New York Journal of Mathematics
New York J. Math. 4 (1998) 97{125.
The Normal Symbol on Riemannian Manifolds
Markus J. Paum
Abstract. For an arbitrary Riemannian manifold
and Hermitian vector
bundles and over we dene the notion of the normal symbol of a pseudodierential operator from to . The normal symbol of is a certain
smooth function from the cotangent bundle
to the homomorphism bundle Hom( ) and depends on the metric structures, resp. the corresponding
connections on , and . It is shown that by a natural integral formula the
pseudodierential operator can be recovered from its symbol. Thus, modulo smoothing operators, resp. smoothing symbols, we receive a linear bijective
correspondence between the space of symbols and the space of pseudodierential operators on . This correspondence comprises a natural transformation
between appropriate functors. A formula for the asymptotic expansion of the
product symbol of two pseudodierential operators in terms of the symbols of
its factors is given. Furthermore an expression for the symbol of the adjoint is
derived. Finally the question of invertibility of pseudodierential operators is
considered. For that we use the normal symbol to establish a new and general
notion of elliptic pseudodierential operators on manifolds.
E
F
X
X
P
E
F
P
T
X
E F
X
E
F
P
X
Contents
1. Introduction
2. Symbols
3. Pseudodierential operators on manifolds
4. The symbol map
5. The symbol of the adjoint
6. Product expansions
7. Ellipticity and normal symbol calculus
References
1.
97
99
101
105
112
116
122
125
Introduction
It is a well-known fact that on Euclidean space one can construct a canonical
linear isomorphisms between symbol spaces and corresponding spaces of pseudodierential operators. Furthermore one has natural formulas which represent a
Received October 14, 1997.
Mathematics Subject Classication. 35S05, 58G15.
Key words and phrases. pseudodierential operators on manifolds, asymptotic expansions,
symbol calculus .
c 1998 State University of New York
97
ISSN 1076-9803/98
98
Markus J. Paum
pseudodierential operator in terms of its symbol, resp. which give an expression
for the symbol of a pseudodierential operator. By using symbols one gets much
insight in the structure of pseudodierential operators on Euclidean space. In particular they give the means to construct a (pseudo) inverse of an elliptic dierential
operator on Rd .
Compared to the symbol calculus for pseudodierential operators on Rn it seems
that the symbol calculus for pseudodierential operators on manifolds is not that
well-established. But as the pure consideration of symbols and operators in local coordinates does not reveal the geometry and topology of the manifold one is working
on, it is very desirable to build up a general theory of symbols for pseudodierential
operators on manifolds. In his articles 13, 14] Widom gave a proposal for a symbol
calculus on manifolds. By using a rather general notion of a phasefunction, Widom
constructs a map from the space of pseudodierential operators on manifolds to
the space of symbols and shows by an abstract argument for the case of scalar
symbols that this map is bijective modulo smoothing operators, resp. symbols. In
our paper we introduce a symbol calculus for pseudodierential operators between
vector bundles having the feature that both the symbol map and its inverse have a
concrete representation. In particular we thus succeed in giving an integral representation for the inverse of our symbol map or in other words for the operator map.
Essential for our approach is an appropriate notion of a phasefunction. Because of
our special choice of a phasefunction the resulting symbol calculus is natural in a
category theoretical sense.
Using the integral representation for the operator map, it is possible to write
down a formula for the symbol of the adjoint of a pseudodierential operator and
for the symbol of the product of two pseudodierential operators.
The normal symbol calculus will furthermore give us the means to build up a natural notion of elliptic symbols, respectively elliptic pseudodierential operators on
manifolds. It generalizes the classical notion of ellipticity as dened for example in
Hormander 7] and also the concept of ellipticity introduced by Douglis, Nirenberg
2]. Our framework of ellipticity allows the construction of parametrices of nonclassical, respectively nonhomogeneous, elliptic pseudodierential operators. Moreover
we do not need principal symbols for dening ellipticity. Instead we dene elliptic
operators in terms of their normal symbol, as the globally dened normal symbol
of a pseudodierential operator carries more information than its principal symbol.
Moreover, the normal symbol of a pseudodierential operator shows immediately
whether the corresponding operator is invertible modulo smoothing operators or
not.
Let us also mention that another application of the normal symbol calculus on
manifolds lies in quantization theory. There one is interested in a quantization map
associating pseudodierential operators to certain functions on a cotangent bundle
in a way that Dirac's quantization condition is fullled. But the inverse of a symbol
map does exactly this, so it is a quantization map. See Paum 8, 9] for details.
Note that our work is also related to the recent papers of Yu. Safarov 11, 10].
The Normal Symbol on Riemannian Manifolds
2.
99
Symbols
First we will dene the notion of a symbol on a Riemannian vector bundle. Let
us assume once and for all in this article that 2 R are real numbers such that
0 < 1 and 1 + . The same shall hold for triples ~ ~ ~ 2 R.
We sometimes use properties of symbols on Rd RN . As these are well-described
in the mathematics literature we only refer the interested reader to Hormander
6, 7], Shubin 12] or Grigis, Sjostrand 5] for a general introduction to symbol
theory on Rd RN and for proofs.
Denition 2.1. Let E ! X be a Riemannian or Hermitian vector bundle over
the smooth manifold X , % : TX ! X its tangent bundle and : T X ! X its
cotangent bundle. Then an element a 2 C 1 ( (E jU )) with U X open is called
a symbol over U X with values in E of order and type ( ) if for every
trivialization (x ) : T X jV ! R2d and every trivialization : E jV ! V RN
with V U open the following condition is satised:
(Sy) For every 2 N d and every K V compact there exists C = CK > 0
such that for every 2 T X jK the inequality
(1)
@ jj @ jj (a( )) C (1 + jj jj)+jj;jj
@x @ is valid.
The space of these symbols is denoted by S (U T X E ) or shortly S (U E ). It
gives rise to the sheaf S ( T X E ) of symbols on X with values in E of order
and type ( ).
T S 1
By dening S;1
(U E ) = 2N S (U E ) and S (U E ) = 2N S (U E ) we
1
get the sheaf S;1
( E ) of smoothing symbols resp. the sheaf S ( E ) of symbols
of type ( ).
In case E is the trivial bundle X C of a Riemannian manifold X we write S
with 2 R f1 ;1g for the corresponding symbol sheaves S ( X C ).
The space of symbols S (U TX E ) consists of functions a 2 C 1 (% (E jU )) such
that a fullls condition (Sy). It gives rise to sheaves S ( TX E ), S1
( TX E )
and S;1
(
TX
E
).
It is possible to extend this denition to one of symbols over conic manifolds but
we do not need a denition in this generality and refer the reader to Duistermaat
3] and Paum 8].
We want to give the symbol spaces S (U E ) a topological structure.
S So rst
choose a compact set K U , a (not necessarily disjunct) partition K = K of K
2J
into compact subsets together with local trivializations (x ) : T X jV ! R2d and
trivializations : E jV ! V RN such that K V U and V open. Then we
can attach for any 2 Rd a seminorm p = p(K (x )) : S (U E ) ! R+ f0g
to the symbol spaces S (U E ) by
8 @ @
< (a())
p(a) = sup : (1@x+ j@
j)+jj;jj
2J
j
(2)
j
j j
9
=
: 2 T X jK :
Markus J. Paum
100
The system of these seminorms gives S (U E ) the structure of a Frechet space
such that the restriction morphisms S (U E ) ! S (V E ) for V U open are
continuous. Additionally we have natural and continuous inclusions S (U E ) S~~~(U E ) for ~ , ~ and ~ . Like in the case of symbols on Rd RN
one can show that pointwise multiplication of symbols is continuous with respect
to the Frechet topology on S (U E ).
Proposition 2.2. Let a 2 C 1( (E jU )). If there exists an open covering of U
by patches V with local trivializations (x ) : T X jV ! R2d and trivializations
: E jV ! V RN such that for every (x ) the above condition (Sy) holds,
then a is a symbol of order and type ( ).
Proof. Let (x ) : T X jV ! R2d , V U open be a trivialization. Then we can
write on V \ V
@ jj @ jj =
@x @ (3)
X
(f
~ ) 0
0
~ +0 @ j~j @ j+ j @x ~ @ +
0
0
1
where f
~ 2 C (V \ V ). As + 1 holds and (Sy) is true for (x ), the
claim now follows.
Example 2.3. (i) Let X be a Riemannian manifold. Then every smooth function a : T U ! C which is a polynomial function of order on every
ber lies in S10 (U ). In particular we have an embedding D0 ! S1
, where
D0 is the sheaf of polynomial symbols on X , i.e., for every U X open
D0 (U ) = ff : T U ! C : f jTx X is a polynomial for every x 2 U g.
(ii) Again let X be Riemannian. Then the mapping l : T X ! C , 7! jj jj2 is
a symbol of order 2 and type (1 0) on X , but not one of order < 2. Next
regard the function a : T X ! C , 7! 1+jj1jj2 . This function is a symbol of
order ;2 and type (1 0) on X , but not one of order < ;2.
(iii) Assume ' : X ! R to be smooth and bounded. Then a' : T X ! R,
7! (1 + jj jj2 )'(()) comprises a symbol of order = supx2X '(x) and type
(1 ), where 0 < < 1 is arbitrary.
The following theorem is an essential tool for the use of symbols in the theory
of partial dierential equations and extends a well-known result for the case of
E = Rd RN to arbitrary Riemannian vector bundles.
Theorem 2.4. Let aj 2 Sj (U E ), j 2 N be symbols such that ( j )j2N is a de
creasing sequence with jlim
j = ;1. Then there exists a symbol a 2 S0 (U E )
!1
P
unique up to smoothing symbols, such that a ; kj=0 aj 2 Sk (U E ) for every
k 2 N . This induces a locally convex Hausdor topology on the quotient vector
;1
space S1
=S (U E ), which is called the topology of asymptotic convergence.
Proof. It is a well-known fact that the claim holds for U Rd and E = Rd RN
(see 7, 12, 5]). Covering U by trivializations (z ) : T X jV ! R2d and :
E jV ! V RN we can nd a partition (' ) of unity subordinate to the covering (V )
Pk
and for every index a symbol bk 2 S (V RN ) such that bk = aj jV 2
0
j =0
The Normal Symbol on Riemannian Manifolds
P
101
Sk (V RN ). Now dene a = (' ) ; 1 bk and check that this a satises
the claim. Uniqueness of a up to smoothing symbols is clear again from a local
consideration.
Polynomial symbols are not aected by smoothing symbols. The following proposition gives the precise statement.
Proposition 2.5. Let X be a smooth manifold, and D0 the sheaf of polynomial
1
;1 the
symbols on X . Then by composition with the projection S1
! S =S
canonical embedding
D0
!
S1
(4)
D0 (U ) 3 f 7! f 2 S1
U X open
(U )
;1 of sheaves of algebras.
gives rise to a monomorphism : D0 ! S1
=S
Proof. If f g 2 D0 (U ) are polynomial symbols such that f ; g 2 S;1(U ), then
f ; g is a bounded polynomial function on each ber of T U . Therefore f ; g is
constant on each ber, hence f ; g = 0 by f ; g 2 S;1 (U ).
3.
Pseudodierential operators on manifolds
Let a be a polynomial symbol dened on the (trivial) cotangent bundle T U =
UP Rd of an open set U Rd of Euclidean space. Then one can write a =
d
(a ) , where : T U ! R is the projection on the \cotangent vectors"
and the a are smooth functions on U . According to standard results of partial
dierential equations one knows that the symbol a denes the dierential operator
A = Op (a):
X
@ jj f 2 C 1 (U ):
C 1 (U ) 3 f 7! a (;i)jj @z
(5)
Sometimes A is called the quantization of a. In case f 2 D(U ), the Fourier transform of f is well-dened and provides the following integral representation for Af :
Z
(6)
Af = Op (a) (f ) = d ei<> a( ) f^( ) d:
R
It would be very helpful for structural and calculational considerations to extend
this formula to Riemannian manifolds. To achieve this it is necessary to have
an appropriate notion of Fourier transform on manifolds. In the following we are
going to dene such a Fourier transform and will later get back to the problem of an
integral representation for the \quantization map" Op on Riemannian manifolds.
Assume X to be Riemannian of dimension d and consider the exponential function exp with respect to the Levi-Civita connection on X . Furthermore let E ! X
be a Riemannian or Hermitian vector bundle. Choose an open neighborhood
W TX of the zero section in TX such that (% exp ) : W ! X X maps
W dieomorphically onto an open neighborhood of the diagonal of X X . Then
there exists a smooth function : TX ! 0 1] called a cut-o function such that
jW~ = 1 and supp W for an open neighborhood W~ W of the zero section in
TX . Next let us consider the unique torsionfree metric connection on E . It denes
a parallel transport : E (0) ! E( 1) for every smooth path : 0 1] ! X . For
102
Markus J. Paum
every v 2 TX denote by exp v or just yx with y = exp v and x = %(v) the parallel
transport along 0 1] 3 t 7! exp tv 2 X . Now the following microlocal lift M is
well-dened:
M : C 1 (U E ) ! S;1 (U TX E )
;1
(7)
v2W
f 7! f = 0 (v) exp v (f (exp v)) for
else:
M is a linear but not multiplicative map between function spaces.
Over the tangent bundle TX we can dene the Fourier transform as the following
sheaf morphism:
F : S;1 (U TX E ) ! S;1 (U T X E )
Z
1
(8)
a 7! a^( ) = (2)n=2
e;i<v> a(v) dv:
T() X
We also have a reverse Fourier transform:
F ;1 : S;1 (U T X E ) ! S;1 (U TX E )
Z
(9)
b 7! !b(v) = (21)n=2
ei<v> b( ) d:
T(v) X
It is easy to check that F ;1 and F are well-dened and inverse to each other.
Composing M and F gives rise to the Fourier transform F = F M on the
Riemannian manifold X . F has the left inverse
(10)
S;1 (U T X E ) ! C 1 (U E ) a 7! F ;1 (a)j0U where j0U means the restriction to the natural embedding of U into T X as zero
section.
By denition M and F do depend on the smooth cut-o function , but
this arbitrariness will only have minor eects on the study of pseudodierential
operators dened through F . We will get back to this point later on in this
section.
The exponential function exp gives rise to normal coordinates zx : Vx ! Rd ,
where x 2 Vx X , Vx open and
(11)
exp;x 1 (y) =
d
X
k=1
zxk (y) ek (y)
for all y 2 Vx and an orthonormal frame (e1 ::: ed) of TVx. Furthermore we receive
bundle coordinates (zx x ) : T Vx ! R2d and (zx vx) : TVx ! R2d . Note that for
xed x 2 X the map
(12)
Vx T Vx 3 (y ) 7! (zy (( )) y ( )) 2 R2d
is smooth and that by the Gau" Lemma
(13)
zxk (y) = ;zyk (x)
for every y 2 Vx .
The Normal Symbol on Riemannian Manifolds
103
Now let a be aPsmooth function dened on T U and polynomial in the bers.
Then locally a = (ax ) x with respect to a normal coordinate system at
x 2 U and functions ax 2 C 1 (U ). Dene the operator A : D(U ) ! C 1 (U ) by
!
Z
1
^
(14) f 7! Af = Op (a) (f ) = U 3 x 7!
a( ) f ( ) d 2 C
(2)n=2 Tx X
and check that
(15)
Af (x) =
=
X
X
@ jj (f exp )] (0 )
ax (x) (;i)jj @v
x
x
jj
@ f (x )
ax (x) (;i)jj @z
x
for every x 2 U . Therefore Af is a dierential operator independent of the choice
of .
However, if a is an arbitrary element of the symbol space S1
(U Hom(E F )),
Eq. (14) still denes a continuous operator A = Op (a) : D(U E ) ! C 1(U F ),
which is a pseudodierential operator but independent of the cut-o function only up to smoothing operators. Let us show this in more detail for the scalar case,
i.e., where E = F = X C . The general case is proven analogously.
First choose 2 R such that a 2 S (U ),Rand consider the kernel distribution
D(U U ) ! C (f g) 7! < A f g > = X g(x) A f (x) dx of A . It can be
written in the following way:
(16)
Z Z Z
< g A f > = (21)n
(v) g(x) f (exp v) e;i<v> a( ) dv d dx
X Tx X Tx X
Z Z Z
= (21)n
(v) g(x) f (exp v) e;i<v> a( ) d dv dx#
X Tx X Tx X
where the rst integral is an iterated one, the second one an oscillatory integral. To
check that Eq. (16) is true use a density argument or in other words approximate
~
the symbol a by elements ak 2 S;1 (U ) (in the topology of S
(U ) with ~ > )
and prove the statement for the ak . The claim then follows from continuity of both
sides of Eq. (16) with respect to a. Let ~ : TX ! 0 1] be another cut-o function
such that ~jO~ = 1 on a neighborhood W 0 W of the zero section of T X . We can
assume W 0 = W~ and claim the operator A ; A~ to be smoothing. For the proof
it su$ces to show that the oscillatory integral
Z
(17) KA ;A~ (v) = (21)n
((v) ; ~(v)) e;i<v> a( ) d v 2 TU
T(v) X
which gives an integral representation for the kernel of A ; A~ denes a smooth
function KA ;A~ 2 C 1 (TU ). The phase function T(v) X 3 7! ; (v) 2 R has
only critical points for v = 0. Hence for v 6= 0 there exists a (;1)-homogeneous
vertical vector
eld L on T X such that L e;i<v> = e;i<v> . The adjoint Ly of
;
y
L satises L k a 2 S;k (U ), where k 2 N . As the amplitude
(v ) 7! ((v) ; ~(v)) a( )
Markus J. Paum
104
vanishes on W 0 U T U , the equation
Z
; 1
(18)
KA ;A~ (v) = (2)n
e;i<v> Ly k a( ) d
T (v ) X
holds for any integer k fullling + k < ;dimX . Note that the integral in
Eq. (18) unlike the one in Eq. (17) is to be understood in the sense of Lebesgue.
Hence KA ;A~ 2 C 1 (TU ) and A ; A~ is smoothing.
Let us now prove that A lies in the space (U ) of pseudodierential operators
of order and type ( ) over U . We have to check that A is pseudolocal and
with respect to some local coordinates looks like a pseudodierential operator on
an open set of Rd .
(i) A is pseudolocal: Let u v 2 D(U ) and supp u \ supp v = . Then the integral
kernel K of C 1 (U ) 3 f 7! v A (uf ) 2 C 1 (U ) has the form
Z
(19)
K (v) = (21)n
(v) a( ) v((v)) u(exp v) e;i<v> d:
T(v) X
There exists an open neighborhood W 0 W of the zero section in TX such that
the amplitude (v ) (u exp ) a vanishes on W 0 U T X . As for all nonzero
v 2 TX the phase function 7! ; < v > is noncritical, an argument similar to
the one above for KA ;A~ shows that K is a smooth function on TU , so C 1 (U ) 3
f 7! v A (uf ) 2 C 1 (U ) is smoothing.
(ii) A is a pseudodierential operator in appropriate coordinates: Choose for
x 2 U an open neighborhood Ux U so small that any two points in Ux can be
connected by a unique geodesic. In particular we assume that Ux Ux lies in the
range of ( exp ) : W ! X X . Furthermore let us suppose that zy (~y) = 1 for
all y y~ 2 Ux. Then we have for f 2 D(Ux ):
(20)
A f (y) =
Z
1
a( ) f^ ( ) d
=
(2)n=2 Ty X
Z ;
;1
= 1n=2
a
T
exp
(
)
det dx (y) F (f exp ) dx (y) ( ) d
x
z
(y )
x
(2)
Tx X | {z }
= dx (y )
Z Z 1
a dx (y) ( ) det dx (y) (f exp y )(v) e;i<dx (y) ()v> dv d
= (2)n
ZTx X ZTy X 1
;1
= (2)n
a dx (y) ( ) det dx (y) det
| (T ({zzy zx })(v)
Tx X Tx X
= Zyx
;i<dx (y) ()zy zx;1 (v)>
(f zx )(v) e
;1
dv d
Now the phase function (v ) 7! ; < dx (y)( ) zy zx;1 (v) > vanishes for 6= 0 if
and only if v = zx (y). Hence after possibly
; shrinking Ux the; Kuranishitrick gives
a smooth function Gx : Ux TxX ! Iso Tx X Ty X Lin Tx X Ty X such that
< dx (y)( ) zy zx;1(v) > = < Gx (y v)( ) v ; zx(y) > for all (y v) 2 Ux TxX and
The Normal Symbol on Riemannian Manifolds
105
2 Tx X . A change of variables in Eq. (20) then implies:
Z Z h
i 1
(21) A f (y) = (2)n
a dx (y) G;x 1 (y v) ( ) det dx (y) Zyx(v)
Tx X Tx X
det G;x 1 (y v) (f zx;1)(v) e;i<v;zx (y)> dv d
Z Z
a~x(y v ) (f zx;1)(v) e;i<v;zx (y)> dv d
= (21)n
Tx X Tx X
h
i where a~x (y v ) = a dx (y) G;x 1 (y v) ( ) det dx (y) Zyx(v) det G;x 1 (y v) is
a symbol lying in S (Ux Ux TxX ), as + 1 and > . Hence A is a
pseudodierential operator with respect to normal coordinates over Ux of order
and type ( ).
Our considerations now lead us to
Theorem 3.1. Let X be Riemannian, E F Riemannian or Hermitian vector bundles over X and exp the exponential function corresponding to the Levi-Civita
connection on X . Then any smooth cut-o function : TX ! 0 1] gives rise to
1
a linear sheaf morphism Op : S1
( Hom(E F )) ! ( E F ), a 7! A dened
by
Z
A f (x) = (21)n=2
(22)
a( ) f^ ( ) d
Tx X
where a 2 S1
(U Hom(E F )), f 2 D(U E ), x 2 U and U X open. This
1
morphism preserves the natural ltrations of S1
( Hom(E F )) and ( E F ).
In particular it maps the subsheaf S;1 ( Hom(E F )) of smoothing symbols to the
subsheaf ;1( Hom(E F )) 1
( Hom(E F )) of smoothing pseudodierential
operators.
;1
1
;1
The quotient morphism Op : (S1
=S )( Hom(E F )) ! ( = )( E F )
is an isomorphism and independent of the choice of the cut-o function .
Proof. The rst part of the theorem has been shown above. So it remains to
;1
1
;1
prove that Op : (S1
=S )(U Hom(E F )) ! ( = )(U E F ) is bijective for
all open U X . Let us postpone this till Theorem 4.2, where we will show that
an explicit inverse of Op is given by the complete symbol map introduced in the
following section.
By the above Theorem the eect of for the operator Op is only a minor one,
so from now on we will write Op instead of Op .
4.
The symbol map
In the sequel denote by ' : X T X jO ! C the smooth phase function dened
by
(23)
X T X jO 3 (x ) 7! < exp ;(1) (x) > = < z() (x) > 2 C where O is the range of (% exp) : W ! X X .
Markus J. Paum
106
Theorem and Denition 4.1. Let A 2 (U E F ) be a pseudodierential operator on a Riemannian manifold X and : T X ! 0 1] a cut-o function. Then
the -cut symbol of A with respect to the Levi-Civita connection on X is the section
(24)
(A) = A : T U ! (Hom(E F ))
h
i
7! % 7! A () () ei'() ()() % (( ))
where for every x 2 X x = exp ;x 1 = zx . (A) is an element of S (U X ).
The corresponding element (A) in the quotient (S =S;1 )(U Hom(E F )) is called
the normal symbol of A. It is independent of the choice of .
Proof. Let us rst check that A lies in S (U Hom(E F )). It su$ces to as-
sume that E and F are trivial bundles, hence that A is a scalar pseudodierential
operator. We can nd a sequence (x )2N of points in U and coordinate patches
U U with x 2 U such that there exist normal coordinates z = zx : U ! Tx X
on the U . We can even assume that the operator A : D(U ) ! C 1 (U ) u 7! AujU
induces a pseudodierential operator on Tx X . In other words there exist symbols
a 2 S (O O Tx X ) with O = z(U ) such that A is given by the following
oscillatory integral:
(25) A u(y) = (21)n
Z Z
Tx X Tx X
a (z(y) v ) (u z;1)(v) e;i<v;z (y)> dvd:
Now let ( ) be a partition of unity subordinate to the covering (U ) of U and for
every index let 1 2 2 C 1 (U ) such that supp 1 U, supp \ supp 2 = and 1 + 2 = 1. Then the symbol A can be written in the following form:
h (26)
i
A ( ) = A ( ) () ei'( ) (( ))
i
Xh
= (21)n
1 A () ( ) () ei'( ) (( ))
+ K ( ) () ei'( ) (( ))
where
(27)
K : D(U ) ! C 1 (U ) f 7!
X
2 A ( f ) 2 T X
is a smoothing pseudodierential operator. By the Kuranishi
; trick we can assume
that there exists a smooth function G : O O ! Lin Tx X Ty X such that
G(v w) is invertible for every v w 2 O and
(28)
;
< G z(y) T (z zy;1 )(v) ( ) v > = < z zy;1(v) ; z(y) >
The Normal Symbol on Riemannian Manifolds
107
for y 2 U, v 2 Ty X appropriate and 2 Tx X . Several changes of variables then
give the following chain of oscillatory integrals for 2 T U :
(29)
h
i
A () ( ) () ei'( ) (y)
= (21)n
Z Z
;
1 (y) a z;1 (y) v (z;1 (v)) (zy zx;1(v))
Tx X Tx X
i<z
e y z 1 (v)> e;i<v;z (y)> dv d
;
= (21)n
Z Z
;
1 (y) a z;1 (y) z zy;1(v) (zy;1 (v)) (v)
Tx X Ty X
i<v>
e
e;i<z zy 1 (v);z (y)> dv d
= (21)n
;
Z Z
;
1 (y) a z;1 (y) z zy;1(v) (zy;1 (v)) (v)
Tx X Ty X
ei<v> e;i<G(z (y)T (z zy 1 )(v))()v> dv d
= (21)n
Z Z
Ty X Ty X
;
;
;
1 (y) a z;1 (y) z zy;1 (v) G;1 z(y) T (z zy;1 )(v) ( )
;
(zy;1 (v)) (v) det G;1 z(y) T (z zy;1)(v) ei<v> e;i<v> dv d
= (21)n
Z Z
Tx X Tx X
b(z (y) v ) e;i<;T (zy z
;1
)( )v ;z (y )>
dv d
where the smooth function b on O O Tx X is dened by
(30)
b(z (y) v ) =
;
= 1 (y) a z;1 (y) z zy;1 T (zy z;1) (v) G;1 z(y) v T (z zy;1 ( )
(zy;1 (T (zy z;1) (v))) (T (zy z;1 ) (v)) det G;1 (z(y) v)
and lies in S (O O Tx X ). Let B 2 (O Tx X ) be the pseudodierential operator on O corresponding
;
to the symbol b. The symbol dened by
(v ) = e;i<v> B ei<> , where v 2 O and 2 Tx X , now is an element of
S (O Tx X ) as one knows by the general theory of pseudodierential operators
on real vector spaces (see 7, 12, 5]). But we have
h
i
;
;
(31) 1 A () ( ) () ei'( ) (( )) = z (( )) T z( ) z;1 ( ) hence Eq. (26) shows A 2 S (U X ) if we can prove that K 2 S;1 (U X ) for
K ( ) = K ( ) () ei'( ) (( )). So let k : U U ! C be the kernel function
Markus J. Paum
108
of K and write K as an integral:
K ( ) =
(32)
Z
U
k(( ) y) ( ) (y) e;i'(y ) dy:
Let further %1 ::: %k be dierential forms over U and V1 ::: Vk the vertical vector
elds on T U corresponding to %1 ::: %k . Dierentiating Eq. (32) under the integral
sign and using an adjointness relation then gives
(33)
j j2m V1 ::: Vk K ( ) =
=
Z
U
k(( ) y) ( ) (y) < %1 z( )(y) > ::: < %k z( )(y) >
i2mn+k
@ 2mn
2m)
@z(2(m:::
)
e;i<z() ()> (y) dy
Z 0 @ 2mn 1y ;
= @ (2m:::2m) A k(( ) ) ( ) () < %1 z( )() > ::: < %k z( ) () > (y)
U
@z( )
i2mn+k e;i<z() (y)> dy:
Therefore j j2m V1 ::: Vk K ( ) is uniformly bounded as long as ( ) varies in a
compact subset of U . A similar argument for horizontal derivatives of K nally
proves K 2 S;1 (U X ).
Next we have to show that A ; A
is a smoothing symbol for any second
~
cut-o function ~ : T X ! 0 1]. It su$ces to prove that ~ with
(34)
~ ( ) = A ( ; ~) z( ) e;i'( )
is smoothing for every 2 D(U ). We can nd a (;1)-homogeneous rst order
dierential operator L on the vector bundle Tx X Tx X ! Tx X such that for all
v 6= z(( )) and 2 Tx X the equation
(35)
Le;i<v;z (( ))> ( ) = e;i<v;z (( ))>
holds (see for example Grigis, Sjostrand 5] Lemma 1.12 for a proof). Because
( ; ~) z( ) z;1 vanishes on a neighborhood of z (( )), the following equational
The Normal Symbol on Riemannian Manifolds
109
chain of iterated integrals is also true:
(36)
~ ( ) =
Z Z ;
1
a z;1(y) v (z;1 (v)) ; ~ (z( ) zx;1 (v))
= (2)n
Tx X Tx X
ei<z() z 1 (v)> e;i<v;z (( ))> dv d
;
= (21)n
Z Z
;
a z;1(y) v (z;1 (v)) ; ~ (z( ) zx;1 (v))
Tx X Tx X
ei<z() z 1 (v)> Lk e;i<v;z (( ))> dv d
;
= (21)n
Z Z ; k ;
Ly a z;1(y) v (z;1 (v)) ; ~ (z( ) zx;1(v))
Tx X Tx X
ei<z() z 1 (v)> e;i<v;z (( ))> dv d:
;
Note that the last integral of this chain is to be understood in the sense of Lebesgue
if k 2 N is large enough. The same argument which was used for proving that the
symbol K from Eq. (32) is smoothing now shows 2 S;1 (U X ).
After having dened the notion of a symbol, we will now give its essential properties in the following theorem.
Theorem 4.2. Let A 2 (U E F ) be a pseudodierential operator on a Riemannian manifold X and a = A 2 S (U Hom(E F )) be its -cut symbol with
respect to the Levi-Civita connection. Then A and the pseudodierential operator Op (a) dened by Eq. (22) coincide modulo smoothing operators. Moreover
;1
1
;1
the sheaf morphisms : (1
= )( E F ) ! (S =S )( Hom(E F )) and
1
;1
1
;1
Op : (S =S )( Hom(E F )) ! ( = )( E F ) are inverse to each other.
Before proving the theorem let us rst state a lemma.
Lemma 4.3. The operator
(37)
K : D(U E ) ! C 1 (U F ) f 7! U 3 x 7! A(1 ; x )f ] (x) 2 F
is smoothing for any cut-o function : TX ! C .
Proof. For simplicity we assume that E = F = X C . The general case follows
analogously. Consider the operators Ka : D(U ) ! C 1 (U ) where Kaf (x) =
(x)A(1 ; a )f ](x) for x 2 U , a 2 U. The open covering (U) of U and a
subordinate partition of unity ( ) are chosen such that for every point a 2 U
there exist normal coordinates za on U . We can even assume that (za (y)) = 1 for
all a y 2 U . Hence supp (1 ; a ) \ supp = , and Ka is smoothing, because A
is pseudolocal. This yields a family of smooth functions k : U U ! C such that
Z
(38)
Kaf (x) = k (x y) (1 ; a (y)) f (y) dy:
U
Now the smooth function k 2 C 1 (U U ) with k(x y) =
is the kernel function of K .
P k (x y) (1 ; (y))
x
Markus J. Paum
110
Proof of Theorem 4.2. Check that for an arbitrary cut-o function ~ : TX ! C
one can nd a cut-o function : TX ! C such that the equation
(39)
x (y) f (y) = (21)n=2
Z
Tx X
x (y) ei<zx(y)> f^~ ( ) d
holds for x y 2 U . Approximating the integral in Eq. (39) by Riemann sums, we
can nd a sequence (ik )k2N of natural numbers, a family (k )k2N1k of elements
k 2 Tx X and a sequence (k )k2N of positive numbers such that
lim = 0
k!1 k
X
nk
(40)
(x )f ] (y) = klim
x (y) ei<k zx (y)> f^~(k ):
n=
2
!1 (2 )
1iik
By an appropriate choice of the k one can even assume that the series on the right
hand side of the last equation converges as a function of y 2 U in the topology of
D(U ). As A is continuous on D(U ), we now have
i
X h
nk
i<k zx ()> (x) f^~ (k )
(41) A(x )f ](x) = klim
A
(
)
e
x
!1 (2 )n=2
1iik
n
X
k
= klim
A (k ) f^~ (k )
!1 (2 )n=2
Z 1iik^~
1
A ( ) f ( ) d
=
(2)n=2
Tx X
= A~f:
On the other hand Lemma 4.3 provides a smoothing operator K such that for
f 2 D(U ) and x 2 U :
(42)
Af (x) = A(x )f ] (x) + Kf (x) = A~ f (x) + Kf (x):
Thus the rst part of the theorem follows.
Now let a 2 S (U Hom(E F )) be a symbol, and consider the corresponding
pseudodierential operator A = A 2 (U E F ). After possibly altering we
can assume that there exists a second cut-o function ~ : T X ! C such that
~jsupp = 1. Then we have for 2 Tx X and % 2 Ex
Z
~ (43)
A ( )% = (21)n=2
a( ) F x ()ei'( ) x()%
( ) d
Tx X
Z
=
=
1
(2)n=2
1
Tx X
(2)n
a( ) F x ()ei'( ) % ( ) d
Z Z
Tx X Tx X
a( )% (v) e;i<v> ei<v> dv d
where the last integral is an iterated one. Next choose a smooth function : R !
0 1] such that supp ;2 2] and j;11] = 1 and dene ak 2 S;1 (U Hom(E F ))
The Normal Symbol on Riemannian Manifolds
111
~
(U Hom(E F ) ) for ~ > . We
by ak ( ) = jkj . Then obviously ak ! a in S
can now write:
(44)
1 Z Z a ( )% (v) e;i<;v> dv d
A ( )% = klim
k
!1 (2 )n
Tx X Tx X
= klim
ak ( )% ;
!1
1
(2)n=2
Z
Z TxX
1
= a( ) ; klim
!1 (2 )n=2
Tx X
(1 ; (v)) F ;1 ak (v) ei<v> dv
h ; i
(1 ; (v)) F ;1 Ly m ak (v) ei<v> dv
= a( ) + klim
Km ( ak )
!1
where m 2 N , L is a smooth (;1)-homogeneous rst order dierential operator on
_ ! TU
_ such that Lei<v> ( ) = ei<v> and
the vector bundle T X U TU
Z
h ; i
(45) Km( ak ) = ; 1n=2 (1 ; (v)) F ;1 Ly m ak (;v) ei<v> dv:
(2)
Tx X
; ~ ;m
_ consists of all nonzero vectors v 2 TU and that Ly m a 2 S
Note that TU
for every ~ . Now xing ~ and choosing m large enough that m >
~ + 2(dimX ; 1) yields
(46)
Km ( a) =: klim
Km ( ak )
!1
1 Z Z ;Ly m a ( ) (1 ; (v)) e;i<;v> d dv
= klim
;
k
!1 (2 )n
Z TZx X Tx;X m 1
= ; (2)n
Ly a ( ) (1 ; (v)) ei<v> e;i<v> d dv
T XT X
x
x
in the sense of Lebesgue integrals. As the phase function 7! ; < v > is
noncritical for every nonzero v and 1 ; (v) vanishes in an open neighborhood of
the zero section of TU , a standard consideration already carried out in preceeding
proofs entails that Km ( a) is a smoothing symbol. Henceforth A~ and a dier
by the smoothing symbol Km ( a). This gives the second part of the theorem.
In the following let us show that the normal symbol calculus gives rise to a natural
transformation between the functor of pseudodierential operators and the functor
of symbols on the category of Riemannian manifolds and isometric embeddings.
First we have to dene these two functors. Associate to any Riemannian manifold
;1
X the algebra A(X ) = 1
= (X ) of pseudodierential operators on X modulo
;1
smoothing ones, and the space S (X ) = S1
=S (X ) of symbols on X modulo
smoothing ones. As Y is Riemannian, we have for any isometric embedding f : X !
Y a natural embedding f : T X ! T Y . So we can dene S (f ) : S (Y ) ! S (X )
to be the pull-back f of smooth functions on T Y via f . The morphism A(f ) :
A(Y ) ! A(X ) is constructed by the following procedure. If f is submersive, i.e., a
112
Markus J. Paum
dieomorphism onto its image, A(f ) is just the pull-back of a pseudodierential
operator on Y to X via f . So we now assume that f is not submersive. Choose
an open tubular neighborhood U of f (X ) in Y and a smooth cut-o function
: U ! 0 1] with compact support and jV = 1 on a neighborhood V U
of f (X ). As U is tubular there is a canonical projection ' : U ! X such that
' f = idX . Now let the pseudodierential operator P represent an element of
A(Y ). We then dene A(f )(P ) to be the equivalence class of the pseudodierential
operator
(47)
E 0 3 u 7! f P ((' u))] 2 E 0 where E 0 denotes distributions with compact support. Note that modulo smoothing
operators f P ((' u))] does not depend on the choice of . With these denitions
A and S obviously form contravariant functors from the category of Riemannian
manifolds with isometric embeddings as morphisms to the category of complex
vector spaces and linear mappings.
The following proposition now holds.
Proposition 4.4. The symbol map forms a natural transformation from the
functor A to the functor S .
Proof. As f is isometric, the relation
(48)
f exp = exp Tf
is true. But then the phasefunctions on X and Y are related by
(49)
'Y (y f ( )) = 'X ('(y) )
where y 2 Y and 2 T X with y and f (( )) su$ciently close. Hence for a
pseudodierential operator P on Y
i
h (50)
(f Y (P )) ( ) = P f (()) () ei'Y (f ) (f (( )))
and
h i
(51)
(X f (P )) ( ) = P f (()) () () ei'X (()) (( )):
Eq. (49) now implies
(52)
f Y (P ) = X f (P )
which gives the claim.
5.
The symbol of the adjoint
In the sequel we want to derive an expression for the normal symbol A of the
adjoint of a pseudodierential operator A in terms of the symbol A .
Let E F be Hermitian (Riemannian) vector bundles over the Riemannian manifold X , and let < >E , resp. < >, be the metric on E , resp. F . Denote by DE ,
resp. DF the unique torsionfree metric connection on E , resp. F , and by D the
induced metric connection on Hom(E F ), resp. (Hom(E F )). Next dene the
function : O ! R+ by
(53) (x y) x = (y) (Tv expx Tv expx) x y 2 O v = exp;x 1 (y)
The Normal Symbol on Riemannian Manifolds
113
where is the canonical volume density on TxX and x its restriction to a volume
density on TxX . Now we claim that the operator Op (a) : D(U F ) ! C 1 (U E )
dened by the iterated integral
(54)
Z Z
1
;1 ;
;1
Op (a)g(x) = (2)n
e;i<v> exp
v a Tv expx ( ) g (exp v ) ~(v )
Tx X Tx X
(expx (v) x) dv d
;1
is the (formal) adjoint of Op (a), where
a ;21 S (UHom(E F )), U X open and ~
is the cut-o function TX 3 v 7! expexpx v (x) 2 0 1]. Assuming f 2 D(V E ),
g 2 D(U F ) with V U open, V V O and to be su$ciently small the
following chain of (proper) integrals holds:
(55)
Z
< f (x) Op (a)g (x) >E d (x) =
X
=1 (21)n
=2 (21)n
=3 (21)n
=4 (21)n
= (21)n
=
Z
X
Z
Z
;
ei<v> < f (x) exp v a Tv exp;x 1 ( ) g(exp v) >
X Tx X Tx X
Z Z
XT V
~(v) ;1 (expx (v) x) d!xn (v )d (x)
ei<T expx ()expx 1 (())> < ()x f (x) a ( ) g(( )) >
Z Z
T V X
(exp;(1) (x)) ;1 (( ) x) d!n ( ) d (x)
e;i<exp() (x)> < ()xf (x) a ( ) g(( )) >
;1
(exp;(1) (x)) ;1 (( ) x) d (x) d!n ( )
Z Z
ZT V TZ X
;
;1
n
e;i<w> < a( ) exp
w f (exp w) g (x) > (w) dw d! ( )
( )
X Tx X Tx X
;1
e;i<w> < a( ) exp
w f (exp w) g (x) > (w) dw d d (x)
< Op (a)f (x) g(x) >F d (x)
In explanation of the above equalities, we have
1. !x ! canonical symplectic forms on Tx X Tx X , resp. T X
2. by T expx : T V ! TxX Tx X symplectic, x 2 V
3. Gau" Lemma and Fubini's Theorem
4. coordinate transformation V 3 x 7! w = exp;(1) (x) 2 T() X .
Recall that S;1 (U Hom(E F )) is dense in any S (U Hom(E F )), 2 R with
~
respect to the Frechet topology of S
(U Hom(E F )) if < ~. By using a partition
Markus J. Paum
114
~
of unity and the continuity of S
(U Hom(E F )) 3 a 7! Op (a)f 2 C 1(U F ),
~
resp. S (U Hom(E F )) 3 a 7! Op (a)g 2 C 1(U E ), for any f 2 D(U E ) and
g 2 D(U F ), Eq. (55) now implies that
Z
(56)
X
< f (x) Op (a)g(x) >E d (x) =
Z
X
< Op (a)f (x) g(x) >F d (x)
holds for every f 2 D(U E ), g 2 D(U F ) and a 2 S (U ) with
Op (a) is the formal adjoint of Op (a) indeed.
The normal symbol A of A = Op (a) now is given by
(57)
Z Z
1
e;i< ;v>
A ( ) % A g( %)] (( )) = (2)n
2 R. Thus
a ;T exp;1 ( ) Tx%X T~x(vX) (v) ;1 (exp (v) x) dv d
exp v
x
v
x
where 2 T X , x = ( ), % 2 Fx and
;1
exp v
1
g(y %) = exp;(1) (y) ei<exp() (y)> y() %
(58)
;
with y 2 X . Thus by Theorem 3.4 of Grigis, Sjostrand 5] the following asymptotic
expansion holds:
X (;i)jj @ jj @ jj
A ( ) 2Nd ! @v v=0 @ =
(59)
;
;1 a T exp;1 ( ) ;1
exp
exp v (expx (v ) x)
v
v
x
2
3
X (;i)j+j 4 Djj+jj 5 @ jj ;1 =
a @z ( x) x x
@zx @x 2Nd ! !
where
Djj+jj a = @ jj a
@ jj a ;1 a ;T exp;1 ( ) exp v
v
x
@v v=0 @ exp v
@zx @x are symmetrized covariant derivatives of a 2 S1 (X Hom(E F )) with respect to
the apriorily chosen normal coordinates (zx x ).
Next we will search for an expansion of and its derivatives. For that rst write
@ = X l (x ) e (61)
l
@zxk kl k
(60)
where (e1 ::: ed ) is the orthonormal frame of Eq. (11), and the kl are smooth
functions on an open neighborhood of the diagonal of X X . Then we have the
following relation (see for example Berline, Getzler, Vergne 1], p. 36):
X
lk (x y) = lk ; 16 Rk mln (y) zxm (y) zxn(y) + O(jzx (y)j3 )
(62)
mn
The Normal Symbol on Riemannian Manifolds
115
where Rk mln (y), resp. their \lowered" versions Rkmln (y), are given by the coe$cients
with respect to normal coordinates at y 2 X ,
of the curvature
P Rk (tensor
i.e., R @z@ym =
mln y ) @z@yk dzyl dzyn . But this implies
kln
;
(x y) = det kl (x y)
XX k
R mkn (x) zxm (y) zxn(y) + O(jzx (y)j3
= 1 ; 16
(63)
k mn
X
;
= 1 ; 16 Ricmn (x) zxm (y) zxn(y) + O jzx(y)j3 :
mn
Using Eq. (13) zxk (y) = ;zyk (x) for x and y close enough we now get
@ ;1 ( x) = 1 X Ric (x) z l (y) + O ;jz (y)j2 (64)
kl
x
x
@zxk y
3 l
@ 2 ;1 ( x) = 1 Ric (x) + O (jz (y)j) :
x
@zxk @zxl y
3 kl
(65)
The above results are summarized by the following theorem.
Theorem 5.1. The formal adjoint A 2 (U F E ) of a pseudodierential operator A 2 (U E F ) between Hermitian (Riemannian) vector bundles E F over
an open subset U of a Riemannian manifold X is given by
Z Z
1
A g (x) = Op (a)g (x) = (2)n
e;i<v>
Tx X Tx X
;1 ;
;1
;1
exp
v a Tv expx ( ) g (exp v ) ~(v ) (expx (v ) x) dv d
where x 2 X , g 2 D(U F ) and a = A is the symbol of A. The symbol A of A
has asymptotic expansion
(66)
A ( ) a ( ) +
X
D2jj
@z() @() a ;
Nd
1
j
2
j
2
3
X X jj 4
D2jj+2
5
; 16
(;i)
@z() @() @k() @l () a Rickl (( )) +
2Nd kl
2
3"
#
jj
X (;i)j + j 4 D2jj+jj
@
;
1
+
a 5 @z ( x)
! !
@z() @(+) () ()
Nd
3
2
j j
= a ( ) ; i
X
2
D2
a
k
k
k @z() @() 3
4
X
; 21 @z k @z l D@ k @ l a ;
kl () () () () 2
X
; 16 4 @ k D@ l a 5 Rickl (( )) + r( )
() () kl
Markus J. Paum
116
with 2 T X and r 2 S;2(;) (U Hom(E F )).
6.
Product expansions
Most considerations of elliptic partial dierential operators as well as applications
of our normal symbol to quantization theory (see Paum 8]) require an expression
for the product of two pseudodierential operators A B in terms of the symbols
of its components. In the at case on Rd it is a well known fact that AB has an
asymptotic expansion of the form
X jj @ jj
@ jj
(67)
AB (x ) (;i)! @
A (x ) @x B (x ):
Note that strictly speaking, the product AB of pseudodierential operators is only
well-dened, if at least one of them is properly supported. But this is only a
minor set-back, as any pseudodierential operator is properly supported modulo a
smoothing operator and the above asymptotic expansion also describes a symbol
only up to smoothing symbols.
In the following we want to derive a similar but more complicated formula for
the case of pseudodierential operators on manifolds.
Proposition 6.1. Let A 2 (U E F ) be a pseudodierential operator between
Hermitian (Riemannian) vector bundles E F over a Riemannian manifold X and
a = (A) its symbol. Then for any f 2 C 1 (U E ) the function
h i
(68)
Af : T U ! F 7! A () f ei'() (( ))
is a symbol of order and type ( ) and has an asymptotic expansion of the form
#
X 1 " Djj # " Djj
(69)
Af ( ) ijj ! @ a @z f :
() () ()
Proof. Let x = (), N 2 N and : T X ! C a cut-o function. Then the
following chain of equations holds:
(70)
h i
Af ( ) = A () f ei'() (x)
1 Z a( ) F ;f ei<> ( ) d
(2)n=2 Tx X
Z Z
;1 f (exp v ) (v ) dv d
= (21)n
ei<;v> a( ) exp
v
Tx X Tx X
Z
Z
;1
= (21)n
e;i<v> a( + ) exp
v f (exp v ) (v ) dv d
Tx X Tx X
X 1 Z Z ;i<v> Djj
;1
= (21)n
e
a( ) exp v f (exp v ) (v ) dv d
!
@
x
T
X
T
X
x
x
jjN
+ rN ( )
X 1 " Djj # Djj =
ijj ! @x a @zx x f + rN ( ):
jjN
=
The Normal Symbol on Riemannian Manifolds
117
Note that in this equation all integrals are iterated ones and check that with Taylor's
formula rN is given by
(71)
j j
X N +1Z Z 1
N D a( + t ) dt F ;f ( ) d:
rN ( ) = (21)n
(1
;
t
)
!
j j=N +1
Tx X
@x
0
Choosing N large enough, the Schwarz inequality and Plancherel's formula now
entail
j j
X N +1 Z 1
1
N D a( + t())dt
jjrN ( )jj (2)n=2
(1
;
t
)
!
@x 0
L 2 ( Tx X )
j j=N +1
(72)
j j
D f
@vx Tx X L2 (Tx X ) :
But for K U compact and t 2 0 1] there exist constants DNK D~ NK > 0 such
that
Djj a( + t ) D (1 + jj jj + jj jj);(N +1)
NK
@x (73)
and
(74)
Z1
0
j j
(1 ; t)k D a( + t())dt
@x
L Tx X )
2(
DNK (1 + jj jj + jj jj);(N +1)
= D~ NK (1 + jj jj)n+;(N +1)
L2 (Tx X )
for all x 2 K , 2 Tx X and all indices 2 N d with j
j = N + 1. Inserting this
in (72) we can nd a CNK > 0 such that for all x 2 K and 2 Tx X
(75) jjrN ( )jj CNK 1 + jj jj
n+;(N +1) X
j jN +1
D j j
sup @z f (y)
x
!
: y 2 supp () :
Because of Nlim
dimX + + (N + 1) = ;1 and a similar consideration for the
!1
derivatives of Af the claim now follows.
Next let us consider two pseudodierential operators A 2 1
(U F G) and B 2
(U E F ) between vector bundles E F G over X . Modulo smoothing operators
we can assume one of them to be properly supported, so that AB 2 1
(U E G)
exists. Dening the extended symbol bext = Bext : U T U ! Hom(E F ) of B by
(76)
bext (x )% =
h i
() (x) e;i'(x) B () () ei'() ()() % (x) x 2 U ( %) 2 (E jU )
1
Markus J. Paum
118
the symbol AB of the composition AB is now given modulo S;1 (U Hom(E G))
by
h i
AB ( )% = AB () ei'() ()()% (( ))
(77)
h i
= A () ei'() Bext ( )% (( )):
By Proposition 6.1 this entails that AB has an asymptotic expansion of the form
#" jj
#
X 1 " Djj
D
ext
(78)
B ( ) :
AB ( ) A @z jj
() ()
2Nn i ! @() As we want to nd an even more detailed expansion for AB , let us calculate the
derivatives @zD() Bext ( ) in the following proposition.
j
j
()
Proposition 6.2. Let B 2 (U E F ) be a pseudodierential operator between
Hermitian (Riemannian) vector bundles over the Riemannian manifold X and b =
(B ) its symbol. Then for any f 2 C 1 (U E ) the smooth function
i
h ext : U T U ! F (x ) 7! (x) e;i'(x) B f ei'() (x)
(79) Bf
()
()
is a symbol of order and type ( ) on U T U and has an asymptotic expansion
of the form
(80)
X
ik;jj (x)
ext (x ) X
Bf
()
::: Nn k !
0 ! ::: k !
k2N +0 :::+k =
"
2
01 j:::jkkj2
Dj j
j
@ j j
#
Djj
b
@x dx '( )
@ jk j '( ) :
'
(
)
:::
@zx 0 x
@zx 1 x
@zxk x
h ext i
( ) (( )) 2 F with 2 N n
Furthermore the derivative T U 3 7! @z@() Bf
is a symbol with values in F of order + jj and type ( ) on T U and has an
0
() f
1
j
j
asymptotic expansion of the form
Djj
ext
@z() Bf ( ) (( )) X X
X
ik;jj!
k!~!0 ! ::: k !
0 ! ::: k !
~ 0 :::k Nn 0 :::k Nn
k2N ~
+ +:::+ = +:::+ =
0
2
k
2
01 j:::jkkj2
8
2
< Dj~j
D j j
4
: @z()~ () @(;)
( j j
" j j
j
39
" j j
#)
= ( Dj j
D
b5 @z f () () @z(;) (;)
'( )
#) ( jk j
" jkj
#)
0
d(
@ 1
@ 1
@z() 1 () @z(;)1
0
0
;)
'( )
(;)
0
::: @z@ k
()
@
@z
(;) k
( )
'( )
(;)
:
Note 6.3. The dierential operators of the form @zD() act on the variables de
noted by (;), the dierential operators of the form @zD( ) on the variables ().
j
j
j j
;
The Normal Symbol on Riemannian Manifolds
119
Proof of Proposition 6.2. Let 2 C 1(U U T U ) be a smooth function such
that
(81)
(x y ) = '(y ) ; '(x ); < dx '( ) zx (y) >
for x y 2 supp () and such that T U 3 7! (x y ) 2 C is linear for all
x y 2 U . Furthermore denote by F 2 C 1 (U U T U E ) the function (x y ) 7!
() (y)f (y)ei(xy). By the Leibniz rule and the denition of we then have
"
(82)
#
Djj F (x ) =
@zx y
"
As (x y ) 7!
X
0k j2 j
#
ik
Dj0 j f ( )
0
0 :::k Nn k !
0 ! ::: k ! @zx y
+:::+ =
2
j
01 j:::jkkj2
#
"
#
@ j1j (x ) ::: @ jk j (x ) ei(xy):
@zx1 y
@zxk y
@ j
@zx j y (x ) , 0 j
j
"
X
j
k is smooth and linear with respect to
D F (x ) is absolutely
, the preceeding equation entails that (x y ) 7! @z
x y
bounded by a function (x y ) 7! C (x y) j j 2 , where C 2 C 1 (U U ) and
C 0. On the other hand we have
h i
ext (x ) = () (x) B () ()f ()ei(x) ei<dx '( )zx ()> (x)
(83)
Bf
j j
j j
and by the proof of Proposition 6.1
(84)
"
ext (x ) = (x)
Bf
()
= () (x)
D j j
(85)
! @x dx'( ) b
j jN
X X
X
j jN 0k jj
2
Djj F (x ) + r (x )
N
@zx x
"
#
D j j
ik;jj
b 0 :::k Nn k !
0 ! ::: k ! @x dx '( )
+:::+ =
2
j
01 j:::jkk j2
@ j1 j '( ) ::: @ jk j '( ) + r (x )
f
N
(
)
@zx 0 x
@zx1 x
@zxk x
0
where
#
X ijj Djj
jjrn (x )jj CNK (1 + jjdx '( )jj)n+;(N +1)
X
j jN +1
sup
("
#
@ j j F (x )
@zx y
: y 2 supp x
)
DNK (1 + jj jj)n+;(;1=2)(N +1)
holds for compact K U , x 2 K , 2 T X jK and constants CNK DNK > 0.
Because > 12 , we have limN !1 dimX + ; ( ; 21 )(N + 1) = ;1. By Eqs. (84)
ext we therefore conclude that
and (85) and similar relations for the derivatives of Af
Eq. (80) is true. The rest of the claim easily follows from Leibniz rule.
The last proposition and Eq. (78) imply the next theorem.
Markus J. Paum
120
Theorem 6.4. Let A B 2 1 (U E F ) be two pseudodierential operators be-
tween Hermitian (Riemannian) vector bundles E F over a Riemannian manifold
X . Assume that one of the operators A and B is properly supported. Then the
symbol AB of the product AB has the following asymptotic expansion.
(86)
"
#"
#
X i;jj
Djj
@z() () B +
2Nd !
X X
X
ik;jj;jj
+
k! ~! 1 ! ::: k !
1 ! ::: k !
~ 1 :::k Nn 1 :::k Nn
k1 ~ +
+:::+ = +:::+ =
AB ( ) "
(
1
Djj @() A
j1 j
2
k
Djj @() A
2
11 j:::jkkj2
2
39
#8
< Dj~j
=
j j
D
4
5
B
: @z()~ () @(;) d '( ) " j j
#) ( jk j
"
j
@
@
@z() 1 () @z(;)1
#)
(;)
1
'( )
(;)
@ jk j
::: @z@ k
() () @z(;) k
'( )
(;)
:
A somewhat more explicit expansion up to second order is given by
Corollary 6.5. If A is a symbol of order and B a symbol of order ~ the
coecients in the asymptotic expansion of AB are given up to second order by the
following formula:
AB ( ) =A ( ) B ( ) ;
X
DB ( ) ; 1 X D2 A ( ) D2 B ( )
(87)
; i @DA ( ) @z
l
2 kl @()k @()l @zk()@zl ()
()l
()
l
X DA
D2 B
(
)
( ) Rk mln (( )) ()k ( ) + r( )
; 121
@
@
@
(
)
n
(
)
l
(
)
m
klmn
where 2 T U , x = ( ), r 2 S+~;3 (U ) and the Rmnkl (y) are the coecients
@ of
the curvature tensor with respect to normal coordinates at y 2 X , i.e., R @zym =
P Rk (y) @ dzl dzn.
mln @zyk
y
y
kln
1 :::k
Proof. First dene the order of a summand s = s
~ 1 :::k in Eq. (86) by ord(s) =
j ; ~ + ; kj. Then it is obvious that s 2 S+~+ord(s) (;) (U ). Therefore we have
to calculate only the summands s with ord(s) 2. To achieve this let us calculate
some
changes and their derivatives.
;k ofcoordinate
; Recall the matrix-valued function
Eq. (61) and construct its inverse )lk . In other words
l
Xl
(88)
k (x y) )mk (x y) = ml
k
holds for every l m = 1 ::: d. We can now write
@ = X l (y z ) )m(x z ) @
(89)
k
l
k
m
@zy z
lm
@zx z
The Normal Symbol on Riemannian Manifolds
121
and receive the following formulas:
@zxk (z ) =k ; 1 X Rk (y) z m (z ) z n(z )
l
x
@zyl
6 mn mln x
(90)
X
;
+ 61 Rk mln (x)zxm (z ) zxn(z ) + O jzx (z ) + zy (z )j3 mn
(91)
@ 2zxk (z ) = ; 1 X ;Rk (y) + Rk (y) z n (z )
mln
nlm
x
@zyl @zym
6 n
X; k
+ 16
R mln (x) + Rk nlm (x) zxn (z )
;n
+ O jzx(z ) + zy (z )j2 (92)
@ 2 zxk (y) = 1 ;Rk (x) + Rk (x) + O (jz (z )j) :
@
mln
nlm
x
@zxn y=z @zyl @zym
6
Next consider the functions
jj
@ jj'( ) (y)
7! @z@ () y=() @ zy which appear in the summands s of Eq. (86). One can write the ' in the form
jj
X
@ jjzk()
@
' ( ) = ()k ( ) @z (y )
(93)
() y=() @zy k
' : T X ! R
Thus by Eq. (91)
'0 ( ) = 0
(94)
holds for every 2 N d with j
j 2. Furthermore check that
X
dy '( ) = ()k ( ) dzk()
(95)
k
y
is true.
Now we are ready to calculate the summands s of order ord(s) 2 by considering
the following cases.
(i) ord(s) = 0:
(96)
s( ) = A ( ) B ( ):
(ii) ord(s) = 1, jj = 1, k = 0, ~ = :
(97)
s( ) = ; i
X DA DB
( ) l ( ):
l @()l @z()
(iii) ord(s) = 1, jj = 0, k = 1, j
j = 2: In this case s = 0 because of Eq. (94).
(iv) ord(s) = 2, jj = 2, k = 0, ~ = :
2
2
X
B ( ):
(98)
s( ) = ; 21 @ D @A ( ) @z kD @z
l
() ()
kl ()k ()l
122
Markus J. Paum
(v) ord(s) = 2, jj = 1, k = 1, j
j = 2: In this case ~ = 0 by Eq. (94). Hence by
Eq. (92)
1 X DA ( )
D 2 B
(99) s( ) = ; 12
( ) Rk mln (( )) ()k ( ):
@
@
@
(
)
n
(
)
l
(
)
m
klmn
Summing up these summands we receive the expansion of AB up to second order.
This proves the claim.
The product expansion of Theorem 6.4 gives rise to a a bilinear map # on the
;1
1
;1
sheaf S1
=S ( Hom(F G)) S =S ( Hom(E F )) by
1
;1
S1 (U Hom(F G)) S1
(U Hom(E F )) ! S =S (U Hom(E G))
(100) (a b) 7! a#b = Op(a) Op(b)
for all U X open and vector bundles E F G over X . The #-product is an
important tool in a deformation theoretical approach to quantization (cf. Paum
8]). In the sequel it will be used to study ellipticity of pseudodierential operators
on manifolds.
7.
Ellipticity and normal symbol calculus
The global symbol calculus introduced in the preceding paragraphs enables us
to investigate the invertibility of pseudodierential operators on Riemannian manifolds. In particular we are now able dene a notion of elliptic pseudodierential
operators which is more general than the usual notion by Hormander 7] or Douglis,
Nirenberg 2].
Denition 7.1. A symbol a 2 S1 (X Hom(E F )) on a Riemannian manifold X
is called elliptic if there exists b0 2 S1
(X Hom(F E ))) such that
(101)
a#b0 ; 1 2 S;" (X ) and b0#a ; 1 2 S;" (X )
for an " > 0. A symbol a 2 Sm
(X Hom(E F )) is called elliptic of order m if it is
elliptic and one can nd a symbol b0 2 Sm
(X Hom(F E )) fullling Eq.(101).
An operator A 2 1
(
X
E
F
)
is
called
elliptic resp. elliptic of order m, if its
symbol A is elliptic resp. elliptic of order m.
Let us give some examples of elliptic symbols resp. operators.
Example 7.2. (i) Let a 2 Sm10(X Hom(E FP
)) be a classical symbol, i.e., assume
;j
that a has an expansion of the form a j2N am;j with am;j 2 Sm
10 (X )
homogeneous of order m ; j . Further assume that a is elliptic in the classical
sense, i.e., that the principal symbol am is invertible outside the zero section.
By the homogeneity of am this just means that am is elliptic of order m. But
then a and Op(a) must be elliptic of order m as well.
(ii) Consider the symbols l : T X ! C , 7! jj jj2 and a : T X ! C , 7! 1+jj1jj2
of Example 2.3 (ii). Then the pseudodierential operator corresponding to l
is minus the Laplacian: Op(l) = ; . Furthermore the symbols l and 1+ l are
elliptic of order 2, and the relation a#(1 + l) ; 1, (1 + l)#a ; 1 2 S;110 (X ) is
satised. In case X is a at Riemannian manifold, one can calculate directly
that modulo smoothing symbols a#l = l#a = 1, hence Op(a) is a parametrix
for the dierential operator Op(1 + l) = 1 ; .
The Normal Symbol on Riemannian Manifolds
123
(iii) Let a' ( ) = (1 + jj jj2 )'(()) be the symbol dened in Example 2.3 (iii),
where ' : X ! R is supposed to be smooth and bounded. Then a is elliptic
and b( ) = (1 + jj jj2 );'(()) fullls the relations (101). In case ' is not
locally constant, we thus receive an example of a symbol which is not elliptic
in the sense of Hormander 7] but elliptic in the sense of the above denition.
Let us now show that for any elliptic symbol a 2 S1
(X Hom(E F )) one can
nd a symbol b 2 S1
(
X
Hom(
F
E
))
such
that
even
(102) a#b ; 1 2 S;1 (X Hom(F F )) and b#a ; 1 2 S;1 (X Hom(E E )):
Choose b0 according to Denition 7.1 and let r = 1 ; a#b0, l = 1 ; b0#a 2 S;1 (X ).
Then the symbols
br = b0 (1 + r + r#r + r#r#r + : : : ) and bl = (1 + l + l#l + l#l#l + : : : ) b0
are well-dened and fulll a#br = 1 and bl #a = 1 modulo smoothing symbols.
Hence br ; bl 2 S;1 (X Hom(F E )), and b = br is the symbol we were looking for.
Thus we have shown the essential part for the proof of the following theorem.
Theorem 7.3. Let A 2 1 (X E F ) be an elliptic pseudodierential operator.
Then there exists a parametrix B 2 1
(X F E ) for A, i.e., the relations
(103)
A B ; 1 2 ;1 (X F F ) and B A ; 1 2 ;1 (X E E )
hold. If A is elliptic of order m, B can be chosen of order ;m. On the other hand,
if A is an operator invertible in 1
(X ) modulo smoothing operators, then A is
elliptic. In case X is compact an elliptic operator A 2 1
(X ) is Fredholm.
Proof. Let a = A. As a is elliptic one can choose b such that (102) holds. If a
is elliptic of order m, the above consideration shows that b is of order ;m. The
operator B = Op(b) then is the parametrix for A. If on the other hand A = Op(a)
has a parametrix B = Op(b), then a#b ; 1 and b#a ; 1 are smoothing, hence A is
elliptic. Now recall that a pseudodierential operator induces continuous mappings
between appropriate Sobolev-completions of C 1 (X E ), resp. C 1 (X F ), and that
with respect to these Sobolev-completions any smoothing pseudodierential operator is compact. Hence the claim that for compact X an elliptic A is Fredholm
follows from (103).
Let us give in the following propositions a rather simple criterion for ellipticity
of order m in the case of scalar symbols.
Proposition 7.4. A scalar symbol a 2 Sm (X ) is elliptic of order m, if and only
if for every compact set K X there exists CK > 0 such that
(104)
jja( )jj 1 jj jjm
CK
for all 2 T X with ( ) 2 K and jj jj CK .
Proof. Let us rst show that the condition is su$cient. By assumption there exists
a function b 2 C 1 (T X ) such that for every compact K X there is CK > 0 with
(105)
a( ) b ( ) = 1 and jjb( )jj CK jj jj;m
Markus J. Paum
124
for 2 T X with ( ) 2 K and jj jj CK . Hence a b ; 1 2 S;1 (X ). After
dierentiating the relation r = a b ; 1 in local coordinates (x ) of T X we receive
(106)
;1 + jjjj2m=2 @ jj @ jj b() C sup a() @ jj @ jj b() sup
@x @ @x @ 2T X jK
8
9
>
>
>
<
=
X @ j j @ j j @ j j @ j j >
C sup >jr( )j +
a
(
)
b
(
)
>
@x @ @x @ 2T X jK >
>
:
2T X jK
1
1
1+ 2=
1+ 2=
j1 +1 j>0
1
2
1
2
2
2
hence by induction b 2 S;m (X ) follows. By the product expansion Eq. (87) this
implies
(107)
a#b ; 1 = r + t1 and b#a ; 1 = r + t2 where t1=2 2 S;" (X ) with " = minf( ; ) 2 ; 1g > 0. Hence a is an elliptic
symbol.
Now we will show the converse and assume the symbol a to be elliptic of order m.
According to Theorem 7.3 there exists b 2 S;m (X ) such that Op(b) is a parametrix
for Op(a). But this implies by Eq. (87) that ab ; 1 2 S;" for some " > 0, hence
ja( )b( ) ; 1j < 21 for all 2 T X jK with jj jj > CK , where K X compact and
CK > 0. But then
1
0
;m
(108)
2 < ja( )b( )j < CK ja( )jjj jj 2 T X jK jj jj > CK which gives the claim.
In the work of Douglis, Nirenberg 2] a concept of elliptic systems of pseudodifferential operators has been introduced. We want to show in the following that this
concept ts well into our framework of ellipticity. Let us rst recall the denition
by Douglis and Nirenberg. Let E = E1 ::: EK and F = F1 ::: FL be direct
sums of Riemannian (Hermitian) vector bundles over X , and Akl 2 1m0kl (X Ek Fl )
with mkl 2 R be classical pseudodierential operators. Denote the principal symbol
kl
of Akl by pkl 2 Sm
10 (X Ek Fl ). Douglis, Nirenberg 2] now call the system (Akl )
kl
elliptic, if there exist homogeneous symbols bkl 2 S;1m
0 (X Ek Fl ) such that
(109)
X
X
pkl blk ; kk 2 S;110 (X Ek Fl ) and
blk pkl ; ll 2 S;110 (X Ek Fl )
0
l
0
k
0
0
for all k k l l . The following proposition shows that A = (Akl ) is an elliptic
pseudodierential operator, hence according to Theorem 7.3 possesses a parametrix.
Proposition 7.5. Let E = E1 ::: EK and F = F1 ::: FL be direct sums
of Riemannian (Hermitian) vector bundles over X . Assume that A = (Akl ) with
Akl 2 Sm10kl (X E F ) comprises an elliptic system of pseudodierential operators
in the sense of Douglis, Nirenberg 2]. Then the pseudodierential operator A is
elliptic.
0
0
The Normal Symbol on Riemannian Manifolds
125
Proof. Write akl = Akl = pkl + rkl with akl 2 Sm10kl (X Hom(Ek Fl) and rkl 2
kl ;1 (X Hom(Ek Fl ). Furthermore let a = (akl ) 2 S1 (X Hom(E F )), b =
Sm
10
10
1
(bkl ) 2 S1
10 (X Hom(E F )) and r = (rkl ) 2 S10 (X Hom(E F )). By Eq. (109) and
the expansion Eq. (87) it now follows
a#b ; 1 = (p#b ; 1) + r#b 2 S;110 (X Hom(E F ))
(110)
b#a ; 1 = (b#p ; 1) + b#r 2 S;110 (X Hom(E F )):
But this proves the ellipticity of A.
So with the help of the calculus of normal symbols one can directly construct
inverses to elliptic operators in the algebra of pseudodierential operators on a
Riemannian manifold. In particular after having once established a global symbol
calculus it is possible to avoid lengthy considerations in local coordinates. Thus
one receives a practical and natural geometric tool for handling pseudodierential
operators on manifolds.
References
1] N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Grundlehren der
mathematischen Wissenschaften, no. 298, Springer{Verlag, Berlin, 1992, MR 94e:58130.
2] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial dierential
equations, Comm. Pure Appl. Math. 8 (1955), 503{538.
3] J. J. Duistermaat, Fourier Integral Operators, Courant Institute of Math. Sci., New York,
1973.
4] Ezra Getzler, Pseudodierential operators on supermanifolds and the Atiyah-Singer Index
Theorem, Comm. Math. Phys. 92 (1983), 163{178, MR 86a:58104.
5] Alain Grigis and Johannes Sjstrand, Microlocal Analysis for Dierential Operators, London
Mathematical Society Lecture note series, no. 196, Cambridge University Press, Cambridge,
1994, MR 95d:35009.
6] Lars Hormander, The Analysis of Linear Partial Dierential Operators I, Grundlehren, no.
256, Springer-Verlag, Berlin, 1983, MR 91m:35001b.
7]
, The Analysis of Linear Partial Dierential Operators III, Grundlehren, no. 274,
Springer-Verlag, Berlin, 1985, MR 95h:35255.
8] Markus J. Paum, Local Analysis of Deformation Quantization, Ph.D. thesis, Fakultat fur
Mathematik der Ludwig-Maximilians-Universitat, Munchen, November 1995.
9]
, A deformation theoretical approach to Weyl quantization on Riemannian manifolds,
Lett. Math. Physics (to appear).
10] Yuri Safarov, Functions of the Laplace-Beltrami operator, Journees \Equations aux derivees
partielles", (Saint-Jean-de-Monts, 1996), Exp. No. XV, 8 pp., E cole Polytech., Palaiseau,
1996, MR 98c:58165.
11]
, Pseudodierential operators and linear connections, Proc. London Math. Soc. (3)
(to appear).
12] M. A. Shubin, Pseudodierential Operators and Spectral Theory, Springer-Verlag, Berlin,
1987, MR 88c:47105.
13] Harold Widom, Families of pseudodierential operators, Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York, 1978, pp. 345{395, MR 81c:58062.
, A complete symbolic calculus for pseudodierential operators, Bull. Sci. Math. (2)
14]
104 (1980), 19{63, MR 81m:58078.
Humboldt Universitat zu Berlin, Mathematisches Institut, Unter den Linden 6,
10099 Berlin
paum@mathematik.hu-berlin.de http://spectrum.mathematik.hu-berlin.de/~paum
This paper is available via http://nyjm.albany.edu:8000/j/1998/4-8.html.