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New York Journal of MathematicsNew York J. Math. 4 (1998) 97â125.
The Normal Symbol on Riemannian Manifolds
Markus J. Pflaum
Abstract. For an arbitrary Riemannian manifold X and Hermitian vectorbundles E and F over X we define the notion of the normal symbol of a pseu-dodifferential operator P from E to F . The normal symbol of P is a certainsmooth function from the cotangent bundle T âX to the homomorphism bun-dle Hom(E,F ) and depends on the metric structures, resp. the correspondingconnections on X, E and F . It is shown that by a natural integral formula thepseudodifferential operator P can be recovered from its symbol. Thus, mod-ulo smoothing operators, resp. smoothing symbols, we receive a linear bijectivecorrespondence between the space of symbols and the space of pseudodifferen-tial operators on X. This correspondence comprises a natural transformationbetween appropriate functors. A formula for the asymptotic expansion of theproduct symbol of two pseudodifferential operators in terms of the symbols ofits factors is given. Furthermore an expression for the symbol of the adjoint isderived. Finally the question of invertibility of pseudodifferential operators isconsidered. For that we use the normal symbol to establish a new and generalnotion of elliptic pseudodifferential operators on manifolds.
Contents
1. Introduction 972. Symbols 993. Pseudodifferential operators on manifolds 1014. The symbol map 1055. The symbol of the adjoint 1126. Product expansions 1167. Ellipticity and normal symbol calculus 122References 125
1. Introduction
It is a well-known fact that on Euclidean space one can construct a canonicallinear isomorphisms between symbol spaces and corresponding spaces of pseudo-differential operators. Furthermore one has natural formulas which represent a
Received October 14, 1997.Mathematics Subject Classification. 35S05, 58G15.Key words and phrases. pseudodifferential operators on manifolds, asymptotic expansions,
symbol calculus .
cŠ1998 State University of New YorkISSN 1076-9803/98
97
98 Markus J. Pflaum
pseudodifferential operator in terms of its symbol, resp. which give an expressionfor the symbol of a pseudodifferential operator. By using symbols one gets muchinsight in the structure of pseudodifferential operators on Euclidean space. In par-ticular they give the means to construct a (pseudo) inverse of an elliptic differentialoperator on Rd.
Compared to the symbol calculus for pseudodifferential operators on Rn it seemsthat the symbol calculus for pseudodifferential operators on manifolds is not thatwell-established. But as the pure consideration of symbols and operators in local co-ordinates does not reveal the geometry and topology of the manifold one is workingon, it is very desirable to build up a general theory of symbols for pseudodifferentialoperators on manifolds. In his articles [13, 14] Widom gave a proposal for a symbolcalculus on manifolds. By using a rather general notion of a phasefunction, Widomconstructs a map from the space of pseudodifferential operators on manifolds tothe space of symbols and shows by an abstract argument for the case of scalarsymbols that this map is bijective modulo smoothing operators, resp. symbols. Inour paper we introduce a symbol calculus for pseudodifferential operators betweenvector bundles having the feature that both the symbol map and its inverse have aconcrete representation. In particular we thus succeed in giving an integral repre-sentation 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 ofour special choice of a phasefunction the resulting symbol calculus is natural in acategory theoretical sense.
Using the integral representation for the operator map, it is possible to writedown a formula for the symbol of the adjoint of a pseudodifferential operator andfor the symbol of the product of two pseudodifferential operators.
The normal symbol calculus will furthermore give us the means to build up a nat-ural notion of elliptic symbols, respectively elliptic pseudodifferential operators onmanifolds. It generalizes the classical notion of ellipticity as defined for example inHormander [7] and also the concept of ellipticity introduced by Douglis, Nirenberg[2]. Our framework of ellipticity allows the construction of parametrices of nonclas-sical, respectively nonhomogeneous, elliptic pseudodifferential operators. Moreoverwe do not need principal symbols for defining ellipticity. Instead we define ellipticoperators in terms of their normal symbol, as the globally defined normal symbolof a pseudodifferential operator carries more information than its principal symbol.Moreover, the normal symbol of a pseudodifferential operator shows immediatelywhether the corresponding operator is invertible modulo smoothing operators ornot.
Let us also mention that another application of the normal symbol calculus onmanifolds lies in quantization theory. There one is interested in a quantization mapassociating pseudodifferential operators to certain functions on a cotangent bundlein a way that Diracâs quantization condition is fulfilled. But the inverse of a symbolmap does exactly this, so it is a quantization map. See Pflaum [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 99
2. Symbols
First we will define the notion of a symbol on a Riemannian vector bundle. Letus assume once and for all in this article that Âľ, Ď, δ â R are real numbers such that0 ⤠δ < Ď â¤ 1 and 1 ⤠Ď+ δ. The same shall hold for triples Âľ, Ď, δ â R.
We sometimes use properties of symbols on RdĂRN . As these are well-describedin the mathematics literature we only refer the interested reader to Hormander[6, 7], Shubin [12] or Grigis, Sjostrand [5] for a general introduction to symboltheory on Rd Ă RN and for proofs.
Definition 2.1. Let E â X be a Riemannian or Hermitian vector bundle overthe smooth manifold X, % : TX â X its tangent bundle and Ď : T âX â X itscotangent bundle. Then an element a â Câ(Ďâ(E|U )) with U â X open is calleda symbol over U â X with values in E of order Âľ and type (Ď, δ) if for everytrivialization (x, Îś) : T âX|V â R2d and every trivialization Ψ : E|V â V Ă RNwith V â U open the following condition is satisfied:
(Sy) For every Îą, β â Nd and every K â V compact there exists C = CK,Îą,β > 0such that for every Ξ â T âX|K the inequalityâŁâŁâŁâŁâŁâŁâŁâŁ â|Îą|âxÎą
â|β|
âΜβΨ(a(Ξ))
âŁâŁâŁâŁâŁâŁâŁâŁ ⤠C (1 + ||Ξ||)Âľ+δ|Îą|âĎ|β|(1)
is valid.
The space of these symbols is denoted by SÂľĎ,δ(U, TâX,E) or shortly SÂľĎ,δ(U,E). It
gives rise to the sheaf SÂľĎ,δ(¡, T âX,E) of symbols on X with values in E of order Âľand type (Ď, δ).
By defining SââĎ,δ (U,E) =âÂľâN SÂľĎ,δ(U,E) and SâĎ,δ(U,E) =
âÂľâN SÂľĎ,δ(U,E) we
get the sheaf SââĎ,δ (¡, E) of smoothing symbols resp. the sheaf SâĎ,δ(¡, E) of symbolsof type (Ď, δ).
In case E is the trivial bundle X ĂC of a Riemannian manifold X we write SÂľĎ,δwith Âľ â R ⪠{â,ââ} for the corresponding symbol sheaves SÂľĎ,δ(¡, X Ă C).
The space of symbols SÂľĎ,δ(U, TX,E) consists of functions a â Câ(%â(E|U )) suchthat a fulfills condition (Sy). It gives rise to sheaves SÂľĎ,δ(¡, TX,E), SâĎ,δ(¡, TX,E)and SââĎ,δ (¡, TX,E).
It is possible to extend this definition to one of symbols over conic manifolds butwe do not need a definition in this generality and refer the reader to Duistermaat[3] and Pflaum [8].
We want to give the symbol spaces SÂľĎ,δ(U,E) a topological structure. So firstchoose a compact set K â U , a (not necessarily disjunct) partition K =
âΚâJ
KΚ of K
into compact subsets together with local trivializations (xΚ, ΜΚ) : T âX|VΚ â R2d andtrivializations ΨΚ : E|VΚ â VΚ Ă RN such that KΚ â VΚ â U and VΚ open. Then wecan attach for any Îą, β â Rd a seminorm p = p(KΚ,(xΚ,ΜΚ)),Îą,β : SÂľĎ,δ(U,E)â R+âŞ{0}to the symbol spaces SÂľĎ,δ(U,E) by
p(a) = supΚâJ
âŁâŁâŁâŁâŁâŁ â|Îą|âxΚι
â|β|
âΜΚβΨΚ(a(Ξ))
âŁâŁâŁâŁâŁâŁ(1 + |Ξ|)Âľ+|Îą|δâ|β|Ď : Ξ â T âX|KΚ
.(2)
100 Markus J. Pflaum
The system of these seminorms gives SÂľĎ,δ(U,E) the structure of a Frechet spacesuch that the restriction morphisms SÂľĎ,δ(U,E) â SÂľĎ,δ(V,E) for V â U open arecontinuous. Additionally we have natural and continuous inclusions SÂľĎ,δ(U,E) âSÂľĎ,δ
(U,E) for Âľ ⼠¾, Ď â¤ Ď and δ ⼠δ. Like in the case of symbols on Rd Ă RNone can show that pointwise multiplication of symbols is continuous with respectto the Frechet topology on SÂľĎ,δ(U,E).
Proposition 2.2. Let a â Câ(Ďâ(E|U )). If there exists an open covering of Uby patches VΚ with local trivializations (xΚ, ΜΚ) : T âX|VΚ â R2d and trivializationsΨΚ : E|VΚ â VΚ Ă RN such that for every (xΚ, ΜΚ) the above condition (Sy) holds,then a is a symbol of order Âľ and type (Ď, δ).
Proof. Let (x, Îś) : T âX|V â R2d, V â U open be a trivialization. Then we canwrite on VΚ ⊠V
â|Îą|
âxÎąâ|β|
âΜβ=
âÎą+Îąâ˛â¤Îą
Κ
(fÎą,Îąâ˛,β ⌠Ď) Μιâ˛
Κ
â|Îą|
âxΚιâ|β+Îąâ˛|
âΜΚβ+ιⲠ,(3)
where fÎą,Îąâ˛,β â Câ(VΚ ⊠V ). As Ď + δ ⼠1 holds and (Sy) is true for (xΚ, ΜΚ), theclaim now follows. ďż˝
Example 2.3. (i) Let X be a Riemannian manifold. Then every smooth func-tion a : T âU â C which is a polynomial function of order ⤠¾ on everyfiber lies in SÂľ1,0(U). In particular we have an embedding D0 â SâĎ,δ, whereD0 is the sheaf of polynomial symbols on X, i.e., for every U â X openD0(U) = {f : T âU â C : f |TâxX is a polynomial for every x â U}.
(ii) Again let X be Riemannian. Then the mapping l : T âX â C, Ξ 7â ||Ξ||2 isa symbol of order 2 and type (1, 0) on X, but not one of order Âľ < 2. Nextregard the function a : T âX â C, Ξ 7â 1
1+||Ξ||2 . This function is a symbol oforder â2 and type (1, 0) on X, but not one of order Âľ < â2.
(iii) Assume Ď : X â R to be smooth and bounded. Then aĎ : T âX â R,Ξ 7â (1 + ||Ξ||2)Ď(Ď(Ξ)) comprises a symbol of order Âľ = supxâX Ď(x) and type(1, δ), where 0 < δ < 1 is arbitrary.
The following theorem is an essential tool for the use of symbols in the theoryof partial differential equations and extends a well-known result for the case ofE = Rd Ă RN to arbitrary Riemannian vector bundles.
Theorem 2.4. Let aj â SÂľjĎ,δ(U,E), j â N be symbols such that (Âľj)jâN is a de-creasing sequence with lim
jââÂľj = ââ. Then there exists a symbol a â SÂľ0
Ď,δ(U,E)
unique up to smoothing symbols, such that a â âkj=0 aj â SÂľkĎ,δ(U,E) for every
k â N. This induces a locally convex Hausdorff topology on the quotient vectorspace SâĎ,δ/S
ââ(U,E), which is called the topology of asymptotic convergence.
Proof. It is a well-known fact that the claim holds for U â Rd and E = Rd Ă RN(see [7, 12, 5]). Covering U by trivializations (zΚ, ΜΚ) : T âX|VΚ â R2d and ΨΚ :E|VΚ â VΚĂRN we can find a partition (ĎΚ) of unity subordinate to the covering (VΚ)
and for every index Κ a symbol bΚ,k â SÂľĎ,δ(VΚ,RN ) such that bΚ,k =kâj=0
ΨΚ ⌠aj |VΚ â
The Normal Symbol on Riemannian Manifolds 101
SÂľkĎ,δ(VΚ,RN ). Now define a =âΚ
(ĎΚ ⌠Ď) Ψâ1Κ ⌠bΚ,k and check that this a satisfies
the claim. Uniqueness of a up to smoothing symbols is clear again from a localconsideration. ďż˝
Polynomial symbols are not affected by smoothing symbols. The following propo-sition gives the precise statement.
Proposition 2.5. Let X be a smooth manifold, and D0 the sheaf of polynomialsymbols on X. Then by composition with the projection SâĎ,δ â SâĎ,δ/S
ââ thecanonical embedding
D0 â SâĎ,δD0(U) 3 f 7â f â SâĎ,δ(U), U â X open(4)
gives rise to a monomorphism δ : D0 â SâĎ,δ/Sââ of sheaves of algebras.
Proof. If f, g â D0(U) are polynomial symbols such that f â g â Sââ(U), thenf â g is a bounded polynomial function on each fiber of T âU . Therefore f â g isconstant on each fiber, hence f â g = 0 by f â g â Sââ(U). ďż˝
3. Pseudodifferential operators on manifolds
Let a be a polynomial symbol defined on the (trivial) cotangent bundle T âU =U Ă Rd of an open set U â Rd of Euclidean space. Then one can write a =âÎą (aÎą ⌠Ď)Μι, where Îś : T âU â Rd is the projection on the âcotangent vectorsâ
and the aÎą are smooth functions on U . According to standard results of partialdifferential equations one knows that the symbol a defines the differential operatorA = Op (a):
Câ(U) 3 f 7ââÎą
aÎą (âi)|Îą| â|Îą|
âzÎąf â Câ(U).(5)
Sometimes A is called the quantization of a. In case f â D(U), the Fourier trans-form of f is well-defined and provides the following integral representation for Af :
Af = Op (a) (f) =âŤRd
ei<Ξ,¡> a(Ξ) f(Ξ) dΞ.(6)
It would be very helpful for structural and calculational considerations to extendthis formula to Riemannian manifolds. To achieve this it is necessary to havean appropriate notion of Fourier transform on manifolds. In the following we aregoing to define such a Fourier transform and will later get back to the problem of anintegral representation for the âquantization mapâ Op on Riemannian manifolds.
Assume X to be Riemannian of dimension d and consider the exponential func-tion exp with respect to the Levi-Civita connection on X. Furthermore let E â Xbe a Riemannian or Hermitian vector bundle. Choose an open neighborhoodW â TX of the zero section in TX such that (%, exp ) : W â X Ă X mapsW diffeomorphically onto an open neighborhood of the diagonal â of XĂX. Thenthere exists a smooth function Ď : TX â [0, 1] called a cut-off function such thatĎ|W = 1 and suppĎ âW for an open neighborhood W âW of the zero section inTX. Next let us consider the unique torsionfree metric connection on E. It defines
102 Markus J. Pflaum
a parallel transport ĎÎł : EÎł(0) â EÎł(1) for every smooth path Îł : [0, 1] â X. Forevery v â TX denote by Ďexp v or just Ďy,x with y = exp v and x = %(v) the paralleltransport along [0, 1] 3 t 7â exp tv â X. Now the following microlocal lift MĎ iswell-defined:
MĎ : Câ(U,E)â Sââ(U, TX,E),
f 7â fĎ,={Ď(v) Ďâ1
exp v(f(exp v)) for v âW0 else.
(7)
MĎ is a linear but not multiplicative map between function spaces.Over the tangent bundle TX we can define the Fourier transform as the following
sheaf morphism:
F : Sââ(U, TX,E)â Sââ(U, T âX,E),
a 7â a(Ξ) =1
(2Ď)n/2
âŤTĎ(Ξ)X
eâi<Ξ,v> a(v) dv.(8)
We also have a reverse Fourier transform:
Fâ1 : Sââ(U, T âX,E)â Sââ(U, TX,E),
b 7â b(v) =1
(2Ď)n/2
âŤTâĎ(v)X
ei<Ξ,v> b(Ξ) dΞ.(9)
It is easy to check that Fâ1 and F are well-defined and inverse to each other.ComposingMĎ and F gives rise to the Fourier transform FĎ = F âŚMĎ on the
Riemannian manifold X. FĎ has the left inverse
Sââ(U, T âX,E)â Câ(U,E), a 7â Fâ1(a)|0U ,(10)
where |0U means the restriction to the natural embedding of U into T âX as zerosection.
By definition MĎ and FĎ do depend on the smooth cut-off function Ď, butthis arbitrariness will only have minor effects on the study of pseudodifferentialoperators defined through FĎ. We will get back to this point later on in thissection.
The exponential function exp gives rise to normal coordinates zx : Vx â Rd,where x â Vx â X, Vx open and
expâ1x (y) =
dâk=1
zkx(y) ¡ ek(y)(11)
for all y â Vx and an orthonormal frame (e1, ..., ed) of TVx. Furthermore we receivebundle coordinates (zx, Îśx) : T âVx â R2d and (zx, vx) : TVx â R2d. Note that forfixed x â X the map
Vx Ă T âVx 3 (y, Ξ) 7â (zy(Ď(Ξ)), Îśy(Ξ)) â R2d(12)
is smooth and that by the GauĂ Lemma
zkx(y) = âzky (x)(13)
for every y â Vx.
The Normal Symbol on Riemannian Manifolds 103
Now let a be a smooth function defined on T âU and polynomial in the fibers.Then locally a =
âÎą (ax,Îą ⌠Ď) Îśx with respect to a normal coordinate system at
x â U and functions ax,Îą â Câ(U). Define the operator A : D(U)â Câ(U) by
f 7â Af = OpĎ(a) (f) =
(U 3 x 7â 1
(2Ď)n/2
âŤTâxX
a(Ξ) fĎ(Ξ) dΞ â C)
(14)
and check that
Af (x) =âÎą
ax,Îą(x) (âi)|Îą| â|Îą|
âvxÎą[Ď (f ⌠exp )] (0x)(15)
=âÎą
ax,Îą(x) (âi)|Îą| â|Îą|
âzxÎąf (x)
for every x â U . Therefore Af is a differential operator independent of the choiceof Ď.
However, if a is an arbitrary element of the symbol space SâĎ,δ(U,Hom(E,F )),Eq. (14) still defines a continuous operator AĎ = OpĎ(a) : D(U,E) â Câ(U,F ),which is a pseudodifferential operator but independent of the cut-off function Ďonly up to smoothing operators. Let us show this in more detail for the scalar case,i.e., where E = F = X Ă C. The general case is proven analogously.
First choose Âľ â R such that a â SÂľĎ,δ(U), and consider the kernel distributionD(U Ă U) â C, (f â g) 7â < AĎf, g > =
âŤXg(x)AĎf(x) dx of AĎ. It can be
written in the following way:
< g,AĎf > =1
(2Ď)n
âŤX
âŤTâxX
âŤTxX
Ď(v) g(x) f(exp v) eâi<Ξ,v> a(Ξ) dv dΞ dx
(16)
=1
(2Ď)n
âŤX
âŤTxX
âŤTâxX
Ď(v) g(x) f(exp v) eâi<Ξ,v> a(Ξ) dΞ dv dx;
where the first integral is an iterated one, the second one an oscillatory integral. Tocheck that Eq. (16) is true use a density argument or in other words approximatethe symbol a by elements ak â Sââ(U) (in the topology of SÂľĎ,δ(U) with Âľ > Âľ)and prove the statement for the ak. The claim then follows from continuity of bothsides of Eq. (16) with respect to a. Let Ď : TX â [0, 1] be another cut-off functionsuch that Ď|O = 1 on a neighborhood W ⲠâW of the zero section of T âX. We canassume W Ⲡ= W and claim the operator AĎ â AĎ to be smoothing. For the proofit suffices to show that the oscillatory integral
KAĎâAĎ (v) =1
(2Ď)n
âŤTâĎ(v)X
(Ď(v)â Ď(v)) eâi<Ξ,v> a(Ξ) dΞ, v â TU,(17)
which gives an integral representation for the kernel of AĎ â AĎ defines a smoothfunction KAĎâAĎ â Câ(TU). The phase function T âĎ(v)X 3 Ξ 7â âΞ(v) â R hasonly critical points for v = 0. Hence for v 6= 0 there exists a (â1)-homogeneousvertical vector field L on T âX such that Leâi<¡,v> = eâi<¡,v>. The adjoint Lâ ofL satisfies
(Lâ )ka â SÂľâĎkĎ,δ (U), where k â N. As the amplitude
(v, Ξ) 7â (Ď(v)â Ď(v)) a(Ξ)
104 Markus J. Pflaum
vanishes on W ⲠĂU T âU , the equation
KAĎâAĎ (v) =1
(2Ď)n
âŤTâĎ(v)X
eâi<Ξ,v>(Lâ )ka(Ξ) dΞ(18)
holds for any integer k fulfilling Âľ + Ďk < âdimX. Note that the integral inEq. (18) unlike the one in Eq. (17) is to be understood in the sense of Lebesgue.Hence KAĎâAĎ â Câ(TU) and AĎ âAĎ is smoothing.
Let us now prove that AĎ lies in the space Ψ¾Ď,δ(U) of pseudodifferential operators
of order Âľ and type (Ď, δ) over U . We have to check that AĎ is pseudolocal andwith respect to some local coordinates looks like a pseudodifferential operator onan open set of Rd.
(i) AĎ is pseudolocal: Let u, v â D(U) and suppuâŠsuppv = â . Then the integralkernel K of Câ(U) 3 f 7â v AĎ(uf) â Câ(U) has the form
K(v) =1
(2Ď)n
âŤTâĎ(v)X
Ď(v) a(Ξ) v(Ď(v))u(exp v) eâi<Ξ,v> dΞ.(19)
There exists an open neighborhood W Ⲡâ W of the zero section in TX such thatthe amplitude Ď (v ⌠Ď) (u ⌠exp ) a vanishes on W ⲠĂU T âX. As for all nonzerov â TX the phase function Ξ 7â â < Ξ, v > is noncritical, an argument similar tothe one above for KAĎâAĎ shows that K is a smooth function on TU , so Câ(U) 3f 7â v AĎ(uf) â Câ(U) is smoothing.
(ii) AĎ is a pseudodifferential operator in appropriate coordinates: Choose forx â U an open neighborhood Ux â U so small that any two points in Ux can beconnected by a unique geodesic. In particular we assume that Ux Ă Ux lies in therange of (Ď, exp ) : W â X ĂX. Furthermore let us suppose that Ď âŚ zy(y) = 1 forall y, y â Ux. Then we have for f â D(Ux):
AĎf (y) =
=1
(2Ď)n/2
âŤTâyX
a(Ξ) fĎ(Ξ) dΞ
=1
(2Ď)n/2
âŤTâxX
a(T âzx(y) expâ1
x︸ ︡︡ ︸= dx(y)
(Ξ)) âŁâŁâŁ det dx(y)
âŁâŁâŁ F(f ⌠exp )(dx(y) (Ξ)
)dΞ
=1
(2Ď)n
âŤTâxX
âŤTyX
a(dx(y) (Ξ)
) âŁâŁâŁ det dx(y)âŁâŁâŁ (f ⌠exp y)(v) eâi<dx(y) (Ξ),v> dv dΞ
=1
(2Ď)n
âŤTâxX
âŤTxX
a(dx(y) (Ξ)
) âŁâŁâŁ det dx(y)âŁâŁâŁ âŁâŁâŁ det (T (zy ⌠zâ1
x )︸ ︡︡ ︸= Zy,x
(v)âŁâŁâŁ
(f ⌠zâ1x )(v) eâi<dx(y) (Ξ),zyâŚzâ1
x (v)> dv dΞ
(20)
Now the phase function (v, Ξ) 7â â< dx(y)(Ξ), zy ⌠zâ1x (v) > vanishes for Ξ 6= 0 if
and only if v = zx(y). Hence after possibly shrinking Ux the Kuranishi trick givesa smooth function Gx : Ux Ă TxX â Iso
(T âxX,T
âyX) â Lin
(T âxX,T
âyX)
such that< dx(y)(Ξ), zy ⌠zâ1
x (v) >=< Gx(y, v)(Ξ), v â zx(y) > for all (y, v) â UxĂTxX and
The Normal Symbol on Riemannian Manifolds 105
Ξ â T âxX. A change of variables in Eq. (20) then implies:
AĎf(y) =1
(2Ď)n
âŤTâxX
âŤTxX
a([dx(y)Gâ1
x (y, v)]
(Ξ)) âŁâŁâŁ det dx(y)
âŁâŁâŁ âŁâŁâŁZy,x(v)âŁâŁâŁ(21) âŁâŁâŁ detGâ1
x (y, v)âŁâŁâŁ (f ⌠zâ1
x )(v) eâi<Ξ,vâzx(y)> dv dΞ
=1
(2Ď)n
âŤTâxX
âŤTxX
ax(y, v, Ξ) (f ⌠zâ1x )(v) eâi<Ξ,vâzx(y)> dv dΞ,
where ax(y, v, Ξ) = a([dx(y)Gâ1
x (y, v)]
(Ξ)) âŁâŁâŁ det dx(y)
âŁâŁâŁ âŁâŁâŁZy,x(v)âŁâŁâŁ âŁâŁâŁ detGâ1
x (y, v)âŁâŁâŁ is
a symbol lying in SÂľĎ,δ(Ux Ă Ux Ă TxX), as Ď + δ ⼠1 and Ď > δ. Hence A is apseudodifferential operator with respect to normal coordinates over Ux of order Âľand type (Ď, δ).
Our considerations now lead us to
Theorem 3.1. Let X be Riemannian, E,F Riemannian or Hermitian vector bun-dles over X and exp the exponential function corresponding to the Levi-Civitaconnection on X. Then any smooth cut-off function Ď : TX â [0, 1] gives rise toa linear sheaf morphism OpĎ : SâĎ,δ(¡,Hom(E,F ))â ΨâĎ,δ(¡, E, F ), a 7â AĎ definedby
AĎf (x) =1
(2Ď)n/2
âŤTâxX
a(Ξ) fĎ(Ξ) dΞ,(22)
where a â SâĎ,δ(U,Hom(E,F )), f â D(U,E), x â U and U â X open. Thismorphism preserves the natural filtrations of SâĎ,δ(¡,Hom(E,F )) and ΨâĎ,δ(¡, E, F ).In particular it maps the subsheaf Sââ(¡,Hom(E,F )) of smoothing symbols to thesubsheaf Ψââ(¡,Hom(E,F )) â ΨâĎ,δ(¡,Hom(E,F )) of smoothing pseudodifferentialoperators.
The quotient morphism Op : (SâĎ,δ/Sââ)(¡,Hom(E,F )) â (ΨâĎ,δ/Ψ
ââ)(¡, E, F )is an isomorphism and independent of the choice of the cut-off function Ď.
Proof. The first part of the theorem has been shown above. So it remains toprove that Op : (SâĎ,δ/S
ââ)(U,Hom(E,F ))â (ΨâĎ,δ/Ψââ)(U,E, F ) is bijective for
all open U â X. Let us postpone this till Theorem 4.2, where we will show thatan explicit inverse of Op is given by the complete symbol map introduced in thefollowing section. ďż˝
By the above Theorem the effect of Ď for the operator OpĎ is only a minor one,so from now on we will write Op instead of OpĎ.
4. The symbol map
In the sequel denote by Ď : X Ă T âX|O â C the smooth phase function definedby
X Ă T âX|O 3 (x, Ξ) 7â< Ξ, expâ1Ď(Ξ)(x) >=< Ξ, zĎ(Ξ)(x) > â C,(23)
where O is the range of (%, exp) : W â X ĂX.
Theorem and Definition 4.1. Let A â Ψ¾Ď,δ(U,E, F ) be a pseudodifferential op-
erator on a Riemannian manifold X and Ď : T âX â [0, 1] a cut-off function. Then
106 Markus J. Pflaum
the Ď-cut symbol of A with respect to the Levi-Civita connection on X is the section
ĎĎ(A) = ĎĎ,A : T âU â Ďâ(Hom(E,F )),
Ξ 7â[Î 7â A
(ĎĎ(Ξ)(¡) eiĎ(¡,Ξ) Ď(¡),Ď(Ξ)Î
)](Ď(Ξ)),
(24)
where for every x â X Ďx = ĎâŚexpâ1x = ĎâŚzx. ĎĎ(A) is an element of SÂľĎ,δ(U,X).
The corresponding element Ď(A) in the quotient (SÂľĎ,δ/Sââ)(U,Hom(E,F )) is called
the normal symbol of A. It is independent of the choice of Ď.
Proof. Let us first check that ĎĎ,A lies in SÂľĎ,δ(U,Hom(E,F )). It suffices to as-sume that E and F are trivial bundles, hence that A is a scalar pseudodifferentialoperator. We can find a sequence (xΚ)ΚâN of points in U and coordinate patchesUΚ â U with xΚ â UΚ such that there exist normal coordinates zΚ = zxΚ : UΚ â TxΚXon the UΚ. We can even assume that the operator AΚ : D(UΚ)â Câ(UΚ), u 7â Au|UΚinduces a pseudodifferential operator on TxΚX. In other words there exist symbolsaΚ â SÂľĎ,δ(OΚ Ă OΚ, T âxΚX) with OΚ = zΚ(UΚ) such that AΚ is given by the followingoscillatory integral:
AΚu(y) =1
(2Ď)n
âŤTâxΚX
âŤTxΚX
aΚ(zΚ(y), v, Ξ) (u ⌠zâ1Κ )(v) eâi<Ξ,vâzΚ(y)> dvdΞ.(25)
Now let (ĎΚ) be a partition of unity subordinate to the covering (UΚ) of U and forevery index Κ let ĎΚ,1, ĎΚ,2 â Câ(U) such that suppĎΚ,1 â UΚ, suppĎΚ⊠suppĎΚ,2 = â and ĎΚ,1 + ĎΚ,2 = 1. Then the symbol ĎĎ,A can be written in the following form:
ĎĎ,A(Îś) =[A(ĎĎ(Îś)(¡) eiĎ(¡,Îś)
)](Ď(Îś))
=1
(2Ď)nâΚ
[ĎΚ,1AΚ
(ĎΚ(¡)ĎĎ(Îś)(¡) eiĎ(¡,Îś)
)](Ď(Îś))
+ K(ĎĎ(Îś)(¡) eiĎ(¡,Îś)
)(Ď(Îś)),
(26)
where
K : D(U)â Câ(U) f 7ââΚ
ĎΚ,2AΚ(ĎΚ f), Îś â T âXΚ(27)
is a smoothing pseudodifferential operator. By the Kuranishi trick we can assumethat there exists a smooth function G : OΚ Ă OΚ â Lin
(T âxΚX,T
âyX)
such thatG(v, w) is invertible for every v, w â OΚ and
< G(zΚ(y), T (zΚ ⌠zâ1
y )(v))
(Ξ), v >=< Ξ, zΚ ⌠zâ1y (v)â zΚ(y) >(28)
The Normal Symbol on Riemannian Manifolds 107
for y â UΚ, v â TyX appropriate and Ξ â T âxΚX. Several changes of variables thengive the following chain of oscillatory integrals for Îś â T âUΚ:
[AΚ
(ĎΚ(¡)ĎĎ(Îś)(¡) eiĎ(¡,Îś)
)](y)
=1
(2Ď)n
âŤTâxΚX
âŤTxΚX
ĎΚ,1(y) aΚ(zâ1Κ (y), v, Ξ
)ĎΚ(zâ1
Κ (v))Ď(zy ⌠zâ1x (v))
ei<Îś,zyâŚzâ1Κ (v)> eâi<Ξ,vâzΚ(y)> dv dΞ
=1
(2Ď)n
âŤTâxΚX
âŤTyX
ĎΚ,1(y) aΚ(zâ1Κ (y), zΚ ⌠zâ1
y (v), Ξ)ĎΚ(zâ1
y (v))Ď(v)
ei<Îś,v> eâi<Ξ,zΚâŚzâ1y (v)âzΚ(y)> dv dΞ
=1
(2Ď)n
âŤTâxΚX
âŤTyX
ĎΚ,1(y) aΚ(zâ1Κ (y), zΚ ⌠zâ1
y (v), Ξ)ĎΚ(zâ1
y (v))Ď(v)
ei<Îś,v> eâi<G(zΚ(y),T (zΚâŚzâ1y )(v))(Ξ),v> dv dΞ
=1
(2Ď)n
âŤTâyX
âŤTyX
ĎΚ,1(y) aΚ(zâ1Κ (y), zΚ ⌠zâ1
y (v), Gâ1(zΚ(y), T (zΚ ⌠zâ1
y )(v))
(Ξ))
ĎΚ(zâ1y (v))Ď(v)
âŁâŁâŁ detGâ1(zΚ(y), T (zΚ ⌠zâ1
y )(v))âŁâŁâŁ ei<Îś,v> eâi<Ξ,v> dv dΞ
=1
(2Ď)n
âŤTâxΚX
âŤTxΚX
bΚ(zΚ(y), v, Ξ) eâi<ΞâTâ(zyâŚzâ1
Κ )(Îś),vâzΚ(y)> dv dΞ,
(29)
where the smooth function bΚ on OΚ ĂOΚ Ă T âxΚX is defined by
bΚ(zΚ(y), v, Ξ) =
= ĎΚ,1(y) aΚ(zâ1Κ (y), zΚ ⌠zâ1
y ⌠T (zy ⌠zâ1Κ ) (v), Gâ1
(zΚ(y), v
)⌠(T â(zΚ ⌠zâ1
y
)(Ξ))
ĎΚ(zâ1y (T (zy ⌠zâ1
Κ ) (v)))Ď(T (zy ⌠zâ1Κ ) (v))
âŁâŁâŁ detGâ1 (zΚ(y), v)âŁâŁâŁ
(30)
and lies in SÂľĎ,δ(OΚ Ă OΚ, T âxΚX). Let BΚ â Ψ¾Ď,δ(OΚ, TxΚX) be the pseudodifferen-
tial operator on OΚ corresponding to the symbol bΚ. The symbol ĎΚ defined byĎΚ(v, Ξ) = eâi<Ξ,v>BΚ
(ei<Ξ,¡>
), where v â OΚ and Ξ â T âxΚX, now is an element of
SÂľĎ,δ(OΚ, TâxΚX) as one knows by the general theory of pseudodifferential operators
on real vector spaces (see [7, 12, 5]). But we have[ĎΚ,1AΚ
(ĎΚ(¡)ĎĎ(Îś)(¡) eiĎ(¡,Îś)
)](Ď(Îś)) = ĎΚ
(zΚ(Ď(Îś)), T â
(zĎ(Îś) ⌠zâ1
Κ
)(Îś)),(31)
hence Eq. (26) shows ĎĎ,A â SÂľĎ,δ(U,X) if we can prove that ĎĎ,K â Sââ(U,X) for
ĎĎ,K(Îś) = K(ĎĎ(Îś)(¡) eiĎ(¡,Îś)
)(Ď(Îś)). So let k : U Ă U â C be the kernel function
108 Markus J. Pflaum
of K and write ĎĎ,K as an integral:
ĎĎ,K(Îś) =âŤU
k(Ď(Îś), y)ĎĎ(Îś)(y) eâiĎ(y,Îś) dy.(32)
Let further Î1, ...,Îk be differential forms over U and V1, ...,Vk the vertical vectorfields on T âU corresponding to Î1, ...,Îk. Differentiating Eq. (32) under the integralsign and using an adjointness relation then gives
|Îś|2m V1 ¡ ... ¡ Vk ĎĎ,K (Îś) =
=âŤU
k(Ď(Îś), y)ĎĎ(Îś)(y) < Î1, zĎ(Îś)(y) > ... < Îk, zĎ(Îś)(y) >
i2mn+k â2mn
âz(2m,...,2m)Ď(Îś)
eâi<Îś,zĎ(Îś)(¡)>(y) dy
=âŤU
â2mn
âz(2m,...,2m)Ď(Îś)
â (k(Ď(Îś), ¡)ĎĎ(Îś)(¡) < Î1, zĎ(Îś)(¡) > ... < Îk, zĎ(Îś)(¡) >)
(y)
i2mn+keâi<Îś,zĎ(Îś)(y)> dy.
(33)
Therefore |Îś|2m V1 ¡ ... ¡ Vk ĎĎ,K (Îś) is uniformly bounded as long as Ď(Îś) varies in acompact subset of U . A similar argument for horizontal derivatives of ĎĎ,K finallyproves ĎĎ,K â Sââ(U,X).
Next we have to show that ĎĎ,A â ĎĎ,A is a smoothing symbol for any secondcut-off function Ď : T âX â [0, 1]. It suffices to prove that ĎΚ with
ĎΚ(Îś) = AΚ
(ĎΚ (Ď â Ď) ⌠zĎ(Îś) e
âiĎ(¡,Îś))
(34)
is smoothing for every ĎΚ â D(UΚ). We can find a (â1)-homogeneous first orderdifferential operator L on the vector bundle T âxΚXĂTxΚX â TxΚX such that for allv 6= zΚ(Ď(Îś)) and Ξ â T âxΚX the equation
Leâi<¡,vâzΚ(Ď(Îś))> (Ξ) = eâi<Ξ,vâzΚ(Ď(Îś))>(35)
holds (see for example Grigis, Sjostrand [5] Lemma 1.12 for a proof). Because(ĎâĎ)âŚzĎ(Îś)âŚzâ1
Κ vanishes on a neighborhood of zΚ(Ď(Îś)), the following equational
The Normal Symbol on Riemannian Manifolds 109
chain of iterated integrals is also true:
ĎΚ(Îś) =
=1
(2Ď)n
âŤTâxΚX
âŤTxΚX
aΚ(zâ1Κ (y), v, Ξ
)ĎΚ(zâ1
Κ (v))(Ď â Ď
)(zĎ(Îś) ⌠zâ1
x (v))
ei<Îś,zĎ(Îś)âŚzâ1Κ (v)> eâi<Ξ,vâzΚ(Ď(Îś))> dv dΞ
=1
(2Ď)n
âŤTâxΚX
âŤTxΚX
aΚ(zâ1Κ (y), v, Ξ
)ĎΚ(zâ1
Κ (v))(Ď â Ď
)(zĎ(Îś) ⌠zâ1
x (v))
ei<Îś,zĎ(Îś)âŚzâ1Κ (v)> Lk eâi<Ξ,vâzΚ(Ď(Îś))> dv dΞ
=1
(2Ď)n
âŤTâxΚX
âŤTxΚX
((Lâ )kaΚ
)(zâ1Κ (y), v, Ξ
)ĎΚ(zâ1
Κ (v))(Ď â Ď
)(zĎ(Îś) ⌠zâ1
x (v))
ei<Îś,zĎ(Îś)âŚzâ1Κ (v)> eâi<Ξ,vâzΚ(Ď(Îś))> dv dΞ.
(36)
Note that the last integral of this chain is to be understood in the sense of Lebesgueif k â N is large enough. The same argument which was used for proving that thesymbol ĎĎ,K from Eq. (32) is smoothing now shows ĎΚ â Sââ(UΚ, X). ďż˝
After having defined the notion of a symbol, we will now give its essential prop-erties in the following theorem.
Theorem 4.2. Let A â Ψ¾Ď,δ(U,E, F ) be a pseudodifferential operator on a Rie-
mannian manifold X and a = ĎĎ,A â SÂľĎ,δ(U,Hom(E,F )) be its Ď-cut symbol withrespect to the Levi-Civita connection. Then A and the pseudodifferential opera-tor OpĎ(a) defined by Eq. (22) coincide modulo smoothing operators. Moreoverthe sheaf morphisms Ď : (ΨâĎ,δ/Ψ
ââ)(¡, E, F ) â (SâĎ,δ/Sââ)(¡,Hom(E,F )) and
Op : (SâĎ,δ/Sââ)(¡,Hom(E,F ))â (ΨâĎ,δ/Ψ
ââ)(¡, E, F ) are inverse to each other.
Before proving the theorem let us first state a lemma.
Lemma 4.3. The operator
K : D(U,E)â Câ(U,F ), f 7â(U 3 x 7â A[(1â Ďx)f ] (x) â F
)(37)
is smoothing for any cut-off function Ď : TX â C.
Proof. For simplicity we assume that E = F = X Ă C. The general case followsanalogously. Consider the operators Ka,Κ : D(U) â Câ(U) where Ka,Κf (x) =ĎΚ(x)A[(1 â Ďa)f ](x) for x â U , a â UΚ. The open covering (UΚ) of U and asubordinate partition of unity (ĎΚ) are chosen such that for every point a â UΚthere exist normal coordinates za on UΚ. We can even assume that Ď(za(y)) = 1 forall a, y â UΚ. Hence supp (1â Ďa)⊠supp ĎΚ = â , and Ka,Κ is smoothing, because Ais pseudolocal. This yields a family of smooth functions kΚ : U Ă U â C such that
Ka,Κf (x) =âŤU
kΚ(x, y) (1â Ďa(y)) f(y) dy.(38)
Now the smooth function k â Câ(U Ă U) with k(x, y) =âΚ kΚ(x, y) (1 â Ďx(y))
is the kernel function of K. ďż˝
110 Markus J. Pflaum
Proof of Theorem 4.2. Check that for an arbitrary cut-off function Ď : TX â Cone can find a cut-off function Ď : TX â C such that the equation
Ďx(y) f(y) =1
(2Ď)n/2
âŤTâxX
Ďx(y) ei<Ξ,zx(y)> f Ď(Ξ) dΞ(39)
holds for x, y â U . Approximating the integral in Eq. (39) by Riemann sums, wecan find a sequence (ik)kâN of natural numbers, a family (Ξk,Κ)kâN,1â¤Îšâ¤Îšk of elementsΞk,Κ â T âxX and a sequence (Îľk)kâN of positive numbers such that
limkââ
Îľk = 0,
[(Ďx)f ] (y) = limkââ
Îľnk(2Ď)n/2
â1â¤iâ¤ik
Ďx(y) ei<Ξk,Κ , zx(y)> f Ď(Ξk,Κ).(40)
By an appropriate choice of the Ξk,Κ one can even assume that the series on the righthand side of the last equation converges as a function of y â U in the topology ofD(U). As A is continuous on D(U), we now have
A[(Ďx)f ](x) = limkââ
Îľnk(2Ď)n/2
â1â¤iâ¤ik
A[Ďx(¡) ei<Ξk,Κ , zx(¡)>
](x) f Ď(Ξk,Κ)(41)
= limkââ
Îľnk(2Ď)n/2
â1â¤iâ¤ik
ĎA(Ξk,Κ) f Ď(Ξk,Κ)
=1
(2Ď)n/2
âŤTâxX
ĎA(Ξ) f Ď(Ξ) dΞ
= AĎf.
On the other hand Lemma 4.3 provides a smoothing operator K such that forf â D(U) and x â U :
Af (x) = A[(Ďx)f ] (x) +Kf (x) = AĎf (x) +Kf (x).(42)
Thus the first part of the theorem follows.Now let a â SÂľĎ,δ(U,Hom(E,F )) be a symbol, and consider the corresponding
pseudodifferential operator A = AĎ â Ψ¾Ď,δ(U,E, F ). After possibly altering Ď we
can assume that there exists a second cut-off function Ď : T âX â C such thatĎ|supp Ď = 1. Then we have for Îś â T âxX and Î â Ex
ĎA,Ď(Îś)Î =1
(2Ď)n/2
âŤTâxX
a(Ξ)F((
Ďx(¡)eiĎ(¡,Îś)Ďx,(¡)Î)Ď)
(Ξ) dΞ(43)
=1
(2Ď)n/2
âŤTâxX
a(Ξ)F(Ďx(¡)eiĎ(¡,Îś)Î
)(Ξ) dΞ
=1
(2Ď)n
âŤTâxX
âŤTxX
a(Ξ)ÎĎ(v) eâi<Ξ,v> ei<Îś,v> dv dΞ,
where the last integral is an iterated one. Next choose a smooth function Ď : Râ[0, 1] such that suppĎ â [â2, 2] and Ď|[â1,1] = 1 and define ak â Sââ(U,Hom(E,F ))
The Normal Symbol on Riemannian Manifolds 111
by ak(Ξ) = Ď(|Ξ|k
). Then obviously ak â a in SÂľĎ,δ(UHom(E,F ), ) for Âľ > Âľ. We
can now write:
ĎA,Ď(Îś)Î = limkââ
1(2Ď)n
âŤTâxX
âŤTxX
ak(Ξ)ÎĎ(v) eâi<ΞâÎś,v> dv dΞ
(44)
= limkââ
ak(Îś)Îâ 1(2Ď)n/2
âŤTxX
(1â Ď(v))[Fâ1ak
](v) ei<Îś,v> dv
= a(Îś)â limkââ
1(2Ď)n/2
âŤTxX
(1â Ď(v))[Fâ1
((Lâ )m
ak
)](v) ei<Îś,v> dv
= a(Îś) + limkââ
Km(Îś, ak),
where m â N, L is a smooth (â1)-homogeneous first order differential operator onthe vector bundle T âX ĂU TU â TU such that Lei<¡,v> (Ξ) = ei<Ξ,v> and
Km(Îś, ak) = â 1(2Ď)n/2
âŤTxX
(1â Ď(v))[Fâ1
((Lâ )m
ak
)](âv) ei<Îś,v> dv.(45)
Note that TU consists of all nonzero vectors v â TU and that(Lâ )m
a â SÂľâĎmĎ,δ
for every Âľ ⼠¾. Now fixing Âľ ⼠¾ and choosing m large enough that Ďm >Âľ+ 2(dimX â 1) yields
Km(Îś, a) =: limkââ
Km(Îś, ak)
(46)
= limkââ
â 1(2Ď)n
âŤTxX
âŤTâxX
((Lâ )m
ak
)(Ξ) (1â Ď(v)) eâi<ΞâÎś,v> dΞ dv
= â 1(2Ď)n
âŤTxX
âŤTâxX
((Lâ )m
a)
(Ξ) (1â Ď(v)) ei<Îś,v> eâi<Ξ,v> dΞ dv
in the sense of Lebesgue integrals. As the phase function Ξ 7â â < Ξ, v > isnoncritical for every nonzero v and 1 â Ď(v) vanishes in an open neighborhood ofthe zero section of TU , a standard consideration already carried out in preceedingproofs entails that Km(¡, a) is a smoothing symbol. Henceforth ĎA,Ď and a differby the smoothing symbol Km(¡, a). This gives the second part of the theorem. ďż˝
In the following let us show that the normal symbol calculus gives rise to a naturaltransformation between the functor of pseudodifferential operators and the functorof symbols on the category of Riemannian manifolds and isometric embeddings.
First we have to define these two functors. Associate to any Riemannian manifoldX the algebra A(X) = ΨâĎ,δ/Ψ
ââ(X) of pseudodifferential operators on X modulosmoothing ones, and the space S(X) = SâĎ,δ/S
ââ(X) of symbols on X modulosmoothing ones. As Y is Riemannian, we have for any isometric embedding f : X âY a natural embedding fâ : T âX â T âY . So we can define S(f) : S(Y ) â S(X)to be the pull-back fâ of smooth functions on T âY via fâ. The morphism A(f) :A(Y )â A(X) is constructed by the following procedure. If f is submersive, i.e., a
112 Markus J. Pflaum
diffeomorphism onto its image, A(f) is just the pull-back of a pseudodifferentialoperator on Y to X via f . So we now assume that f is not submersive. Choosean open tubular neighborhood U of f(X) in Y and a smooth cut-off functionĎ : U â [0, 1] with compact support and Ď|V = 1 on a neighborhood V â Uof f(X). As U is tubular there is a canonical projection Î : U â X such thatΠ⌠f = idX . Now let the pseudodifferential operator P represent an element ofA(Y ). We then define A(f)(P ) to be the equivalence class of the pseudodifferentialoperator
E Ⲡ3 u 7â fâ [P (Ď(Î âu))] â E â˛,(47)
where E Ⲡdenotes distributions with compact support. Note that modulo smoothingoperators fâ [P (Ď(Î âu))] does not depend on the choice of Ď. With these definitionsA and S obviously form contravariant functors from the category of Riemannianmanifolds with isometric embeddings as morphisms to the category of complexvector spaces and linear mappings.
The following proposition now holds.
Proposition 4.4. The symbol map Ď forms a natural transformation from thefunctor A to the functor S.
Proof. As f is isometric, the relation
f ⌠exp = exp âŚTf(48)
is true. But then the phasefunctions on X and Y are related by
ĎY (y, fâ(Ξ)) = ĎX(Î (y), Ξ),(49)
where y â Y and Ξ â T âX with y and f(Ď(Ξ)) sufficiently close. Hence for apseudodifferential operator P on Y
(fâĎY (P )) (Ξ) =[P(Ďf(Ď(Ξ))(¡) eiĎY (¡,fâΞ)
)](f(Ď(Ξ)))(50)
and
(ĎXfâ(P )) (Ξ) =[P(Ďf(Ď(Ξ))(¡)Ď(¡) eiĎX(Î (¡),Ξ)
)](Ď(Ξ)).(51)
Eq. (49) now implies
fâĎY (P ) = ĎXfâ(P )(52)
which gives the claim. ďż˝
5. The symbol of the adjoint
In the sequel we want to derive an expression for the normal symbol ĎAâ of theadjoint of a pseudodifferential operator A in terms of the symbol ĎA.
Let E,F be Hermitian (Riemannian) vector bundles over the Riemannian man-ifold 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 theinduced metric connection on Hom(E,F ), resp. Ďâ(Hom(E,F )). Next define thefunction Ď : O â R+ by
Ď(x, y) ¡ νx = ν(y) ⌠(Tv expxà ¡ ¡ ¡ Ă Tv expx), x, y â O, v = expâ1x (y),(53)
The Normal Symbol on Riemannian Manifolds 113
where ν is the canonical volume density on TxX and νx its restriction to a volumedensity on TxX. Now we claim that the operator Opâ(a) : D(U,F ) â Câ(U,E)defined by the iterated integral
Opâ(a)g(x) =1
(2Ď)n
âŤTâxX
âŤTxX
eâi<Ξ,v> Ďâ1exp v
[aâ(T âv expâ1
x (Ξ))g(exp v)
]Ď(v)
Ďâ1(expx(v), x) dv dΞ
(54)
is the (formal) adjoint of Op (a), where a â SÂľĎ,δ(U,Hom(E,F )), U â X open and Ď
is the cut-off function TX 3 v 7â Ď(
expâ1expx v
(x))â [0, 1]. Assuming f â D(V,E),
g â D(U,F ) with V â U open, V Ă V â O and Âľ to be sufficiently small thefollowing chain of (proper) integrals holds:
âŤX
< f(x),Opâ(a)g (x) >E dν(x) =
=1 1(2Ď)n
âŤX
âŤTxXĂTâxX
ei<Ξ,v> < f(x), Ďexp v
[aâ(T âv expâ1
x (Ξ))g(exp v)
]>
Ď(v) Ďâ1(expx(v), x) dĎnx (v, Ξ)dν(x)
=2 1(2Ď)n
âŤX
âŤTâV
ei<Tâ expx(Ξ),expâ1
x (Ď(Ξ))> < ĎĎ(Ξ),xf(x), aâ(Ξ) g(Ď(Ξ)) >
Ď(expâ1Ď(Ξ)(x)) Ďâ1(Ď(Ξ), x) dĎn(Ξ) dν(x)
=3 1(2Ď)n
âŤTâV
âŤX
eâi<Ξ,expâ1
Ď(Ξ)(x)>< ĎĎ(Ξ),xf(x), aâ(Ξ) g(Ď(Ξ)) >
Ď(expâ1Ď(Ξ)(x)) Ďâ1(Ď(Ξ), x) dν(x) dĎn(Ξ)
=4 1(2Ď)n
âŤTâV
âŤTĎ(Ξ)X
eâi<Ξ,w> < a(Ξ) Ďâ1expwf(expw), g(x) > Ď(w) dw dĎn(Ξ)
=1
(2Ď)n
âŤX
âŤTxXĂTâxX
eâi<Ξ,w> < a(Ξ) Ďâ1expwf(expw), g(x) > Ď(w) dw dΞ dν(x)
=âŤX
< Op (a)f(x), g(x) >F dν(x)
(55)
In explanation of the above equalities, we have
1. Ďx, Ď canonical symplectic forms on TxX Ă T âxX, resp. T âX2. by T â expx : T âV â TxX Ă T âxX symplectic, x â V3. GauĂ Lemma and Fubiniâs Theorem4. coordinate transformation V 3 x 7â w = expâ1
Ď(Ξ)(x) â TĎ(Ξ)X.
Recall that Sââ(U,Hom(E,F )) is dense in any SÂľĎ,δ(U,Hom(E,F )), Âľ â R withrespect to the Frechet topology of SÂľĎ,δ(U,Hom(E,F )) if Âľ < Âľ. By using a partition
114 Markus J. Pflaum
of unity and the continuity of SÂľĎ,δ(U,Hom(E,F )) 3 a 7â Op (a)f â Câ(U,F ),resp. SÂľĎ,δ(U,Hom(E,F )) 3 a 7â Opâ(a)g â Câ(U,E), for any f â D(U,E) andg â D(U,F ), Eq. (55) now implies thatâŤ
X
< f(x),Opâ(a)g(x) >E dν(x) =âŤX
< Op (a)f(x), g(x) >F dν(x)(56)
holds for every f â D(U,E), g â D(U,F ) and a â SÂľĎ,δ(U) with Âľ â R. ThusOpâ(a) is the formal adjoint of Op (a) indeed.
The normal symbol ĎAâ of Aâ = Opâ(a) now is given by
ĎAâ(Ξ) Î âź [Aâg(¡, Ξ,Î)] (Ď(Ξ)) =1
(2Ď)n
âŤTxX
âŤTâxX
eâi<ÎśâΞ,v>
Ďâ1exp v
[aâ(T âv expâ1
x (Îś))Ďexp vÎ
]Ď(v)Ď(v) Ďâ1(expx(v), x) dv dÎś,
(57)
where Ξ â T âX, x = Ď(Ξ), Î â Fx and
g(y, Ξ,Î) = Ď(
expâ1Ď(Ξ)(y)
)ei<Ξ,expâ1
Ď(Ξ)(y)>Ďy,Ď(Ξ)Î(58)
with y â X. Thus by Theorem 3.4 of Grigis, Sjostrand [5] the following asymptoticexpansion holds:
ĎAâ (Ξ) âźâÎąâNd
(âi)|Îą|Îą!
â|Îą|
âvÎą
âŁâŁâŁâŁv=0
â|Îą|
âΜι
âŁâŁâŁâŁÎś=Ξ( [
Ďâ1exp v ⌠aâ
(T âv expâ1
x (Îś)) ⌠Ďexp v
]Ďâ1(expx(v), x)
)=
âÎą,βâNd
(âi)|Îą+β|
ι!β!
D|Îą|+|β|âzβx âΜιx
âŁâŁâŁâŁâŁÎž
aâ
[ â|Îą|
âzxÎą
âŁâŁâŁâŁx
Ďâ1(¡, x)],
(59)
where
D|ι|+|β|
âzβx âΜιx
âŁâŁâŁâŁâŁÎž
aâ =â|β|
âvβa
âŁâŁâŁâŁv=0
â|Îą|
âΜιa
âŁâŁâŁâŁÎž
[Ďâ1exp v ⌠a
(T âv expâ1
x (Îś)) ⌠Ďexp v
],(60)
are symmetrized covariant derivatives of a â Sâ(X,Hom(E,F )) with respect tothe apriorily chosen normal coordinates (zx, Îśx).
Next we will search for an expansion of Ď and its derivatives. For that first write
â
âzkx=âk,l
θlk(x, ¡) el,(61)
where (e1, ..., ed) is the orthonormal frame of Eq. (11), and the θlk are smoothfunctions on an open neighborhood of the diagonal of X à X. Then we have thefollowing relation (see for example Berline, Getzler, Vergne [1], p. 36):
θkl (x, y) = δkl â16
âm,n
Rkmln(y) zmx (y) znx (y) +O(|zx(y)|3),(62)
The Normal Symbol on Riemannian Manifolds 115
where Rkmln(y), resp. their âloweredâ versions Rkmln(y), are given by the co-efficients of the curvature tensor with respect to normal coordinates at y â X,i.e., R
(â
âzmy
)=âk,l,n
Rkmln(y) ââzkyâ dzly â dzny . But this implies
Ď(x, y) =âŁâŁdet
(θlk(x, y)
)âŁâŁ= 1â 1
6
âk
âm,n
Rkmkn(x) zmx (y) znx (y) +O(|zx(y)|3
= 1â 16
âm,n
Ricmn(x) zmx (y) znx (y) +O(|zx(y)|3) .
(63)
Using Eq. (13) zkx(y) = âzky (x) for x and y close enough we now get
â
âzkx
âŁâŁâŁâŁy
Ďâ1(¡, x) =13
âl
Rickl(x) zlx(y) +O(|zx(y)|2) ,(64)
â2
âzkxâzlx
âŁâŁâŁâŁy
Ďâ1(¡, x) =13
Rickl(x) +O (|zx(y)|) .(65)
The above results are summarized by the following theorem.
Theorem 5.1. The formal adjoint Aâ â Ψ¾Ď,δ(U,F,E) of a pseudodifferential oper-
ator A â Ψ¾Ď,δ(U,E, F ) between Hermitian (Riemannian) vector bundles E,F over
an open subset U of a Riemannian manifold X is given by
Aâg (x) = Opâ(a)g (x) =1
(2Ď)n
âŤTâxX
âŤTxX
eâi<Ξ,v>
Ďâ1exp v
[aâ(T âv expâ1
x (Ξ))g(exp v)
]Ď(v) Ďâ1(expx(v), x) dv dΞ
where x â X, g â D(U,F ) and a = ĎA is the symbol of A. The symbol ĎAâ of Aâ
has asymptotic expansion
ĎAâ(Ξ) âź aâ(Ξ) +âÎąâNd|Îą|âĽ1
D2|Îą|
âzÎąĎ(Ξ) âΜιĎ(Ξ)
âŁâŁâŁâŁâŁÎž
aâ â
â 16
âÎąâNd
âk,l
(âi)|Îą| D2|Îą|+2
âzÎąĎ(Ξ) âΜιĎ(Ξ) âÎś
kĎ(Ξ) âÎś
lĎ(Ξ)
âŁâŁâŁâŁâŁÎž
aâ
Rickl(Ď(Ξ)) +
+âÎą,βâNd|β|âĽ3
(âi)|Îą+ β|
ι!β!
D2|ι|+|β|
âzÎąĎ(Ξ) âΜι+βĎ(Ξ)
âŁâŁâŁâŁâŁÎž
aâ
[ â|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁĎ(Ξ)
Ďâ1(¡, x)
]
= aâ(Ξ) â iâk
D2
âzkĎ(Ξ) âÎśkĎ(Ξ)
âŁâŁâŁâŁâŁÎž
aâ â 12
âk,l
D4
âzkĎ(Ξ) âzlĎ(Ξ) âÎś
kĎ(Ξ) âÎś
lĎ(Ξ)
âŁâŁâŁâŁâŁÎž
aâ â
â 16
âk,l
D2
âÎśkĎ(Ξ)âÎślĎ(Ξ)
âŁâŁâŁâŁâŁÎž
aâ
Rickl(Ď(Ξ)) + r(Ξ)
(66)
116 Markus J. Pflaum
with Ξ â T âX and r â SÂľâ2(Ďâδ)Ď,δ (U,Hom(E,F )).
6. Product expansions
Most considerations of elliptic partial differential operators as well as applicationsof our normal symbol to quantization theory (see Pflaum [8]) require an expressionfor the product of two pseudodifferential operators A,B in terms of the symbolsof its components. In the flat case on Rd it is a well known fact that ĎAB has anasymptotic expansion of the form
ĎAB(x, Ξ) âźâÎą
(âi)|Îą|Îą!
â|Îą|
âΜιĎA(x, Ξ)
â|Îą|
âxÎąĎB(x, Ξ).(67)
Note that strictly speaking, the product AB of pseudodifferential operators is onlywell-defined, if at least one of them is properly supported. But this is only aminor set-back, as any pseudodifferential operator is properly supported modulo asmoothing operator and the above asymptotic expansion also describes a symbolonly up to smoothing symbols.
In the following we want to derive a similar but more complicated formula forthe case of pseudodifferential operators on manifolds.
Proposition 6.1. Let A â Ψ¾Ď,δ(U,E, F ) be a pseudodifferential operator between
Hermitian (Riemannian) vector bundles E,F over a Riemannian manifold X anda = Ď(A) its symbol. Then for any f â Câ(U,E) the function
ĎA,f : T âU â F, Ξ 7â[A(ĎĎ(Ξ)f e
iĎ(¡,Ξ))]
(Ď(Ξ))(68)
is a symbol of order Âľ and type (Ď, δ) and has an asymptotic expansion of the form
ĎA,f (Ξ) âźâÎą
1i|Îą|Îą!
[D|Îą|
âÎśĎ(Ξ)Îą
âŁâŁâŁâŁÎž
a
] [D|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁĎ(Ξ)
f
].(69)
Proof. Let x = Ď(Ξ), N â N and Ď : T âX â C a cut-off function. Then thefollowing chain of equations holds:
ĎA,f (Ξ) =[A(ĎĎ(Ξ)f e
iĎ(¡,Ξ))]
(x)
(70)
=1
(2Ď)n/2
âŤTâxX
a(Îś)[F (fĎ ei<Ξ,¡>)](Îś) dÎś
=1
(2Ď)n
âŤTâxX
âŤTxX
ei<ΞâÎś,v> a(Îś) Ďâ1exp vf(exp v)Ď(v) dv dÎś
=1
(2Ď)n
âŤTâxX
âŤTxX
eâi<Îś,v> a(Ξ + Îś) Ďâ1exp vf(exp v)Ď(v) dv dÎś
=1
(2Ď)nâ|Îą|â¤N
1Îą!
âŤTâxX
âŤTxX
eâi<Îś,v>D|Îą|
âÎśxÎą a(Ξ) Μι Ďâ1
exp vf(exp v)Ď(v) dv dÎś
+ rN (Ξ)
=â|Îą|â¤N
1i|Îą| Îą!
[D|Îą|
âÎśxÎą
âŁâŁâŁâŁÎž
a
]¡[D|ι|
âzxÎą
âŁâŁâŁâŁx
f
]+ rN (Ξ).
The Normal Symbol on Riemannian Manifolds 117
Note that in this equation all integrals are iterated ones and check that with Taylorâsformula rN is given by
rN (Ξ) =1
(2Ď)nâ
|β|=N+1
N + 1β!
âŤTâxX
⍠1
0
(1â t)N D|β|
âÎśxβa(Ξ + tÎś) dt Μβ
[F (fĎ)](Îś) dÎś.
(71)
Choosing N large enough, the Schwarz inequality and Plancherelâs formula nowentail
||rN (Ξ)|| ⤠1(2Ď)n/2
â|β|=N+1
N + 1β!
âŁâŁâŁâŁâŁâŁâŁâŁâŤ 1
0
(1â t)N D|β|
âÎśxβa(Ξ + t(¡))dt
âŁâŁâŁâŁâŁâŁâŁâŁL2(TâxX)âŁâŁâŁâŁâŁâŁâŁâŁD|β|âvxβ
fĎâŁâŁTxX
âŁâŁâŁâŁâŁâŁâŁâŁL2(TxX)
.
(72)
But for K â U compact and t â [0, 1] there exist constants DN,K , DN,K > 0 suchthat âŁâŁâŁâŁâŁâŁâŁâŁD|β|âÎśx
βa(Ξ + tΜ)
âŁâŁâŁâŁâŁâŁâŁâŁ ⤠DN,K(1 + ||Ξ||+ ||Îś||)ÂľâĎ(N+1)(73)
and
âŁâŁâŁâŁâŁâŁâŁâŁâŤ 1
0
(1â t)k D|β|
âÎśxβa(Ξ + t(¡))dt
âŁâŁâŁâŁâŁâŁâŁâŁL2(TâxX)
⤠DN,K
âŁâŁâŁâŁâŁâŁ(1 + ||Ξ||+ || ¡ ||)ÂľâĎ(N+1)âŁâŁâŁâŁâŁâŁL2(TâxX)
(74)
= DN,K (1 + ||Ξ||)n+ÂľâĎ(N+1)
for all x â K, Ξ, Îś â T âxX and all indices β â Nd with |β| = N + 1. Inserting thisin (72) we can find a CN,K > 0 such that for all x â K and Ξ â T âxX
(75) ||rN (Ξ)|| â¤
CN,K
(1 + ||Ξ||
)n+ÂľâĎ(N+1) â|Îł|â¤N+1
sup{âŁâŁâŁâŁâŁâŁâŁâŁD|Îł|âzxÎł
f(y)âŁâŁâŁâŁâŁâŁâŁâŁ : y â supp ĎĎ(Ξ)
}.
Because of limNââ
dimX + Âľ + Ď(N + 1) = ââ and a similar consideration for the
derivatives of ĎA,f the claim now follows. ďż˝
Next let us consider two pseudodifferential operators A â ΨâĎ,δ(U,F,G) and B âΨâĎ,δ(U,E, F ) between vector bundles E,F,G over X. Modulo smoothing operatorswe can assume one of them to be properly supported, so that AB â ΨâĎ,δ(U,E,G)exists. Defining the extended symbol bext = ĎextB : U Ă T âU â Hom(E,F ) of B by
bext(x, Ξ)Π=
ĎĎ(Ξ)(x) eâiĎ(x,Ξ)[B(ĎĎ(Ξ)(¡) eiĎ(¡,Ξ)Ď(¡),Ď(Ξ)Î
)](x), x â U, (Ξ,Î) â Ďâ(E|U )
(76)
118 Markus J. Pflaum
the symbol ĎAB of the composition AB is now given modulo Sââ(U,Hom(E,G))by
ĎAB(Ξ)Î =[AB
(ĎĎ(Ξ)e
iĎ(¡,Ξ)Ď(¡),Ď(Ξ)Î)]
(Ď(Ξ))
=[A(ĎĎ(Ξ) e
iĎ(¡,Ξ) ĎextB (¡, Ξ)Î)]
(Ď(Ξ)).(77)
By Proposition 6.1 this entails that ĎAB has an asymptotic expansion of the form
ĎAB(Ξ) âźâÎąâNn
1i|Îą|Îą!
[D|Îą|
âÎśĎ(Ξ)Îą
âŁâŁâŁâŁÎž
ĎA
][D|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁĎ(Ξ)
ĎextB (¡, Ξ)].(78)
As we want to find an even more detailed expansion for ĎAB , let us calculate thederivatives D|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁĎ(Ξ)
ĎextB (¡, Ξ) in the following proposition.
Proposition 6.2. Let B â Ψ¾Ď,δ(U,E, F ) be a pseudodifferential operator between
Hermitian (Riemannian) vector bundles over the Riemannian manifold X and b =Ď(B) its symbol. Then for any f â Câ(U,E) the smooth function
ĎextB,f : U Ă T âU â F, (x, Ξ) 7â ĎĎ(Ξ)(x) eâiĎ(x,Ξ)[B(ĎĎ(Ξ)f e
iĎ(¡,Ξ))]
(x)(79)
is a symbol of order Âľ and type (Ď, δ) on U ĂT âU and has an asymptotic expansionof the form
ĎextB,f (x, Ξ) âźâkâN
âβ,β0,...,βkâNnβ0+...+βk=β|β1|,...,|βk|âĽ2
ikâ|β|
k!β0! ¡ ... ¡ βk!ĎĎ(Ξ)(x)
[D|β|
âÎśxβ
âŁâŁâŁâŁdxĎ( ,Ξ)
b
]
[D|β0|
âzxβ0
âŁâŁâŁâŁx
ĎĎ(Ξ)f
] [â|β1|
âzxβ1
âŁâŁâŁâŁx
Ď(¡, Ξ)]¡ ... ¡
[â|βk|
âzxβk
âŁâŁâŁâŁx
Ď(¡, Ξ)].
(80)
Furthermore the derivative T âU 3 Ξ 7â[
â|Îą|âzĎ(Ξ)
ÎąĎextB,f (¡, Ξ)
](Ď(Ξ)) â F with Îą â Nn
is a symbol with values in F of order Âľ + δ|Îą| and type (Ď, δ) on T âU and has anasymptotic expansion of the form[
D|Îą|
âzĎ(Ξ)ÎąĎextB,f (¡, Ξ)
](Ď(Ξ)) âź
âźâkâN
âÎą,Îą0,...,ÎąkâNnÎą+Îą0+...+Îąk=Îą
âβ,β0,...,βkâNnβ0+...+βk=β|β1|,...,|βk|âĽ2
ikâ|β|Îą!k!Îą!Îą0! ¡ ... ¡ Îąk!β0! ¡ ... ¡ βk!
D|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁĎ(Ξ)
D|β|
âÎś(â)β
âŁâŁâŁâŁâŁd(â)Ď( ,Ξ)
b
{
D|Îą0|
âzĎ(Ξ)Îą0
âŁâŁâŁâŁĎ(Ξ)
[D|β0|
âz(â)β0
âŁâŁâŁâŁ(â)
f
]}¡
{â|Îą1|
âzĎ(Ξ)Îą1
âŁâŁâŁâŁĎ(Ξ)
[â|β1|
âz(â)β1
âŁâŁâŁâŁ(â)
Ď(¡, Ξ)]}¡ ... ¡
{â|Îąk|
âzĎ(Ξ)Îąk
âŁâŁâŁâŁĎ(Ξ)
[â|βk|
âz(â)βk
âŁâŁâŁâŁ(â)
Ď(¡, Ξ)]}
.
Note 6.3. The differential operators of the form D|Îą|âzĎ(Ξ)
Îą act on the variables de-
noted by (â), the differential operators of the form D|β|âz(â)
β on the variables (¡).
The Normal Symbol on Riemannian Manifolds 119
Proof of Proposition 6.2. Let Ρ â Câ(U ĂU ĂT âU) be a smooth function suchthat
Ρ(x, y, Ξ) = Ď(y, Ξ)â Ď(x, Ξ)â < dxĎ( , Ξ), zx(y) >(81)
for x, y â supp ĎĎ(Ξ) and such that T âU 3 Ξ 7â Ρ(x, y, Ξ) â C is linear for allx, y â U . Furthermore denote by F â Câ(U ĂU ĂT âU,E) the function (x, y, Ξ) 7âĎĎ(Ξ)(y)f(y)eiΡ(x,y,Ξ). By the Leibniz rule and the definition of Ρ we then have[
D|β|
âzxβ
âŁâŁâŁâŁy
F (x, ¡, Ξ)]
=â
0â¤k⤠|β|2
âβ0,...,βkâNnβ0+...+βk=β|β1|,...,|βk|âĽ2
ik
k!β0! ¡ ... ¡ βk!
[D|β0|
âzxβ0
âŁâŁâŁâŁy
ĎĎ(Ξ)f
]¡
[â|β1|
âzxβ1
âŁâŁâŁâŁy
Ρ(x, ¡, Ξ)]¡ ... ¡
[â|βk|
âzxβk
âŁâŁâŁâŁy
Ρ(x, ¡, Ξ)]eiΡ(x,y,Ξ).
(82)
As (x, y, Ξ) 7â[â|βj |
âzxβj
âŁâŁâŁyΡ(x, ¡, Ξ)
], 0 ⤠j ⤠k is smooth and linear with respect to
Ξ, the preceeding equation entails that (x, y, Ξ) 7â[D|β|âzxβ
âŁâŁâŁyF (x, ¡, Ξ)
]is absolutely
bounded by a function (x, y, Ξ) 7â Cβ(x, y) |Ξ| |β|2 , where Cβ â Câ(U Ă U) andCβ ⼠0. On the other hand we have
ĎextB,f (x, Ξ) = ĎĎ(Ξ)(x)[B(ĎĎ(Ξ)(¡)f(¡)eiΡ(¡,x,Ξ) ei<dxĎ( ,Ξ),zx(¡)>
)](x)(83)
and by the proof of Proposition 6.1
ĎextB,f (x, Ξ) = ĎĎ(Ξ)(x)â|β|â¤N
i|β|
β!
[D|β|
âΞxβ
âŁâŁâŁâŁdxĎ( ,Îś)
b
] [D|β|
âzxβ
âŁâŁâŁâŁx
F (x, ¡, Ξ)]
+ rN (x, Ξ)
(84)
= ĎĎ(Ξ)(x)â|β|â¤N
â0â¤k⤠|β|2
âβ0,...,βkâNnβ0+...+βk=β|β1|,...,|βk|âĽ2
ikâ|β|
k!β0! ¡ ... ¡ βk!
[D|β|
âΞxβ
âŁâŁâŁâŁdxĎ( ,Îś)
b
]¡
[D|β0|
âzxβ0
âŁâŁâŁâŁx
ĎĎ(Ξ)f
]¡[â|β1|
âzxβ1
âŁâŁâŁâŁx
Ď(¡, Ξ)]¡ ... ¡
[â|βk|
âzxβk
âŁâŁâŁâŁx
Ď(¡, Ξ)]
+ rN (x, Ξ),
where
||rn(x, Ξ)|| ⤠CN,K(1 + ||dxĎ( , Ξ)||)n+ÂľâĎ(N+1)
â|Îł|â¤N+1
sup
{âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁ[â|Îł|
âzxÎł
âŁâŁâŁâŁy
F (x, ¡, Ξ)]âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁ : y â supp Ďx
}⤠DN,K(1 + ||Ξ||)n+Âľâ(Ďâ1/2)(N+1)
(85)
holds for compact K â U , x â K, Ξ â T âX|K and constants CN,K , DN,K > 0.Because Ď > 1
2 , we have limNââ dimX + Âľâ (Ďâ 12 )(N + 1) = ââ. By Eqs. (84)
and (85) and similar relations for the derivatives of ĎextA,f we therefore conclude thatEq. (80) is true. The rest of the claim easily follows from Leibniz rule. ďż˝
The last proposition and Eq. (78) imply the next theorem.
120 Markus J. Pflaum
Theorem 6.4. Let A,B â ΨâĎ,δ(U,E, F ) be two pseudodifferential operators be-tween Hermitian (Riemannian) vector bundles E,F over a Riemannian manifoldX. Assume that one of the operators A and B is properly supported. Then thesymbol ĎAB of the product AB has the following asymptotic expansion.
ĎAB (Ξ) âźâÎąâNd
iâ|Îą|
Îą!
[D|Îą|
âÎśĎ(Ξ)Îą
âŁâŁâŁâŁÎž
ĎA
][D|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁĎ(Ξ)
ĎB
]+
+âkâĽ1
âÎą,Îą,Îą1,...,ÎąkâNnÎą+Îą1+...+Îąk=Îą
âβ,β1,...,βkâNnβ1+...+βk=β|β1|,...,|βk|âĽ2
ikâ|Îą|â|β|
k! ¡ ι! ¡ ι1! ¡ ... ¡ ιk!β1! ¡ ... ¡ βk!
[D|Îą|
âÎśĎ(Ξ)Îą
âŁâŁâŁâŁÎž
ĎA
] D|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁĎ(Ξ)
D|β|
âÎś(â)β
âŁâŁâŁâŁâŁd(â)Ď( ,Ξ)
ĎB
¡{â|Îą1|
âzĎ(Ξ)Îą1
âŁâŁâŁâŁĎ(Ξ)
[â|β1|
âz(â)β1
âŁâŁâŁâŁ(â)
Ď(¡, Ξ)]}¡ ... ¡
{â|Îąk|
âzĎ(Ξ)Îąk
âŁâŁâŁâŁĎ(Ξ)
[â|βk|
âz(â)βk
âŁâŁâŁâŁ(â)
Ď(¡, Ξ)]}
.
(86)
A somewhat more explicit expansion up to second order is given by
Corollary 6.5. If ĎA is a symbol of order Âľ and ĎB a symbol of order Âľ thecoefficients in the asymptotic expansion of ĎAB are given up to second order by thefollowing formula:
ĎAB(Ξ) =ĎA(Ξ) ¡ ĎB(Ξ) â
â iâl
DĎAâÎśĎ(Ξ),l
(Ξ)DĎBâzlĎ(Ξ)
(Ξ) â 12
âk,l
D2ĎAâÎśĎ(Ξ),kâÎśĎ(Ξ),l
(Ξ)D2ĎB
âzkĎ(Ξ)âzlĎ(Ξ)
(Ξ)(87)
â 112
âk,l,m,n
DĎAâÎśĎ(Ξ),n
(Ξ)D2ĎB
âÎśĎ(Ξ),lâÎśĎ(Ξ),m(Ξ)Rkmln(Ď(Ξ)) ÎśĎ(Ξ),k(Ξ) + r(Ξ)
where Ξ â T âU , x = Ď(Ξ), r â SÂľ+Âľâ3Ď,δ (U) and the Rmnkl(y) are the coefficients of
the curvature tensor with respect to normal coordinates at y â X, i.e., R(
ââzmy
)=â
k,l,n
Rkmln(y) ââzkyâ dzly â dzny .
Proof. First define the order of a summand s = sβ,β1,...,βkι,ι,ι1,...,ιk
in Eq. (86) by ord(s) =
|Îąâ Îą+βâk|. Then it is obvious that s â SÂľ+Âľ+ord(s) (Ďâδ)Ď,δ (U). Therefore we have
to calculate only the summands s with ord(s) ⤠2. To achieve this let us calculatesome coordinate changes and their derivatives. Recall the matrix-valued function(θkl)
of Eq. (61) and construct its inverse(θkl). In other wordsâ
k
θlk(x, y) θkm(x, y) = δlm(88)
holds for every l,m = 1, ..., d. We can now write
â
âzky
âŁâŁâŁâŁz
=âl,m
θlk(y, z) θml (x, z)â
âzmx
âŁâŁâŁâŁz
(89)
The Normal Symbol on Riemannian Manifolds 121
and receive the following formulas:
âzkxâzly
(z) =δkl â16
âm,n
Rkmln(y) zmx (z) znx (z)
+16
âm,n
Rkmln(x)zmx (z) znx (z) + O(|zx(z) + zy(z)|3) ,(90)
â2zkxâzlyâz
my
(z) =â 16
ân
(Rkmln(y) +Rknlm(y)
)znx (z)
+16
ân
(Rkmln(x) +Rknlm(x)
)znx (z)
+ O(|zx(z) + zy(z)|2) ,
(91)
â
âznx
âŁâŁâŁâŁy=z
â2zkxâzlyâz
my
(y) =16(Rkmln(x) +Rknlm(x)
)+ O (|zx(z)|) .(92)
Next consider the functions
Ďιβ : T âX â R, Ξ 7â â|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁy=Ď(Ξ)
â|β|Ď(¡, Ξ)âβzyβ
(y)
which appear in the summands s of Eq. (86). One can write the Ďιβ in the form
Ďιβ(Ξ) =âk
ÎśĎ(Ξ),k(Ξ)â|Îą|
âzĎ(Ξ)Îą
âŁâŁâŁâŁy=Ď(Ξ)
â|β|zkĎ(Ξ)
âzyβ(y)(93)
Thus by Eq. (91)
Ď0β(Ξ) = 0(94)
holds for every β â Nd with |β| ⼠2. Furthermore check that
dyĎ(¡, Ξ) =âk
ÎśĎ(Ξ),k(Ξ) dzkĎ(Ξ)
âŁâŁâŁy
(95)
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:
s(Ξ) = ĎA(Ξ)ĎB(Ξ).(96)
(ii) ord(s) = 1, |Îą| = 1, k = 0, Îą = Îą:
s(Ξ) = â iâl
DĎAâÎśĎ(Ξ),l
(Ξ)DĎBâzlĎ(Ξ)
(Ξ).(97)
(iii) ord(s) = 1, |ι| = 0, k = 1, |β| = 2: In this case s = 0 because of Eq. (94).(iv) ord(s) = 2, |ι| = 2, k = 0, ι = ι:
s(Ξ) = â 12
âk,l
D2ĎAâÎśĎ(Ξ),kâÎśĎ(Ξ),l
(Ξ)D2ĎB
âzkĎ(Ξ)âzlĎ(Ξ)
(Ξ).(98)
122 Markus J. Pflaum
(v) ord(s) = 2, |ι| = 1, k = 1, |β| = 2: In this case ι = 0 by Eq. (94). Hence byEq. (92)
s(Ξ) = â 112
âk,l,m,n
DĎAâÎśĎ(Ξ),n
(Ξ)D2ĎB
âÎśĎ(Ξ),lâÎśĎ(Ξ),m(Ξ)Rkmln(Ď(Ξ)) ÎśĎ(Ξ),k(Ξ).(99)
Summing up these summands we receive the expansion of ĎAB up to second order.This proves the claim. ďż˝
The product expansion of Theorem 6.4 gives rise to a a bilinear map # on thesheaf SâĎ,δ/S
ââ(¡,Hom(F,G))Ă SâĎ,δ/Sââ(¡,Hom(E,F )) by
SâĎ,δ(U,Hom(F,G))Ă SâĎ,δ(U,Hom(E,F ))â SâĎ,δ/Sââ(U,Hom(E,G)),
(a, b) 7â a#b = ĎOp(a) Op(b)
(100)
for all U â X open and vector bundles E,F,G over X. The #-product is animportant tool in a deformation theoretical approach to quantization (cf. Pflaum[8]). In the sequel it will be used to study ellipticity of pseudodifferential operatorson manifolds.
7. Ellipticity and normal symbol calculus
The global symbol calculus introduced in the preceding paragraphs enables usto investigate the invertibility of pseudodifferential operators on Riemannian man-ifolds. In particular we are now able define a notion of elliptic pseudodifferentialoperators which is more general than the usual notion by Hormander [7] or Douglis,Nirenberg [2].
Definition 7.1. A symbol a â SâĎ,δ(X,Hom(E,F )) on a Riemannian manifold X
is called elliptic if there exists b0 â SâĎ,δ(X,Hom(F,E))) such that
a#b0 â 1 â SâÎľĎ,δ(X) and b0#aâ 1 â SâÎľĎ,δ(X)(101)
for an Îľ > 0. A symbol a â SmĎ,δ(X,Hom(E,F )) is called elliptic of order m if it iselliptic and one can find a symbol b0 â SmĎ,δ(X,Hom(F,E)) fulfilling Eq.(101).
An operator A â ΨâĎ,δ(X,E, F ) is called elliptic resp. elliptic of order m, if itssymbol ĎA is elliptic resp. elliptic of order m.
Let us give some examples of elliptic symbols resp. operators.
Example 7.2. (i) Let a â Sm1,0(X,Hom(E,F )) be a classical symbol, i.e., assumethat a has an expansion of the form a âźâjâN amâj with amâj â Smâj1,0 (X)homogeneous of order mâ j. Further assume that a is elliptic in the classicalsense, 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. Butthen a and Op(a) must be elliptic of order m as well.
(ii) Consider the symbols l : T âX â C, Ξ 7â ||Ξ||2 and a : T âX â C, Ξ 7â 11+||Ξ||2
of Example 2.3 (ii). Then the pseudodifferential operator corresponding to lis minus the Laplacian: Op(l) = ââ. Furthermore the symbols l and 1+ l areelliptic of order 2, and the relation a#(1 + l)â 1, (1 + l)#aâ 1 â Sâ1
1,0(X) issatisfied. In case X is a flat Riemannian manifold, one can calculate directlythat modulo smoothing symbols a#l = l#a = 1, hence Op(a) is a parametrixfor the differential operator Op(1 + l) = 1ââ.
The Normal Symbol on Riemannian Manifolds 123
(iii) Let aĎ(Ξ) = (1 + ||Ξ||2)Ď(Ď(Ξ)) be the symbol defined in Example 2.3 (iii),where Ď : X â R is supposed to be smooth and bounded. Then a is ellipticand b(Ξ) = (1 + ||Ξ||2)âĎ(Ď(Ξ)) fulfills the relations (101). In case Ď is notlocally constant, we thus receive an example of a symbol which is not ellipticin the sense of Hormander [7] but elliptic in the sense of the above definition.
Let us now show that for any elliptic symbol a â SâĎ,δ(X,Hom(E,F )) one canfind a symbol b â SâĎ,δ(X,Hom(F,E)) such that even
a#bâ 1 â Sââ(X,Hom(F, F )) and b#aâ 1 â Sââ(X,Hom(E,E)).(102)
Choose b0 according to Definition 7.1 and let r = 1âa#b0, l = 1âb0#a â Sââ(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-defined and fulfill a#br = 1 and bl#a = 1 modulo smoothing symbols.Hence br â bl â Sââ(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 â ΨâĎ,δ(X,E, F ) be an elliptic pseudodifferential operator.Then there exists a parametrix B â ΨâĎ,δ(X,F,E) for A, i.e., the relations
AB â 1 â Ψââ(X,F, F ) and BAâ 1 â Ψââ(X,E,E)(103)
hold. If A is elliptic of order m, B can be chosen of order âm. On the other hand,if A is an operator invertible in ΨâĎ,δ(X) modulo smoothing operators, then A iselliptic. In case X is compact an elliptic operator A â ΨâĎ,δ(X) is Fredholm.
Proof. Let a = ĎA. As a is elliptic one can choose b such that (102) holds. If ais elliptic of order m, the above consideration shows that b is of order âm. Theoperator 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#bâ 1 and b#aâ 1 are smoothing, hence A iselliptic. Now recall that a pseudodifferential operator induces continuous mappingsbetween appropriate Sobolev-completions of Câ(X,E), resp. Câ(X,F ), and thatwith respect to these Sobolev-completions any smoothing pseudodifferential oper-ator is compact. Hence the claim that for compact X an elliptic A is Fredholmfollows from (103). ďż˝
Let us give in the following propositions a rather simple criterion for ellipticityof order m in the case of scalar symbols.
Proposition 7.4. A scalar symbol a â SmĎ,δ(X) is elliptic of order m, if and onlyif for every compact set K â X there exists CK > 0 such that
||a(Ξ)|| ⼠1CK||Ξ||m(104)
for all Ξ â T âX with Ď(Ξ) â K and ||Ξ|| ⼠CK .
Proof. Let us first show that the condition is sufficient. By assumption there existsa function b â Câ(T âX) such that for every compact K â X there is CK > 0 with
a(Ξ) ¡ b (Ξ) = 1 and ||b(Ξ)|| ⤠CK ||Ξ||âm(105)
124 Markus J. Pflaum
for Ξ â T âX with Ď(Ξ) â K and ||Ξ|| ⼠CK . Hence a ¡ b â 1 â Sââ(X). Afterdifferentiating the relation r = a ¡ bâ1 in local coordinates (x, Ξ) of T âX we receive
(106)
supΞâTâX|K
âŁâŁâŁâŁ(1 + ||Ξ||2)m/2 â|Îą|
âxÎąâ|β|
âΞβb(Ξ)
âŁâŁâŁâŁ ⤠C supΞâTâX|K
âŁâŁâŁâŁa(Ξ)â|Îą|
âxÎąâ|β|
âΞβb(Ξ)
âŁâŁâŁâŁ â¤â¤ C sup
ΞâTâX|K
|r(Ξ)|+â
ι1+ι2=ιβ1+β2=β|ι1+β1|>0
âŁâŁâŁâŁ â|Îą1|
âxÎą1
â|β1|
âΞβ1a(Ξ)
â|Îą2|
âxÎą2
â|β2|
âΞβ2b(Ξ)
âŁâŁâŁâŁ ,
hence by induction b â SâmĎ,δ (X) follows. By the product expansion Eq. (87) thisimplies
a#bâ 1 = r + t1 and b#aâ 1 = r + t2,(107)
where t1/2 â SâÎľĎ,δ(X) with Îľ = min{(Ď â δ), 2Ď â 1} > 0. Hence a is an ellipticsymbol.
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 â SâmĎ,δ (X) such that Op(b) is a parametrixfor Op(a). But this implies by Eq. (87) that ab â 1 â SâÎľĎ,δ for some Îľ > 0, hence|a(Ξ)b(Ξ)â 1| < 1
2 for all Ξ â T âX|K with ||Ξ|| > CK , where K â X compact andCK > 0. But then
12< |a(Ξ)b(Ξ)| < Câ˛K |a(Ξ)|||Ξ||âm, Ξ â T âX|K , ||Ξ|| > CK ,(108)
which gives the claim. ďż˝
In the work of Douglis, Nirenberg [2] a concept of elliptic systems of pseudodif-ferential operators has been introduced. We want to show in the following that thisconcept fits well into our framework of ellipticity. Let us first recall the definitionby Douglis and Nirenberg. Let E = E1 â ...âEK and F = F1 â ...â FL be directsums of Riemannian (Hermitian) vector bundles over X, and Akl â Ψmkl
1,0 (X,Ek, Fl)with mkl â R be classical pseudodifferential operators. Denote the principal symbolof Akl by pkl â Smkl1,0 (X,Ek, Fl). Douglis, Nirenberg [2] now call the system (Akl)elliptic, if there exist homogeneous symbols bkl â Sâmkl1,0 (X,Ek, Fl) such that
âl
pkl ¡ blkⲠâ δkkⲠâ Sâ11,0(X,Ek, Fl) and
âk
blk ¡ pklⲠâ δllⲠâ Sâ11,0(X,Ek, Fl)
(109)
for all k, kâ˛, l, lâ˛. The following proposition shows that A = (Akl) is an ellipticpseudodifferential operator, hence according to Theorem 7.3 possesses a parametrix.
Proposition 7.5. Let E = E1 â ... â EK and F = F1 â ... â FL be direct sumsof Riemannian (Hermitian) vector bundles over X. Assume that A = (Akl) withAkl â Smkl1,0 (X,E, F ) comprises an elliptic system of pseudodifferential operatorsin the sense of Douglis, Nirenberg [2]. Then the pseudodifferential operator A iselliptic.
The Normal Symbol on Riemannian Manifolds 125
Proof. Write akl = ĎAkl = pkl + rkl with akl â Smkl1,0 (X,Hom(Ek, Fl) and rkl âSmklâ1
1,0 (X,Hom(Ek, Fl). Furthermore let a = (akl) â Sâ1,0(X,Hom(E,F )), b =(bkl) â Sâ1,0(X,Hom(E,F )) and r = (rkl) â Sâ1,0(X,Hom(E,F )). By Eq. (109) andthe expansion Eq. (87) it now follows
a#bâ 1 = (p#bâ 1) + r#b â Sâ11,0(X,Hom(E,F )),
b#aâ 1 = (b#pâ 1) + b#r â Sâ11,0(X,Hom(E,F )).
(110)
But this proves the ellipticity of A. �So with the help of the calculus of normal symbols one can directly constructinverses to elliptic operators in the algebra of pseudodifferential operators on aRiemannian manifold. In particular after having once established a global symbolcalculus it is possible to avoid lengthy considerations in local coordinates. Thusone receives a practical and natural geometric tool for handling pseudodifferentialoperators on manifolds.
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Humboldt Universitat zu Berlin, Mathematisches Institut, Unter den Linden 6,10099 Berlin
[email protected] http://spectrum.mathematik.hu-berlin.de/Ëpflaum
This paper is available via http://nyjm.albany.edu:8000/j/1998/4-8.html.