyah
where
Vis
is the covariant derivative in the
direction ei
and
multiplication.
ei(
denotes Clifford
On the level of symbols D is
clearly Skew-adjoint (cf. the coiputation in 1 ~ ~ ~ ) . Hence D
- D*
is an algebraic invariant of the
metric and it only involves the first derivatives Use of normal coordinates shows that
of the gij.
any such invariant is zero and so D = D
* , i.e.
D
is self-adjoint. As in Euclidean space the restriction of D to the half-spinors E+ will be denoted by
B
:
c~(x,E+) + c-(x,E-).
The index of B is given by index B = dim,&+
-
dirng-
where H * denote the spaces of solutions of Du = 0 for u
a section of Ef.
Since D is elliptic and
self-adjoint these spaces are also the solutions of D ~ =U 0 and are called harmonic spinors. 52.
The 7-operator On a compact complex manifold X of dimension k
the (complex-valued) exterior differential forms can be written in terms of dzi and dzi get a decomposition:
.
Thus we
M. P. Atiyah
where the
npsq involves
dzi
.
p
of the dzi
and
q
of
This is just the decomposition (1.3)
of Lecture I.
Moreover the exterior derivative d
is of the form d = a +T where
a 3
:
:
nPyq + Q ~ * P A ,*P,9+1
~
~
,
~
.
A function is holomorphic if and only if it satisfies
the Cauchy-Riemann equation F f = 0. a form w
-
aw
=
E
More generally
Q ~ " is holomorphic if and only if
o. If we introduce a Hermitian metric then we can
*
define the adjoint 7 of 7 and consider the operator
7 + T*.
For fixed p this acts on
np'*
Its square is the complex Laplacian (since T~ = (%*I
2
0
=
= 6 nPyq 9
.
m* +
= 0) and hence is elliptic.
*
Restricting 3 t'3 to even q we get an operator *P 9 ev * *P ,odd C~ ' whose index is given by k index C = C (;ldqhP'q p q=O where hPsq = dim fiP9q is the dimension of the space
.
of complex-harmonic forms of type (p,q), i.e. solutions of the equation a u = 0. The complex analogue of the Hodge-de Rham theorem
is the Dolbeault theorem which asserts that
& Pyq where
@P
5
H~(x,
denotes the sheaf of holomorphic p-forms.
Thus index C = X(~,@P) P where x is the Euler characteristic (of the sheaf
.
In particular index C o = y,(X,@ 0
where
@
0
is the sheaf of holomorphic functions.
This is called the arithmetic genus of X. If the Hermitian metric is compatible with the complex structure (parallel transport preserving the complex structure) it is called a Kshler metric. In this case
D = 18
and the complex harmonic forms
are also harmonic so that C hpsq = Br p+q=r is the r-th Betti number.
This applies notably to
projective algebraic varieties. 3.
Relation of 7 and the Dirac operator In Euclidean space all our operators are constant
coefficient operators and so by Fourier transforms are given by linear algebra.
In particular the
relation between the complex exterior algebra and the Spin representation explained in Lecture I1 shows that, in C k, the Dirac operator essentially coincides with the operator 7
*
+7
acting on Q
0
*.
M. F. Atiyah Suppose now that X
is a compact complex
manifold of dimension k. on
In order to have Spinors
X we must assume that X has a Spin-structure.
In terms of the complex structure this means that the canonical bundle
K = Ak :T* )
has a square root L.
In fact one has a bijection between Spin-structures on X and holomorphic square-roots L of K (cf. [I] or [ll]). Spinor fields on X can now be identified with sections of L
Equivalently
A"*(T*).
be identified with sections of
* L 8E
a0 '*
can
E is
where
the total Spin-bundle. If L
is trivial (which can only happen if K
is trivial) then D and 7
+ %*
act on the same
bundle and their symbols agree.
In general to
compare them we have to "twist" one of them by L Jr
or L
.
More generally if V is any complex vector bundle over X with a connection we can define acting on S @ V, or
* (7+ 7
acting on A'*(T
*
4,, 63 V
.
The symbol of DV is obtained from the symbol of D by multiplying by the identity of V.
In this
notation we see that D has the same symbol as
(r + 7* I L
*
1
.
D has the same symbol as 7 + 7 L For a Kghler manifold we have a natural connection and
M. F. Atiyah on K, hence on operators 7 Note -
t
L
and one can check that the
F* and D
L
*
actually coincide.
The operator
can be defined without DL* assuming existence of a Spin-structure. All we need is a spinc-structure. The formulae in Lecture I1 also show that the operator DE, i.e. the Dirac operator acting on E @ E , has the same symbolasd
t
*
d..
In fact,
* , it
by an argument similar to that proving D = D follows that DE = d
+ d
*.
Moreover E* 4 E
correspond to the eigenspaces of the involution on forms.
T
Thus, if k = 2L,
Sign X = index A = (-1).9 index BE Similarly the Euler-characteristic x(X) is
We see therefore that the Dirac operator appears to play
a basic role for computing indices.
The
underlying reason for this will be clarified in the next lecture.
M. F. Atiyah Lecture V Introduction We shall now study the stable homotopy of the unitary group, explaining the Bott periodicity theorem and its implications.
The symbols of
elliptic operators play a basic role. 1.
The Bott periodicity theorem Suppose that P
is any constant coefficient
elliptic operator on IR", acting on functions with values in
ern.
Then its symbol ap(c) gives
a mapping up : sn-l
+
GL(m,C)
For example the restricted Dirac operator B on
R~~ gives a map OB
.
s*~-' +
GL( 2k-1 ,c)
The Bott periodicity theorem asserts that this is a stable generator. 1) ai(GL(m,C))
+
More precisely we have
mi(GL(m+1,a))
is an isomorphism
for large m (in fact m > iI2). 2)
The limit &oup
ai(GL(m)
given by 1) is 0 if
i is even and is 2, generated by aB for i odd. 1) is a fairly trivial dimension argument.
The
essential point is 2). There is also a global version of the Bott theorem,
M.F. Atiyah which is in fact a fairly routine consequence of the local theorem, asseeting that the symbol of the Dirac operator on a compact manifold is a generator, in a suitable sense, for all elliptic symbols. More precisely let X
be a compact oriented 2k-
dimensional manifold with a Spin structure. elliptic differential operator P :
c~(x,v) +
An c~(x,w)
has a symbol a p which is an isomorphism J(
IT
V
+ IT
*W ,
? : S(X) + X
being the projection of
the unit tangent sphere bundle. is one where V = W and
a
A trivial symbol
is the identity.
Two
symbols are called equivalent if they become homotopic after the addition of trivial symbols. Then the global Bott theorem says that every symbol is equivalent to the symbol of BV, for some V, where BV is the restricted Dirac operator extended to act on E+
@ V.I
The interest of this for index problems
is that two operators with equivalent symbols have the same index.
Thus if we can find the index of
BV for all V we will get a formula for the index of all elliptic operators on X. 12.
K-theory To handle and prove the Bott periodicity theorem
it is best to introduce the K-groups.
In this section
M. F. Atiyah
we shall give the basic definitions and elementary properties.
The periodicity theorem will be dealt
with in the next lecture. We consider complex vector bundles V over a compact space X.
Their isomorphism class forms
an abelian semi-group under direct sums and the associated abelian group is denoted by K(X).
Thus
elements of K(X) can be viewed as virtual bundles [V]
-
[W].
The following lemmas are quite
elementary (for proofs see [Z]): LEMMA (2.1)
Homotopic vector bundles are isomorphis
LEMMA (2.2)
Every vector bundle V has a complement W,
i.e. a bundle such that V 8 W is isomorphic to a trivial bundle
X
eN .
x
For a space X with base point xo we define
:(XI
= Ker {K(X)
have K(X)
= C(X)
we put K(X,Y) =
+
K(xo))
2. -K(XIy) @
Since K(point)
Z, we
For a closed subspace Y of X where XIy is obtained by
collapsingY to a point (taken then as base point). For a locally compact space X we put K(X) =
K(x+),
where X+ is the one-point compactification of X (the new point being taken as base point).
We then have
another elementary lemma (proved using (2.1) and (2.2)) LEMMA (2.3)
If Y
K(X,Y)
C
+
X
K(X)
is an exact sequence.
is a compact pair +
K(Y)
'M. F. Atiyah
The tensor product of vector bundles induces a ring structure in K(X) and K(X,Y)
becomes a
K(X)-module. A map
X
+
GL(m,C) defines a vector bundle over
the suspension of X. map on
s1 + U(1) s2 = .'C
C
In particular the identity
GL(1,E) defines a line-bundle H
We denote by b
E
K(C) the class of H
-
1.
Then the Bott periodicity theorem may be formulated as follows: THEOREM ( 2 . 4 )
Multiplication by b gives an isomorphism
B : K(X)
+
K(C
XI
x
Taking X = C" and applying induction on n 2 0 we deduce that
K(c")
e
I.
Moreover it is not
difficult to identify the symbol of the restricted Dirac operator B with the element bn.
This neaovers
the local statements in 51. For the global statements we first show that an elliptic symbol a where X.
T
defines an element
[a] of K(TX),
denotes the cotangent bundle of the manifold
Moreover the element
equivalence class of a
[a]
is essentially the
in the sense of 51.
The
global version is then THEOREM ( 2 . 5 )
For a Spin manifold of even dimension
multiplication by aB gives an isomorphism K(X)
+
K(TX).
M. F. Atiyah If X to (2.4).
is a parallelizable manifold (2.5) reduces In the general case we take a covering
of X, over which TX is trivial, and apply induction using the long exact sequence which extends (2.3). This long exact sequence involves the higher groups
K-"(x)
defined for n 3 0
and extended to all n If X
by
K-"(x)
= K(W"
x
X)
by periodicity.
is not a Spin manifold we do not have a
Dirac operator. operator A.
However we have the Bignature
The computations of Lecture I1 show
that locally the symbol of A is equivalent to a multiple of the symbol of B: uA = 2 k aB
(2.6)
In fact when
X
has a Spin structure we have
globally aA = uB Using ( 2 . 6 )
.E E
K(TX)
.
one obtains
THEOREM (2.7)
For a 'compact oriented manifold X
of
dimension 2k multiplication by the symbol oA of the Signature operator gives an isomorphism K(X) €3 Q
+
K(TX) @ Q
.
Thus to get the index formula for general elliptic operators on X it will be sufficient to compute index AV for all V.
21. F. Atiyah Lecture VI Introduction
I shall now give the essentials of the proof of the Bott periodicity Theorem. details I refer to [3] 51.
or [4]
For fuller
.
Structure of the prpof We shall explicitly construct a map a : K(C
X)
x
+
K(X)
which will be the required inverse of B formal properties of a
.
The
which will be easily
established from its definition are as follows: (1)
naturality in X,
( 2 ) multiplicativity, i.e. for any Y the obvious
diagram K(C
XI 8 K(Y)
x
+
+
-4
K(a
K(X) @ K(Y)
x
X
x
Y)
+
K(X
x
Y)
is commutative, (3) a(b) = 1
(for X = point, b
E
K(C) the
Bott element). Taking X = point and Y = X in ( 2 ) we see that, for any x
E
K(X)
,
aB(x) = a(bx) = a(b) x = lx = x
(using (3)).
M. F. Ativah Thus a6 = 1. for 5
E
K(C
Now apply (2) with Y = C and we get, XI,
x
a(5b) = a(5)b
.
= @a(()
Now use the fact that we can switch the two factors 62 in K(C x X
C) by a map homotopic to the identity.
x
This shows that a(6b) = Ba(6) = 5 using the fact that Ba = 1 Hence Ba(5) = 5
and so a
, (applied on the space CxX). is a 2-sided inverse of @
as required. 02.
Construction of a The basic idea is best explained by considering
fiEst the case X = point. K(g)
= -K(S 2
Then
sl(GL(m$))
We want to associate to any map
s1 +
f :
GL(m,C)
an integer a(f) in a way which will allow the introduction later of the parameter space X.
We
proceed as follows. 1) Approximate. f
by a finite Fourier series of
the form zvn where 2)
p
~(2)
is a matrix with polynomial entries.
Consider p
as a homomorphism Am
+
Am where
M.F. Atiyah A = C[ zl , and let M = coker p 3)
since p
is non-singular for (z1 = 1 we can
decompose M in the form M = M+ Q Mwhere M+
Iz1
in 4)
put
corresponds to the zeros of det p( z) = 0
.
< 1
a(f) = dim M+
-n
It is not hard to check that this is independent of the approximation chosen and that a(f) = 1 if f
= z. We now generalize to include X by considering
fCz ,x) and approximating by zcn p( z ,XI. Then
M = Coker p is an A-module depending continuously on x , and
M+ gives a vector bundle over X.
a(f) = [M+]
-
[X
x
cn]
E
K(X)
We then put
.
Again it is easy to check independence of the approximation.
The functoriality and multiplicativity
are straightforward and
a(b) = 1 has already been
noted above. This completes the construction of a
and hence
the prod of the periodicity theorem. There is an alternative and more analytical way of defining a.
This is based on the fact that the index
of a Fredholm operator can be generalized to families parametrized by X:
the index of such a family being
M. F. Atiyah riow an element of K(X) Now let on
(see the Appendix in [ 2 ] ) .
# denote she Hilbert space of L' functions
sl, and
let 9(' denote the closed subspace spanned
by the functions zn for n > 0. the projection onto
+.
Let P denote
For any
we let Tf be the operator on
4 ' €4
f :
s1 +
GL(m,G)
trn given by
Tf($) = P Q 1 (f$) This is a Fredholm operator and if a parameter x
E
X
f
depends on
we put
a(f) = index (Tf)
E
K(X).
Again it is easy to check all required properties of a. The relation between the two definitions of a becomes clearer once one notices that multiplication
-
by
2
1x1
> 1.
X is invertible on
4'
if and only if
M. F. Atiyah Lecture VII Introduction I shall now explain the heat equation approach to the index theorem.
This will be applied to
the classical operators introduced earlier.
The
success of the method depends on the invariance theory expounded in Bott's Lectures.
For fuller
details see [6]. 51.
The heat equation formula Let
P
be an elliptic differential operator
on the compact manifold X, and let adjoint.
eigenfunction of
* P PI$
PI$ is an
* * P P
*.
P and
If I$
with eigenvalue
* = XI$ , hence (PP )P$ * eigenfunction of PP
provided X f 0.
* P
*
be its
We can then consider the two self-
adjoint operators P P and PP
have
P
is any h
we
= hPI$ showing that with eigenvalue A ,
We can interchange the roles of
in this argument, hence
*
A
P P and PP have
the same non4z~roeigenvalues. Now we form the operators e-tP*P and e-tPP*5Y fundamental solutions of tve corresponding "heat equations".
For t > 0
so are of trace class. -tX trace e-tP*P = trace e-tPP* =
.-tv
these have cW kernels and Their traces are given by
M. F. Atiyah where
h,p
run over the respective eigenvalues.
Since the non-zero h
coincide with the non-zero p
and since h = 0 corresponds to the null space of P while p = 0 corresponds to the null space of P
*,
we get
-
i n d e x P = t r a c e e"*'
(1.1)
trace e-tPP*
On the other hand for any non-negative selfadjoint elliptic operator A of order m there is an asymptotic expansion trace e-tA
-
00
C a.t j/m j=-n 3
n = dim X
where the coefficients a are given explicitly in j integral form a = aj The integrands o j .. depend on the coefficients of the operator A and
4
their derivatives.
.
Applying this to (1.1), and
equating coefficients of to, leads to a formula index P =
(1.2)
where
a
SX
(ao
-
BO)
B . refer to the two operators P*P and PP*.
j' J
Application of invariance theory
52.
The explicit formula ( 1 . 2 )
is in general much
too complicated to be of any use.
However, for the
classical operators of Riemannian geometry it turns out that ( 1 . 2 ) size.
simplifies down to a recognizable
Although a.
and B 0 individually are very
complicated, their difference is simple because
M.F. Atiyah tremendous cancellation takes place.
The best
approach to this is by invariance theory.
We
find out the obvious formal properties of the integrand in (1.2) and then use invariance theory to show that it must be of a very special form. As a prototype let us consider the Signature operator A on a manifold of dimension U .
Since
A depends only on the meteic g and the orientation the integrand in (1.2) is a measure depending on the metric and changing sign with the orientation.
It is therefore a 4&-form w(g). The first crucial property of w(g) is that
Z = kLg, k a positive constant. To see this let (Q,A,?) denote the space of forms,
w(Z) = w(g) where
the Signature operator and the involution of Lecture I11 relative to g. by E(+) ET
=
?E
= kP+ if and
+
SIP.
-E
Define Then
E
E : 0 +
-
Q
is an isometry,
EA = ~ A E . The same formula holds
for A* and hence "
-
AA* = k-2(~AA*~-1) Since conjugation by
E
does not affect traces,
and since multiplication by k-2 does not affect the constant term in the heat expansion it follows
M.F. Atiyah that w(g) = w(g). The next
thing Me need to know about w(g)
is
the nature of its dependence on the components gij of g in any coordinate system.
To apply the
Gilkey Theorem (see Bottts Lectures) we must show that w(g) is given by an expression w(g) =
(2.1)
C aa(x,g(x)) ma 9 where aa(x,y) are C functions and ma denotes a
monomial in the partial derivatives of the g..(x). 13
In view of the way the asymptotic expansion depends on the operator it is sufficient to show that the coefficients of A*A and AA* are each of the form (2.1).
Now the coefficients of the
total Laplace operator A
on i-2
certainly are
of the form (2.11, and A*A is simply A to the subspace Q+.
restricted
To get an explicit matrix
expression for A*A we have then to choose a basis for Q+.
We can do this by applying the Gram-
Schmidt orthogonalization process to the basic forms 4, of i-2 1 at dx1 , , dxn to get a basis
...,
...
each point.
where
Then the forms
$K is any monomial in
a base of Q+.
,@2,
..., @n-l, give
Using this basis (and a similar one
M. F. Atiyah f o r Q-1
it i s easy t o check t h e c o r r e c t form f o r
t h e c o e f f i c i e n t s of A*A and AA*. The c o n c l u s i o n of a l l t h i s i s t h a t index A i s given by a formula
SX
index A =
(2.2)
where
~ ( g )
w(g) i s some u n i v e r s a l polynomial i n t h e
P o n t r j a g i n forms of g.
Thus index A (which
i s e q u a l t o Sign (XI) i s i d e n t i f i e d w i t h some P o n t r j a g i n number.
To f i n d which p a r t i c u l a r
combination of t h e b a s i c P o n t r j a g i n numbers i s 'involved we must check s u f f i c i e n t l y many s p e c i a l cases.
For example i n dimension 4 we have only
p1 and s o Sign X = kpt k.
Taking
one f i n d s
k
X
=
f o r some u n i v e r s a l c o n s t a n t
t o be t h e complex p r o j e c t i v e p l a n e
.
M. F. Ativah Lecture VIII Introduction
I shall now generalize the treatment of the Signature operator to cover all the classical operators.
From this the general index theorem
will follow.
51.
The generalized Si~natureoperator For any vector bundle with connection we have
the operator AV generalizing the Signature operator A. Treating this like we did A in Lecture VII leads to a formula for index Av as an integral
JU(~,V).
The integrand is now a form depending on the metric g and also on the vector bundle V (with metric and connection).
The generalized Gilkey Theorem then
enables us to assert that w(g,V) is a universal polynomial in the Pbntrjagin forms of g and the Chern forms of V.
Note that X can now be of any even
dimension (not necessarily a multiple of
4).
Taking sufficiently many suitable examples one can now identify this universal polynomial. Combined with the K-theory results of Lecture IV this leads to a completely general formula for the index of an elliptic operator in terms of characteristic classes
-
provided dim X is even.
On an odd-dimensional manifold it is easy to show
M. F. Atiyah
that the index is always zero for differential operators.
However one can consider pseudo-
differential operators and this case can be treated by relating X to X
*
sl, but I shall
not pursue this here. The Dirac operator and the operator 7
+ T*
are of course included in the above, but they can also be treated directly by the same method as the Signature operator (see [6]). operator
* +7
The
leads to the Hirzebruch-
Riemann-Roch theorem, expressing the holomorphic Euler characteristic of a complex manifold X in terms of its Chern classes.
More generally one
gets a formula for a holomorphic vector bundle on X. Note -
One can also define x(X,S) for any coherent
analytic sheaf on S.
If either
i) S is locally
free (hence corresponds to a vector bundle) or ii) X is algebraic (then S can be resolved by locally free sheaves) is known to hold. unsolved.
the Hirzebruch-RR theorem
In general the problem is still
As a first step one has to define the
Chern classes of S, but this can be done for example by the method of [8]. 12.
Lefschetz formula There is a natural generalization of the index
M.F. Atiyah which can be treated also by heat equation methods.
Moreovec in the simplest case one
can use this to complete the calculations for the inded without producing the necessary examples as I shall now explain. Assume first that
P
is an elliptic
differential operator and that f
:
X
+
X is a
self-mapping (also on the appropriate bundles) commuting with
P
and
*.
P
Then one can
*
consider the action induced by f on Ker P and Ker P and define the Lefschetz number L(P,f) = Trace (fl~erP) when
f
*
-
L(P,f) = Trace e-tfP f
Trace (fl~erP
*)
is the identity L(D,l) = index D.
We clearly have
where
-
*
Trace e -tfPP
on the right stands for the operator
induced by f.
Now as t
+
0
the right side has
an expansion involving integrals over the fixedpoint set of f.
When
f
is an isometry and
P
is a classical operator one can extend the invariant theory approach and derive the explicit formulae of [ 9 ] .
The situation is particularly simple when
f has only isolated fixed points.
In this case
only the leading term in the asymptotic expansion contributes and one obtains the formula of [ 5 ] .
M. F. Atiyah If we now have a one-parameter group of isometries f(u) having only isolated fixed points we can apply this simple Lefschetz formula to compute L(A,f(u)). fixed points.
It will be given by a sum over the Letting u
+
0
f(u)
+
identity we
then get by continuity a formula for index A.
On
the other hand Bott in [ l o ] proved certain residuetype formulae relating sums over fixed points to Pontrjagin numbers.
Combining all this one gets
the desired formula for index A on any manifold admitting such a one-parameter group of isometries. In particular this holds for homogeneous spaces of compact Lie groups and these are sufficiently numerous to characterize our universal polynomials. This method has the advantage of not having to compute indices and also it produces the various universal polynomials in a direct and elegant fashion. We could of course compute indices on homogeneous spaces directly in terms of representation theory. However this might be thought slightly circuitous because the best way to derive the character formula is by using the above Lefschetz formula.
M.F. Atiyah References M.F. Atiyah, Riemann surfaces and spin structures, Ann.Sci.Ecole Norm.Sup. 4 (19711, 47-62.
, K-theory, , Bott
Benjamin 1967.
periodicity and the index of
elliptic operators, Quart.J.Math. 19 (19681, 113-140. and R. Bott, On the periodicity theorem for complex vector bundles, Acta Math. 112 (19641, 229-247.
,A
Lefschetz fixed-point
formula for elliptic complexes 11, Ann. of Math. 88 (19681, 451-491. and V.K. Patodi, On the heat equation and the index theorem, Inventiones Math. 19 (19731, 279-330 (and Errata, 28 (19751, 277-280). and A. Shapiro, Clifford modules, Topology, 3 (19641, Suppl. 1, p. 3-38.
M.F. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (19621, 25-45.
M. F. Atiyah
9.
M.F. Atiyah and I . M .
S i n g e r , The index
of e l l i p t i c o p e r a t o r s 111, Ann. of Math. 87 (19681, 546-604.
10. R. Batty Vector f i e l d s and c h a r a c t e r i s t i c numbers, Mich. Math. J. 14 (19671,
231-244. 11. N. H i t c h i n , Harmonic S p i n o r s , Advances i n Math. 14 (19741, 1-55.
C. I.M.E.
Some Aspects of Invariant
?heory in Differential
Geometry.
.
R BOTT
C. I. h4.
E
Lectures delivered at Varenna, August 1975.
Table of Contents
Introduction Some basic theorems in differential geometry 7lhe group GL(n, IR) The Group Diff (M) The Jet-spaces JkM The natural thickenings of
M
The first main theorem of Invariant Theory Characteristic classes of foliations
R . Bott Some Aspects of Invariant Theory in Differential Geometry Raoul Bott 1.
Introduction. A central theme in the development of topology and geometry has been the
relation between Riemannian curvature and the topological shape of a manifold. Roughly speaking, certain universal polynomials in the curvature tensor Riemannian structure g on M
4k
--
the Pontriagin forms of
deRham theory of Pi(M)
.
give r i s e to differential forms
M
,
g
a r e independent of
--
P.(g)
R of a
of dimension
whose cohomology classes in the g
,
and s o can be denoted by
These classes a r e therefore invariants of the differentiable manifolds
and play a central role in all aspects of global differential topology, and analysis. My first aim in these lectures will be to present, in a self-contained manner, a relatively new way of getting at these classes, which a s noted only recently, say 1972, by several of u s (Bernstein, Rosenfeld, Haefliger, Malgrange) essentially simultaneously,
as a consequence of the work of Gelfand-Fuks and
Godbillon-Vey.
Of course once observed it was also clear to all geometers
concerned with
G
- structures
and would have been no surprise to E. Cartan.
The essential point is the following one. Traditionally the theory of characteristic classes is developed f o r Lie groups and bundles in general, and then specialization to the orthogonal sub-bundle
*
O(M) c F(M) of the frame
I wish to thank the NSF f o r partial support under Grant MPS 74-11896 of the work reported here.
R. Bott bundle
F(M) of a manifold yields the characteristic classes of
M
.
On the
other hand, a smooth manifold naturally c a r r i e s more structure than its tangent frame bundle, f o r instance f o r each k
>1 , M
h a s a natural k-th e
r
frame bundle J (M) associated to it, and our observation is simply that already k in J (M) 2 Pi(M)
one finds a natural curvature class
can be constructed without choice.
R(M) of
M
,
out of which the
This R(M) is furthermore the
universal curvature of a Riemannian structure on M
,
in the sense that under
the natural map Levicivita OW)
> Jk(M)
furnished by the Levicivita connection of a Riemannian structure g
,
R(M)
goes over into R(g) I myself learned about [ 45
1,
R(M)
many years ago in a lecture by Kobayashi,
however, i t s significance became clear to m e only in the framework of
Gelfand-Fuks, whose theory surveys all the possible invariants of might be culled from the higher order contact bundles of
M
.
M which
Of course their
conclusion is that although "interesting" classes can be found in J M k
, their
dimension is too high to yield new invariants of manifolds. On the other hand, a s we will s e e in the last section, these invariants can also be applied to foliations of codimension nontrivial.
n
and there indeed they can be
R. Bott I will try to keep these lectures a s self-contained a s possible, but do expect some experience with manifold theory in my audience. Basically, I have in mind an audience, say of analysts who feel ill at ease in global differential geometry and especially Lie groups etc..
1 will therefore start with deRham
theory and the basic integrability criterion in Section 2
. Thereafter in Section
3 I will construct and discuss the structure equation for some of the classical
in
groups in considerable detail. From there we move on to the group Diff(M)
Section 4 and in Sections 5, 6, 7, I will outline the computation of the characteristic classes cohomology from this point of view of jet-spaces.
l l e s e are
essentially the computations initiated by Gelfand-Fuks. Section 8 deals with foliations and vanishing phenomena in general. By and large these lectures then, are a pedestrian account of a point of view developed jointly with Andre Haefliger over the past few years. A superb but rather condensed version can be found in Haefliger's Bourbaki Seminar Lecture [ 36
1. For a short survey of this material see our note [ 10 1. For an account
of some of this material, but slanted more to topology than geometry see [ 0 ]
.
Finally, I have appended the bibliography from [ 0 ] , to related material, which attests to the interest in this general subject in recent years.
August 1975 Harvard University Cambridge, Mass.
R. Bott
2.
Some basic theorems in differential geometry. I will assume the basic facts of deRham theory.
Thus every
coo-manifold comes equipped with a complex of differential operators
where M
/f T*
, and r
denotes the i-th exterior power of the cotangent bundle of
denotes the space of coo-sections of a vector bundle.
Furthermore d'
(2.2)
= 0 and
H4de~am (M)
We also write
coordinates
Ker d/Im d
n q ( ~ )for
space of q-forms on M 1 x ,
=
, xn
. If ,
in
rAq T*
.
~ A T * ( Mand ) refer to this
space a s the
U CM i n an open subset of M covered by
then
w E nq(u)
by
a cm-function on U , alternating i n its q-tupled indices, 9 and the differential operator d has the local representation:
with LOil..
,
The formula d2 = 0 then follows f r o m the symmetry of the second partials, and the naturality of (2.4) follows easily from the chain rule.
R. Bolt In short one may think of u E
nqM,
a s a collection of.q-forms defined
on each open set of U of an atlas for M
, and agreeing on the overlaps
of two open sets of the atlas. Given a map f : M A
N
,
the deRham groups behave contravariant-
is invariant under a smooth homotopy of f
. In particular a s
lRn is
contractible H"(IRn) = IR in dim 0
.
This i s of course the well known ~ o i n c a r ; lemma. its infinitesimal version. then the Lie derivative
where
Namely,
Finally recall
if X is a smooth veotor field on M
E(X) in the direction X , acts on
nM
by
L(X) i s the antideriv5tion which on 1-forms is given by the
duality L(X)w =
4x1
.
To a topologist L(X) thus plays the role of a homotopy operator for the operator E(X) showing that E(X) acts trivially on H*nM
.
Next let me recall some of the most basic existance theorems which a r e the standard tools of the geometer:
,
R. Bott 1) Implicit function theorem:
p EM
If
fi,
,
then they can be completed to a coordinate system near p by
,f
4
a r e smooth functions with dfl A
the addition of n -q function fq+l,
2)
, fn
A
df
4
# 0 at
.
The fundamental existence theorem of ordinary differential equations : If X i s a vector field on M with X # 0 , then there P exists a coordinate system xl, xn near p so that a ,
near p,
a ax.,
X
Recall that to create this coordinate system one may proceed a s follows.
centered at p
,
then i s transversal to X in the vicinity of p
,
, yn
Choose some coordinates yl, y2,
and by a possible renumbering arrange i t so that dyl(Xp) f 0 The plane P : y1
=0
, xn
and we can define the desired coordinates xi, xl(q)
= time needed to
Next, letting
P
3)
,
set x.(q) 1
q
by :
flow along X from q to P
.
be the intersection of this flow line with
= y.(q)
for i > l
.
A fundamental extension of this theorem due to Deana-Clebsch and known a s the Frobenius theorem i s the following: Assume that Xlr
-.., Xm
are vector fields on M
span a q-dimensional space near p EM
{xl,
, x ~ at ) each
. Assume further that near
p
, which point q
R. Bott
Then there exist coordinates
centered at p
, such that near
p
There a r e many equivalent formulations of this result all of them useful in various contexts. VARIANT 1 X, Y
.
Let me state some of them:
Let V CTM be a sub-bundle of TM such that if
E r(V) then [X, Y] c r(V)
. Then, locally
V = Ker df where f
i s a submersion onto a manifold of dimension T M/V M P P
VARZANT 2.
1 Suppose 8 ,
,
such that dei i s contained in the ideal s
h n(M)
a r e independent 1-forms on M (
&,
. Then there exist coordinates
such that sufficiently close to p 1
or
as the 8 . * * * ,
er
VARurrT 3.
1 the dx ,
*,
3,
or)
. ., xr
, dxr
spanned by the near any p E M
span the same space
. Given B ~ , * = * , 6 with
u=dA0**
and dWAW= 0 near p . Then the conclusions of Variant 2 hold.
.
R. Bott This theorem i s proved by induction and the following type of argument:
Suppose X and Y a r e two vectorfields satisfying our
hypotheses
.
Then we may, by Theorem I, assume that in a suitable
coordinate system centered at p
and after subtracting an appropriate multiple of X from Y that Y i s of the form
in the same region. It follows that
and therefore does not involve follows that [XYYI where
i s a smooth function.
--a
b"l
. Being i n the span of
= XY,
X, Y i t
near p
Hence for any smooth function cP
and we can clearly choose a .(p with
-.--
with
by simply integrating i n
In short we can span the same s p a c e a s the original cP X and Y did, with a new X and Y ( = e Y) such that
the xl direction.
R. Bott simply restricts to forms from g to g'
. F o r instance, in our case
this procedure leads, to the-following results: EXAMPLE 1
.
The Special Linear G r o u ~SL(n,lR)
.
This i s the subgroup of A E GL(n, lR) with det A A
+
=1
.
Because
det A i s a homomorphism of GL(n, lR) onto GL(1, lR) the induced
map on forms sends the left invariant form 8 :
of GL(1, IR) into a left
invariant form 8 in GL(n, lR) which will vanish on SL(n, LR).
To find
this 8 we need to understand the map det only near the identity;
and
there: d d t =
dxi
,
as follows directly from Krarner's rule. Hence
8 = Ci
EXAMPLE 2.
and the structure equations of SL(n, lR) a r e
The Orthogonal Group O(n)
.
This i s the subgroup of matrices A with A ~ * A = l .
In our coordinates this implies the relation Za x: xy = 6i j matrix notation: (3.14)
xt*x
Thus on O(n) the matrix of 1-forms
'
o r in
= 1 . 8 = y dx i s given by xt
dx
.
R. Bott We now plan to change the coordinates x2 so that Y = -F2
x2, . * * ,
*
, xn
. F o r this purpose we again let
of p on the hypersurface because Y
,
.5 = 0
near p
2=0.
.
- leaving
5
alone-
P be a neighborhood
The field Y i s tangent to P a t p
Hence we can choose new coordinates
xn on P near p so that
It follows that near p
with
and,in view of (2.7)
Hence in this new coordinate system
Y
- a
-2
a s was to be shown.
R. Bott 3.
The full linear group. In this Section I would liki to construct the left invariant forms on
the full linear group GL(n, lR) from scratch because they play such a fundamental role in all phases of modern geometry.
The underlying space
of GL(n,lR) i s of course the set of real n x n matrices with determinant
3
0
lRn\
. Thus
GL(n, lR) i s naturally an open subset of the Euclidean space
and a s such has a natural structure of a real analytic manifold.
Under matrix multiplication, which i s a clearly real analytic law GL(n, B) i s furthermore a group and in fact therefore a Lie group, i. e., a group object in the category of differentiable o r real analytic manifolds. Now for any such object G
, the
deRharn complex RG, contains a natural
finite dimensional subcomplex formed by the left invariant forms InvG *n(G) on G
for every g E G
,
, i.e.,
those forms w E RG for which
l g denoting the map
given by
Because G acts transitively on itself, it is'clear that an invariant t
form w i s determined by its value at one point and therefore for instance by its value at the identity e E G tangent space to G at e Ey g to.
g
. Hence
, then i t i s
clear that 'restriction of tdq
imbeds InvG SIq G into
i f we denote the
B* ,
R. Bott and it i s not hard to show that this map i s onto, so that as a vector space over lR
On
Thus the particular nature of G i s not apparent at this stage. the other hand because each I g commutes with d
,
InvG fi G i s stable
under the deRham d, and so furnishes us with a local invariant of G
.
In particular this local invariant can be put i n concrete form a s follows.
*.
Let r a be a base for g
Then we must have
where the cQ E IR a r e skew symmetric in
Br
a sense capture the local nature of G
8, Y , and these
numbers in
. Traditionally one of course
favors the dual point of view. Namely, one extends the elements of g = T G to left-invariant vector fields.
be closed under the bracket so that algebra, "the Lie algebra of G
",
These a r e furthermore seen to
g becomes a finite dimensional Lie and in t e r m s of a dual base X
OL
Eg
to our r Q the structure constants of G now appear in the form
Thus (3.3) and ( 3 . 4 ) a r e duals of each other, and in this context the dual of the Jacobi identity
i s seen to be
K. Bott
Of course f r o m the topologists point of view the complex ( A gjr, d) which we will denote by fl( g)
, has the advantage that we a r e naturally
*
lead to consider i t s cohomology H ( t2 g) ~ " ( f l ~ i s) related to H*(G)
.
,
and to the question of how
But before we turn to these m a t t e r s l e t u s
construct the left invariant forms and the structure constants for
GL(n, IR)
- the arche-type of all Lie groups. F o r this purpose i t i s convenient to think of the matrix A a s a smooth function x? (A), i j = 1, J
a r e then global coordinates on G
*-,
n
(i,j)-th
.
entry of a
These functions
= GL(n, lR) and they satisfy the rule
Hence under P A these functions transform by :
Their differentials therefore transform similarly
and in particular a r e not left invariant.
But i t i s not difficult to c o r r e c t
Indeed, l e t yf be the function on G given by y f ( ~ ) = x ! ( ~ - \ , J J J s o that the y's and x's satisfy the identity
for this here.
R. Bott
yk.x;=
(3.10)
€$
a!
where
bi i s the usual Kronnecker delta j
It follows that
so that
Hence if we set (3.11) then
Thus the
e!
a r e n Z left invariant 1-forms on G 3 with dx! at 1 E G , and so form a basis for g*
, which
agree
.
J
To differentiate our forms, l e t me rewrite this development i n matrix shorthand
. Thus
x stands for the matrix of function
~ I xiJ.I/
etc.
R. Bott (and i s therefore ultimately just the identity map !) Then we have
whence
but which gives ybrytdy = 0
de =
- ydx Ay dx
- -
In terms of our
& J
8
~
.8
this amounts to
and these a r e the "structure equations" or "Maurer-Cartan equations" for GL(n,lR)
.
The corresponding structure constants a r e given by
but by and large (3.12) i s of course a better way of recording these than (3.13)
. Once the structure equations of a Lie group G a r e written down
it i s an easy matter to deduce them for a subgroup G' C G
.
One
R. Bott Differentiating therefore yields d t 9 = 0
(3.14)
on
O(n)
.
Hence the structure equations for O(n) a r e given by (3.15)
d $ + x e i
EXAMPLE 3.
a.ith
~ 8 : = 0,
ejt@=
The Syrnplectic Subgroups Sp(J; IR)
0
.
.
Let n = 2m and let J be a nondegenerate skew form on RZm The subgroup preserving this form 9 i. e.
9
.
the matrices A with
A ~ J A= J
(3.16)
then define the symplectic subgroup Sp(J; IR) of GL(n; IR). Proceeding a s before one obtains the relations
whence
8 = y dx
= J- I x t J dx on Sp(J)
.
This time differentiating (3.17)
yields the "symmetry":
characteristic of Sp(J; lR)
.
If we choose J to be given by the matrix
each entry indicating an m x rn block, then J~ = -1J the form:
so that (3.18) takes
R. Bott
Pictorially this implies that the matrix 8 has the symmetries:
and modulo these, the structure equations a r e given by the old formulae (3.12)
. EXAMPLE 4.
The Conformal Group CO(n) c GL(n, lR)
,
The. conformal group CO(n) c GL(n, lR) i s given by the subgroup generated by O(n) and the scalar multiples of 1 in GL(n, lR) CO(n) = O(n) x
IR* , and the defining equations a r e
.
Thus
:
Proceeding a s before, this leads to the relation t 8
+ 8-=
2 (Trace 8 ) 1
,
so that here a baais i s given by
~ f i, f j , EXAMPLE 5.
and
6 =
+(G
ea)
.
The Subgroup Preserving a Subspace; GL(n, p; lR).
As a final instance consider the subgroup GL(n, p; lR) preserving IRP clRn under the standard action of GL(n, lR) on lRn
. If
el,
., e n
R. Bott i s the standard basis then this action i s given by:
so that the defining relations of GL(n,.p; lR) a r e simply given by
XP
= o
for
a> p i_p,
iL
8PL J
for I 0 on G L ( ~P) ,
n.
These examples should suffice for u s and I will therefore continue towards our main goal, by constructing the corresponding structure equations for the Lie groupoids which underlie ~ ~ - ~ e o mine tthe r ~ next sections. Let me close this section however, by briefly discussing cohomology of these complexes of left invariant forms, and how
the this
concept might be of use in geometry. F i r s t a remark on H
* (ng) . In genera1,computing
this space from
the structure constants of g i s a difficult task and usually has to be attacked indirectly.
F o r the "reductive Lie-algebras
g
this problem
R . Rott can however be reduced to a problem in invariant theory.
The
argument takes the following form : First observe that right translation rg by an element g E G commutes with all left translations. on
n( g )
which preserves d which is the dual of the usual adjoint
representation of
. The infinitesimal right translations
G on g
are given by the Lie-derivatives in g
Hence rg induces a representation
C(v) , of left-invariant vector fields v
. We may therefore consider the sub-complexes
given by the infinitesimally and globally invariant elements under the adjoint action of G on
n(9 ) ,
and of course these coincide if G
is
connected. In any case determining these spaces i s clearly a question in the realm of invariant theory. On the other hand, I claim that the d these spaces of left and right
invariant forms inherit from n(9) i s trivial, so that all homology questions disappear on them. To see this, recall that for any X E g
, (considered now as
a left-
invariant vector field on G !) C(X) = L ( X )+~d L(X) on all of n(G)
(3.22)
. On
nl(g)
c n(G) ,
S(X) 6
=
'(X)
L (XIde
,
i s constant, so that there
,
which enables one to express d on 01( g ) in terms of the C(xa) where
R. R o t t
'X
ranges over a base for
8
.
Indeed if Xa and
,rDL
are dual bases
we see from (3.22) that
for any 8 E R*(g). 8 , which means E(X) 8 = 0 for all X
Hence invariance of certainly implies d e = 0
,
.
Now then the reductive action of
g
ng
on
implies that 0 g breaks
up into a direct sum of subcomplexes
ranges over the irreducible representations of g
where
on ( 0 g ) * only via A
.
On the other hand g(X) acts trivially
on H(0 g ) in view of the homotopy formula (2.6).
~nv'n(g)
.
X
with A
In short then, one has PROPOSITION 3.1. (3.24)
and g acts
It follows that
has a corresponding decomposition.
homotopy type of n g
,
- the trivial
g
Hence fl g has the
- module,
i. e.,
E
For a reductive Lie algebra :
~ * ( n g ) 1 lnvg n(g)
.
with
R. Bott Of course this still leaves a lot of work to be done, and in fact although (3.24) goes back to E.' Cartan, the actual computations of the invariants in n ( g ) when G i s one of the classical simple Lie groups was only carried out by R. Brauer in the 1930's
. We
will return to this
problem later; here let me rather turn to the geometric analogues of (3.24) and the unitary trick.
Here the problem i s to compare the cohomology of n g with that of nG
,
and the result analogous to ( 3 . 2 4 ) i s given by:
PROPOSITION 3 . 2 .
If
G i s a compact connected Lie group then
The argument for ( 3 . 2 5 ) i s to decompose left translation
- into a direct sum of
nG qua G module under
subcomplexes
ranging this time over the irreducible representations of G
. As
G
i s compact this i s possible and'the projection on Cl G being given by an
X
integral operator, and in particular
n g i s precisely
nX G
with
the
trivial representation and the projection of !J G to fig i s simply given by averaging a form 8 over G :
This result then brings the Lie algebra cohomology into line with the topological cohomology for compact groups.
For non-compact ones,
*
however, there i s in general very little relation between H ( n g ) a n d
R. Bott
H"( R
. Here it i s only for the semi-simple groups that one can still
get at H( Rg ) via (3.24) and a principle H. Weyl calls the unitary trick. First of all, given a Lie algebra g over 1R
, we may
construct a
new Lie algebra over the complex numbers Q: simply by extending the Q:
base field and we denote this new Lie algebra by g @ Q: o r simply g
lR
Thus if the X a generate
then
and 8 has structure constants
ga is also generated by
now however being linear over naturally a complex of Q;
c:e
:
Xa over Q: and (3.26) still holds, the [ Q;.
.
,]
Q: The complex R (g ) = R ( g ) @ Q: is
- modules and
Q: H*(R g ) also.
lR
Therefore it
follows from
n(gQ;) = R(*) @Q:
,
that
H*(RgC) = H b ( n g ) @ Q :
(3.27)
1R
.
In short, complexification commutes with homology and hence two Lie algebras with isomorphic complexifications have isomorphic cohomology. Now the semi-simple Lie algebras 8 have the marvelous property that their complexification
gQ: are simultaneously the complexification of
a Lie algebra k( g ) of a certain compact connected Lie group K( g ) so that for these Lie algebras one has :
,
R. Bott This trick also works for gl (IR) because
where u(n) denotes the Lie algebra of the unitary group. Indeed, using the same type of analysis a s in our subgroups of GL(n; lR)
, we
identify U(n) E GL(n;
(C)
with the complex matrices A
subject to
P-A = 1 , where the bar denotes conjugation, and hence the Lie algebra of U(n) with the skew Hermitian matrices satisfying:
K ~ + A= It follows that iA, i s Hermitian for any
0
.
A E u(n) and hence, as every
matrix i s the sum of a Hermitian and a skew Hermetian matrix, d(n) O (C
gl (n, (C)
that
. The upshot i s therefore that
and this i s how, by a d large, topologists prefer to compute the left hand side.
In any case let me conclude this section by recording the answer
here for future reference: THEOREM :
where -
*
H
{n gr (n. IR)}= E(hlr
-
,hn)
denotes the exterior algebra generated by the elements
R. Bott
.
hi
Representative forms for the hi can be given to be hi
=
Trace (8 A 0 A * * * A e }
2i-1 factors
-8 and
=
11 (IiJ 11
is the matrix of our left invariant forms.
R. Bott 4.
The Group Diff M
.
We turn now to the analogues of these classical constructions for certain function spaces and in particular, the group Diff (M) of diffeomorphisms of a COD-manifold. Here Gelfand-Fuks proposed the complex OCLM forms on the Lie algebra analogue of h6g
, of continuous
LM of all COD-vectorfields a s the proper
in the finite dimensional case.
Precisely then, let L(M) be given the CODtopology, so that Xn
-(
Y , if and only i f on every compact K c M
, the components of Xn
relative to a finite set of coordinate systems covering K
, a s well a s
all their derivatives, converge to the components and corresponding derivatives of Y
. With this understood
alternating continuous q-forms on LM clearly a continuous map LM x LM
+
h6:
LM is defined a s the
, and a s
LM
X, Y
-
[X ,Y]
is
, the old formula for d on
n g preserves this complex. Recall that explicitly:
To check that this complex which we also denote by A(M) is the proper analogue of
a W A G ,
W' then (f
(4.2)
0
g)
#
= g"
0
f
#
where g* denotes the deRham homomorphism induced by g b)
where P
c)
,
Left invariant: that is for any diagram
g
denotes left translation by g , one has
Universal i n this context
to each map f :W
-'
G
.
That i s , given a function @ which
, assigns a form @(f)E
(W) subject to a) and
b) : g*@(f) = @ ( g f ) , then there exists a unique cp
and
@ ( f o I g )= @(f)
E Sd Q with
,
R. Bott Mf) = Indeed to construct
Q
,
.
PCP
just consider the identity map 1
C+G and set
=
Q
.
Now these criteria also make perfect sense i n the infinite dimensional case, once we decide
when a map W
3
Diff (M)
i s to be considered a s smooth, and here the natural criterion i s clearly that the adjoint map f'
,
given by
be smooth.
Note by the way that this criterion works equally well for
the function spaces of all
maps from M to N
, as
well a s for many
of i t s natural subspaces. To return to our main task we have to construct a map
subject to (4.2) and (4.3) for any map f :W
Diff (M)
.
R. Bott The construction of f # proceeds as follows:
Consider the map
ft: W x M M ' and l e t us think of W X M a s a fibering over W
denote the natural projection.
, with fiber
M
, and let
The product structure furthermore gives
us a natural decomposition of the tangent space to W XM into a vertical and horizontal space
Now consider the map ft
, and let
Hf be the sub-bundle of the t tangent space to W x M which i s in the kernel of df : Hf
(4.6)
.
= Ker dft
Restricted to a fiber of n
,
ft i s a diffeomorphism.
vertical space intersects Hf only in 0
. Furthermore dft i s an Tn M, n = f(w) . m . In short,
isomorphism of T M at (w,m) onto m Hf is indeed a sub-bundle of dimension equal to dim W to the vertical tangent space in W x M described by:
Hence the
, and transversal
. Pictorially the situation
is
R. Bott
Figure 4.1
the slanted l i n e s indicating the sub-bundle Hf
.
In this situation then note that every vector field X on W has two natural liftings to W x M , namely the lift HX given by
the
horizontal lift relative to the product structure, and the lift H r relative to the horizontal complement Hf
.
Thus H$
by the fact that it maps t o X under
is uniquely characterized
, and i s
TI
contained in Hf
. Now
then define
a s the vertical com@onent of
Hp
i n our product decomposition.
i s a pointwise operation sending a tangent vector Xw f Thus to a vector field V F w on W X M , tangent to the fiber w X M Note that V
.
Vf i s a construction f r o m smooth vector fields an X to smooth vector fields tangent to the fibers on W x M
.
R. Bott Now the continuity of a 0 E A ~ ( L M )implies that we may interpret such a
e
a s a q-multilinear function from the cross-sections of the
tangent bundle along the f i b e r s of r to
functions on W
interpreted we define the desired construction f
Therefor*
it remains only to prove that f
,
and so
'
by :
'
commutes with
d
.
i s natural and invariant. L e t me explain thetlcommuting with d
o n 1-forms
6 E A1(M)
.
The higher ones proceed analogously. We have, by definition.
while, again by definition,
To bring these into accord we therefore need to recall f i r s t of all that Hf i s an integrable sub-bundle of W x M Hence [HfXy HfYl and similarly of course,
= Hf[Xs Y]
.
R. Bott It now follows from H p
Hf[X, Y]
V p
+ HX
= [ VfX,
that
VfY]
+ [HX, VfY] + [HY, V p ] +H[X,Y] ,
whence,
Finally, it follows again from the continuity of
for any section U of the bundle along the fibers of
"
8 that
. Hence using
(4.12) and (4.13), (4.10) takes the form
The proof of naturality i s quite straightforward, and of invariance also. Indeed, if
=
dIV* xkl
k
* i
a!
= (rp yk)ua* dx
.
i
Next consider y dxk k j
. We have
On the other hand
P)
9 ,
i k m * 1 kxax~ymdx~ = Yk xjm + 'P (yk) U* j 6
,
R. Bott
R. Bott Hence both the expressions considered failed to be invariant by an equal e r r o r and so their difference is invariant.
8'
8:
Q. E. D.
I will leave it a s an instructive exercise to the reader to identify the
of this section with our earlier ones
6 in A P
.
Let me finally discuss the relation of these formulas to those one finds
in the text on connections. We have already remarked that J2M i s naturally "fibered over J M and JIM over M 1
the fibers being diffeomorphic to GL(n, lR) for r1 and to a Euclidean space n of dim n(2) for
r2
.
Actually each of the spaces Jk(M) should be thought of a s a "Lie group over
M"
,
in the sense that there is a Lie group
GL(n, R)'~) which
naturally acts on the right in J (M) ; Furthermore locally the map k k I I M <JJM) always has a section, and if
s : u->
J (M) k
is such a section over u then the map
$s: UxGL(n,lR) (k)->
d(u)
R. Bott given by
in a diffeomorphism. Such "Lie groups over principal bundles over
M " a r e technically called
M and occur in many places in geometry. Here let me
just obselve that the group GL(n ,lRjl) = GL(n, IR) and that the right action of
1 11
on J ~ C I R is ~ )simply given by: e
(5.25)
*
i i * i k x = x ; a x i = x a k j
The higher groups GL(n, lRjO) k > 2 are the groups formed by the k
- jets at
0 of invertible diffeomorphisms of IRn
.
Now with this in mind, consider an arbitrary smooth section
.
s : JIM->J2M
Topologically there is no obstruction to such a section because these
In any case we have the following:
fibers are contractible.
Let
PROPOSITION 4.2. Then the forms
s* kIi
1
s : JIM r J M be any smooth section. 2
restrict to the left invariant forms denoted by
section 2, on any fiber G L ( ~lR) ,
,
JIM
.
together with the
i
in
f -
9. determine a framing of J and that the field of planes 3C determined by the kernel of the
It follows that the T JIM
ei
of
i
9
R. Bott
X(s) = Ker s* Eli
f
i s transversal to the fibers of J M 1
Proof :
.
If one applies s* to the local expression (5.23) for
&1
one
obtains
because pulls the
* i i * i i s x = x and s x. = x in our notation conventionland so s*
J
em
on J2M to the
em
annihilates the
j
em
on JIM
.
Now restriction to a fiber
and the first term clearly restricts to our ,\eft invariant
forms of section 2. At this stage we are very close to the theory of connections a s it is treated in most contemporary texts. Indeed one definition of "connection" for a manifold M is the following. Definition. of dimension n (5.25)
2
A connection on the manifold
JIM
M is a smooth plane field 3C
, which is:
transversal to the W r s
,
and (5.26)
invariant under the action of Hence the horizontal space X(s) of a section
GL(n, IR) on J 1(M)
s : JI(M) r
.
J2(M)
R. Bott
,
nearly constitutes a connection on .M not satisfy (5.26)
. If
in that it satisfies (5.25),
but need
K(s) does satisfy (5.26) a s well, then we call
s
equivariant, and these now yield all "torsion free" connections on M a s we will see in a moment.
In the present context the torsion of a connection X is
defined a s follows. The horizontal space X enables u s to extend the left invariant forms on J€!
J
iij
GL(n, R) to forms
on all of JIM hg the
requirement that
annihilates X
.
iij
ei1
restrict to
on each fiber and
iij
It follows that the expressions
l-i = dBi
+
,:A&
a r e "horizontal" forms in the sense that they a r e in the span of the these a r e called the torsion forms of the connection M
d ,
. Now in view of
and (5.24)
we have d& s o that X(s)
+
, if it is invariant,
=
si{nd
0
,
is automatically torsion free.
Let u s now look in detail at what the equivariant sections of J2 M look
*
i
like. Under the transformation a x. =
J
a*
ef
= A;
Xkaj
t
,
we have
R. Bott where
is the inverse of
a
. Similarly
Hence X(s) will be equivariant if and only if
*
i
and this condition implies a restriction on the functions s x jk i far were arbitrary sn~oothfunctions of the xi and x j
,
which so
.
In fact substitution (5.23) and (5.27) yields the relation :
whence
*
i
s (x ) must have the form jk
where now the f
are arbitrary smooth functions of the x's
symmetric in
the lower indices. In any case it is clear that locally such sections can be found and they can then be patched together w e r
M by a partition of unity without trouble. This
development is now completed by the following
R. Bott PROPOSITION 5.3 the form X(s) where Proof : LEMMA.
.
Every torsion f r e e connection X
s is an equivariant section of J M 2
isof over J 1M . J M 1
This follows directly from the folloying
Let
sl
and
s
2
be two sections of J2(M)
ri
.Then
,
a r e smooth function on J M symmetric in jk jk I versely given, a section s and a system of, functions where the
ri
jk
=
fkl ,
. Con-
Tijk
1
then there is a unique section s with 2 -
This in turn follows directly from the local representation (5.23)
,
and
k the fact that the s* (x. ) a r e quite arbitrary except f o r the symmetry in j Jm and m
.
To prove the proposition, consider a torsion free connection X on JIM
,
and let
-i
8
j
be the corresponding extension of the invariant forms on
the fiber. Also let s be a section of J2(M)
is horizontal and hence there exist
. Then
cW functions Ti
jk
such that
R. Bott Now because X is torsion free dBi
+
dei
+
,
= 0
J
and in any case S*$nd
=
.
o
Hence subtracting these two we obtain the condition
Therefore (k and s
1
is symmetric in j
and k
,
and so by the lemma, there is
with
Remarks. Connections a r e often introduced a s differential operators on tensor fields over M and a word concerning the relationship of that definition and our present one is appropriate.
?he essential point linking them
is that the J1 construction enables one to define all tensor-fields on M functions on JIM
as
subject to an equivariance condition which is characteristic
of the tensor field. Precisely, given a representation p : GL(n, 1R)
the tensors of type D on
M
,
- End V 7
can be defined to be the function
R. Bott subject to
With this understood, let X be a connection on JIM field on JIM
M
. Clearly
X
whichprojects to X
,
X a vector
% M ,
lifts to a univque horizontal vector field
. Hence if
f
is any tensor fieldon
is a new function which because of the equivariance of 3C again satisfies This is the covariant derivative the connection X
vXf
,
of
f . in the direction
on
5.33.
X vis-;-vis
.
In conclusion let m e now interpret the curvature of connection X a s the obstruction to the integrability of X PROPOSITION 5.4
.
Let
.
X be a connection on JIM
be the corresponding extension of the invariant form on the fiber.
;i j
and let
X
is integrable if and only if the forms
vanish identically. Proof : -
Integrability of X is equivalent to the requirement that
-i d 8 . b e in the ideal 9 generated by J
"8f
a s we learned in chapter 2. Now
R. Bott d
3J
fd
ndi (X) differ by an element in 9 so the integrability criterion is
J
equally expressed by the requirement that
i R. (3C) 3
be in
d
.
However,
i R. (X) has the virtue of being a horizontal form: That is, there exist functions 3
Ri jke
skew in k and e
granting
(5.35)
mentary to the
To prove
(5.35)
,
such that
the proposition is immediate, because the
9 ' s a r e comple-
-i
8 . 's : so that 3
it is sufficient to show that i
L(V)R. 3
5
0
for any vertical vector-field on J ~ ( M ), and a s these a r e spanned by the infinitesimal right translations, of (5.37)
GL(n, IR) it i s sufficient to prove
L(X)R~=O j
NOIVthe equivariance of X under
.
GL(n, IR) implies that
But then by the structure equations on each fiber = -r(X) ?l:A
"k
ej , yielding
(5.37)
.
P(X)
@j =
J(X)
"8 c d .
J i L(x) d 8 . 3
=
Q. E. D.
R. Bott As an immediate corollary we now have: COROLLARY
in -
.
The Invariant forms
JZ(M) play the role of a universal obstruction to the integrability of a
torsion free connection on J 1M in the sense that if
in the section defining the torsion free connection '3C if and only if
then
X i s integrable
R. Bott 6.
The natural thickenings of
M
.
Let me from now on take the position that JkM is defined a s the space of k
- jets a t
0 of imbeddings f : U-
(6.1) where
U is some neighborhood of
>IR
n
ucIRn
0 in a fixed lRn
This corresponds to chosing fixed coordinate Diff M/ Diff (M; p
k+l
)
p
. in our model
, and in the "honest" approach this means that we
associate to f the point of
Jk(M) with coordinates
relative to local coordinates a t f(o)
.
The present version has the advantage of clearly indicating both the left action of Diff M and the right action of
GL(n, IR) on the JkM ' s
they a r e both given by composition:
It follows that we can naturally form the spaces JkM / G L(n, IR)
.
Indeed
R. Bott where we identify points under the right action of
GL(n, IR)
,
and more gen-
erally the spaces
for G a subgroup of GL(n, IR) the orthogonal subgroup of 1IXn
,
. In particular if
O(n) c GL(n, IR) denotes
then the space
i s of especial interest, because O(n) is a maximal compact subgroup of GL(n, IR)
.
For k = 1 the space JIM/ GLb, JR)
reduces to M a s GL(n, IR) is the fiber of the principal bundle J M over M. 1 On the other hand the space
is already interesting. First of aN it fits into the sequence of maps
where w'
is a principal fibering with group
over M with fiber GL(n,lR) / O(n) " Recall here that a map
qn)
, and
n is a "fibering
.
E> M is called a fibering with fiber F
R. Bott if
M admits a covering by open sets Ir,
such that over Ira
product decomposition into b, x F compatible with
E admits a
w : That is, there exist
diffeomorphism
-1 which map the slices p x F to w (p) ; p fiber of
E
Ira
. In the present instance the
w is the homogeneous space GL(n, IR) / q n ) which i s contractible.
Indeed the well known Gramm Schmidt construction of an orthogonal base from an arbitrary one can be interpreted a s a product decomposition
where
T(n) denotes the triangular matrices with positive diagonal entries.
Hence
which is clearly contractible
- one simply lets the off-diagonal entries go to 0 ,
and the diagonal ones to
.
1
Another proof of this fact can be based on the spectral theorem from which one deduces that (6.4 )
GL(n, Ill) = ~'(n) x a n )
where
S (n) denotes the convex set of positive definite symmetric matrices,
+
R. Bott and using this point of view it is easy to see that a section s t M > JI(M) / O(n) corresponds to describing a Riemannian structure on M
,
, with
(v1)'l s c J (M) playing the role of the oorre.$ponding sub-bundle 1
furnished by the orthogonal frames of the structure. Note here that from this point of view the existence of a Riemannian structure on M follows from the quite general proposition that a fibering with contractible fibers admit sections. Note also that such a section can be used to construct a homotopyequivalence of JIM / O(n) Jk-l
k >, 2
with
M
,
and using the fact that the fibers of Jk over
, are in any case contractible we see that for all
PROPOSITION 6.1
9Diff (M)
.
k
,
The manifolds JkM / O(n) are naturally acted
and their projections: M ~k i mH* ( J ~ M/ o(n)) = H*(M)
.
To sum up, the Gelfand-Fuks point of view, leads us to the THEOREM : The inclusion d the Dlff M DeRham theory of the k-th order thickenings of M
x
: H* Clc (a,
, On) >-
,
invariant forms the leads to a natural map
H*(MI
and therefore to natural cohomology invariants on M
.
The next question is, of course, to compute the universal object
H* nc(an , On) and in the next section I will outline how such computations proceed with the aid of some classical theorems of Invariant theory.
R. Bott 7.
The "first main theorem" of invariant theory and some of its
applications. In his remarkable book on the classical groups, Herman Weyl explicitly constructs a basis for the invariants of certain representation of the classical groups. For GL(n,lR) = Aut(V) his "First Main theorem" then asserts that all the invariants of
Aut(V) acting on the tensor spaces
a r e generated by contractions
-- that is -- by the natural pairing of
V @ V* -. lR
Explicitly one can state this result in the following form: THEOREM 7.1 : There a r e invariant maps
only if
k = 4.
, and then these a r e generated by the map vkc+ei....
or -
ep com~osedwith an arbitram ~ermutation U
that is, by maps (7.2)
@$)
Qo0
= n$(,,)
,
of the factors in
given by
Q O D ( V , @ . . .~
3
k .
0 8 )9= l=l .n 61(vD(i,)
1 ~ @.-. ~ ~ 3
k @
V ;
.
R. Bott A dual and quite equivalent formulation reads:
THEOREM 7.2 : The Aut(V) module k C BvBv* has no invariant subspaces when k
-If
k=C
, and
{vi
,d)
+8 .
i s a set of d u d bases for
V and V*
, then
the subspace of invariants in
k
k
BvBv* is generated by elementary invariants of the form
where o is some permutation of the factors in
k @
V
. + , the
These theorems are also valid if Aut(V) is replaced by Aut(V)
automorphisms of det > 0 of V but are not correct for the Special linear orthongonal groups, where additional determinental factors appear. hand if
On the other
G c Aut(V) is the subgroup presendng a symmetric or skew symmetric
form, then Herman Weyl also shows that
THEOREM 7 . 3 : TI : V x
V
-
HZ
, then
G
If G
is the subgroup preserving the form
- invariants linear forms k rp : @V-
> IR
R. Bott exist only if
k
is even, and they are then generated by the elementary forms: cp(vl@...
(7.4)
@ V2n
,
= ~ v ( v v~ ~~ ~, + ~ )
and forms obtained from cp by permutating the factors arbitrarily. Note that in this case V and V* are isomorphic G
- modules ,
so that (7.4) may be thought of a s an elementary invariant coming from
.
with V* 's
GL(n ;R) via an identification of half the V's
Essentially both these theorems are easy to comprehend in that they assert that the obvious invariants generate all invariants.
However, the proofs are
nontrivial and I will not have time to discuss them here. Weyl's book still seems to me the best reference.
By and large, Herman
See also [ 2
1.
Armed with these results one can already make a large number of interesting computations. Let me illustrate by computing
where g-C the Lie algebra of n
GL(n ; IR)
. From our present point of view
it i s best to identify GL(n ; R) with Aut(V) correspon~gly
Hence
- from our discussions in section 3 -
,
dim V = n
,
and then
R. Bott
Furthermore our forms
& B vj ,
Bf
E
1 D (gkn) a r e a basis for V* B V of the form
so that the invariants of
Aut(V) on
9
@(v' @ V)
a r e generated by
expressions
where the j'th
a r e a permutation of the i's
admits an Aut(V) - invariant
. Now the natural projection
section given by the skew symmetric tensors.
Hence the invariants below a r e all images of invariants above. The skew symmetry of the wedge now makes it clear that the basic "potential" invariants a r e given by the cyclic contractions:
Further observe that the ones with an even number of c&etituents vanish, again because A i s skew: For example
R. Bott It folIows then that our invariants are generated by the classes hi = trace (8 A
(7.6)
. A 8)
(21 - 1 factors)
This is about a s far a s the first main theorem takes one. What is
needed to complete the computation of H* (ghn) is the following.
LEMMA
.
of the hi , i 5 n -
For
i > n the forms
hi
can be expressed a s in terms
. Further
and so represent the volume form on GL(n ;lR)
.
Unfortunately I know of no simple proof of this fact purely in the framework of representation theory. CIearIy this is a result in the framework of the "second fundamental theorem" of Herman Weyl dealing with relations between the basic invariants, and I refer you to his book for details. Actually a detailed proof
- going back to the original computations of R.
Brauer a r e found on page
140 of Herman Weyl's I. A. S. Lectures of 1934-35, purely in the context of
invariant theory. As another iIlustration let me now turn to the cohomology of (an, g.tn) where the "First main theorem" enables one to eliminate all the jets of order greater than two.
,
R. Bott Here of course g& c a denotes the linear vector-fields. 'Ibis subn n algebra acts reductively on n(a n) and considered a s the corresponding Aut(V)* module we clearly have
where S denotes the kth k
Symmetric product of
symmetric algebra. Further our old classes
1%)
V
, and
S(V)
the
are a basis for the space.
Hence the elementary invariants under the action of AU~(V)+are again given by total contractions of our
( are invariants in Q(an)
and
6. & elj
Thus for instance
A
. Now recall that by definition
generated by only those invariants which do not involve any
,
Cl(an gtn) is
$j 's
. Hence
our first example is not in this complex while the second one is. 'Ibis restriction on the
n ( a ;g& ) involving a n n
&
makes it impossible to create invariants in
with &(a)> 2
the answer is zero, because the i j in
f$k
. Indeed if we try,
say
is symmetric, while
Oi A
ei
is
of course skew, and this phenomenon is easily seen to persist in any elementary
R. Bott
contraction of
in which some
s
.
has length >2
Thus our complex consists entirely of forms created by contraction out of the
8i
and the
8ijk . Further the symmetry of the lower indices in
seriously restricts the non-zero invariants.
For example
is out. In fact using this symmetric it is now easy to come to the following
conclusion: THEOREM 7.6
.
The complex O(an, g& ) vanishes in dim > 2n n
has only even dimensional elements, and is generated by the 2k Z1
k
k
i where R. J
= Trace (R )
denotes the universal curvature
R =
/ I R i~ / /
,
- forms
kLn
- form
of section 4. COROLLARY (7.8)
. H* {O(an, g.Cn)} = a ( a n , gtn) = ECcl,
where g[cl,. , cn] -
"
, cnl
denote the truncated polynomial ring of proposition (4.4).
R. Bott The first equality is immediate because Q(an; gan) has only even forms
- whence
dr0
. To see that the
c 's generate a s well a s the 1
Z.
1
recall that det (1
+ R)
ci (R)
=
i=o
defined our Chen-forms while
k
Z = Trace R k
Hence these are related by the Newton formulae:
which show that over Q these forms span the same space. Over the integers the c's
are preferable because they generate the B's
over
Z
, h t not
vice versa. Geometrically Diff M
H* {O(on, g.Cn)} corresponds to the cohomoIogy of the
- invariant forms on
R. Bott so that (7.8) evaluates this ring and also implies that the map
J, M-
> J2M
induces a r e isomorphism in the cohomology of the GL(n, IR)
-
Mff M
-
invariant forms. At this stage then, one could expect 2i ci (M) r H~~(J2M/ GL(n, R ) s H (M)
.
M to have "Chern-classes"
Ho.wever, half of these
- the odd ones -
disappear under the natural map:
whose algebraic counterpart is the homomorphism (7.10)
H* 0tan ;gtn) >H*(U~:W~))
To study this arrow one uses a standard spectral sequence argument, which then easily leads to the formula (4.32) of section 4
.
R. Bott 8.
Characteristic classes of Foliations. At this stage we have seen that every
and that in the dimensions
dim n
M naturally gives rise to a map
, this map takes the elements
c
2i
of the
L. H. S. to the Pontrajgin classes of M on the right hand side. Thus our construction yields all the classical characteristic classes. On the other hand H*(an, O(n)) also has classes in much higher dimensions and they seem to be completely extraneous at this stage. For instance for n = 1
,
how is the class
to be interpreted? The correct answer here seems to be that one should consider an n
- manifold
M to be a special case of a foliation 3 of codimension n
.
Indeed for such a foliation all our constructions go through and one ends up with a natural map H*(an, a n ) )
"'
> H*{M(~}
of our universal L. H. S. into the cohomology of the manifold M(9 on which 5
is defined. As the dim of M(5) can be any integer
2
n all the class of
H*(a ;O(n) ) become potential characteristic classes, and for instance hl cl n
R. Bott becomes the Godbillon-Vey class for foliation of
codim 1
. This is, of course,
the class that sparked much of the recent work in this subject. To obtain this development recall that a foliation 3 of a manifold M(3) bundle
can be defined a s an integrable sub-bundle
, which has
TM of M Thus taking
that is why every n
E=0
codim q
E3
on
of the tangent
.
, defines the (trivial) foliation of
- manifold
codim q
trivially defines a foliation of
M by points and codimension n
On the other hand, by the Frobenius theorem, any such an
E
,
defines
a whole class of submersions
u >of open sets which U c M to IF?
IR
n
, with the property that
Ker df = EIu
.
Hence 3 can also be described by a collection {ba, pa) of a cover lra
of
M
, and submersions Qa 'it-
such that on ba (8.4)
n lr
> lRn
B Ker d% = Ker d p
B
*
.
consisting
From this point of view a codimension
- manifold
much like an n
-n
foliation 5
, looks very
with the r ~ o , playing the role of the coordinate
charts. Following this line of reasoning one defines J (5) to be the space of k
k
- jets of maps
submersions of foliation 5
cp with source in
M and target 0 in lRn which are in the
5 in the sense that near the source of ep Ker dcp defines the
.
Note that if 5 is the trivial foliation of Jk(3
M
,
then
= Jk(M)
and one merely assigns to each cp its inverse to pass from one side to the other. In short then, replacing Jk(M) by Jk(3 one obtains a "thickening"
Jk(5) / O(n) of
M
, and the invariant forms on this thickening give rise to the natural
extension of our map (8.1) :
H*(~~,o~)-H*(J~(~/o(~)-H*(M)
.
Furthermore, note that J (3) corresponds to the frame-bundle of the 1 dual of the quotient bundle Q defined by
0->E->TM->Q->O
5 : ?bus if
R. Bott defines Q then
i. e.
, the fibers of
-
1 forms at p
Jl(a) over p c M correspond to frames 9;
, which are independent and vanish on
Hence under ep(v the teristic classes of H* ( a , 0
Q* on M
c2i
,
E P
,..., $
of
.
of the L. H. S. go over into the charac-
and in particular, the truncation of the c's
) none implies the Vanishing theorem which I noted many years
ago. The Pontriagin ring of the normal bundle to a foliation of dimension n vanishes in dim > 2n
.
At present the map (8.5) essentially summarizes the general state of affairs concerning the characteristic classes of foliations.
It expresses the
vanishing condition just described and at the same time predicts new characteristic classes. However, there is of course no assurance that all the classes of H* (an, O(n)) can be distinguished by foliations, or that no other characteristic classes of foliations exist, and by and large these two aspects of the subject are much harder to treat. To answer the first question one needs a large arsenal of examples of foliations and here the first step was taken by Godbillon-Vey-Rousarie, who
,
R. Bott considered the following very simple, and beauflfkl homogeneous foliation. On SL(2,IR)
,
let 5 be the foliation given by 1 e2)
5 = {Ker
where SL(2,lR)
e12
.
is the left invariant form of that name from section 3, restricted to
. We have
by our structure equations, whence
so that 5 is integrable, and indeed the "leaves" of
5 are, of course, the
cosets of the subgroup
of
1 SL(2,IR) given by the condition x2 3 0
.
Now this foliation in itself is not yet interesting in the sense that h c (3) is necessarily zero because 1 1
SL(2, IR) is a non-compact 3 manifold
and so has no cohomology in dimension 3
.
example on the left by a discrete subgroup
r c SL(2, R)
However, if we divide the whole @
compact, then the example becomes interesting and in particular
r\ SL(2, IR)
R. Bott
Let me close by showing you why this is so. What we need is the following algorithm for computing hl c1(q
, which - read the other way around -
was the original definition of this characteristic class in Godbillon-Vey.
PROPOSITION
.
Let
Ker where
e
is a non-vanishing
condition there exists
e
-
1 form
-
1 form 71
.
on 8
M
In terms 71 the class hl c1(3)
.
is then represented by q Ad q :
h 1 c 1(3) = {q To see this, recall that in 0(a1, 01)
whence h
1
=el1 ,
dq) we have
1 1 1 1 and c = d e + e l h e = d e 1 1 1 1
It follows that hl c1 =
(8.8) in n ( a l , o l >
.
Then from the integrability
with
d @ + q A e= 0
(8.7)
M given by
3 be the foliation on
el1 Adel1
'
.
R. Bott Now one simply interprets
8 a s defining a section
se : M-
>~
~
(
3
with
and q a s a section
sq : (M) >
* e21 = q
with sq
, to identify
(8.8) with (8.7)
.
Q. E. D.
Note here that in higher codimension we would not be able to describe a foliation by a single 8
, nor would we have a section
s
and that is why it
took some time to unravel the proper generalizations of the class
hl
'
Armed with this algorithm we return to the example 5 on SL(2, R) Clearly
in view of the identity
on SL(2, IR)
. Hence, computing again by our structure equations, yields
.
R. Bott
and
Now this
3
- form
is clearly
# 0 on SL(2,R) and being left
,
invariant comes from the corresponding form on r\SL(2,IR) represents h c of r\Z 1 1
r\ SL(2, R)
which clearly
. Integrating over the compact manifold
now implies that
This construction can clearly be extended to higher dimensions and this yields corresponding nontriviality results for certain charactertistic classes, see for instance [ 10 ] , and the work of Kamber-Tondeur [ 44 ]
.
However, these are
far from sufficient to settle the matter and what is really needed at this stage i s an arsenal of nonhomogeneous examples. The master par excellence of such foliations at present is W. Thurston who first of all built examples of foliations of sim I
, whose invariant
/M
hl cl (3) takes on any
recently proved that every manifold with Euler number 0 codimension 1
codimen-
value, and more
, admits a foliation of
. For these developments see [ 66 1, [ 67 1, also [ 48 ] .
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Lawson, H. B. [ 48 ]
"Semisimpliziale algebraische Topologie. " Springer, Berlin.
, Jr. 9 369-418.
1974
Foliations. Bull. AMS,
1973
Loops and foliations. Proc. of Int. Conf. on Topology and Related Topics, Univ. of Tokyo, Japan, 1973.
1967
'Simplicia1 Objects in Algebraic Topology. " Van Nostrand, Princeton.
[ 51 ]
1956
Construction of universal bundles, I, 11. Ann. Math., 272-284 and 430-436.
[ 52 ]
1957
The realization of a semi-simplicial complex' Ann. Math. , 5 357-362.
[ 53
1957
"Notes on Characteristic Classes. " Princeton Univ. Press, Princeton.
Mather, J. [ 49
1
May, J. P. [ 50 ]
Milnor, J.
1
Mosher, R., and Tangora, M: [ 54 ]
1968
"Cohomology Operations. " Harper and Row.
Schwartz, L. [ 55 ]
1950
" ~ h 6 o r i edes Distributions. " Hermann, Paris.
[ 56 ]
1968
Classifying spaces and spectral sequences. Inst. des Hautes Etudes Scientifiques, Publ. Math., 34, 105-112.
[ 57 ]
1974
"Categories and cohomology theories. " Topology,
Segal, G.
13, 293-312.
Serre, J-P. [ 58 ]
1951
Homologie singulikre des espaces fibrks. Ann. of Math, 425-505. "Lie Algebras and Lie Groups. " Benjamin, New York
"On characteristic classes. " Thesis, Univ. of Cal., Berkeley.
Spanier, E. [ 61 ]
1966
"Algebraic Topology. " McGraw Hill, New York.
Stasheff, J. [ 62 ]
1963
Homotopy associativity of H-spaces 11. Trans. AMS,
108,293-312.
Steenrod, N. [ 63 ]
1943
Homology with local coefficients. Annals of Math., 4 4 , 610-627.
[ 64 ]
1951
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tom Dieck, T. [65 ]
1974
"On the homotopy type of classifying spaces. " Manu. Math., 4 1-49.
Thurston, W. [ 66 ]
1972
Noncobordant foliations of 511-514.
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.
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The theory of foliations of codimension greater than one. Proc. of Stanford U. Conf. on Wf. Geom. 1973. Foliations and groups of diffeomorphisms. Bull. AMS,
3 304-312.
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1973
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Weil, A. [ 71 ]
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Combinatorial Homotopy, I. Bull. AMS, 55, 213-245.
(C.I.M.E.)
SINGULAR INTEGRAL OPERATORS AND NILPOTENT GROUPS
E. M. Stein
Lectures given at C.1.M,E., Varenna, Italy, August-September 1975
E. M. Stein SINGULAR INTEGRAL OPERATORS AND NILPOTENT GROUPS
E. M. Stein
The Hilbert transform and the Cauchy integral Homogeneous distributions in
5"
'L theory Sobolev and Lipschitz spaces Pseudo-differential operators Parametricies and estimates for elliptic operators Entr'acte Cauchy-Szego integral and Heisenberg group Homogeneous groups The sb-complex on the Heisenberg group The Lewy equation Application to hypoelliptic operators Literature
E. M. Stein 91.
The H i l b e r t transform and Cauchy i n t e g r a l The H i l b e r t transform and i n f a c t the whole develop-
ment of s i n g u l a r i n t e g r a l s begins with t h e theory of functions of one complex v a r i a b l e .
The s t a r t i n g point
i s , i n e f f e c t , t h e Cauchy i n t e g r a l :
-m
f
E
Suppose f o r s i m p l i c i t y t h a t we a r e dealing with 2 1 L (E ) . What i s t h e meaning of the mapping f + F?
The elements Hardy space
F
H ~ . The Hardy space i s t h e H i l b e r t space
of functions y>'o,
of the form (1.1) a l l belong .to t h e
F holomorphic t o t h e upper half-plane
vanishingat
(i.e., as
m
lim
t h a t t h e boundary values i n the H'
~ ~ ( 2 'norm. )
i s the
y - , ~ ) , and s o
F(x+iy) = Fb(x)
Y-jO
The norm of an element
L2(g1) norm of
Fb.
Thus
i d e n t i f i e d with a ( c l o s e d ) subspace of
F
H~
may be
L2.
As a
r e s u l t of these considerations it may be shown t h e mapping
f
+
Fb,
exists in
that
a r i s i n g from ( 1 . 1 ) , i s t h e
orthogonal p r o j e c t i o n of
L~
5
H ~ . For f u r t h e r
reference we s h a l l use t h e notation: f 5 Fb = Cb(f)
.
The Fourier transform i s very u s e f u l i n shedding f u r t h e r l i g h t on a l l of t h i s .
In fact i f
E. M. Stein
then a well-known calculation shows that (1.1)
may be
rewritten as
As a consequence
-a
and hence the operator Cb is a multiplication on the Fourier transform side by the characteristic m c t i o n of the interval ( - ~ ~ 0 )(This . multiplier will later be called the "symbol" of the operator Cb.) The Hilbert transform, H(f),
is defined by
That the limit in M.3) exists (in the L~ norm, if f
E
L~), and its basic relation to the Cauchy
integral, given by
can be obtained as follows: Write Then E
1 /x+ie
[zl1 -
= l/x E [ l/xl
if
1x1
= EC~(X/E),
~zrp(x/~)dx= L~c~(x) dx =
-
ni.
2
E, =
0, otherwise.
where The relation ( 1.4)
then follows by using standard properties of approximations to the identity.
E. M . Stein From ( 1.4) and ( 1.2) one concludes that
Thus the Hilbert transform corresponds t o t h e multip l i e r ( i . e . , symbol)
i sign f .
This i d e n t i f i c a t i o n of
the symbol o f ' t h e Hilbert transform of course may be seen more d i r e c t l y by using t h e f a c t t h a t
' 1 2im.EG= m
P.V.
7
E + O
-m
= i
m
1im2:J
e
sign E
lim
E + O
$J
From ( 1.5) i t i s c l e a r t h a t
00
&
dx s i n 2m.1 Si-
&
s i n x -ax = X
i sign
i s u n i t a r y on
H
E.
L ~ .
We want t o make one f u r t h e r remark about t h e H i l b e r t transform.
Let
K
denote i t s kernel K(x)
=
=.
This
can not be viewed a s a function ( s i n c e i t i s l o c a l l y integrable), but a s a d i s t r i b u t i o n , whose action on a t e s t function cp
i s given by ~ ( c p )= P.V.
lE K ( ~ ) ~ ( x ) d x .
Now the Hilbert transform i s w r i t t e n a s H(f) = f + K
(see ( 1 . 3 ) ) ,
and so commutes with translations.
point we wish t o make i s t h a t dilations.
I n f a c t f o r any
a l s o commutes with
H
8
B U ~the
>
0,
lef
z8(f)(x)
=
Then
T
-
This i s a l s o equivalent with the f a c t t h a t the
f(8x).
1 H ~ 8= H. This follows from the f a c t t h a t 8(T,K)(x) = ~ - ' K ( x ) , i . e . , K i s homogeneous of degree I.
m u l t i p l i e r corresponding t o 0,
i.e.,
i Sign
(0 =
H
i s homogeneous of degree
i sign (E/8), 8
> 0.
Note i n
E. M. Stein t h i s connection t h a t i f
3 denotes t h e one-dimensional
Fourier transform, then
Rs = 8-'T
3. 8-I
To summarize, we l i s t the t h r e e basic properties of the Hilbert transform we have j u s t discussed.
These w i l l
provide t h e themes t h a t w i l l be developed i n the f i r s t p a r t of these lectures: (i)
The intimate r e l a t i o n of the Hilbert transform
with the Cauchy i n t e g r a l . (ii)
The simple expression of t h e Hilbert trans-
form i n terms of t h e Fourier transform. ( i i i ) The f a c t t h a t t h e Hilbert transform commutes with d i l a t i o n s ( i n addition t o t r a n s l a t i o n s ) . $2.
xn
Homogeneous d i s t r i b u t i o n s i n
We generalize p r o p e r t i e s ( i i ) and ( i i i ) t o
E~.
We make the following d e f i n i t i o n : Definition. (I.
A distribution
K
i s of
X,
a complex number) i f
f o r a l l t e s t i n g functions agrees with a Here
COO
a l s o we assume t h a t
cp,
K
function away from the origin.
og(?)(x)
= ~(tix).
Examples (1)
K
i s the distribution
preceding section. (2)
Suppose
Then K k(x)
of t h e
P.v.(&)
i s of type
0.
i s a homogeneous f i n c t i o n of
E. M. Stein
-
degree
n
that Re(X)
+ X, > 0,
As a result k bution K,
i.e.,
k( 8x)
=
cm
and k is
8-nt-Xk(x). Assume away from the origin.
is locally integrable, and the dlstri-
defined by K(cp)
=
$
k(x)cp(x)dx
,Rn
distribution of type X.
is a
The Dirac delta function is a distribution of
( 3)
type 0. More generally, the distribution cp -, ((x)acp)a
-
is of type
la[.
(4) Let k(x) of degree
-
(0)
n,
be a function which is homogeneous
i.e.,
k( 8x) =
6-%(x) .
Assume that k
is smooth away from the origin, and satisfies the vanishing mean-value property (2.2)
k(x)dx a 0,
Re(X)
the
~ ' ( 5 ~ )Therefore .
away from the o r i g i n , f o r every
a.
This concludes the proof of Theorem 1 . To prove p a r t ( a ) of Theorem 2, w r i t e function agreeing w i t h k
K
k
f o r the
away from t h e o r i g i n .
i s homogeneous of degree
i t follows t h a t
k
-
n
+
X,
and i f
Then
Re(X)
>0
i s l o c a l l y i n t e g r a b l e , and s o
defines a d i s t r i b u t i o n
KO
by i n t e g r a t i o n .
Now it i s
E. M. Stein easy t o v e r i f y t h a t
i s a d i s t r i b u t i o n of type
KO
Consider, therefore,
1.
it i s a d i s t r i b u t i o n
K :KO;
supported a t t h e origin, and hence a l i n e a r combination of t h e Dirac d e l t a function and i t s derivatives. by Example ( 3 ) above, we see t h a t d i s t r i b u t i o n s of type that
K
-
( 0,
i s of type
KO
K
-
Hence
i s a sum of
KO
which contradicts the f a c t
a,
unless
K
-
KO = 0.
TO
prove p a r t ( b ) one can use t h e same kind of argument if one shows t h a t (2.2). ko
k
s a t i s f i e s t h e mean-value property
To do t h i s w r i t e
s a t i s f i e s (2.2).
k(x)
=
alx1-11
+
We must sh& t h a t
a
ko(x), =
0.
where Now
consider the d i s t r i b u t i o n i( defined by
K
-2
i s supported.at t h e origin, hence transforms
under d i l a t i o n (replacing cp(x) nomial i n
6.
However K
by
cp(
a)) a s
a poly-
i s invariant under d i l a t i o n
(by assumption), and s o i s t h e distribuMon
P.V.(ko).
Finally, i t i s easy t o v e r i f y t h a t
transforms logarithmically i n
6,
and so we have a
E.M. Stein contradiction unless
a = 0.
We come t o an important consequence of these two .neorems.
We consider two classes of l i n e a r operators
(each I n i t i a l l y defined on t e s t i n g Amctions).
The
f i r s t class consists of singular i n t e g r a l operators (2.3) where k
J'
P.V.
f
zn
f(x-y)k( y)dy
+
af
which s a t i s f i e s the mean-value property (2.2); constant.
-
i s a homogeneous cm function of degree a
n
is a
The second c l a s s i s composed of pseudo-
d i f f e r e n t i a l operators of a specific kind, namely (2.4)
f
-s
e-2'ix"a(!)?(E)dl
gn
where
?( l )
e2"LX"f(x)dx,
=
E~ and
cm
a(l)
i s a homogeneous f'unction of degree
0,
and
away from the origin. Corollary.
The two classes of operators, ( 2 . 3 )
and ( 2.4), a r e identical. This i s the extension of the Hilbert transform representation ( 1 . 5 ) t o n-dimensions. n =
1
Notice t h a t when
the class of operators described by the corollary
i s spanned by the I d e n t i t y and the Hilbert transform.
E.M.Stein There are two formulas, in the nature of explicit realizations of Theorem 1 , which are worth recording. We state them without proof.
Suppose first that k
is a
- n, cm away from the
homogeneous function of degree
origin, satisfying the mean-value property (2.2), and let
denote the homogeneous function of degree 0,
o(5)
cW away from the origin, which is the Fourier transform of the distribution P.V.(k).
Then
a
k are
and
related by the identity (2.5) a ( ! )
=
f
(+ign(~.x)
+
log ~/~i.xl)k(x)dcr, 1 1 1
Sn- 1
Now the operators (2.3) Or (2.4), L ~ , since a
is bounded.
are bounded on
The relation (2.5) shows
that this boundedness on L~ holds if we merely assume that k
is bounded on the unit sphere, or even less,
if as before k
is homogeneous of degree
-
n and has
vanishing mean-value. For the purposes of the second formula we assume that 0
< Re(X) < n.
distribution K
This will assure that both the
and its Fourier transform are given
by locally integrable functions. Suppose K by the homogeneous function of degree - n
where
+
is given X
Pk is a homogeneous harmonic polynomial of n
degree k. Then K
is give? by the homogeneous
= 1.
E.M. Stein m c t i o n of degree
-
X.
where
The formula actually holds f o r a l l
X,
when suitably
interpreted t o take i n t o account the zeros and poles of the
function a r i s i n g i n
P
c ~ , ~ .
Here a r e some examples of operators whose kernels a r e distributions of homogeneous type, t h a t a r i s e a s inverses of constant coefficient e l l i p t i c p a r t i a l d i f f e r e n t i a l equations.
The kernels In question a r e "funda-
mental solutions" f o r these equations.
We write
Example ( i ) Here the d i f f e r e n t i a l operator i s
when n
1 3.
Then K(x) = cn(xl-%",
R(E.1
=-(nn1
!.t2)-'
.
This i s an instance of the formula (2.6' ) , when X = 2 ,
k=O. Of course AT = TA = I,
when acting. on s u f f i c i e n t l y
smooth flmctions of campact support.
E. M. Stein Example ( i i ) The cases
n = 2,
(or
t h e above example because
-
longer l o c a l l y i n t e g r a b l e . case
n = 1) (
a r e excluded from
26 1 )
i s then no
The appropriate
K(x) = 1
i s , a s i s well-known,
( n = 2)
in this
K
l o g 1x1.
This follows from formula ( 2.6' ) by a passage t o the
limit. Example ( i i i ) Here
a ai
-
l
a
1 ) .
=
ax
ay
formula ( 2 . 6 0 ,
a aB
T -a=
-.T=
the d i f f e r e n t i a l operator i s
n = 2,
Then
K(z) =
.
(See a l s o
x( X + ~ Y
n = 2, X = 1 , k = 1.)
We have
I.
az
Example ( i v ) We a r e dealing i n e f f e c t with t h e over-determined system
au = f ,j ax
can w r i t e
j
3
= I,.
..,n,
with
1 4, Tj
where
one
3
T ( f ) = f x Kj
3
j=1
L'
afk . ax
n
I =
$3.
a taxk
4=
Theory
We have already pointed out ( s e e t h e remarks a f t e r ( 2 . 5 ) ) t h a t operators bution of type
T(f) = f
+t
k
0, , a r e bounded on
with
K
a distri-
L ~ . A basic result
E. M. Stein
of the ~alderdn-~ygmund theory is that these operators are also bounded on ,'L
1
< p < m;
(this extends the
classical theorem of M. Riesz for the Hilbert transform). A
somewhat more general formulation is Theorem 3.
X, with R~(x)
Suppose K is a distribution of type Let T(f)
= 0.
to a bounded operator on LP, 1 Proof.
=
f + K.
Then T extends
< p < m.
Because of Theorems 1 an0 2 we can write
(as in (2.4))
= J e-2dx50(0f(~)d5
~(f)
(3.0)
,
zn where
a( 5 ) =
i(b )
is a homogeneous thnction of degree
- X, which is cm away from the origin. Since ~e( x).
= 0,
it follows that
CJ
is bounded, and hence
by Plancherelts theorem T is bounded on L2. The 'L theory of T would be simple if T were bounded on L', but unfortunately this is not the case. The operators T, nevertheless, satisfy an L1 estimate of "weak type" which together with the L2 estimate already proved gives the desired inequalities for 1
01
t h e r e a r e two groups of holomorphic self-mappings a c t i n g on i t which were of importance t o u s . t r a n s l a t i o n s , (isomorphic t o $ I ) ,
F i r s t the
which a c t e d i n a
simply t r a n s i t i v e manner on t h e boundary and which were responsible f o r the f a c t t h a t t h e Cauchy o p e r a t o r
Cb
and t h e H i l b e r t transform were convolution operators on $ 1
.
Secondly, t h e d i l a t i o n s
z + 82, 8
>
0,
which determined the homogeneity of t h e k e r n e l s of these operators. significant.
The analogues of these a r e e q u a l l y
The Heisenberg group
t r a n s l a t i o n s of the domain Hn
=
cn x n1
=
A. ((f,f);
The a c t i o n of an element
gives the
Hn
3
(C ,E )
E
cn, I
z'
=
(zl,
...,zk).
M u l t i p l i c a t i o n on
R)
on (8.1 ) i s given
by
where
F
K, i s given by
E.M. Stein The group Y, acts on the boundary bB in a simplytransitive manner. The mapping of An
to
:*',
identifies the group Hn with the boundary (8.2). The second group consists of dilations. The action on the domain (8.1) is given by
Observe that these are automorphisms of the group structure of Hn
.
Now an analogue of the Cauchy integral formula ( 1 . 1 ) is the expression
The elements of the formula (8.7) are as follows. is the Euclidean (i.e., bivariant) measure on
do,
Hn
=
$ n 2'
induced to 'bB via (8.5).
appropriate function (say L~(H,)).
f is an
S(z,m)
is the
Cauchy-Szego kernel given by n
with c When
f range8 over L'(H,),
=
2n-1r(w1)
P1 F ranges over
E. M. Stein ~ ~ ( 8 )The . l a t t e r c o n s i s t s of those holomorphic functions F
in
R which in a s u i t a b l e sense have boundary values
( o r bB = Hn) which belong t o L'(H,). values a r e given on Hn
The boundary
by
khere the convolution i n t h e sense of t h e Heisenberg group s t r u c t u r e , and the l i m i t i s t h e k E ( ( ) )= c
2
+
E
-
iE)nl.
L~
and
C
addition charcterized by t h e f a c t t h a t
norm. Cb
Cb
Here
are in
is s e l f -
a d j o i n t and has t h e reproducting property: Whenever i s a ( s u i t a b l e ) holomorphic function i n boundary values
f
on
bB,
then
F
=
F
8, with
C( f ) , and
Finally, t h e analogue of the Hilbert trans-
f = Cb(f).
form (1.3) i s given by t h e principal value convolution ~ ( f ) ( x =) l i m
(8.9)
E'O
where
with
c' =
-
sic,
One has, a s i n $1, 9.
J
f(xy-')k(y)dy
1~12~
and t h e norm-function i s defined by
Cb =
I + iH 2 -
Homogeneous groups A basic f e a t u r e of the above formalism a r e t h e
d i l a t i o n s (8.6), which as we pointed out, a r e automorphisms
E. M. Stein of t h e Heisenberg group.
These "non-isotropic" d i l a t i o n s
a r e the analogues of the usual ("isotropic") d i l a t i o n s x
=
...,xn)
(X
+
( 8x l , Ex2,
. ..,8xn)
$,
on
played such an important r o l e before.
which
Observe t h a t the
kernel (8.10) a s well a s t h e "norm-functionff (8.11) a r e homogeneous with respect t o t h e d i l a t i o n s (8.6). To generalize t h i s we consider a simply-connected group
whose Lie algebra
N
g i s homogeneous i n the
following sense: There i s a d i r e c t sum decomposition k and a sequence of p o s i t i v e numbers 2 = c j,l @ k a a 1,a2,.. ,ak, s o t h a t the d i l a t i o n s x + t j = 8l jx j, k where x = Cj,,xj, xj E n a r e automorphisms of -3' the Lie algebra E , f o r a l l 8 > 0. Such Lie alge-
.
ej,
bras a r e automatically n i l p o t e n t , but not a l l n i l potent algebras a r e homogeneous i n t h i s sense. exponential map n
to
CU
N
exp:z
+
N
The
gives a diffeomorphism of
and allows one t o i d e n t i f y
g with
N.
On such groups there always e x i s t s a norm function: x
-t
1x1.
identity
It i s non-negative, vanishing only a t the
e, continuous, and
cm on
N
-
(el;
finally
i t i s homogeneous i n t h e sense t h a t
I a 8 ( ~ ) I= 61x1,
where
zkj= 1 8
a
is the dilation,
z kj = l ~ j ,x j
Let Q n -3the homogeneous dimension of
x
=
E
Heisenberg group
%
a8(x) = =
a
jxj,
with
zkj = l a j
N.
dim(n ) denote -3 Observe t h a t f o r t h e
i s homogeneous i n t h i s sense,
(8.11) i s a norm f'unction, and
Q = 2n
+
2.
O f course
E. M. Stein
gn
with its usual dilations is also homogeneous; in
that case the standard Euclidean norm is a norm function, and clearly Q
=
n.
By analogy with what was considered in 92, a f'unction k on a homogeneous group N of
&=
X, X
2
0, if
(= group identity
k
is said to be
is cW away from the origin
e) and k(ag(x))
=
€i'Q+%c(x).
If
X = 0 we make the additional assumption that k has vanishing mean-value in the sense that
More generally, a distribution K to be of
X,
if K(a8cp)
=
8-hK(cp),
on N
is said
for all test
functions cp; and K agrees with a cm function away from the origin. When
X
>0
all distributions of
type X are given by functions of type X. When
X
such distributions are given by functions of type 0 plus a multiple of the Dirac delta function. For such k we define the convolution operator
When
X
> 0,
the integral in (9.2) converges
absolutely when (say) f is bounded and has compact support. When X
= 0
the integral must be taken in
the principal-value sense
= 0
E. M. Stein
which e x i s t s i f (say) Theorem 1 1 . Then the operator
f
i s smooth and has compact support.
i s a Arnction o f ' t y p e
Suppose k T,
given by (9.2' )
extends t o a
bounded operator on L P ( ~ ) ,1
< p < -.
holds, more generally, i f
i s given by T( f ) = f
where K
T
Is a distribution of type
Proof.
X,
0.
The same r e s u l t
* K,
with Re(X) = 0.
The main difference between the present
r e s u l t and the case when
gn
is
N
treated in Theorem
3 of 63 i s t h a t the Fourier transform i s not available
t o deal with the
L~ theory.
One proceeds a s follows.
For each integer
we
j
write Tj(f) = 2j 0. If f is locally bounded !3+2 in n, then u E P2(n loc.).
u
E
Theorem 14'.
Pp(n loc.)
C A ~ ~ ~ ( ~ ~ oBc>. 0. ),
For the proofs of these two theorems we refer the reader to the literature. 11.
The LeWY equation
We have excluded
q),
cussion of
W = 0,
or
=
n in our dis-
since the above results break down
in these cases, ( these correspond to a = f n, for day see
( 1 0.5)).
and take q
= 0.
We shall now consider these cases. A
simple calculation using ( 10.1' )
and (10.6) shows that
do)
therefore annihilates the boundary values of
holomorphic functions on Hny and so is far from hypoelliptic. Moreover (we shall see this below) the equation
is in generaly not even locally solvable. This is related to the Lewy equation, which arises when n
= 1
.
The equation is
E. M. Stein
where ( s e e $10) Z l
=
1
$Xl-iY1)
a =+
=
iil
1
a
.
We s h a l l give the necessary and s u f f i c i e n t cond i t i o n s f o r t h e s o l v a b i l i t y of ( 11 .2) o r ( 11.3) i n t h e neighborhood of a given point Suppose
f
P
E
Hn.
is a given function ( o r d i s t r i b u t i o n )
defined i n a neighborhood of
P.
Modify
away from
f
P s o t h a t t h e r e s u l t i n g function ( o r d i s t r i b u t i o n ) has compact support. of
f
Consider
C( f ) , t h e Cauchy i n t e g r a l
given by (8.7), which i s holomorphic in
P may be i d e n t i f i e d with a point of property
"C( f )
Now t h e
bB.
i s a n a l y t i c a l l y continuable past
depends only on the behavior of neighborhood of
8.
P,
t i o n we mave made of
f
P"
i n t h e immediate
and not on t h e a r b i t r a r y modif i c a f
away from
P.
This condition
i s a l s o equivalent with t h e boundary values
being real-analytic in a neighborhood of
Cb(f)
P.
Theorem 15. (a) borhood of
The equation ( 1 1 .2) i s solvable i n a neighP i f and only i f
continuable past (b)
C(f)
is analytically
P.
The L e v equation ( 1 1 . 3 ) i s solvable i n a
neighborhood of continuable past
P
i f and only if C(f)
is analytically
P.
The main s t e p i n t h e proof of t h i s theorem i s t h e
E. M. Stein following lemma which gives a r e l a t i v e fundamental \
solution f o r Let
with
c = 2n-2 r ( n )
type 2.
Let
.
0
i s a homogeneous function of
denote the operator
K'
K(f)
= f
0.
it
Theorem 1 6.
4( 0 ) (Kf) = K ( q( 0 )f ) = f - C b ( f )
(11.4) f o r smooth
with compact support.
f
The i d e a of t h e proof of Theorem 16 i s t o d i f f e r e n t i a t e t h e i d e n t i t y (10.8) with respect t o
a
set
has a
a = n.
(Observe t h a t t h e constant
zero of order one a t
a = n.)
One observes a l s o t h a t
(0)
C,,
b Thus the orthogonal projection Cb The meaning of (11.4) i s t h a t
K
ca
and
(0)
= 0. b commutes with
= C$
inverts
(0) '
on
-
the subspace orthogonal t o t h e boundary values of holomorphic flmctions.
We make several remarks about
these r e s u l t s . ( 1 ) The necessary condition (on f )
for the
s o l v a b i l i t y of the LeW equation (11.3) depends on t h e a n a l y t i c i t y of t h e Cauchy-Szego kernel (8.7' ) away from the diagonal; i t thus does not r e a l l y need Theorem 16.
An i n t e r e s t i n g s p e c i a l case i s a s follows.
E. M. Stein Suppose
i s a holomorphic function i n t h e domain
F
( 8 . 1 ) ~when
n = 1.
Suppose
F
i s smooth up t o t h e
boundary
bB,
and i s small a t i n f i n i t y .
also that
F
i s nowhere continuable p a s t
f
be the r e s t r i c t i o n - o f
t o bB.
F
B
We assume bB.
Let
Then t h e L ~ W Y
equation (11.3) i s not l o c a l l y solvable anywhere. (2)
When t h e necessary conditions on
s a t i s f i e d in an open s e t u
are
f
then t h e r e e x i s t solutions
Sl,
of the equation (11.2) which s a t i s f y r e g u l a r i t y
properties of the same kind as described in Theorem 13, 13' of
14, and 14'.
coj 0 u1 =
-
n = 1 and
If
- Z 1Z 1u 1 = f ,
u,
thenclearly
i s a solution
u =
-
ilul i s a solution of the Levy equation, and s a t i s f i e s h
corresponding r e g u l a r i t y properties. 12.
Application t o hypoelliptic operators The next s t e p i s analogous t o t h e passage from
constant coefficient p a r t i a l d i f f e r e n t i a l equations t o We intend t o describe a
v a r i a b l e coefficient ones.
c l a s s of singular i n t e g r a l operators modeled on convolution operators f o r homogeneous groups, somewhat In the same yay a s t h e pseuclo-differential operators of
zn.
$5 a r e m d l e d on convolution operators on
To make t h i s point c l e a r e r consider the operators (5.1 ) ( o r ( 5 . 2 ) ) corresponding t o symbols respectively.
u
and
R e s t r i c t a t t e n t i o n t o an immediate
neighborhood of a point
x
0
.
Freeze
a(x,f )
to
r
E. M. Stein a ( x O , and
r(x,l)
t o T(x',E).
For
Tu t h i s has
the e f f e c t of passing t o ' t h e genuine convolution operator
T is a(xO,~)~(xO,r) exactly t h e product of T and T ; the a(xO,r) *(xO,t g i s t of Theorem 6 i s t h a t when x0 i s unfrozen, then and s i m i l a r l y f o r
TT.
Now
i s a good first approximation t o
TaT
Ta.T,.
Thus one might say t h a t the e f f e c t of t h e symbolic calculus i s t o allow one t o c a l c u l a t e by approximating
at each point with an appropriate convolution operator on
zn.
Our generalization w i l l take the following shape:
We s h a l l be given a manifold group
N.
For each point
neighborhood of
M
x0
E
centered a t
M
M
x
and & homogeneous
we s h a l l i d e n t i f y a 0
,
with a neighbor-
hood of the o r i g i n (group i d e n t i t y ) of
N.
This w i l l
allow us t o construct a 'class of operators on M modeled on convolution operators on N. our first goal.
That w i l l be
Our second goal w i l l be t o describe
some a p p l i c a t i o n t o c e r t a i n "natural" h3ipoelliptic d i f f e r e n t i a l operators. Basic t o t h e construction i s the existence of a smooth mapping O:M x M -, N, ( 12.1)
0(x,y) =
-
O(y,x),
so that ( - i s group inverse i n N)
E. M. Stein
and for each x (12.2)
E
M
y -r 0(x,y) is a diffeomorphism of a neighborhood of x in M to a neighborhood of 0 in N.
0 takes the role of x
-
y in the case of the standard
singular integrals. Besides the properties several others come into play. Let
0
be the "pseudo-metric" defined on M by
~(x,y)= I0(x,y)
1,
where
I
1
is the norm function
for the homogeneous group. We shall have to assume that P
satisfies the "triangle inequality"
and in fact a stronger and more technical variant,
for some fixed a, a
> 0.
With these assumptions we can proceed to study a class of singular operators. These will be of the form
The kernel K will be of
X (X 2 0) in
the following sense. For any positive integer can write:
a , we
E. M. Stein
with (a)
E~
(b)
ai
(c)
For each
defined on
C;')(M x M I ;
e
and
N
smoothly on
e
i,
c:(M); the function
2
is of type
u
+
( s e e §9),
X
( 1 )( u ) k, and depends
x.
Theorem 17. i s of type
bi
0.
Suppose Then
L ~ ( M )t o i t s e l f , when
T
T
i s given by (12.4) and
K
i s a bounded operator of 1
)
2.3.
=
on obtient
E (v)
Martingales. Revenant aux notations du paragraphe 1,si h
est une fonction harmonique born6e on pose fs(u) = h(\gs(w)) alors
Cette formule n'est rien d'autre que 1.4. dans laquelle on a substitug xo par
cQ
.
=o La formule 2.3.1. est fondamentale. On dit que la famille Ift] une martingale relativement 2 la famille
{&'I.
.
ef I[ 91 .85]
3. Egalit6 d16nergie,temps dlarre^tet convergence %martingales. I1 r6sulte de 2.3.1. que
forme
P. Malliavin
3.1.
- fz)
E((ft+]
(fel+l- ftl)) = 0 si
~
'
2 7
.
En effet d'aprss 2.1. cette integrale s'lcrit
ce qui ltablit ,3.1. I1 rlsulte de 3.1. l'identitl fondamentale d'lnergie pour une martingale
Nous allons localiser cette identitl sur un temps d'arrtt, ceci signifiant que l'entier mesurable
sur JL
figurant dans 3.2. sera une fonction
q
; q = q(W).
$Ondit qu'une fonction q 1 valeurs
entisres positives est un teqs d1arr9t si notant Al = [ w ; q(u
hl]
;
alors
c'est-l-dire si l'ensemble
AQ
est un cylindre sur
est sur les 1 -premiires coordonnles Thlorsme : Soit q(d)
A , dont la base
.
un temps dlarr^et, f
mar tingale, alors
Preuve : le premier membre s'lcrit
et tout revient l dlmontrer la relation d'orthogonalitb si
1 4 1 '
P. Malliavin
E((f1+]
-
fi) (fl'+]
- fA1)nA
= 0
OAA
I
.
1'
Utilisant 2.2. ceci slBcrit
Remarquons que PA
, llA
- fp) .
que
sont des fonctions d
1'
L'
1
mesurables ainsi
" qui se rbduit ainsi L
D'aprSs 2.1. elles sortent de llespEranceE
3.4. Corollaire :
Soit 3.4.1.
Ifr(
~ n martingale. e Supposons que
M = sup r /Ifr - fo/lL20L)c +a
alors notant --
Preuve : Soit k un entier fixi5. Introduisons le temps d'arrct qk(0) 16qk(w ) -L k
dBfini
par les conditions suivantes
S'il n'existe pas un tel qk (w)
nous posons qk ( w ) = k.
P. Malliavin
Alors A
e
4 ,k
est ~'qesurable. Pour
cet 6vSnement est
impossible. Appliquons l'identit6 3 . 3 .
Utilisons maintenant l'identit6 3 . 2 . on obtient
d'o3 3 6 6 . entraine
3 . 7 . Th6orPme (Doob) Supposons que f
soit une martingale vgrifiant r--
2 3 . 4 . 1 . et telle que foE L ( A ) .
Alors 3.7.1.
lim r-+m
fr(U)
= fa ( w )
existe presque partout
Preuve : La preuve sera fond6e sur la remarque que si martingage alors pour tout s relativement a la famille
sO+r & I
trjr,, est une
est une martingale
. On notera
eS
f
0
cette nouvelle martingale. I1 r6sulte de 3 . 4 . 1 . et de 3 . 2 . que
.
P. Malliavin
Par suite pour
s
assez grand
Utilisant 1 nouveau l'identitb 3 . 2 .
La suite fa par
forme ainsi une suite de Cauchy dans
L 2 (A). Notons
salimite. Appliquons B
(es f)
llin&galitL 3 . 4 . 2 . . On obtient
0
d'oG la convergence presque partout des
fr(w)
. (cf
pg
p. 11) pour
1
J'extension de ce rLsultat si f e L ). Ce thi5orPme de convergence des martingales de Doob est un analogue abstrait du thLorSrne de Fatou de convergence angulaire des fonctions h a m n i q u e s .
3.8. Formule de Poisson et balayage Revenons B la signification concrete de la martingale dbfinie dans le premier paragraphe. bornLe sur sur A.
0
h
h(yr(u))
btant une fonction harmonique
on obtient en notant presque surernent le presque partout
P. Malliavin 3.8.1.
Thlorlme de Fatou - h o b . Presque surement lorsque
r--+
+a,
converge pour t o u t e f o n c t i o n harmonique
Preuve : e t 3.4.2. (cfL53 pour l a r e l a t i o n avec l e thBorlme de Fatou de
3.7.
convergence non angulaire)
3.8.2.
. r
Corollaire. Lorsque
7
vr ( ~ 0 )converge presque
+a ;
surement vers un p o i n t a p p a r t e n a n t a s f r o n t i s r e Preuve : S o i t
Btant born& h l
converge. Ainsi presque
vr(W )
0.
l a premisre f o n c t i o n coordonn~e,'alors
h l ( x) = x l
e s t harmonique e t , 0
&
e s t bornce. Par s u i t e
h1 ( v r ( w ) )
surement
converge v e r s un point
zb 5
.
En p a r t i c u l i e r
v
r -Y
r
+
,
0
.
Mais
w
- v r +1
P
r
(
P(z) = o
, d'oci
= p(Yr(w))
Ainsi 0
d'oci
Notation. On pose 3.9.
~66.
QOD(w)= l i m
.
if (o)
ThBorsme (PoincarB) Pour t o u t e fonction
continuesur --
-0
on 5
h
harmonique
hl
sur
0,
P. Malliavin
h (xo) =
1. h (x) dv (x) 0
Preuve : Le passage 1 la limite sur 1.5. lorsque r+
+CO
est lBgitime en
vertu de la continuit6 de h et d h n t r e le thBorSme.
3.1 0. Conclusion La mlthode de balayage de Poincars conduit 2 une construction explicite de la mesure harmonique. Cette construction peut Stre Bventuellement utilisBe pour obtenir des estimations de cette mesure.
4. Passage a une variBtB riemannienne
:
Soit V une variQtB riemannienne, xo EV. On notera par unite dans l'espace tangent Tx (V)
t
la sphire
et JL notera encorezN munie
0
de la mesure produit naturelle
. Si
x E V on note par n X : T~(V)+
V
dLfinie par la carte nol;plale (A tout vecteur z du plan tangent on associe ltarc de gBodBsique tangente B ce vecteur et de mcme longueur, nx(z) est alors ltextrBmitlde cet arc). Kxant dtapplicationsVI :A
+V
8 70
dlfinissons la suite
par les formules de rBcurrence
'y do = xo
dans cette derniere formule on convient d'identifier la sphire unit6
P. Malliavin
de T
(V)
a v e c x utilisant le transport parallsle le long
Q , (w)
b B Ql(w). (On suppose que V possBde
de la glodlsique joignant x
un rayon d'injectivitd global et on a fix6 6 d' injectivitl)
inflrieur B ce rayon
.
Ce transport parallBle sera contiiud le long de la glodlsique
S
6
et ainsi de suite. On dlfinit ainsi
Q1(w),lf2(tU)
Soit h une fonction haonique sur V pour l'opdrateur de Laplace
AV
Beltrami 1
de V.
h o nil
n'est pas en glndral une fonction haonique sur V
0
par suite
Plus prdcisdment si on alors
AN
'
(V)
lit l'opdrateur A V
+ L 03
AT
dans la carte normale,
,est le laplacien sur l'espace
X
0
et oii L est un opdrateur elliptique B coefficients
euclidien Tx (V) 0
variables sur Tx (V)
dont tous les coefficients sont ? I en zlro.
0
Ainsi
Cette formule entraine que si (AVh) (xo) = o
On se propose d'exploiter 4.1. pour construire, en faisant tendre
&+ o , une "martingale" il partir de h. I1 est alors ndcessaire de prlvoir une renormalisation sur les
indices de
et B la limite de les faire varier continument.
\P,
Nous dgfinirons un chemin continu 4.2.
6 6 U,(t)-yn(w) 2
Si
4t 4.
si 2
to, 6.2.8.
yt ( 0
6.3.9.
vt (t,$)
a2) .
t-t
= exp (-
2
0
Calculons la d6rivbe en t de 4 0
yt(t+E,?)-yt(ts2>=E 0
-
4
E
0
E4
0
exp
(ich (Zg(t)
exp (i9(zW(t+€)
-
ZW (to)>) E
- ~~(t))>- ]I .
Utilisons le d6veloppement de Taylor limit6 B 1,'ordre 2 de l'exponenrielle on trouve d'aprzs 6 . 3 . 7 .
P. Malliavin
d'ob
Yt
satisfait l'gquation diffsrentielle
ce qui d b n t r e 6.3.9.. Indiquons rapidement la dsmonstration du thbrgme de structure dans le cas'gsngral, Posons
Alors on vgrifie facilement
6.3.11.
do ([z:
E -I E
lim
(to+€ )
& -0
- 2k(t0)J2)
=
(dw(to))
2
.
la fonction positive strictement croissante dsfinie par
Soit yEU(t)
Posons 2 ZJt) 2 Alors Zw
1 = ZcU(Yw(t)).
satisfait les hypothsses 6.3.7. donc peut d'aprss la
premiOre partie de la dsmonstration peut ttre identifie 1 un brownien ,b
adapts 1 la famille dt.
Posons t
'~(~1
uw
(t)
a
Z~ (0)
- zw
(t)
fw(E )dg
+
-
Yw(t>
.
Alors il rssulte de 6.2.1. et 6.2.2. 6.3.12.
E
cr"t(@(t+&)
+
0
- ,U
(t)I2)
0
4(U,(tl))
E
-0
(E).
(5)dbW(~ )
-
UW(t)
si tr t'
P. Malliavin Posons
rp (t) (t+8
=
-
E([u~
(t)] 2).
E(Edt(U,(t+&
)12)
9 (t) est constante et come
dVaprSs 6.3.12 on a
(9 (0) = 0 on a
WE
0 et le theorPme
est dlmontr6. (cf [lfl
page 11).
6.4. Consequence & th6orSme & structure : le calcul~difflrantiel de Ito -On convient dVBcrire la relation 6.3.1.
sous la forme
symbolique 6.4.1.
dY,
=awdbw + Pwdt
on appelle le second membre la differentielle stochastique de Ito de Y. Soit F une application de classe C* de R dans R, posons
ThEorSme UJt)
s16crit sous la forme 6.3. I . .
& difflrentielle
stochastique vaut 6.4.2.
-
dU, (t) = (F1(Y,
(t)) d , ( t )
Preuve : I1 suffit de verifier que
db,+
,U
(t)
satisfait les hypothsses
P. Malliavin du thlorsme de structure pour forme 6.3.1..
obtenir que U w
slQcrit sous la
Cette vsrifjcation s'cffectue en effectuant le dsveloppe-
ment Tayloried de F
E
4 0
limits
l'ordre
2
(U,,(to+E)-Uy(to))
7. Equations intggrales stochastiques Soit
0
un ouvert de
IR"
donnons nous sur 0
un operateur
diffCrentie1
.. on dit que
L est semi elliptique si la matrice alJ
est une matrice
rQelle symgtrique, positive (on ne suppose pas dgfinie positive),(on utilisera les conventions de sonmation-de la g6omLtrie diffgrentielle ) b On note par i la matrice
la matrice symgtrique pesitive racine 'carrge de
aij
.
On se propose dc construire une "diffusion associQe1'3 L. On considsre le systsme d1Bquations diffsrentielles stochastiques.
7. I . d& oii ,b
r af
(X (t)) db (t) + ci(x ,(t)) dt j est le brovnien sur I R ~ .Les Qquations intQgrales
(t) =
correspondantes slQcrivent.
P. Malliavin i r.(x) 1
7.3. ThdorPme : Supposons que les
, Ci(x)
soient des fonctions
localement lipchitziennes. Donnons nous il existe une ---
xu (0). Alors presque
surement bw
Btant donnB,
solution unique de l'dquation intBgrale 7.2., dlfinie pour
t > o assez petit. MBthode de --identit&
Preuve : On utilise la mathode dtit8rationde Picard. Les
dtBnergie 6.2.2. conduiraient B des majorations en moyenne
quadratique, inliuffisant pour le resultat cherchB (du type p.s pour tout t assez petit).
en
cd
I1 convient donc d'utiliser les estimBes
auxquel il est fait allusion en 6.2.7. cf.
[2g p
52,Lla p.39, bf) d Bd 1 page 207
ConsBquence : L'application mesure de Wiener sur I!? (R") permet de dBfinir Ex (f (x,,
bw
+ xW
permet de transporter la
en une mesure sur &?
X
(t))
(01, mesure qui
.
0
7z.4. ThBorSme : Soit f de classe
c2 a
( E 1)
- f(x0))
E~ (f(x,
lim &-I E-7 0
support compact dans 0. @ = (Lf) (x0).
0
Preuve : Utilisons 7.1. i x~(& C;(X~) b;(~ j
-
+
cicxor
On obtient en utilieant 6.2.1. et 6.2.2. = c1(Xo) 0
+
o(1)
P. Malliavin
'
E
~
o
E
x
j
o On
i aj(xo)
=
k J
T . ( x ) +oh 0
utilise pour cette derniPre relation le fait
D'autre part
Ecrivons le d6veloppement taylorien de f limit6 21 l'ordre 2
Remarque : Sur une vari6t6 riemannienne V nous avons ainsi deux manisres de construire une diffusion adapt6e 21 110p6rateurde Laplace Beltrami, d'abord celle d6duite du balayage de Poincar6, dlautre part la m6thode des Bquations int6grales stochastiquas en choisissant un atlas et en construisant de cartes en cartes les solutions des lquations int6grales de Ito correspondant 21 la lecture du Laplacien dans chaque carte et en raccordant les cartes suivant une m6thode qui sera explicit6 chapitre I. Les deux mesures ainsi construites sur
\
(V) sont 6gales en vertu d'un th6orPme
0
d1unicit6 d'une diffusion>associ6eau sens du th6orame 7.4.;21 un op6rateur diffsrentiel L. (cf 1121~.Bd I p. 54).
P. Malliavin 7.5. Calcul diffgrentiel de Ito : Soit
..
(bl .bq) un brownien abstrait sur R, adaptl B une
famille croissante l'espace
A
dtd8alp8bres de
fonctions mesurables sur
.
7.5.1. Considlrons
ct
un systsme de diffgrentielles stochastiques les
M~
anticipante, relativement B
des fonctions F : x-
, ci Btant soit
Y
une application de classe
c2
de Rn dans 'R
ThBorZme : Avec les notations ci-dessus y = F(x)
vgrifie le
systZme de diffgrentielle stochastique analogue 5 7.5.1. -7.52. oii -
db&
dyr =
rf
dbj +
suivant
hr dt
est la mzme trajectoire & b,bgo(Rq)
apparaissant
qu'en
7.5.1. gtt
7.5.3. Remarque : Dans cette expression les expressions 02 interviennent la matrice jacobienne
3; F~
sont famili8resen
gBom6trie diffgrentielle. Par contre les dBrivles secondes
a i3
Fr
constituent un ph8nomZne nouveau. On peut rlsumer ce changement de variable en disant que l'on dBveloppe F B l'ordre 2 par la formule
P. Malliavin de Taylor e t que l ' o n remplace dans c e t t e expression dt2
par z d r o
d t dbJ p a r zero p a r z6ro s i
d b j dbk
( d l 2 par
dt
j # k
.
Ces r s g l e s c o n s t i t u e n t l e c a l c u l d i f f e r e n t i e l s t o c h a s t i q u e de I t o . (cf [25] p.44.)
La mdthode de di5monstration du th6orSme 7.5.2.
repose s u r l e th6orSme
de s t r u c t u r e s u i v a n t
de s t r u c t u r e
7.6. Thi5orlme mesure --
.
& fonctions
que l ' o n n o t e r a
,z
(t)
dt
mesurables. S o i t une a p p l i c a t i o n
. Supposons que s d t crf-mesurable . Supposons d ' a u t r e zo(t)
l i m & " eLPt(% g+ 0
( t + h)-z,(t))
qui s o i t
7.6.3.
dt
l i m E-' E+ 0
E
p a r t que
= C(OY , t )
e t q u ' i l e x i s t e une forme quadratique 9w-- d 6 f i n i e s u r ----IRq
d e sous
q'
JL K R+
7.6.2.
ensemble a b s t r a i t muni d'une
de p r o b a b i l i t i 5 e t d'une f a m i l l e c r o i s s a n t e
algibres
7.6.1.
:S J 2
( R ~ ) *l e dual de
mesurable t e l l e que &t
(12 (zw
(t+C )-zw(t)))
= q,
(d)
(En coordonni5es
Alors il e x i s t e ---ti5 ZI l a ----
famille
sur
brownien a b s t r a i t 5 v a l e u r s dans
dt,t e l l e
que
R ~ , adap-
P. Malliavin
R6ciproquement x d6fini par
7.5.1. satisfait
et 7.6.3..
7.6.2.
Ceci
implique 11unicit6de la representation 7.6.4. Preuve : Ospeut donner une preuve parallgle 2 celle donn6e ci-dessus en dimension 1 , avec toutefois certaines complication.
7.6.5. Commentaire : La condition 7.6.2.
a plusieurs variables de
est la transcription naturelle
6.3.4. La condition 7.6.3. parait plus
nouvelle, mais au fond n'est guSre diff6rente de 6.35. 7.7. ThborSrne (Ito) : Soit f
rgelles, ---d6finie sur un ouvert -
& R".
0
+&
(adt) , llespace& probabilit6 de
il existe un ---
L
c2, 2 valeurs op6rateur
la diffusion associ6e 5 L au sens de
elliptique sur O,x,(t) 7.4, Soit
une fonction de elasse
cette diffusion. Alors
brownien abstrait 2 valeurs dans
(R,
adapt6 3
oY*,
tel que --
05 si --
L = L d i j diaj + p i a i 2 posi~ived6finie par
,
PfIC = a i j (aif)cajf) Preuve : -
Notons par
1:
.
la racine ear&
th6orPme 7.4. la diffusion x w
1vflL notela fonction
associ6e 2 L v6rifie les 6quations
diff6rentielles stochastiques dxi
-~lf
..
de dlJ ,alors suivant le
dbj +
pi dt .
P. Malliavin Par suite d'apres le calcul diff6rentiel de Ito, posant f (xw(t))=Z,
(t)
,
dZy= (Jif)df
dbJ + M I
.
Calculonslesinvariants 6 . 3 . 4 . et 6 . 3 . 5 . du theoreme de structure on a
4[e-'(~,(t+g)
-
7.7.1. lim E E+o
lim E;rt [E-~(z, E-0 0
-
' u "
(t+~ )
lim ESr €30
-
ZW(t))]
=
(Lf)(xy(t))
~ ~ ( t ) ) ~ =] ( aif) i.
[E-l(bj(t+~
)
j
- bj(t))(bs(t+~
d'aprss 7.4.
(a
8: )
.jys
- bs(t))]
Cette derniere expression valant 1 si j = s et z6ro sinon ,on obtient
La formule de Ito rgsulte alors de l'application du theoreme de structure 6 . 3 . Come r6f6rences g6n6rales sur ce chapitre indiquons 141 p7]¶
b9]y[2]
y[2gy[2d-
,cq a,
.
P. Malliavin CHAPITRE I Equations de comparaison et diffusion.
L16tude qualitative d'un op6rateur semi elliptique peut 8tre entreprise B partir de l'6tude qwlitative des trajectoires de la diffusion associce ;
ce dernier point de vue a une signification gLom6trique
directe etles m6thodes B utiliser pourront ftre sugg6r6es par les m6thodes classiques dans l'intcgration des 6quations diff6rentielles ordinaires, 1'. C6mparaison avec un problsme
en dimension 1.
Soit V une vari6t6 diff6rentielle connexe munie d'un op6rateur semi elliptique L. En une carte locale L = -2 aij
ai aj + ci Ji .
Etant donn6 une fonction f de classe ,'c
on pose
et on introduit l'invariant de conduction de la chaleur
Soit p une fonction d'exhaustion de V c'est-B-dire une application
-1
c2 de V dans lR+ telle que p (compact) soit compact. Introduisons
a-(6)
=
inf a(x)
et soient L;,L;
les deux 6qurtions diff6rentielles ordinaires sur constitug par l'image de p :
l'intervalle
I de
Soit x,
la diffusion sur V associse B
x , ( O )
=
) t ( , x
(t)
Y-
Y ; ( o )
=
p (xo)
.
+ , Lp-
les diffusions associCes B
w-
a)+
de V si bien que
p(d) =+&I
est dgfinipour tout t. Onpose
=
L, et v6rifiant
xo. On introduit le point B l'infini
Notons par Y+
)o(:Y
R
L
avec
.
come valeur initiale
Si I n'est pas born6 convenons de le compactifer en lui adjoignant + a
. Si
appartient B la
Y+^(t )E @+-
frontisre de I convenons de poser ~+,(t)
0
= y++(tO) w-
W
pour
t7
to.
Ainsi Y* (t) w*
est dsfinie pour toute valeur de t. Posons enfin
1.3.
(t) = p (x,(t))
,Y
Lenune de comparaison. --
-
I1 existe des applications ---
u,
>-
w+, w
-
-3 W
prgservant
Tes probabilit6s telles que pour tout t> o
1.4. 1.5.
-
Y- (yw(t))c cu 06 ~_(t)
, Y
- iO
(t)6Yw+
&
(Y,(t))
t
lvpjt (xu ( 5 1146 r +a
.
Preuve : Ecrivons la fomule de Ito 0.77.
={ 0 ~ v P I I ~ ( x J ~ ) ~ & ~ ( ~I0 (LP).(X@(~ t
1.6. YJt)-YJO)
Soit V,(t)
t
+
la fonction inverse de la fonction
vu(t)
1) dS
.
P. Malliavin (on supposera gue
Ilpp
l(x,
( 5 )) f 0
Vw (t)
trajectoire ce qui entraene que
presque surement sur chaque est strictement croissante)
.
Posons
zu
1.7.
*
(t 1
Yw (qw(t
*
*
Nous voulons Bcrire
Zw (t*)
de structure
: calculons donc
0-6.3
sous la forme donnee dans le thloreme
-
Utilisons le theoreme de structure 0-6.3. 3. 1 dimension
t
*)
bw
* (t )
0 =
I1 existe un brownien abstrait
adapt6 3, la famille
+[
t* -
dbYI(S) 0
lim ecU',{~-l(~~(t+€)-&(t$ E-?o
(lim -)E E-0 E*
dtt* tel que
t*
La premisre int6grale se calcule explicitement et vaut ,b la formule
I.I I . z ~ t * )-2.~0)
+I
*
t
= b~t*)
aW$)
d~ avec
%(s
*
*
(t )
d'oa
= af (xJ\~(c*)
0
En rlsumB, par un changementd'horlogela sous la forme 1.1
.
af ( X ~ ( V ~ ( E * ) d) )5 * 0
formule de Ito. 1.6. s'lcrit
1. qui ne contient plus d'intlgrale stochastique.
1)
P. Malliavin
Ecrivons 1'6quation diffsrentielle stochastique associee B
L+ on
obtient
-
Convenons d'effectuer l'application des espaces de probabilitss
JL
A+ en choisissant le m h e Qchantillon du brownien
a 1 dimension dans 1.1
1 . et 1.12.
On obtient en resolvant 1.12
la valeur initiale Y++(o) = Zo (0) W
1.13. Y+ (t*) W+
- zW (t*)
lo
l'lquation
t*
=
(at($+$*))
- au
( 8 *I) d 5 *
Le seul rensedgnement que l'on connaisse sur ,a pour co
f ixQ, am ( g*)
1.14. Y+ (t*) = Zi(tz)
u+
O
avec
(
5*)
.
c'es t que
est continue et d'autre part que entrsine aW ( t 3 4 a+(Y+ (t*)). w+ 0
On peut dscrire le mouvement de ~'(t*)
et
*
ZU) (t )
c o m e 6tant
celui de deux voitures sur une m h e route, lorsque la seconde voiture cherche Zi doubler le premiere voiture alors la vitesse de la seconde voiture est plafonnQe par celle de la premisre. I1 est clair que la seconde voiture ne parviendra jamais B doubler,bien
que
lorsqu'elle slQloigne de la position de depassement sa vitesse puisse ttre tres supgrieure B la vitesse de la premiere voiture considQrle au mEme instant. I1 est facile d'obtenir 1.4. en mettant ce raisonnement sous une forme mathQmatique 1303
2. Theoreme
Soit V
.
cornparaison 2 geometrie Riemannienne semi-locale. une variGt8 riemannienne BT(V)(~o,r)
=PET (V) "0
xobV, ; (lzl)2r?
.
P. Malliavin
v
On note nx
la carte normale
du rayon dl?njectivitE
(0-4)
pV
restreint IB T(V) (Xo'r)
P(xo) = sup {r ; nx
soit un difflomorphisme sur son image
On pose
en xo. On rappelle la dlfinition 0
3.
inf pV(xO). xo
Pour tout r6PV(xo)
on dlfinit la m g l o d l s i q u e BV(xo,r) = nx
(BT(V) (xo,r))
.
0
La carte normale permet d'identifier une boule glodlsique avec une boule euclidienne. La courbure sectionnelle en
xQV
est une fonction \px rlelle
dlfini sur la grassmannienne des deux-plans de
Tx (V). 0
On pose :K = max
-
qx
Kx=dncpx
:K a+(xo,r) = max x€B(xo,r) Si b
, a-(xo ,r)
= min
xEB(x ,r)
est une constante on no,te par
Vb
Kx
5 variltl
riemannienne B
courbure sectionnelle constante Egale 3 b. Ei
r L h(xo),
d'identifier
r
o
la chaleur,
dependant que des bornes
P. Malliavin
Msthode de preuve --
:
On dgfinit la suite de temps dfarre^t
Les hypothises sur la courbure donnent que lfon possede des Squations de comparaison universellespour les
Tn. On peut montrer ainsi qufil existe une constante c 4.4.
P(T~LE)4 exp (-c
Soit 2N(o)
E-~).
le plus grand entier m
existe au moins N<w) 1
L'Qvinement
N(w)
=
tel que
I. Alors il
Tn
termes de cette some qui sont infdrieurs 1
1 . DfaprPs 4.4. N(w )
P(N(U))
telle que
=
ceci implique s)
4
2s 2 (s ) exp (-cs )
.
s implique que d(xo,xw(t))
(s+l)
PV
quelque soit o L t < 1 d'oii finalement
4.5.
P
(
sup d(xo.xW(t))
> ('+I)
f y )c exp
(-CIS2)
o orttl
.
Cette insgalits majore la rapidit6 de propagation de la chaleur sur V. combine/ ensuite avec des estimQes paraboliques uniformes sur les boules B(x,PV)
et une mdthode deva et vient pour Qtudier les
probabilitQs de retour dans le boule B(xo, ddmontrer le th6orPme.
Pv)
t€[0,,l],
elle permet de
P. Malliavin : Sous les hypothPses & theorsme 4.1.
--
4.6. ThBorPme de vie sur ---
V est infini cf (
GI ,[40],
le temps
14q).
Preuve : L'inegalite 4.5. -
montre x,(t)
presque surement t 6@,1]
come 1 peut ttre remplace par n'importe
quel nombre
7
reste 2i distance finie
0 on obtient le resultat.
5. Applications
2 ltanalyse complexe.
Les mBthodes developpees dans le paragraphe 4. permettent 5.1. d'obtenir une estimation de la fonction de Green d'un ouvert
strictement pseudo convexe de an, ouvert muni d'une metrique g.hlBrienne du type de celle de Bergmann C43 5.2. D'obtenir des estimations 'L
'1.
pour des opBrateurs de Green sur
les vari6tBs de Stein C433. Le point de d6part est de la definition suivante de la fonction de Green g:
-IO
+uJ
5.3.
g(x,y)
~ ~ ( ~ ,dty )
. On combine cette definition avec
des estimations du pt ; ces estimations peuvent ttre conduites projetant le processus par une fonction d'exhaution dBfinissant la metrique kahldrienne
p g. On
trouve Bgalement dans[77! une etude
probabiliste de l'existence de fonctions harmoniques dans .'L 6. Autres Ltudes asymptot'iques [45] contient une $remi~reetude de pertubation de diffusion sur un espace homogsne. L'Btude du mouvement de la diffusion sur un espace symetrique en P9Ia.
CHAPITRE I1 Annulations ergodique
& cohomologie
propri8tes diffusion horizontale.
La thgorie classique de Bochner
L3 ]
d'annulation des formes
harmoniques sur une vari6t6 riemannienne cornpackrepose sur la positivite d'une intggrale de Dirichlet. On se propose de remplacer dans ce qui suit la methode par "un principe du maximum" pour les formes harmoniques. Ce prineipe du maximum restera valable sous des hypothlses de positivite en moyenne qui Blargissent C32] le champ d'application de la theorie de Bochner.
1. Formule de la moyenne pour les formes harmoniques. Le principe du maximum proviendra de formules de la moyenne. Pour pouvoir additionner une forme differentielle prise en des points differents il est utile de scalariser les formes diffgrentielles en les remontant B un fib& 1.1.
Le fibre On
O(V)
de replres.
.
note par O(V)
le fibre principal des replres orthonormes
de V. Un Blgment r t O(N) isometrique r
: Tx (V)-----+
est par definition une application
R~ (m
= dim V)
la projection ; pourOtout xo € v, p-I le groupe orthogonal en m droite sur O(V) Soit
a1
O(m)
variables). De plus
x
= p(ro)
(on note O(m)
0(d agit
B
.
la representation naturelle de O(m)
reprgsentation contragrgdiente sur Soit 7 L A ' ( V )
. On note
sur R~ ;
1;
la
(R~)*.
une forme diffgrentielle de degri5
1
sur V. On lui
P. Malliavin dBfinie sur 0(V) , 21 valeur dans
associe la fonction fw
*
(R~)
dsfinie par 1.1.1.
fn (r).
-1 ('217 . 5 €R=.
= C r, ( , )r
Alors pour tout gCO(3
Rsciproquement une fonction satisfaisant 1.1.2. provient d'une forme diffgrentielle de degrBs8 sur V. En ce qui concerne les formes diffgrentielles de degrBs,l.l.l. et 1.1.2. se gBnBralisent de fapon Bvidente en remplacant
2
par sa puissance extBrieure s-Sme
1.2. Laplacien horizontal:
,,..
On note par e .,em la base canonique de R ~ .Si r est -I un repsre ro (es) est un vecteur de Tx (V) , (xo = p(ro)). On 0
pose -1
us(t> = nx (tro
(es))
0
Effectuons le transport parallele de replre ro le long de l'arc t,
,
u (t), on obtient
IQ (t) E
o(v),
on pose
On dBfinit ainsi a c h a w de vecteur HI,.
..,dn
sur O(V)
qui sont
appel6,champs de vectem horizontaux canoniques. On note par
"CH"
la dBrivBe suivant le vecteur IIs. On pose
Ao(v)
est un opBrateur semi elliptique appelg le laplacien horizontal
ou laplacien de Bochner.
P. Malliavin 1.3. Fomule de ~eitzenback
Soit d le cobord sur les formes diffgrentielles d* adjoint
a = dd+ + d*d.
ThSorBme: && 08
[oil J(r)
son
Alors on sait que
~tl\l(V)
=-40(v)fir+J~
est le tenseur de Ricci lu dans le replre r et oil
(J.fip (r) = J(r)
(rg
.
1.4. Formule de la moyenne Soit ru(t)
la diffusion sur O(V)
1.2.2. Notons par %(t)
associSe B lropQrateur
la solution de llBquationdiffgrentielle
matricielle. I .4. I .
t
=
- -2I X w (r)
J(ru (t)),
X , ( O )
= Identit6.
Th6orBme : Soit TeJ\l(~),C I T T = ~ fir (to) = Er (Xu (t) +(rO
(t))).
0
Preuve : I1 s'agit de la formule de Feynmann-Kac C20 1, ggnSralis& par Pinsky B t b c a u cadre matriciel. L'analogie avec le cas scalaire sera plus frappante si on convient d'utiliser la notation des ,intBgrales multiplicatives qui permet dlBcrire 1.4.2.
(t) = exp(
*
+ ~ ( r ~(E
Les Stages de la preuve consiste B poser pour toute fonction f d6finie sur O(V) 1.4.3.
(Qtf)(rO)
= E r (XW(t). 0
f(rU)(t)))
P. Malliavin 1 . 4 . 4 . -L
dBfinit un semi-grouve
Q
t
Preuve : On utilise la formule de Chasles pour les integrales multiplicatives
combing avec la proprigti5 de Markov pour la diffusion rw (t)..
1.4.5. Lemme : Soit f Q E -1
lim (-,f) E E-90
c2 alors
fonction de classe 1
(rO) =
P
( AO(M)f)
(ro)
- qI
J(ro> f(ro)
Preuve :
( 5 ) ) d5
e~(*[o,i~(r~ D'autre part f(r,(E
oG o( f )
))
-
f(ro) +
=
( 3 Hsf)
I
- 5 J(ro)
(rO) b;(E)
tend vers zero p .s. en
lorsque
est un Bchantillon du brownien sur R ~ . 1 . 4 . 6 . Corollaire :
+
o(E)
.
-
+ P1 (dO(M)f)(ro)E+ E
o(E)
o, et ou b&
solution & problsme de Cauchy pour 1'Bquation
de la chaleur sur les formes -----a'rt I = - T O V ~ ,To i) t
do"nQ s'obtient par
Ce corollaire ltablit la formule de la moyenne. En effet si est une forme harrmnique
Wt= To
To
.
1.4.7. Remarque : Le dgplacement parallPle stochastique de Ito, Dynkin
ni/Gq[ 13 est implicitement contenu dans l1op8rationde
P. Malliavin scalarisation suivie de l'utilisation de la diffusion rw(5). 2. Principe du maximum et calcul de pertubation. 2.1. Proposition : Supposons que pour tout
-
r0 -il existe to tel
m (la norme Qtait prise au sens des endomorphismes de B ) forme harmonique -
est
identiquement
alors toute
(on notera alors
I H (V) = 0)'.
1% I
Preuve : Soit xo un point 03 atteint son maximum, soit -1 (xo) Bcrivons la formule de la moyenne
reap
2.2. Noyau rQsolvant : Soit J1 une application continue,de O(V)
dans les d m matrices. Posons si A est une partie borQlienne de 0 (V)
-st (ro,A) e dt, Res assez grand. Uors 2. I . sTOcrit:pour tout Ats,:'~;R
continue en
S
6
1/2 J (ro,A) est une fonction Si pour tout ro et pour tout A, 61 continue & s [0,+m[ alors H (V) = o. 2.2.1.
sur
P. Malliavin 2.3. Equation depertubation Soient J1, J2 deux fonctions continues alors on a pour Res assez grand
Preuve : I1 s'agit de transposer au cas matriciel un calcul classique de Darling-Siegert-Kac C 1 d . Posons
Alors
Calculons
Calculons en appliquant la propriGt6 de Markov en
5
.
Effectuant la transformation de Laplace sur la variable t qui transEorme un produit de convolution en produit ordinaire on obtient 2.3.1.
-1 + 3. Conclusion : On pose 2 J=-J +J
-
oii 25'
de la matrice de Ricci. On prend dans 2.3.1.
-
J2-J1=J
.
3.1. S'il existe ro tel que ~+(r ) ~~2 Rs (ro,dr) sera continu jusqu'en 3.2. Si
est la partie positive -1 J,= 7 J,J~=-J+ ; alors
11.1-&,
est dEfinie positive alors
s=o.
est assez petit alors l'gquation integrale en
c1
P. Malliavin 2 . 3 . 1 . sera inversible jusqu'en s=o et on pourra ainsi obtenir le
CI
prolongement analytique de
jusqu'en
R 1 (V) = O.Les
sOo d'ol 1
hypotheses 3 . 1 . et 3 . 2 . signifient que l'on obtient H (V)
sous des
hypotheses de positivite en moyenne. Pour un exemple d'un tel 6nonc6 slBcrit en detail
cf .C323.
3 . 3 . Remarque :Milgram et Rosenbloom 07 t
---+
a,
7 ont dtudie la dlformation
par le semi groupe de la chaleur d'une forme 8 fermBe
vers une forme harmonique. Cette horntopie demontre en particulier le thBorSme de Hodge. La construction de Milgram Rosenbloom repose sur la connaissance de la d6composition spectrale L~
de ltop6rateur 0
. Dans l'approche qui pr6cSde on utilise la
formule explicite 1 . 4 . 6 . du semi groupe Q
t
spectre de 0
pour montrer que le
ne contient pas z6ro.
4. Cas des espaces homogenes : Les matrices J apparaissant dans les formules de weitzenbGck sont des matrices constantes. Par suite l'gquation diffsrentielle
1 . 4 . 2 . determinant Xg(t)
est une Bquation
1 coefficients constants, qui se rLsoud par une diagonalisation de J. Le calcul des valeurs propres de la matrice J
est ainsi la premiSre
Ctape. Un algorithme est donne dans L28] pour le cas des espaces sym6triques. Le second point pour avoir des conditions n6cessaires et
suffisantes d'annulation de cohomologie & valeurs dans des
fibres homogenes est la connaissance de la plus petite valeur propre du Laplacien horizontal
A 0(V)
sur des espaces des fonctions
Bquivariant. Ce programme a St6 entrepris dans calcul de minoration
[29]
. Enfin le
spectres de laplacien horizontaux a partir
P. Malliavin
d'estimles hypoelliptiques se transporte au cas des variltes riemannienne
homoglne, compacte. Ceki donne de nouveaux thlorsme d'annulation
de cohomologie B partir d'hypothlees infinitleimales portant sur le tenseur de courbure L 2 Q 5. Cohomologie relative
des varietes
linlaires
.
B bord :
La theorie de Bochner a une traduction dans la thlorie de la cohomologie relative utilisant les conditions de Neumann-Spencer (cf Yano C441). La construction des semi groupes associEes aux conditions de Neumann-Spencer peut ltre entreprise. Alors apparaissent les Qllments nouveaux suivant a) Une diffusion reflgchie sur le bord technique classique pour l'etude du problSme de Neumann. b) De nouvelles integrales multiplicatives au dessus sur la diffusion rlflbchie. Ces integrales sont d'un type nouveau. , elles correspondent B des pertubations ,. raidee
de syst2mes diffgrentiel-
les dont certain8 coefficients tendent vers l'infini ; les resolvantes associees B ces lquations tendent vers des operateur de projection.
I1 convient enfin de maitriser cette situation tenant compte du fait que le processus touche le bord en des temps formant un ensemble parfait. Cette thgorie gst menle .3 bien ; elle culmine dans la definition de nouvelles fonctionnelles multiplicatives au dessus du temps local
C
13
.
6. Cas de fibrls localement plats Des conditions suffisantes d'annulation de la cohomologie peuvent ltre obtenues B partir de proprietes asymptotiques d'un
P. Malliavin processus de Markov
B temps continu d6fini sur le groupe.de
Poincar6 de la base b 4
.
CHAPITRE 111
Formules de Poisson.
0.
La m6tHode de Poincar6 de balayage expos6 en 0 .
a 6t6 Etendue
par Doob et Brelot-Doob C 5 3 1 une situation trPs g6n6rale. Soit h est une fonction harmonique bornle relativement B un op6rateur semi elliptique L, bur simplifier supposons que le temps de vie de la diffusion associ6e B L soit infini h(xo) = E x (h(xw(t)))
0.1.
. Alors
pour tout
tC+W
.
0
Utilisant le thLorPme de convergence de Doob des martingales h(xo (t))
converge ps lorsque
t -7
+a)
.
Dans le cas gEn6ral se pose le problsme de rEaliser une frontisre abstraite compactifiant V, soit xu (t)
& a€aV
de he3V
b V , et ensuite de montrer que
p. s. puis de dEfinir une extension naturelle
par un th6orPme de Fatou adapt6,
Dans les cas qui nous occuperont ici cette compactification se trouvera d6jB r6alis8e et le th6orSme de Fatou sera d6ja connu. Alors passant 5 la limite 0.1. s16crit
P. Malliavin
oG p (da) est la loi de sortie sur a V de la diffusion partant xo de a. On se propose d'ltudier ces mesures de sorties dans des cas splcifiques. conjecture & Bremermann : Notons par D un
1 . D6monstration
ouvert born6 de
c3
classe
cn.
On suppose qu'il existe une fonction q de
-
sur un voisinage de D tel que
de la frontiare
Vq #
o pour tout point
3 D de D, et telle que
Pour tout z fix6 on considere la forme hermitienne
on suppose que pour tout z 6 5 , QZ(& point mE3D on note Tm(a D)
a
)
est positive. Etant donna un
l'hyperplan rlel tangent en m B
D et ~a ~ D)( l'hyperplan 3 complexe contenu dsns Tm(a D) Lon 0
a (,T
3 D) = (~~(3 D))
(
.
Gi T,(J D)~J
On note
I.
a
asD = [m C ~ ;D % restreinte 1 T ( 3 D) est dlfinie positive m 1.1,
ThlorGme [ j o ] : Considlrons 2 m6trique riemannienne
3D
dlfiaie par
Alors pour tout -
zo E D la mesure de sortie de la diffusion associ6e
-B ltop6rateurde Laplace Beltrami pour 1.2.
est portle par
-
aSD.
1.3. Cons6quence : La mltrique 1.2. ltant une mltrique kghllrienne les
P. Malliavin fonctions holomorphes sur D
sont harmoniques pour la diffusion
Ceci permet de demontrer la conjecture suivante de
associ6e
Bremerman C 63: la frontisre de Silov de llalgSbreuniforme A(D) contaue dans
asI) . Un travail de
est
H. Rossi L413 donne sous des
hypotheses plus strictes, l'inclusion inverse. MBthode de dhonstration : -1.4. L
z : Le temps de vie de la diffusion est infinie.
Preuve : Posons f (z) = fonction p(z) =
- log
112
I*.
On projete + f (z)
(-q(z))
Alors l'invariant
la diffusion par la
.
est compris entre deux constantes stricte-
ment positives- et le leme de comparaison du chapitre I permet de conclure.
-
1.5. Lenrme : Soit mo f DSD alors il existe un voisinage A dans -
&m
et une constante c o tel que sur ltint6rieur& A
ait A f > c -
on
.
Preuve : cfQo]. Fin de la demonstration & th6orPme : --Ecrivons la formule de Ito (0.7.7) alors :
Prenons les esplrances des deux membres utilisant (0.6.2.1.)
ez
(f(z,(t),
- f(zo)
=
I
e
0
Lorsque t--+ +a D'autre part
, come
(Io t
(A f)(zu ( 5 ))dS
.
D est born6 le second membre est born6.
(A f) (z)> o quelque soit z 6D. Supposons qu'avec une
probabilit6 positive on touche un voisinage de mo
lorsque t
7
+W.
P. Malliavin Notons par U l'ensemble de ces trajectoires aboutissant dans un voisinage oh le leme
pour tout w E U t 7 R(W)
1.5. s'gpplique. Alors pour tout
il existe
R(w )
tel que
t
( A f) (2, (t))Z
c si
dl03 1.6. entrake que pour tout t
ce qui entrarne E(%)
1.7. Remarque
= 0 et le thlorime est dBmntrB.
El07 : contient Bgalement une
construction, par une
mBthode analogue, d'un noyau de Poisson pour tout polysdre analytique. 2. Une formule de Poisson sur le demi-plan de Siege1 de rang
2.
2.0. La formule de Poisson envisage dans le paragraphe 0 correspond 1 la reprgsentation des solutions d'un oplrateur elliptique
. Dans
h
le paragraphe 1 Btait Btudil la formule de Poisson pour les solutions du systime de Cauchy-Riemann, ~ystsmesurdlterminl, ce qui a permis de rlduire la dimension de la frontisre. La mlthode pour traiter ce systSme surdBtermin.5 Btait encore de se ramener 1 un oplrateur unique elliptique
associB B une mltrique kahllrienne adapts. On se
propose ici d'indiquer, dans un cas spBcifique, une mlthode d'approche pour construire la frontisre adaptaaux solutions h d!un systsme d'lquations elliptiques. 2.0.1.
$h=
4 h P O
h dlfini sur l'owert
0.
La mgthode consietera B utiliser un processus de diffusion B vitesses la premiire vitesse sera la diffusion associle 1 8
a ,
la seconde vitesse la diffusion associee B A
2 ' La loi de changement de vitesse sera effectus suivant des temps d'arrft, B la situation gBomBtrique considBrBe. Dans une partie
adapt& 0
O1
de
on choisira la premiere vitesse come Btant celle la plus adaptee
B la rgduction de la frontisre, de mzme pour la seconde vitesse. S'il s'agissait de processus dgterministe dBfinie par deux champs de vecteurs, il est clair que le fait de pouvoir choisir le champ de vecteur suivant lequel on Bvolue permet de mieux diriger la trajectoire vers l'ensemble de sortie choisi. Bien que l'on trouve dans un contexte stochastique il en est de & m e . 2 . 1 . Le systsme de Hua
Soit
x2 les matrices
2 ~ 2 B, coefficients complexes
€4
symBtriques (ai = a ). Si Z on ecrit Z = U t iV 03 U ji V sont des matrices rBelles. Le demi plan de Siege1 slBcrit
% =[Z sur 2.1.1.
EG%~ ; V
est dBfinie positive]
et
. La mStrique de Bergmann
est la mBtrique GhlBrienne. ds2 = 3 5
log CdSt
(v)]
.
La frontisre F de Furstenberg fonctions
permet de reprcsenter les
harmoniques bornBes pour la dtrique 2 . 1 . 1 . Elle
s'identifie essentiellement B
F = { z Q & ~ ; v = 03 XT 03 T est le cercle. La frontisre de Silov S de
s = [zeA2 ;v
=
s'identifie
03
Ainsi on a une fibration naturelle F . y S
. Lee fonctions
holomorphes bornSes se representent sur F par des valeurs frontisres qui sont constantes sur les fibres de cette fibration
Plus gengrale-
P. Malliavin ment on considere la classe
&
des fonctions bornses sur
harmoniques au sens de la metrlque de Bergmann 2.1.1.
%
possEdant des
valeurs frontieres constantes sur ces fibres. Le systeme de Hua est un systPme d'oplrateurs difflrentiels du second ordre ainsi form6 : Notons par
JZ
la matrice formde d'opsrateurs
differentiels du ler ordre
-
i)
et par
la matrice obtenue en
prenant les opdrateurs conjuguds
3 321
etc.... Considlrons la matrice
(on convient d'effectuer ce produit en utilisant les rsgles du
a a azs vk '*: "k 7'.
produit de matrice et en convenant Alors m
,,
m2, m3, m4
sont quatre operateurs diffsrentiels du
second ordre. 2.2. Thdoreme de reprlsentation. 0 3 1 Soit h ngcessaire Mthode --
fonction bornee sur
% , alors une
suff isante pour que h 6 g
condition
est que ms h = 0,1< ~ $ 4 .
de d6mnstration. La ndcessitd de la condition se verifie Par un
calcul direct de dSrivation sur le noyau de Poisson. Pour montrer l'implication inverse on extrait du systPme de HIM les deux opdrateurs elliptiques suivants :
P. Malliavin d l = Trace V.
3.v. aZ
42 = Trace sZ.v.az et on note
oii a(V) Q;
+
A,.
- a(v)A,
est la fonction positive d6termin6e par la condition que
soit
elliptique mais nan elliptique.
Finalement on d h n t r e que si h est une fonction born6e sur satisfaisant 2.2.1. alors h
Aih = A2h = 0
€8
diffusions
. Pour cela on Btudie le mouvement asymptotique des 2 z?, zd2
A;
associ6e 1
et A
en appliquant un
certain nombre de fois le leunne de comparaison. I1 r6sulte de cette Btude que lorsque, t j + O D , z
"'2
(t)
a tendance
de "s'enrouler ind6finiment" autour d'une fibre de la frontisre de F~stenberg.Ma1heureusement cette bonne tendance est inutilisable
(t) quike % (Les op6rateurs zW2 ne sont pas invariants sous l'action
parce que au bout d'un temps fini de Hua et en particulier
A2
du groupe de transformation de
1;&
qui en fait un espace homogPne).
de s16chapper de Cfd peut stre 2 qui a tendance 1 sV6carterdu bord de Tde Le jeu
Cette habitude vicieuse de zw
.
corrig6 par Z~
est alors le suivant, on fait fonctionner bord ; on appelle alors
a 11int6riaur de
(t), on arrive prSs du z@2 3 notre secours zU (t) qui nous ramSne 1
; lorsque
'w,(t)
on fait fonctionner 1 nouveau zw2(t)
nous a assez 6loign6 du bord etc... Finalement on r6alise
ainsi un grocessus compos6 za (t) tel que si h 3
satisfait 2.2.1.
P. Malliavin 2.2.2.
h(z
(t))
est une martingale de temps de vie infini.
"'3 2 . 2 . 3 . La trajectoire z
(t)
s'enroule indlfiniment lorsque t---S+m
"'3
vers une fibre de la fibration de Furstenberg. Come lim
existe d'aprPs le thloreme de Doob de
h(zu ( t)) 3
convergence des martingales ceci entracne que la valeur frontisre de h sur la frontiere de Furstenberg est constante le long des fibres.
APPENDICE 1.
Solution Illmentaire & l'lquation de la chaleur sur les groupes
nilpotents d'ordre 2. Cette solution peut Ctre calculle explicitement B une transformation de Fourier sur pn prCs L183. Par int4gration en t on retrouve dans le cas particulier du groupe de Heisenberg la solution BlImentaire du Laplacien de Folland-Stein L14 1
.
2. Principe de minimum d'action etpropagation de la chaleur en
hypoelliptique. Pour t-zo
et conditionnant lee extrlaitls, les trajectoires
de la diffusion associl B un oplrateur hypoelliptique d'ordre 2, s'accumulent sur les courbes minimisant 'l'action lagrangienne classique L 18 ] 3 . Paradtrix trajectorielle. SystCmes diagonaux dans le symbole
principal. On obtient
L 333
un dlveloppement limit6 pour t petit de la
P. Malliavin diffusion associ6e a un op6rateur hypoelliptique d'ordre 2 :
Dans ce diiveloppement apparaissent les expressions :
On reconnait une intlgrale Qtudile par Paul Levy et M. Kac. Les systsmes d'ordren diagonaux dans leur symbole principal via une fonctionnelle multiplicative matricielle
s'intbgrent
stochastique convenable L34-J.
4. InCgalitQs de Sobolev fines. Un ouvert fin est un ensemble borllienB tel que pour tout xo€ B
, alors
I1 existe des ouverts fins sans points intgrieures. Dans l 7 ] 116galit6 d'lnergie pour les martingales penqet de d6montrer la diffsrentiabilitl des fonctions finement harmoniques au sens de Fuglede 53. 5. Algbbres
& Fonctlon etpotentiel fin
Le potentiel fin donne un cadre nature1 pour Etudier des "structures analytiques" sur des spectres d'algsbres de fonctions ainsi que pour construire des enveloppes d'holomorpie de certains ensembles parfaits L8
1, et [8],
,
Le potentiel fin s'applique B l'application rationnelle sur C,
L
83,,[8]b
, il permet
de caractlriser la frontisre de Jensen de
l'algbbre uniforme la plus g6n6tale.
P. Malliavin REFERENCES
H. Airault : a) R6solution stochastique & problPmes de Neumann Spencer,. Comptes Bendus 1975 (Tome 280) , p. 781 b) S6minaire Jean Leray 1975. (Expos6 111) 25 pages. c) Journal de Mat%matiques
P ~ ~ P s -.AppliquBes et
janvier 1976. M. Berger
, Gauduchon, E. Mazet
: Lecture Notes no 194.
S. Bochner-Yano : Curvature and Betti numbers Princeton 1951.
Blumenthal-Getoor : Markov Process and Potential Theory. Academic Press 1968. Brelot-Doob : Convergence angulaire
convergence fine. Annales
de Fourier 1963. Bremermann : &&generalized
Dirichlet problem for plurisubhar-
monic function T.A.M.S. -
1959, p.261.
A. Debiard, B. Gaveau, E. Mazet : Th6orPmes & cornparaison gSom6trie a) ComptesRendusI974 (278) p.723 et p. 795 b) Ibidem 1975 (281) p.455. d) Preprint de 35p.(thSse de Gaveau juin 75). A. Debiard, B. Gaveau : Potentiel fin et $lgSbres
& fonctions
a) Journal of Functional Analysis 1974 (16) p.289-304. b) Ibidem 1974 (17) p.296-310. c) Ibidem janvier 1976. d) Comptes Rendus 1975 (280) p.85.
P. Malliavin
191
A. Debiard, B. Gaveau : Diffdrentiabilit& & fonctions finement harmoniques. Inventiones Mathematicae 1975 (29) p.lll-123.
[ld
A. Debiard, B. Gaveau : Frontilre de Silov des domaines faiblement pseudo-convexes a) Comptes Rendus 1974 (279) p.407-410 b) Preprint 15p. (thlse de Gaveau 6.75).
El]
Darling-Siegert : Proc.
Nat. Acad. of Sciences 1956 p.525.
E2] a E.B. Dynkin : Markov Process Grundlehren Band 121-122. p2Ib E.B. Dynkin : Diffusion of tensors :Sov. Doklady 1958 p.532-536.
fig
Eells-Ellworthy : Seminaire & gGom6trie diffdrentielle stochastique Warwick.
b4]
Folland-Stein : Heisenberg group and
a
complex.Communication in
pure and appl. mathematics ddcembre 1974. c5]
Fuglede : Finely Harmonic functions,Iecture Notes n0289.
E6]
R. Gangolli : Notices of the American Math. Soc. 1964.
7] [Is]
Gihman,Skohorod
1
Stochastic differential equations Ergebnisse
B. Gaveau : Principe de moindre a~tion~propagation de la chaleur
a) Comptes Rendus 1975 (280) p.571. b) Comptes Rendus 1975 (281)p. c)
09)
PO]
Preprint de 90 pages.(thSse
G.A. Hunt : Martingales
327. de B. Gaveau juin 1975).
Processus de Markov. Monographies de la
socidt6 Mathdmatiques de France. 1967. M. Kac : Probability & Integral Equations ,Second Berkely symposium
P. Malliavin
K. Ito : Stochastic parallel1 displacement a) Proc. Int. Congres Math. (Stockholm 1962) p.536-539. b) Lecture Notes 451 p.1-7.
K. Ito
:
a) Lectureon Stochastic Process Tata Institute n024. b) Stochastic differential equations,Memoir of the A.M.S. 1951.
A. Koranyi-P. Malliavin : Poisson formulae on the Siege1 half plane of rank ---
Acta Mathematica
1975, vol. 134, p.185-209 D. LatremoliOre :
condition non iinGaire dlannulation,Comptes Rendus, 1974 (279), p.413.
H.P. Mc Kean Jr : Stochastic Integrals. Academic Press 1969. H.P. Mc Kean Jr : An upper bond to the spectrum of the 1aplacia.n on a manifold of negative curvature.J. differential geometry 1970, p.359. P.A. Meyer : Probabilitss spotentiel, Hermann 1966. M.P. Malliavin-P. Malliavin : Diagonalisation du systsme de de Rham Hodge au dessus dlun espace riemannien homogene. Lecture Notes 11'466,
M.P. Malliavin-P. Malliavin : PropriGtGs argodiques
diffusion
horizontale et spectre du Laplacien au dessus d'un espace riemannien sym6trique a) Lecture Notes 404,p.155-210 b) Comptes Rendus 1974 (279) p. 185-188 c) Comptes Rendus 1975 (280) p.793.
P. Malliavin
[30]
1.
Malliavin : Asymptotic of the Green's function of 5 Riemannian manifold and Ito's stochastic integrals. Proc. Nat. Acad. 1974 (71) p.381-383.
]I,[
Ed
P. Malliavin : Ponctions de Green d'wn ouvert pseudo-convexe et classe de Nevanlinna --
ComptesRendus 1974 (278) p.114
P. Malliavin : Formules de la moyenne pour lea formes harmoniques, calcul de pertubation g --
thdorsmes d'annulation.
Journal of Functional Analysis 1974 (17) p.274-291. &3]
EL]
P. Malliavin : Paramgtrix trajectorielle. Comptes Rendus 1975 (281) p.241 P. Malliavin : a) Comptes Rendus 1975 (280)p.941. b) Comptes Rendus 1975 (280) p. 1009. c) Annales de 1'Institut Fourier fascicule 3 et 4
[35]
P. Mallihin : a) Proc.
of the conference df Probabilistic
Methods in Differential EquationsLecture Notes
b) Proc. of the Summer Institute of the American Mathematical Society on Complex Analysis Williamstown 1975 [36]
8. Mazet : Construction & & diffusion sur une varietg riemannienne
Sgminaire de g6omLtrie riemannienne. Institut H. PoincarO janvier 1973. [37]
Milgram-Roaenbloom : Harmonic forms and Heat conduction Proc. Nat. Acad. 1951 (37) p. 180-184.
[38]
.
Milgram-Rosenbloom : Proc. Nat Acad. 1951 (37) p. 435-438.
P. Malliavin
M. Pinski : Multiplicative operator functional and their asymptotic propeitfes. a) TAMS 1972 (167) p.89-164. b) Advance in Probability I11 Marcel Dekker New-York 74, c) Lecture Notes 451 p. 89.99. d) Isotropic transport process. A para'itre TAMS 1976. J.J. Prat : Convergence angulaire & ladiffusion g r une variGt6
simplement connexe B courbure nlnative. a) ComptesRendus 1971 (272) p.1586-1589.
H. Rossi : Holomorphically convex set Annals of Mathematics 1962. S. Varadhan : Diffusion process in small time intervalls Comm. in
Pure and appl. Math. 1966. J. Vauthier : a) 'L estim6es sur une vari6t6 @ Stein ComptesRendus
1974 (279) p.409-411. b) Sdminaire J. Leray Janvier 1974. c) Unicitd de l'dquation d_e la chaleur Comptes
endu us
1975 (281) p.41 et preprint de 15 pages.
Yano : Integral formulae &Riemannian
geometry. Marcel Dekker 1972.
J. Ibero : a) Calcul differentiel stochastiaue intrinssaue sur 1 s
groupes de Lie. Compte Rendus 1975 (280) p.13. b) Calcul de pertubation sur la diffusion horizontale de SL(2,R) Comptes Rendus 1975 (280) p.985.
R. Azencott
:
---
Behaviourofdiffusion semi group at infinity.Bul1. Sot. Math. de France 1974 (102) p.193-240.
E. JORGENSEN
Central limit for Geodesic Randon w& (1975).p.1.64.
:
2. Warhsch 32
(C.I.M.E.)
SOLVABILITY OF INVARIANT DIFFERENTIAL OPERATORS ON HOMONOGEOUS MANIFOLDS
S. Helgason
Varenna. August 25 / September 2, 1975
S. Helgason SOLVABILITY OF INVARIANT DIFFERENTIAL OPERATORS O N HOMOGENEOUS MANIFOLDS S i g u r d u r Helgason MIT Cambridge Mass. 51.
I n t r o d u c t i o n and summary. Over a 100 y e a r s ago S. L i e [18] r a i s e d t h e
following question: Given a d i f f e r e n t i a l e q u a t i o n , how c a n knowledge a b o u t i t s i n v a r i a n c e group be u t i l i z e d t o w a r d s i t s s o l u t i o n ? Special case.
i n t h e plane, group
Ot
Consider t h e d i f f e r e n t i a l e q u a t i o n
It i s c a l l e d s t a b l e under a one-parameter
( t 4 1R)
o f diffeomorphisms i f e a c h
t h e i n t e g r a l curves
Ot
permutes
( a l l concepts a r e h e r e l o c a l ) .
Example.
The l e f t hand s i d e r e p r e s e n t s t h e t a n g e n t o f t h e a n g l e between t h e r a d i u s v e c t o r and t h e i n t e g r a l c u r v e .
Thus
t h e e q u a t i o n shows t h a t t h e i n t e g r a l c u r v e s i n t e r s e c t e a c h circle
x2
+ y2
= r2
under a f i x e d angle.
Hence t h e group
S. Helgason o f r o t a t i o n s around t h e o r i g i n permutes t h e i n t e g r a l c u r v e s , i.e.,
leaves t h e equation, s t a b l e . F o r a one-parameter group
i n the plane l e t
Theorem ( L i e ) .
of transformations
d e n o t e t h e induced v e c t o r f i e l d
U
E q u a t i o n ( " ) i s s t a b l e under
only i f t h e v e c t o r f i e l d [U,Z]
-
In t h i s case
(Xn
equation
- Ydx
Xdy
($t)
Z = X-
a a ax + Y- ay
= hZ
(A
($t)
i f and
satisfies
a function)
Y C ) - ~ i s an integrating f a c t o r f o r the = 0.
Thus knowing a s t a b i l i t y group f o r a d i f f e r e n t i a l e q u a t i o n p r o v i d e s a way t o s o l v e i t .
For t h e example above
t h e e q u a t i o n i s s t a b l e under t h e group $ t ( ~ , y )=
(X
cos t
-
y sin t, x sin t
+ y cos t )
f o r which
and t h e theorem g i v e s t h e s o l u t i o n y = x tan($(x2
+
y2)
+ c).
I n s p i t e o f e x t e n s i o n s o f t h i s theorem t o h i g h e r o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s t h e s e methods have had
S. Helgason a l i m i t e d i n f l u e n c e on t h e g e n e r a l t h e o r y o f d i f f e r e n t i a l equations.
N e v e r t h e l e s s , L i e ' s q u e s t i o n h a s been of enormous
importance i n m a t h e m a t i c s , i n f a c t l e a d i n g t o t h e t h e o r y of Lie groups. Aiming f o r l e s s g e n e r a l i t y we c a n pose a few problems i n L i e ' s s p i r i t .
Let
be a c o s e t s p a c e ,
G/H
b e i n g a c l o s e d subgroup o f p L i e group
G
d i f f e r e n t i a l o p e r a t o r on t h e manifold i n v a r i a n t under
for a l l
+
origin A.
eH
of
of
B.
Let
o
denote t h e
We can t h e n a s k t h e f o l l o w i n g q u e s t i o n s .
2
T
on
G/H
denoting t h e d e l t a - d i s t r i b u t i o n a t D
is
locally solvable, i.e.,
f o r some neighborhood
2
D
V
D, i . e . ,
such t h a t
DC"(V) 3 c ~ ( v )
C.
which i s
Does t h e r e e x i s t a fundamental s o l u t i o n f o r
6
be a
~ ( g )d e n o t i n g t h e t r a n s l a t i o n
onto i t s e l f .
G/H
G/H.
distribution
D
i n t h e sense t h a t
G
g 6 G , t h e mapping
~ ( g ) : x H gxH
G/H
and l e t
H
of
o?
globally solvable, i.e., DC-(G/H) = c"(G/H) ?
is
o
on
G/H?
a
S . Helgason Let
D(G/H)
denote t h e a l g e b r a of a l l G-invariant
d i f f e r e n t i a l o p e r a t o r s on" G/H. group of t r a n s l a t i o n s
of
F o r example, i f and
IRn
H = 10)
is the
G
then
D(G/H)
consists of a l l d i f f e r e n t i a l operators with constantc o e f f i c i e n t s ; by r e s u l t s of E h r e n p r e i s and Malgrange, questions A) case.
B)
and
C)
have p o s i t i v e answers i n t h i s
We s h a l l s e e however t h a t i n g e n e r a l t h e a n s w e r s
depend on t h e s p a c e
G/H.
We s h a l l now s u r v e y
-
i n chronological order
-
some
of t h e p r i n c i p a l r e s u l t s which have been o b t a i n e d f o r t h e q u e s t i o n s above. First i f
X = G/K
i s a symmetric s p a c e o f t h e
noncompact t y p e ( i . e . ,
G
noncompact, connected s e m i s i m p l e
w i t h f i n i t e c e n t e r and
K
a maximal compact s u b g r o u p ) t h e n
a s proved i n [ l l ] e a c h
D
D(G/K)
has a fundamental solu-
t i o n , and c o n s e q u e n t l y , by c o n v o l u t i o n , solvable.
D
Since i n solving t h e equation
by t h e K - i n v a r i a n c e o f K-invariant,
D
and
6 , assume
is locally
DT = 6 T
we may,
t o be
t h e problem can be handled w i t h i n F o u r i e r
a n a l y s i s of K-invariant
f u n c t i o n s on
X.
The g l o b a l
s o l v a b i l i t y d i s c u s s e d l a t e r d o e s n o t seem t o b e a c c e s s i b l e by t h i s method. It was proved i n P e e t r e [lg] ( c f . a l s o Harmander
[15],
p . 170) t h a t a d i f f e r e n t i a l o p e r a t o r of c o n s t a n t
strength is necessarily locally solvable.
Without e n t e r i n g
into the precise definition, a d i f f e r e n t i a l operator of c o n s t a n t s t r e n g t h i s i n a c e r t a i n s e n s e a bounded p e r t u r b a -
S. Helgason t i o n of a c o n s t a n t c o e f f i c i e n t o p e r a t o r .
One might wonder
whether an i n v a r i a n t d i f f e r e n t i a l o p e r a t o r i s not n e c e s s a r i l y of c o n s t a n t s t r e n g t h .
This can not be s o because t h e n it
would be l o c a l l y s o l v a b l e and t h e o p e r a t o r
a + ax
I-
a + ix- a az
ay
which i s e s s e n t i a l l y t h e non-locally s o l v a b l e Levi-operator i s ( c f . [ 4 ] ) a l e f t i n v a r i a n t d i f f e r e n t i a l o p e r a t o r on t h e Heisenberg group
Thus we s e e t h a t answer.
B
has i n g e n e r a l a n e g a t i v e
I n view of t h i s , one can modify t h e problem and a s k :
For what L i e groups
L
a r e a l l l e f t invariant d i f f e r e n t i a l
o p e r a t o r s ( i . e . , a l l members of solvable.
D(L) = D(L/Ie))
locally
Using H6rmander1s necessary c o n d i t i o n f o r t h e
l o c a l s o l v a b i l i t y of a d i f f e r e n t i a l o p e r a t o r [15], p. 157, he, and independently Csrezo-Rouvisre [ 4 ] , o r d e r t h a t each
D Q D(L)
proved t h a t i n
should be l o c a l l y s o l v a b l e it i s
necessary and s u f f i c i e n t t h a t t h e L i e a l g e b r a of
L
be
e i t h e r a b e l i a n o r isomorphic t o t h e a l g e b r a of
n
m a t r i c e s where each row except t h e f i r s t i s
(and t h u s has
0
x
n
an a b e l i a n i d e a l of codimension 1). The c o n t r a s t between t h i s n e g a t i v e r e s u l t f o r and t h e p o s i t i v e r e s u l t f o r t h e symmetric space
G/K
L
S. Helgason d i s a p p e a r s when we view spaces.
A c o s e t space
L i n t h e context o f symmetric
(B a L i e group,
B?C
C
a closed
sfibgroup) i s c a l l e d a symmetric c o s e t space i f t h e r e e x i s t s an i n v o l u t i v e automorphism C.
The spaces
a
of
with f i x e d p o i n t s e t
B
G/K
considered above have t h i s p r o p e r t y .
L
a s a homogeneous space under l e f t and
Now we c o n s i d e r
r i g h t t r a n s l a t i o n s simultaneously, i . e . , we l e t t h e product group
L x L
a c t on
L
by
The subgroup l e a v i n g
e
fixed i s t h e diagonal
L*
L
so
we have t h e c o s e t space r e p r e s e n t a t i o n
Then t h e a l g e b r a the algebra
Z(L)
D(L
x
L/L*)
i s c a n o n i c a l l y isomorphic t o
of b i - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on
L and t h e n a t u r a l problem becomes: l o c a l l y ( g l o b a l l y ) s o l v a b l e on
Given
D E Z(L), Is i t
L?
This problem was considered by Rafs f o r simply connected n i l p o t e n t L i e groups D E Z(L)
L.
He proved t h a t each
has a fundamental s o l u t i o n and hence i s l o c a l l y
solvable.
The method i s based on harmonic a n a l y s i s on
L.
S h o r t l y a f t e r w a r d s C131 I proved l o c a l s o l v a b i l i t y f o r each D E Z(L)
f o r t h e c a s e when
L
i s semisimple.
The proof was
an easy consequence of a s t r u c t u r e theorem of HarishuChandra for the operators i n a n a l y s i s on
L.
Z(L)
b u t d i d not r e q u i r e any harmonic
S. Helgason A t a b o u t t h e same t i m e I proved g l o b a l s o l v a b i l i t y ,
D C ~ ( G / K=) c ~ ( G / K ) f o r e a c h i n v a r i a n t d i f f e r e n t i a l o p e r a t o r D
on t h e symmetric s p a c e
The p r o o f [13] i s based on
G/K.
a c h a r a c t e r i z a t i o n of t h e image o f t r a n s f o r m on
under t h e Radon
c:(G/K)
G/K.
What a b o u t g l o b a l s o l v a b i l i t y on
L?
A remarkable
example o f Cerkzo-Rouvikre [5] shows t h a t f o r any complex semisimple L i e g r o u p
L
t h e r e e x i s t s a n element
which i s n o t g l o b a l l y s o l v a b l e .
w
This
w € Z(L)
c a n be t a k e n a s t h e
imaginary p a r t of t h e complex C a s i m i r o p e r a t o r on L
L.
viewed a s a r e a l L i e g r o u p h a s a C a s i m i r o p e r a t o r
which t h e y showed, u s i n g harmonic a n a l y s i s on solvable.
C
L, i s g l o b a l l y
T h i s g l o b a l s o l v a b i l i t y was more r e c e n t l y proved
by Rauch-Wigner [21], Lie groups methods.
But
f o r a l l non compact r e a l semisimple
L, w i t h f i n i t e c e n t e r , u s i n g e n t i r e l y d i f f e r e n t
The main s t e p i n t h e i r proof i s v e r i f y i n g t h a t no
n u l l b i c h a r a c t e r i s t i c of
C
l i e s o v e r a compact s u b s e t o f
G.
The c a s e of a s o l v a b l e L i e g r o u p l e f t out.
h a s s o f a r been
~ a i s lmethod f o r t h e n i l p o t e n t c a s e was e x t e n d e d
by ~ u f l o - ~ a ? [s7 ] t o s o l v a b l e D E Z(L)
L
is locally solvable.
found by R o u v i s r e [22],
L
and t h e y proved t h a t e a c h
An a l t e r n a t i v e proof was
m o d i f y i n g i n a n i n t e r e s t i n g manner
methods which Hzrmander had i n t r o d u c e d f o r c o n s t a n t coefficient operators.
H i s method, b e i n g l o c a l , does n o t
g i v e t h e supplementary r e s u l t of [ 7 ] t h a t i f exponential s o l v a b l e L i e group then
D
L
i s an
h a s a fundamental
S. Helgason solution. The r e s u l t s I have quoted on t h e L i e g r o u p s on t h e symmetric s p a c e
G/K
L
and
s u g g e s t t h e f o l l o w i n g more
general question. Let B/C B-invariant
be a symmetric c o s e t s p a c e and
d i f f e r e n t i a l o p e r a t o r on i t .
Is
D
D
5
necessarily
locally solvable?
52.
Solvability results.
I n d i c a t i o n s of p r o o f s .
We s h a l l now i n d i c a t e t h e p r o o f s of t h e p r i n c i p a l r e s u l t s already stated.
We emphasize t h a t t h e p r o o f s of
Theorems 1-5 a r e q u i t e independent and socan be r e a d i n any order. We have a l r e a d y mentioned t h e f a c t t h a t a l e f t i n v a r i a n t d i f f e r e n t i a l o p e r a t o r on t h e Heisenberg group i s not necessarily l o c a l l y solvable.
I n order t o describe the
proof of ~ a y s 'p o s i t i v e r e s u l t f o r n i l p o t e n t g r o u p s I r e c a l l t h e main r e s u l t s from harmonic a n a l y s i s on n i l p o t e n t L i e groups 1171. Let
9
G
be a simply connected n i l p o t e n t L i e group,
i t s Lie algebra;
G
a c t s on
9
v i a t h e adjoint repre-
s e n t a t i o h and by d u a l i t y on Oj*, t h e d u a l v e c t o r s p a c e t o
.
To each
f
representation
E
9" nf
is associated a unitary irreducible
a s follows:
Let
H
be a c l o s e d
connected subgroup w i t h t h e p r o p e r t y t h a t i t s L i e a l g e b r a
S. Helgason
3
satisfies
f([$,$])
= 0.
with t h e s e properties.
xf
of
Assume
Then we can d e f i n e a homomorphism
i n t o t h e c i r c l e group T
H
xf(exp X ) = e and l e t
sf
G
fl,f2 E
q*,t h e n
a c t i n g on
i d e n t i f i e d with of
6,
fl
q*.
9
sf
1
and
and
induced by
G
n
f2
xf.
H;
are unitarily
2
a r e i n t h e same
Thus t h e o r b i t s p a c e %*/G
t h e d u a l space o f
i s a d i s t r i b u t i o n on
sf
2 n i f (X)
i s independent o f t h e c h o i c e of
nf
e q u i v a l e n t i f and o n l y i f orbit of
by
denote t h e representation of
It i s known t h a t
moreover i f
of marimum~dimension
H
The c h a r a c t e r
G.
G (Dixmier
is
Tf
[6]) which i s g i v e n
by t h e f o r m u l a
where
-
d e n o t e s t h e E u c l i d e a n F o u r i e r t r a n s f o r m and
a G - i n v a r i a n t measure on t h e o r b i t P l a n c h e r e l formula:
G-f.
X
on %*/G
such t h a t
denoting the o r b i t
( o
enp =
G-f.
is
Finally there is a
t h e r e e x i s t s a measure
fo
IJ
Using ( 2 ) and ( 1 ) f o r
$ we g e t for a Haar measure d f on
S. Helgason
9
For d g e n e r a l Lie a l g e b r a s
of t h e symmetric a l g e b r a
enveloping a l g e b r a
U(7);
i s determined by t h e p r o p e r t y
s
. (m E I*, X E f ) .
a c t i o n of
G
Also
U(9)
s
commutes with t h e
s o maps t h e r i n g of G-invariants
o n t o t h e r i n g of G-invariants i n
U(g).
onto t h e u n i v e r s a l
S(q)
s(P) =
there i s a bijection
I ( q ) C S(9)
Z(9)
U(v), t h e c e n t e r
of
R e c a l l now t h a t i n terms of d i f f e r e n t i a l o p e r a t o r s , and
= D(G)
Z(9) = Z(G).
e i g e n d i s t r i b u t i o n of each
and t h e eigenvalue evaluated a t
f
The c h a r a c t e r
Tf
i s an
Z E Z(G)
2(f)
i s j u s t t h e polynomial
s-~(z)
.
Let u s a l s o r e c a l l t h e following g e n e r a l f a c t ([2],
R". $I
[3],
[20]):
Let
Then i f Re s > 0, +
P 2 0 PS
~ ~ ( x ) $ I ( x ) don x R",
kin meromorphic f u n c t i o n on
be a p o s i t i v e polynomial on
defines a distribution
t h e mapping C
s
+
PS
extends t o a
with v a l u e s i n t h e space
S' (Eln)
of tempered d i s t r i b u t i o n on JRn, and t h e p o l e s occur among t h e numbers on
P).
- 1 , - 2 , .
Moreover
(N
being an i n t e g e r depending
S. Helgason T h i s c a n b e used t o c o n s t r u c t a fundamental s o l u t i o n f o r a constant coefficient d i f f e r e n t i a l operator Taking F o u r i e r t r a n s f o r m P
i s a polynomial,
D
by
DD*
f
+
?
we have
D
on lRn:
( D f ) - = P?
where
which we may assume p o s i t i v e , r e p l a c i n g
i f necessary ( * = adjoint).
The tempered
distribution
T1 n Ts = Ts+l which i s holomorphic n e a r Expanding Ts i n a L a u r e n t s e r i e s n e a r t h i s p o i n t
then s a t i s f i e s s = -1.
OD
we g e t
Ts =
lak(s+l)
where each
-r
The r e l a t i o n above t h e n g i v e s factor,
a.
T1
is a distribution.
ak
*
a.
and up t o a
= 6
i s a fundamental s o l u t i o n f o r
D.
We c a n now p r o v e t h e f o l l o w i n g r e s u l t ( c f . [ 2 0 ] ) . Theorem 1.
Each b i - i n v a r i a n t
n i l p o t e n t L i e group
G
We may assume n o t a t i o n above. polynomial
the distribution
41
G
simply c o n n e c t e d and u s e t h e Z
by
ZZ we may assume t h e
i n ( 4 ) t o be p o s i t i v e . +
Z
is locally solvable.
Replacing
P = ?(f)
d i f f e r e n t i a l operator
(Z*+)(e)
(t =
adjoint).
Let
5
denote
Then by ( 2 )
(4)
I n analogy with
Ts
above we d e f i n e
Ss
f o r Re s > 0
by
-
S. Helgason
By t h e r e s u l t a b o u t
P
quoted,
meromorphic d i s t r i b u t i o n - v a l u e d
s
+
cs
extends t o a
f u n c t i o n on
and by ( 5 )
C
and ( 6 1 ,
Moreover, by (1) and ( 3 ) ,
whence bo
= 5s+1,
ZEs
i.e.,
Es
5 Cs
i n t h e Laurent s e r i e s
fundamental s o l u t i o n f o r
=
=
ESt1.
Again t h e term
7 b * ( ~ t l ) g~i v e s
a
-r
Z.
Duflo and ~ a i s[ 7 ] extepded t h i s argument t o solvable groups giving Theorem 2 .
A bi-invariant
d i f f e r e n t i a l operator
D
s o l v a b l e L i e group i s l o c a l l y s o l y a b l e . A c o m p l e t e l y d i f f e r e n t proof was found by Rouvisre
We s k e t c h h i s argument below.
[22].
Let and l e t
1
5
j
2n
D%
...,X,
X1,
be a b a s i s of any L i e a l g e b r a
denote t h e derived algebra let
a
j
1
d e n o t e t h e endomorphism of
.
T'
F o r any
U( )
9
j,
given
S. Helgason
Lemma 1. U(
Assume
and
4
-
Xj
4
DT.
Then 3
i s a d e r i v a t i o n of
J
ajZ(cj) C Z(%).
aj
I n f a c t t h e r e s t r i c t i o n of on IR
and e x t e n d s t o a d e r i v a t i o n
T(q).
Since
d
d([Lj,Cf])
= 0
induces a d e r i v a t i o n of
immediately by i n d u c t i o n , s o
,
and
d
to
d =
- Y
8 X) = 0
Then ( 8 ) f o l l o w s f o r
aj '
d
Finally, i f
Z E Z O
'I t h e n
Remark.
The p r o o f i n [ 2 2 ] i s q u i t e d i f f e r e n t and p r o c e e d s
by showing t h a t i n a s u i t a b l e neighborhood o f
where e.
vanishes
of t h e tensor algebra
d(X 8 Y
U(4).
q,,+ IR
fj
e
in
G,
i s a s u i t a b l y c h o s e n f u n c t i o n which v a n i s h e s a t
T h i s w i l l be used below. Next we have t h e g e n e r a l i d e n t i t y f o r t h e
L~
norm,
S. Helgason U C G,
on a n open s e t
u
d i f f e r e n t i a l o p e r a t o r on P1 = [ P , f ] ,
c ~ ( u ) f C c"(u),
C
with adjoint
U
p'l = [ P V , f ] .
P
any
P*, where
Using t h i s on
P E Z(q),
f = f
. I
i n (9) we g e t e a s i l y by Lemma 1 and ( 9 )
where
C = suplfjl.
T h i s i m p l i e s by i n d u c t i o n on t h e o r d e r
Let r be a n i n t e g e r 2
Lemma 2 . P
z(Dr0f).
Q €
z(Dr?) A Z ( D ~ - ' ~ ) , Q
in -
G
1
and
Then t h e r e e x i s t s a n o p e r a t o r and a neighborhood
0
f e
U
such t h a t
for all
u E c:(u). T h i s i s proved by i n d u c t i o n on
comes from ( 1 0 ) and Lemma 1. j = dim?
-
dim D T
Then
nonzero d e r i v a t i v e o f Then
P C Z(%),
Q
E Z(9)
and
t h e p a s s a g e from t h e a n a l y t i c group
r
P
j
2
H C G
1 and we t a k e f o r
of maximal o r d e r i n
rtl
so
X1,
...,Xj.
Q E Z(%) A Z(D(1P.
one u s e s t h e c a s e
with Lie algebra
G
a
Q
4 = Dr9.
is solvable l e t
r
For
r = 1 on
y i e l d s ( 1 1 ) a f t e r some a n a l y s i s on t h e f a c t o r g r o u p Now i f t h e group
r = 1
The c a s e
In fact, l e t
Q 6 U(Dql)
to
r.
This
G/H.
be such
S. Helgason
that
D r 9 = 101
SO
7
Z(D' ) = C .
Then Lemma 2 g i v e s an
inequality
which a s w e l l known [14] i m p l i e s l o c a l s o l v a b i l i t y , i . e . , Theorem 2 . Next we c o n s i d e r t h e semisimple c a s e . by
Here we have
[lo],
Theorem 3.
Each b i - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r
s e m i s i m p l e L i e group
of
0
9
in
i s locally solvable.
G
For t h i s l e t
b e a connected open neighborhood
U
?'
such t h a t
exp
o n t o a n open neighborhood
i s a diffeomorphism of
UG
assume, a s we can ( c f . [g])
of
that
t h a t i s , f o r e a c h compact s u b s e t Cl(Ad(G)*C) C U
9'
Let
D
TI
e U
9
in
G.
I n a d d i t i o n we
i s completely invariant, t h e closure
C C Uv
denote t h e Jacobian of t h e
e x p o n e n t i a l mapping, i . e . ,
Then a c c o r d i n g t o Harish-Chandra t h e r e e x i s t s a s p e c i f i c Ad(G)-invariant d i f f e r e n t i a l o p e r a t o r
pD
On
constant c o e f f i c i e n t s such t h a t ( ~ f 0) exp = n-1/2pD(n1/2f
(1.2) whenever
f
is a
cm
f u n c t i o n on
under i n n e r automorphisms.
0
%
with
exp)
UG, l o c a l l y i n v a r i a n t
( T h i s formula i s proved i n [ g ]
S. Helgason by computing the radial parts of the operators D
and
pD;
for an analogous global forniula for the symmetric space G/K
(with G
complex) see [Ill).
Denoting by
A
the mapping
we have by (1.21, ADf = pDAf
(13) If tions on
G is compact, the
Uq Ad(G)-invariant given by n1l2S
if
functions in
(n1/2~)exp,
distrtbu-
ca
Cc(U9).
The adjoint of
A
S is an Ad(G)-invariant
the image of the distribution
[g],
distribution on
Harish-
u9.
p.' 4771, that (14) is true even if
is noncompact; since the process of averaging over
G
not available, entirely different methods are required. (pD*)*
has a fundamental solution
take Ad(G)-invariant Theorem 1). on
is
exp, so by (13)
Chandra proved ([8], G
Ad(G)-invariant
form in a natural way the dual to the space of
A*S =
under
.
S
on
is Now
which we may
(cf. the construction of
a .
before
Using (14) we get a fundamental solution for D
UG, DT = 6, and Theorem 3 follows by convolution. Global solvability does not in general hold in
Theorem 3 as shown by the following example of CerszoRouviSre [ 5 ] .
Let
G
be a complex semisimple Lie group,
S. Helgason
9 , fi
a maximal compact subgroup and
K
Then
Lie algebras.
9.
structure of
9 = k + JC
Let
where
t h e i r respective
i s t h e complex
J
be a b a s i s o f
(Ti)
4,orthonormal
f o r t h e n e g a t i v e o f t h e K i l l i n g form, and p u t
w = Then i f
~ e b ,[T,Til
CT,ol =
SO
Similarly w =
1 cijTj,
=
i
[JT,w] = 0
1 Ti(JTi).
Thus
where
(cij)
(JT~).CT,T~I
since w
w
can a l s o b e w r i t t e n
is 8 bi-invariant d i f f e r e n t i a l
Since it a n n i h i l a t e s a l l
G.
1s Skew
J
1 CT,JT~I.T~+
i o p e r a t o r on G
1 (JTi).Ti.
i
which a r e r i g h t i n v a r i a n t under
K
C-
f u n c t i o n s on
it i s not globally
solvable. N e v e r t h e l e s s we have by [5] f o r complex [21] f o r r e a l Theorem 4 .
& e J
G
and by
G, G
be a connected noncompact semisimple L i e
group w i t h f i n i t e c e n t e r .
Then t h e Casimir o p e r a t o r
C
5
globally solvable. T h i s f o l l o w s from g e n e r a l r e s u l t s of H'Clrmander[16] once t h e f o l l o w i n g t h r e e f a c t s have been e s t a b l i s h e d C211.
S. Helgason
(I) C
is one-to-one on c~(G).
rCG r1C G
(11) For any compact set another compact set
r C Int
I" and if u
€
there exists such that
E1(G)
(distributions
on G with compact support) with then
supp u
C
supp(Cu)Cr
l".
(111) No null characteristic of compact subset of
C
lies over a
G.
The Holmgren uniqueness theorem implies for the analytic operator C that if a hyper-surface non-characteristic at
S
in G
is
xo then a solution u to Cu = 0
which vanishes on one side of S must vanish in a neighborhood of xo
in G. This implies (I) and.(II) without
difficulty because it is easy to determine plenty of noncharacteristic surfaces for C. In order to describe the proof of (111) let us recall a few facts concerning null bicharacteristics. Let M If
...,xn )
(xl,
be a manifold, T*M
is a local coordinate system on M
is a local coordinate system T*(M).
its cotangent bundle.
(xl,
...,xn,Fl, ...,6,)
then
on
The two-form
is called the canonical 2-form on T*M.
It has the property
S. Helgason that i f
4
i s a diffeomorphism of
diffeomorphism o f
T*M
Given a 1-form
on
on
T*M
If
w = Caidxi
If
w = df
w
preserves T*M
let
then t h e l i f t e d
M
(cf. e.g.
R
111, p . 9 7 ) .
denote t h e vector f i e l d
Xu
g i v e n by
+ Cbjd&j
where
then
f E C-(T*M)
then
is called a
Xdf
Hamiltonian v e c t o r f i e l d ; t h e f u n c t i o n be c o n s t a n t on t h e i n t e g r a l c u r v e s of
f
i s e a s i l y seen t o I n fact, i f
Xdf.
( ~ ~ ( t ) . . . , ~ ( t i)s ) a n i n t e g r a l c u r v e of
Xdf
t h e n by
(16),
I n particular, l e t symbol o f
C, i . e . ,
c:TYG * lR
c(S,() =
1
be t h e principal if
gij
m e t r i c t e n s o r d e t e r m i n e d by t h e K i l l i n g form.
The
i,j
g
5 5 iJ i j
is the
bicharacteristics of
C
curves i n
t o t h e Hamiltonlan v e c t o r f i e l d
T*G
-
(0)
I f t h e constant value of is
a r e by d e f i n i t i o n t h e i n t e g r a l
c
on a n i n t e g r a l c u r v e o f
Xdc. Xdc
0, t h i s c u r v e i s c a l l e d a n u l l b i c h a r a c t e r i s t i c . By t h e i n v a r i a n c e p r o p e r t y of
R
s t a t e d above i t
S. Helgason is clear that
Since
i s bi-invariant
C
f u n c t i o n on T*G.
a , a s a 2-form on T*G, i s b i - i n v a r i a n t . T*G
so
on
c
G,
i s a bi-invariant
is a bi-invariant
Xdc
Using i n a d d i t i o n t h e f a c t t h a t
Xdc
v e c t o r f i e l d on i s Hamiltonian
t h e a u t h o r s p r o v e t h a t i t s i n t e g r a l c u r v e s have t h e form ~ ( t =) ( a exp t X , ua exp t X 1 where
a 6 G, X L Y
and
Now suppose
u
y
a l e f t i n v a r i a n t 1-form on
G.
is a null bicharacteristic for
l y i n g o v e r a compact s u b s e t o f
G.
C
Then t h e same i s t r u e of
t h e curve y 0 ( t ) = (exp t X , uexp t X 1 so
c(yo(t) s 0
group c
of
K
G.
and
exp t X
But t h e n
l i e s i n a maximal compact subB(X,X) < 0
i s s t r i c t l y n e g a t i v e on
implies e a s i l y that
y o , which i s a c o n t r a d i c t i o n .
F i n a l l y , we have by [ll] and 1131, Theorem 5.
Let
compact t y p e , D
X = G/K
b e a symmetric s p a c e of t h e non-
G - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r on
X.
Then (i)
D
h a s a fundamental s o l u t i o n .
(ii)
D
is globally solvable.
T h i s theorem can be proved e i t h e r by means o f t h e F o u r i e r t r a n s f o r m on on
X.
X
o r by means of t h e Radon t r a n s f o r m
A f t e r ( i ) i s proved, ( i i ) f o l l o w s on t h e b a s i s o f
g e n e r a l f u n c t i o n a l a n a l y s i s methods p r o v i d e d one c a n p r o v e f o r e a c h ball
B C X:
S . Helgason
Here "supp" s t a n d s f o r "support". We now s k e t c h how ( i ) and (17) can be proved by means of t h e Radon t r a n s f o r m on Iwasawa decomposition of
G
nilpotent.
A horocycle i n
a point i n
X
group
C0
5
i s unique and
Thus t h e space
i d e n t i f i e d with manifold of
k c K
m
X
E
(K/M)
N-o
(o = origin i n
5 = ka-E0 where
i n the centralizer
A.
x
Each horocycle
M
5
s o i n h e r i t s a volume element f
on
of
in
A
i s a sub-
do
from
X;
i s d e f i n e d by
X
= IP(x)~u(x),
6
d i f f e r e n t i a l operator
X)
i s unique up t o a r i g h t m u l t i -
5 whenever t h i s i n t e g r a l e x i s t s .
where
The
N.
of a l l horocycles i s n a t u r a l l y
P(E)
(19)
conjugate t o
G
can be w r i t t e n
t h e Radon t r a n s f o r m of a f u n c t i o n
moreover
N
i s by d e f i n i t i o n an o r b i t of
X
d e n o t e s t h e horocycle
p l i c a t i o n by a n element K.
i s a b e l i a n and
A
under a subgroup of
t h e n each h o r o c y c l e a € A
where
be a n
G = KAN
permutes t h e h o r o c y c l e s t r a n s i t i v e l y ; more p r e c i s e -
G
ly, i f
Let
X.
(5
E a)
There e x i s t s a G-invariant
on t h e manifold
E
such t h a t
6 h a s t h e form ( 6 $ ) ( k a * 5 0 ) = ~ ~ ( q ( k a . 5 ~ )k) € K,a E A),
P
i s a translation-invariant
t h e Euclidean space
A
(cf.
d i f f e r e n t i a l o p e r a t o r on
[ l o ] , 54-51.
Thus t h e Radon
S. Helgason transform converts t h e operator
T h i s p r i n c i p l e g i v e s a fundament-
ent d i f f e r e n t i a l operator. a l solution for
D
i n t o a constant c o e f f i c i -
D
by a method s i m i l a r t o t h e one used f o r
Theorem 3 ( c f . [ l l ] ) . I n o r d e r t o use t h i s p r i n c i p l e t o deduce ( 1 7 ) t h e f o l l o w i n g lemma i s d e c i s i v e .
Let f E c:(x)
Lemma 3.
?(5) = 0
that
Then
B.
f
and
BC X
a closed b a l l .
whenever t h e h o r o c y c l e
R
denote i t s r a d i u s .
( 1 8 ) and (19) and
Now
d(o,ka-So)
d(o,a-o)
denoting distance.
B.
centered at
B
supp(Df)
~ ~ ( ? ( k a - 5 ~= )0) i f d
i s d i s j o i n t from
vanishes i d e n t i c a l l y outside
To v e r i f y ( 1 7 ) we may assume
Let
5
cB
o.
i m p l i e s by
that
d(0,a.o)
But t h e f u n c t i o n
Assume
a
> R, +
?(ka*co)
has
compact s u p p o r t s o by t h e Lions-Titchmarsh c o n v e x i t y theorem we conclude t h a t
? ( k a - ~ ~= ) 0
if
Lemma 3 i m p l i e s
supp(f) C gR(g)
d ( o , a - o ) > R.
But t h e n
s o (17) i s v e r i f i e d ,
p r o v i n g Theorem 5. P l a u s i b l e a s Lemma 3 l o o k s , i t s p r o o f ( c f . [13])
i s t o o t e c h n i c a l t o descITibe h e r e .
I n s t e a d , I ~ s k e t c ha proo'f
of t h e a n a l o g u e o f Lemma 3 f o r Fin.
Here t h e Radon t r a n s f o r m
of a f u n c t i o n
F(x)
i s d e f i n e d by
fi(5) = J ~ ( x ) d u ( x ) , 5 b e i n g a n a r b i t r a r y hyperplane and do
(20 5
element on i t .
being t h e s u r f a c e
S. Helgason Lemma 4 .
Let
F € Cm(Rn) s a t i s f y t h e c o n d i t i o n s :
(i)
For e a c h i n t e g e r
k 2 0,
~ ( x ) l x l ~
bounded. (ii)
There e x i s t s a c o n s t a n t s(5) = 0
for d(O,S)
such t h a t
A
d
> A,
denoting
distance. Then
F
vanishes identically outside the b a l l
1x1 2 A . Proof ( c f . [ 1 2 ] ) . function, i.e.,
Suppose f i r s t
F(x) = $ ( l x l )
where
$ 6
Then t h e r e e x i s t s a n e v e n f u n c t i o n $(d(0,6)) = g(5).
where
F
is a radial
c-(R)
Cm(R)
i s even.
such t h a t
Now ( 2 0 ) t a k e s t h e form
i s t h e a r e a o f t h e u n i t s p h e r e i n R"'~.
An-l
By a
s i m p l e change o f v a r i a b l e s ( 2 1 ) i s t r a n s f o r m e d i n t o A b e l l s i n t e g r a l e q u a t i o n and i s i n v e r t e d by
where 0 < u
c
is a constant.
\
5 A-I s o
s
lemma f o r t h e c a s e when
Now b y ( i i ) , = 0
F
for
0 < s ( A'~,
for
proving t h e
is radial.
Consider now t h e g e n e r a l c a s e . consider the function
6(u-l) = 0
Fix
x
f
R"
and
S. Helgason
i s t h e n o r m a l i z e d Haar-measure on t h e o r t h o g o n a l
where
dk
group
Otn); g x ( y )
center
where
x
i s t h e a v e r a g e of
and r a d i u s
lyl.
d(O,x+k0S) so i f
d(O,S) > 1x1
+
A
on t h e s p h e r e w i t h
We have
x+kaS i s t h e hyperplane
Clearly
F
k*S
t r a n s l a t e d by
,
-
we have
z x ( C ) = 0.
d(O,E)
x.
1x1 But
gi
is a
r a d i a l f u n c t i o n s o by t h e f i r s t p a r t o f t h e p r o o f ,
T h i s means t h a t i f
S
i s a sphere with s u r f a c e element
dw
then
if
S
encaoses t h e s o l i d b a l l
radius
A),
Let
F(x) lxlk
i s bounded.
B ~0 )(.
(with center
0
and
Lemma 4 w i l l t h e r e f o r e f o l l o w from a n o t h e r lemma.
Lemma 5.
solid ball
A B (0)
F
c-(R")
such t h a t f o r each
Assume ( 2 2 ) whenever
~ ~ ( 0 )Then . F
k
S
2
0,
encloses t h e
vanishes identically outside
S. Helgason The i d e a f o r p r o v i n g t h i s i s t o p e r t u r b
S
in
( 2 2 ) a b i t and differentiate w i t h r e s p e c t t o t h e parameter of p e r t u r b a t i o n , thereby o b t a i n i n g a d d i t i o n a l r e l a t i o n s . S = s R ( x ) and
g R ( x ) t h e c o r r e s p o n d i n g b a l l , t h e n by ( 2 2 ) ,
that is
Using t h e d i v e r g e n c e theorem on t h e v e c t o r f i e l d V(Y) = F ( y ) e i we o b t a i n from ( 2 3 )
Combining t h i s w i t h ( 2 2 ) we deduce
By i t e r a t i o n
If
S. Helgason
if
p(s)
Remark.
i s a n a r b i t r a r y polynomial, s o t h e lemma f o l l o w s . The f u n c t i o n
F(z) = 1
( z € C)
smoothed o u t
Z
n e a r t h e o r i g i n s a t i s f i e s (by C a u c h y t s theorem) a s s u m p t i o n ( i i ) i n Lemma 4 . S i m i l a r remark
Thus c o n d i t i o n ( i ) c a n n o t b e weakened. a p p l i e s t o Lemma 5.
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