
LECTURE 19:DIFFERENTIAL OPERATORS ON MANIFOLDS1.DIFFERENTIALOPERATORS ON MANIFOLDSWe are aiming at extending the definition of semiclassical pseudodifferentialoperatorsfromIRntomanifolds.Let'sstartbythesimplestclassof pseudodifferentialoperators:the differential operators.For simplicity we consider h =1 only.Onecan easily extend to the semiclassical setting. Differential operators under coordinate change.Let's assume U, V are open sets in Rn and letf : Uc R - Vc Rnbe a diffeomorphism.We can easily“"transplant" a differential operator defined forr-functions to a differential operator defined for y-functions via f: IfP=aa(r)D(1)lal<mis a differential operator acting on Co(Rn), then when restricted to U, P is also adifferential operator Plu acting on Co(U), and Plu induces a differential operatorP acting on C(V) as follows: for any u e C(V), we just definePu := (f-1)"Pluf*u.Let's calculate P in coordinates: for any u = u(y) e Co(V) we have(f*u)(r) =u(f(r))and thus[Pluf*u(r) = aa(r)D[u(f(r)]lal<mSinceoyir [u(f(r)] == or (Op u)(f(r),by induction it is easy toget[u(f(r)] :u(f(r) +1.o.t.,where l.o.t. denotes terms that encounter less Oy-derivatives on u. It follows(2)Pu(g) = aa(f-1(g)u(y) + 1.o.t.[a|=m
LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS 1. Differential operators on manifolds We are aiming at extending the definition of semiclassical pseudodifferential operators from R n to manifolds. Let’s start by the simplest class of pseudodifferential operators: the differential operators. For simplicity we consider ~ = 1 only. One can easily extend to the semiclassical setting. ¶ Differential operators under coordinate change. Let’s assume U, V are open sets in R n and let f : U⊂ R n x → V ⊂ R n y be a diffeomorphism. We can easily “transplant” a differential operator defined for x-functions to a differential operator defined for y-functions via f: If (1) P = X |α|≤m aα(x)D α x is a differential operator acting on C ∞(R n x ), then when restricted to U, P is also a differential operator P|U acting on C ∞(U), and P|U induces a differential operator Pe acting on C ∞(V ) as follows: for any u ∈ C ∞(V ), we just define P ue := (f −1 ) ∗P|U f ∗u. Let’s calculate Pe in coordinates: for any u = u(y) ∈ C ∞(V ) we have (f ∗u)(x) = u(f(x)) and thus [P|U f ∗u](x) = X |α|≤m aα(x)D α x [u(f(x))]. Since ∂xi [u(f(x))] = ∂yj ∂xi (∂y ju)(f(x)), by induction it is easy to get ∂ α x [u(f(x))] = " ∂y ∂xT ∂y #α u(f(x)) + l.o.t., where l.o.t. denotes terms that encounter less ∂y-derivatives on u. It follows (2) P ue (y) = X |α|=m aα(f −1 (y)) " ∂y ∂xT ∂y #α u(y) + l.o.t. 1

2LECTURE19:DIFFERENTIALOPERATORSONMANIFOLDS Gluing differential operators on manifolds.Now suppose M is a smooth manifold, [(a,Ua, Va, r,..., rn)) is a coordinatechart. For simplicity we assume M is compact, and the coordinate chart is finite.Recall that a function u defined on M is smooth if for any Q,uopalis smooth.This can be expressed in another way: if we have smooth functionsu E Coo(U), (or equivalently, smooth functions ua o -l Co(Va), and ifa0p-l= (ug0B1)(0p-l)ona(U&nUp),then we can glue all these uas defined on Uas, (or equivalently, glue all these ug op.)defined on Vas,) to one smooth function u defined on M: we just letu(r) := ua(r)for r U..The above condition tells us that u= ug on U&nUg-Now suppose P: Co(V)→C(Va) be differential operators defined on Vas,PaB=BOP:P(UnUe)CVPB(U&nUB)CVBbe the coordinate transition diffeomorphism. Assume that(Pa)*Palpa(UanUe)Paβ = Ppleg(UanUe) on C(P(U&nUp)Then we can “glue" Pa's via as's to get a differential operator on M:for anyu ECo(M) and r E U. C M, we just letPu(r) := p,Pa((p-l)*u)(r)We check this P is well-defined:if r eU&nUs, thenPgPp((B")u)(r) = PPpl(UanUe)(pB1)u)(r)= [()*Pal(UanU)PaB] ((B)*)(r)= P [()-IpPala(UanUe)())*) (Bl)*)(r)= P*Palea(UanUe)(P-)*u(r).In the above constructions,the most important propertywe used toglue localfunctions or local differential operators to global ones is the locality of functions ordifferential operators themselves:in the case of differential operators, it is crucialthat we can restrict a differential operator P on an open subset U to a differentialoperator Pi on its open subset Ui;moreover, this restriction is“universal"in thesensethatifP2istherestrictionofPiontoanopensubsetU2ofUi,thenP2isalsotherestriction of P onto theopen subsetU2 of U
2 LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS ¶ Gluing differential operators on manifolds. Now suppose M is a smooth manifold, {(ϕα, Uα, Vα, x1 α , · · · , xn α )} is a coordinate chart. For simplicity we assume M is compact, and the coordinate chart is finite. Recall that a function u defined on M is smooth if for any α, u ◦ ϕ −1 α is smooth. This can be expressed in another way: if we have smooth functions uα ∈ C ∞(Uα), (or equivalently, smooth functions uα ◦ ϕ −1 α ∈ C ∞(Vα), and if uα ◦ ϕ −1 α = (uβ ◦ ϕ −1 β ) ◦ (ϕβ ◦ ϕ −1 α ) on ϕα(Uα ∩ Uβ), then we can glue all these uαs defined on Uαs, (or equivalently, glue all these uα ◦ϕ −1 α defined on Vαs,) to one smooth function u defined on M: we just let u(x) := uα(x) for x ∈ Uα. The above condition tells us that uα = uβ on Uα ∩ Uβ. Now suppose Pα : C ∞(Vα) → C ∞(Vα) be differential operators defined on Vαs, ϕαβ = ϕβ ◦ ϕ −1 α : ϕα(Uα ∩ Uβ)⊂ Vα → ϕβ(Uα ∩ Uβ)⊂ Vβ be the coordinate transition diffeomorphism. Assume that (ϕ −1 αβ) ∗Pα|ϕα(Uα∩Uβ)ϕ ∗ αβ = Pβ|ϕβ(Uα∩Uβ) on C ∞(ϕβ(Uα ∩ Uβ)). Then we can “glue” Pα’s via ϕαβ’s to get a differential operator on M: for any u ∈ C ∞(M) and x ∈ Uα ⊂ M, we just let P u(x) := ϕ ∗ αPα((ϕ −1 α ) ∗u)(x). We check this P is well-defined: if x ∈ Uα ∩ Uβ, then ϕ ∗ βPβ((ϕ −1 β ) ∗u)(x) = ϕ ∗ βPβ|ϕβ(Uα∩Uβ)((ϕ −1 β ) ∗u)(x) = ϕ ∗ β (ϕ −1 αβ) ∗Pα|ϕα(Uα∩Uβ)ϕ ∗ αβ ((ϕ −1 β ) ∗u)(x) = ϕ ∗ β (ϕ ∗ β ) −1ϕ ∗ αPα|ϕα(Uα∩Uβ)(ϕ −1 α ) ∗ϕ ∗ β ((ϕ −1 β ) ∗u)(x) = ϕ ∗ αPα|ϕα(Uα∩Uβ)(ϕ −1 α ) ∗u(x). In the above constructions, the most important property we used to glue local functions or local differential operators to global ones is the locality of functions or differential operators themselves: in the case of differential operators, it is crucial that we can restrict a differential operator P on an open subset U to a differential operator P1 on its open subset U1; moreover, this restriction is “universal” in the sense that if P2 is the restriction of P1 onto an open subset U2 of U1, then P2 is also the restriction of P onto the open subset U2 of U.

3LECTURE19:DIFFERENTIALOPERATORSONMANIFOLDSTDifferentialoperators onmanifolds:an abstract definition.Here is another way to express the locality of differential operators: for a differ-ential operator P, the values of the function Pu on an open set U depends only onthevaluesofuonU.Equivalently,Definition 1.1.We say a linear operator P:C(M)→ Co(M)is local if foranyuE C(M),(3)supp(Pu) C supp(u).It is this "locality"that allows us to"glue"differential operators defined onlocal charts to a differential operator on the whole manifold:By definition it is easyto see that if P is a local operator on M, and if U C M is an open subset, then the"restriction operation"Pluu:= (Pu)ludefines a "restricted operator" Plu : C(U) -→ C(U). Moreover, such restrictedoperators satisfies the property that for any open sets Ui C U,(Plu)lu, = Plur:Now we can give an abstract definition of a differential operator on a smoothmanifold:Definition 1.2. Let M be a smooth manifold. A differential operator on M oforder at most m is a local linear operator P : Co(M) → C(M) such that whenrestricted to each coordinate chart {a,Ua, Va,r',..., rn], the operatorPa := (p-1)* 0 Plua 0ais a differential operator on V.of order at most m (namely,is of the form (1))Erample. Any smooth vector field V on M is a differential operator of order 1.Conversely one can prove (exercise): any differential operator of order 1 on M hasthe form V+ mf, where V is a vector field, and f is a smooth function and mf isthe operator“multiplication by f" (which is a differential operator of order O),Erample. In general, if V's are a finite collection of smooth vector fields, thenP=Vir...Vik0<k≤nis a differential operator on M of order atmost m (wherek =O representsamultiplication operator).Conversely, at least for compact manifold, we can write any differential operatorin this form.To see this, we just use a partition of unity subordinate to a coordinatecovering, so that in each coordinate chart P has the form (1)
LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS 3 ¶ Differential operators on manifolds: an abstract definition. Here is another way to express the locality of differential operators: for a differential operator P, the values of the function P u on an open set U depends only on the values of u on U. Equivalently, Definition 1.1. We say a linear operator P : C ∞(M) → C ∞(M) is local if for any u ∈ C ∞(M), (3) supp(P u) ⊂ supp(u). It is this “locality” that allows us to “glue” differential operators defined on local charts to a differential operator on the whole manifold: By definition it is easy to see that if P is a local operator on M, and if U ⊂ M is an open subset, then the “restriction operation” P|U u := (P u)|U defines a “restricted operator” P|U : C ∞(U) → C ∞(U). Moreover, such restricted operators satisfies the property that for any open sets U1 ⊂ U, (P|U )|U1 = P|U1 . Now we can give an abstract definition of a differential operator on a smooth manifold: Definition 1.2. Let M be a smooth manifold. A differential operator on M of order at most m is a local linear operator P : C ∞(M) → C ∞(M) such that when restricted to each coordinate chart {ϕα, Uα, Vα, x1 , · · · , xn}, the operator Pα := (ϕ −1 α ) ∗ ◦ P|Uα ◦ ϕ ∗ α is a differential operator on Vα of order at most m (namely, is of the form (1)). Example. Any smooth vector field V on M is a differential operator of order 1. Conversely one can prove (exercise): any differential operator of order 1 on M has the form V + mf , where V is a vector field, and f is a smooth function and mf is the operator “multiplication by f” (which is a differential operator of order 0). Example. In general, if Vi ’s are a finite collection of smooth vector fields, then P = X 0≤k≤m Vj1 · · · Vjk is a differential operator on M of order at most m (where k = 0 represents a multiplication operator). Conversely, at least for compact manifold, we can write any differential operator in this form. To see this, we just use a partition of unity subordinate to a coordinate covering, so that in each coordinate chart P has the form (1)

ALECTURE19:DIFFERENTIALOPERATORSONMANIFOLDS Distributions and Sobolev spaces on manifolds.Inwhatfollowsweassume(M.g)isacompactRiemannianmanifold.sothatthereisa well-defined Riemannian volume form using which we can define L?(M)(We can also develop the theory without a Riemannian metric, in which case we canuse the space of half densities).As in theEuclidean case,one candefine,for eachnon-negative integerk,the Sobolev spaceHk(M)by(4)H*(M) = (u E L?(M) / Vi ... Veu E L?(M) for all smooth vector fields Vi, ... V).Since M is compact, one can choose a family of vector fields Wi, ..:, W on M thatspanTM at each point &.The Sobolev norm onH(M)isdefined to be IWa..Waull2(M)Ilull Hk(M)=01awhile the semi-classical Sobolev norm on Hk(M) is defined to be1/2 ?'Wa .. Warul/(M)lullH(M)[=0 1≤aj≤NTo define Sobolev spaces H*(M)for negativek,onehas to extend the concep-tion of distributions to manifolds. Again the idea is to quite simple: we pull-backeverything to Euclidian space via coordinate charts. Suppose (Pa, Ua, V) is a coor-dinate chart. Then given any u:Coo(M)→ C, we want to "transplant uto bealinear functional u on (Rn) via the chart map, so that we say u is a distributionif theinduced linearmapisanelementingi:Definition1.3.Let Mbe a smooth compactmanifold.We saya linearmapu:Co(M)→ C is a distribution on M if for every coordinate chart (pa,Ua,V)and every x e Co(V), the mapping defined for (IRn) by(5)p-u((x))belongs to (Rn). The space of distributions on M is denoted by '(M)Remark.In the case of noncompact manifolds, one can also definethe space ofdistributions in a similar way. A more rigorous way: first define a topology onCo(M),then realize '(M) as the dual space of Co(M). Here is how we definesuch a topology on Co(M): first we can always write M = Unint(Kn), where eachKn is compact and Kn C int(Kn+i) for any n.Since each Kn is compact, it iscontained in finitely many coordinate charts.Using coordinate charts we can definea locally convex topologyl on Co(int(Kn)) via local semi-norms (c.f. Lecture 4).Now we get a sequence of locally convex topological spaces Co(int(Kn)), so that1A topological vector space is called locally conver if the origin has a neighborhood basis con-sisting of convexsets
4 LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS ¶ Distributions and Sobolev spaces on manifolds. In what follows we assume (M, g) is a compact Riemannian manifold, so that there is a well-defined Riemannian volume form using which we can define L 2 (M). (We can also develop the theory without a Riemannian metric, in which case we can use the space of half densities). As in the Euclidean case, one can define, for each non-negative integer k, the Sobolev space Hk (M) by (4) H k (M) = {u ∈ L 2 (M) | V1 · · · Vku ∈ L 2 (M) for all smooth vector fields V1, · · · Vk}. Since M is compact, one can choose a family of vector fields W1, · · · , WN on M that span TxM at each point x. The Sobolev norm on Hk (M) is defined to be kukHk(M) = X k l=0 X 1≤αj≤N kWα1 · · · Wαluk 2 L2(M) 1/2 . while the semi-classical Sobolev norm on Hk (M) is defined to be kukHk ~ (M) = X k l=0 X 1≤αj≤N ~ 2l kWα1 · · · Wαluk 2 L2(M) 1/2 . To define Sobolev spaces Hk (M) for negative k, one has to extend the conception of distributions to manifolds. Again the idea is to quite simple: we pull-back everything to Euclidian space via coordinate charts. Suppose (ϕα, Uα, Vα) is a coordinate chart. Then given any u : C ∞(M) → C, we want to “transplant” u to be a linear functional ue on S (R n ) via the chart map, so that we say u is a distribution if the induced linear map ue is an element in S 0 : Definition 1.3. Let M be a smooth compact manifold. We say a linear map u : C ∞(M) → C is a distribution on M if for every coordinate chart (ϕα, Uα, Vα) and every χ ∈ C ∞ 0 (Vα), the mapping defined for ϕ ∈ S (R n ) by (5) ϕ 7→ u(γ ∗ (χϕ)) belongs to S 0 (R n ). The space of distributions on M is denoted by D0 (M). Remark. In the case of noncompact manifolds, one can also define the space of distributions in a similar way. A more rigorous way: first define a topology on C ∞ 0 (M), then realize D0 (M) as the dual space of C ∞ 0 (M). Here is how we define such a topology on C ∞ 0 (M): first we can always write M = ∪nint(Kn), where each Kn is compact and Kn ⊂ int(Kn+1) for any n. Since each Kn is compact, it is contained in finitely many coordinate charts. Using coordinate charts we can define a locally convex topology1 on C ∞ 0 (int(Kn)) via local semi-norms (c.f. Lecture 4). Now we get a sequence of locally convex topological spaces C ∞ 0 (int(Kn)), so that 1A topological vector space is called locally convex if the origin has a neighborhood basis consisting of convex sets

5LECTURE19:DIFFERENTIALOPERATORSONMANIFOLDSeach Co(int(Kn)) is a topological subspace of Co(int(Kn+i). Finally we define atopology on C(M)=UnCo(int(Kn))to be thefinest locally convex topology sothat each inclusion In : Co(int(Kn) Co(M) is continuous. Such a topology isknown as the strict inductive limit topology, which turns Co(M) into an LF space.For details of this construction, c.f. Reed-Simon Vol 1, Section 5.4.By locality, if P is a differential operator, then P maps Co(M) to Co(M). Soby duality, P maps '(M) to '(M). Now one can define(6) H-k(M) = span[uE 9(M) I u= Vi... Vif for f e L2(M),<I<k)with SobolevnormIlull-k(M) = inf(E llfllz2(M) I u = War .. Wa, fa).or semiclassical SobolevnormIlull *(M) = inf(Ilfll (M) I u = h'Wa. . Wa fa).aObviously if P is a differential operator of order m, then P maps Hk(M) to Hk-m(M)2.SYMBOLICCALCULUSOFDIFFERENTIALOPERATORSDifferential operators on manifolds:principle symbols.For differential operators on Rn, say, the operatorP= aD°,Jal<mwe can define its full Kohn-Nirenberg symbol to beaKN(P)(r,E) := aa(r)s,la|<mwhich is, of course, a function on T*Rn = Rn × Rn. Similarly one can define theWeyl symbol of P = /al<m aaDa, which is given by (c.f. Lecture 9)Ow(P)(r, E) = e0-0e(7) aa(r)sa aa(r)s°+1.o.t.,(lal<m[a|=mwhere l.o.t. represents a polynomial in whose degree is at most m -1. So althoughokn(P)(r,$)+ ow(P)(r,s), they are both polynomials in $ of degree m,andtheir leading terms are the same. (Of course the same conclusion holds for anyt-quantization.)It is natural to ask: can we define the Kohn-Nirenberg or Weyl symbol fordifferential operators on manifolds? Unfortunately the answer is no, because thefull symbol, as a function on T*M, is not well-defined. Let me remind you that
LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS 5 each C ∞ 0 (int(Kn)) is a topological subspace of C ∞ 0 (int(Kn+1)). Finally we define a topology on C ∞ 0 (M) = ∪nC ∞ 0 (int(Kn)) to be the finest locally convex topology so that each inclusion ιn : C ∞ 0 (int(Kn)) ,→ C ∞ 0 (M) is continuous. Such a topology is known as the strict inductive limit topology, which turns C ∞ 0 (M) into an LF space. For details of this construction, c.f. Reed-Simon Vol 1, Section 5.4. By locality, if P is a differential operator, then P maps C ∞ 0 (M) to C ∞ 0 (M). So by duality, P maps D0 (M) to D0 (M). Now one can define (6) H −k (M) = span{u ∈ D 0 (M) | u = V1 · · · Vlf for f ∈ L 2 (M), 0 ≤ l ≤ k} with Sobolev norm kukH−k(M) = inf{ X α kfαkL2(M) | u = X α Wα1 · · · Wαl fα}. or semiclassical Sobolev norm kukH −k ~ (M) = inf{ X α kfαkL2(M) | u = X α ~ lWα1 · · · Wαl fα}. Obviously if P is a differential operator of order m, then P maps Hk (M) to Hk−m(M). 2. Symbolic calculus of differential operators ¶ Differential operators on manifolds: principle symbols. For differential operators on R n , say, the operator P = X |α|≤m aαD α , we can define its full Kohn-Nirenberg symbol to be σKN (P)(x, ξ) := X |α|≤m aα(x)ξ α , which is, of course, a function on T ∗R n = R n x × R n ξ . Similarly one can define the Weyl symbol of P = P |α|≤m aαDα , which is given by (c.f. Lecture 9) (7) σW (P)(x, ξ) = e i 2 ∂x·∂ξ X |α|≤m aα(x)ξ α = X |α|=m aα(x)ξ α + l.o.t., where l.o.t. represents a polynomial in ξ whose degree is at most m−1. So although σKN (P)(x, ξ) 6= σW (P)(x, ξ), they are both polynomials in ξ of degree m, and their leading terms are the same. (Of course the same conclusion holds for any t-quantization.) It is natural to ask: can we define the Kohn-Nirenberg or Weyl symbol for differential operators on manifolds? Unfortunately the answer is no, because the full symbol, as a function on T ∗M, is not well-defined. Let me remind you that

6LECTURE19:DIFFERENTIALOPERATORSONMANIFOLDSthe coordinate change on the cotangent space T*M induced by a coordinate change y = f(r) on the base manifold is given by(= f(),n= (1)-1)Ts(r,E) ~So if a function is well-defined on T*M, and it has the form o(r,) in one chart(c,), then it should has the form(s-1(),(%)n)in the other chart (y,n). However, as we have seen, if P = aa(r)Da in onechart, then in the other chart, it has the form P given by (2).Because of thecomplicated nature of l.o.t. (which also contains derivatives of the coordinate changediffeomorphism),wehaveKN(P)(α, E) +0KN(P)(f-1(y), (TnHowever, if one stare at the formula (2), one can easily see that the terms withJa=m do satisfy the correct"change of variable"formula! Thus wedefineDefinition 2.1. The principal symbol of a differential operator P of order m on asmooth manifold M is defined to be the smooth function m(P) e Co(T*M) so thaton a coordinate chart (s, U, r,... ,a"), if P has the form P = Ejal<m aa(r)D,then0m(P)(r, E) := aa(r)s%.lal=mRemark. In view of (2), the principal symbol of a differential operator is a well-defined smooth function on T*M. Moreover, we don't need to distinguish the prin-cipal symbol in different t-quantizations, since, as we just explained in (7), they areall the same at the “principal" level!Erample. Let V be a smooth vector field V, viewed as a differential operator oforder1.SoifwewriteV=a,D,inalocal chart,and recall =sda,weget,thenalocal computation yieldsoi(V)(r,s) =ajsj=a,s(0,) =s(ajo,)Inotherwords,weget0(V)(,) = E(iV) = (V)More generally, the principal symbol of P-EVi. ... Vi on M is o(Vi) ..o(Vi)
6 LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS the coordinate change on the cotangent space T ∗M induced by a coordinate change x y = f(x) on the base manifold is given by (x, ξ) y = f(x), η = ([∂y ∂x] −1 ) T ξ . So if a function σ is well-defined on T ∗M, and it has the form σ(x, ξ) in one chart (x, ξ), then it should has the form σ f −1 (y),( ∂y ∂x) T η in the other chart (y, η). However, as we have seen, if P = Paα(x)Dα x in one chart, then in the other chart, it has the form Pe given by (2). Because of the complicated nature of l.o.t. (which also contains derivatives of the coordinate change diffeomorphism), we have σKN (P)(x, ξ) 6= σKN (Pe)(f −1 (y),( ∂y ∂x) T η) However, if one stare at the formula (2), one can easily see that the terms with |α| = m do satisfy the correct “change of variable” formula! Thus we define Definition 2.1. The principal symbol of a differential operator P of order m on a smooth manifold M is defined to be the smooth function σm(P) ∈ C ∞(T ∗M) so that on a coordinate chart (ϕ, U, x1 , · · · , xn}, if P has the form P = P |α|≤m aα(x)Dα x , then σm(P)(x, ξ) := X |α|=m aα(x)ξ α . Remark. In view of (2), the principal symbol of a differential operator is a welldefined smooth function on T ∗M. Moreover, we don’t need to distinguish the principal symbol in different t-quantizations, since, as we just explained in (7), they are all the same at the “principal” level! Example. Let V be a smooth vector field V , viewed as a differential operator of order 1. So if we write V = PajDj in a local chart, and recall ξ = Pξjdxj , we get, then a local computation yields σ1(V )(x, ξ) = Xaj ξj = Xaj ξ(∂j ) = ξ( X j aj∂j ) In other words, we get σ1(V )(x, ξ) = ξ(iVx) = iξ(Vx). More generally, the principal symbol of P = PVj1 · · · Vjk on M is Pσ(Vj1 )· · · σ(Vjk )

7LECTURE19:DIFFERENTIALOPERATORSONMANIFOLDSErample.Themost important example is theLaplace-Beltramioperator g on aRiemannian manifold (M,g),whichlocallyhas theformZa:(gVgla).Ag=VgIt is a second order differential operator on M withprinciple symbol(g)(,)= g= Symbolic calculus for differential operators.Let's denote the set of differential operators on M of order no more than m byDm(M). Then the principal symbol gives a mapm : Dm(M) → C(T*M).Then by definition, wehaveProposition 2.2. If P e Dm(M) and om(P) = 0, then P e Dm-1(M).Since differential operators are special cases of pseudodifferential operators, ac-cording to the symbolic calculus for pseudodifferential operators, we haveProposition 2.3. If P e Dmi(M), Q E Dm2(M), then(1) PoQ E Dmi(M) andOm1+m2(P 0 Q)= 0m(P)om2(Q).(2) [P,Q] e Dm1+m2-1(M) and 20m1+m2-1([P,Ql) = [αmi(P),0m2(Q)Remark.Althoughthefull symbol of adifferential operator isnotwell-defined onT*M, there is a sub-principal symbol which is well-defined on T*M. More precisely,suppose Pis a diferential operator on M which has the form P-jal<m a(s)Dain local charts. As we have seen, the principal symbol om(P) = Zjal=m aa(r)ea iswell defined on T*M. The next term,m-1(P)(r,s) = Z aa(r)saal=m-is only locally defined and is not well defined on T*M. However, one can check thatthe function (exercise)02Osub(P)(r, S) := 0m-1(P) +OriaEim(r,s)is a well-defined function on T*M. It is called the sub-principal symbol of P.2Here, (, } is the Poisson bracket on T*M which locally has the form {f, g) =E(e, f0r,g -Oa, foe,g). One can check that it is a well-defined function on T*M
LECTURE 19: DIFFERENTIAL OPERATORS ON MANIFOLDS 7 Example. The most important example is the Laplace-Beltrami operator ∆g on a Riemannian manifold (M, g), which locally has the form ∆g = − 1 p |g| X∂i(g ijp |g|∂j ). It is a second order differential operator on M with principle symbol σ(∆g)(x, ξ) = Xg ij ξiξj = |ξ| 2 g . ¶ Symbolic calculus for differential operators. Let’s denote the set of differential operators on M of order no more than m by Dm(M). Then the principal symbol gives a map σm : D m(M) → C ∞(T ∗M). Then by definition, we have Proposition 2.2. If P ∈ Dm(M) and σm(P) = 0, then P ∈ Dm−1 (M). Since differential operators are special cases of pseudodifferential operators, according to the symbolic calculus for pseudodifferential operators, we have Proposition 2.3. If P ∈ Dm1 (M), Q ∈ Dm2 (M), then (1) P ◦ Q ∈ Dm1 (M) and σm1+m2 (P ◦ Q) = σm1 (P)σm2 (Q). (2) [P, Q] ∈ Dm1+m2−1 (M) and 2 σm1+m2−1([P, Q]) = {σm1 (P), σm2 (Q)}. Remark. Although the full symbol of a differential operator is not well-defined on T ∗M, there is a sub-principal symbol which is well-defined on T ∗M. More precisely, suppose P is a differential operator on M which has the form P = P |α|≤m aα(x)Dα in local charts. As we have seen, the principal symbol σm(P) = P |α|=m aα(x)ξ α is well defined on T ∗M. The next term, σm−1(P)(x, ξ) = X |α|=m−1 aα(x)ξ α is only locally defined and is not well defined on T ∗M. However, one can check that the function (exercise) σsub(P)(x, ξ) := σm−1(P) + i 2 X j ∂ 2 ∂xj∂ξj σm(x, ξ) is a well-defined function on T ∗M. It is called the sub-principal symbol of P. 2Here, {·, ·} is the Poisson bracket on T ∗M which locally has the form {f, g} = P(∂ξj f ∂xj g − ∂xj f ∂ξj g). One can check that it is a well-defined function on T ∗M