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