《半经典微局部分析 Semiclassical Microlocal Analysis》课程教学资源（英文讲义）Lec19 DIFFERENTIAL OPERATORS ON MANIFOLDS.pdf_大学文库

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.

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

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