《半经典微局部分析 Semiclassical Microlocal Analysis》课程教学资源（英文讲义）Lec20 SEMI-CLASSICAL PSDOS ON MANIFOLDS.pdf_大学文库

LECTURE 20: SEMI-CLASSICAL PsDOs ON MANIFOLDS1.SEMICLASSICALPsDOsUNDER COORDINATECHANGE Pseudolocality.As we have mentioned several times, we want to extend the theory of semiclas-sical pseudodifferential operators from Rn to smooth manifolds. So in some sense,we have to study how to “glue" local pseudodifferential operators to global ones.As we have seen last time, for differential operators it is easy to get global oper-ators from local ones, because of the locality of differential operators. In general,pseudodifferential operators does not satisfy locality.For example, one can look atthe Dirichlet-to-Neumann operator A in PSet 2, which is by definition the map thatsends the Dirichlet boundary value of a Harmonic function on Rn+1 to its Neumannboundary value. It is easy to find f e C(Rn) with f = 0 on an open subset U inIRn, while A(f) + 0 on U.Fortunately, although locality fails, we do have a weaker version of locality forpseudodifferential operators, namely,thepseudolocality.Roughly speaking,localitytells us that the value Au(r) at a point r is determined by the values u(y) for y nearr,while pseudolocality tells us that the value Au(r)is determined, modulo O(h)(which is negligible in semiclassical analysis), by the values u(y) for y near r. Tosee the pseudolocality of A = aw, we just start with the definition of aw, namely1et(r-w)sa(“,)u(y)dyaWu(r) :(2元h)nJR2and noticethat away from the diagonal,namely in theregion r-yl> C (whichcan be produced by multiplying uby a cut-off function that is supported away fromthe point r), one can produce as many h's as we want via integration by parts usinget(c-)s =h(α-) Dset(a-)s.[-y12Pseudolocality can also be explained as follows. Suppose a E S(m) and supposeX1, X2 E Co(IR") are two cut-off functions such thatsupp(x1) n supp(x2) = 0.Then (c.f. PSet 3)XiaWx2 = O(h~).So if we take Xi to be a cut-off function supported near (which equals 1 in a smallneighborhood of ) and take X2 to be a cut-off function supported away,then theeffect of the value of u in the support of x2 is negligible in computing awu(r)

LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS 1. Semiclassical PsDOs under coordinate change ¶ Pseudolocality. As we have mentioned several times, we want to extend the theory of semiclassical pseudodifferential operators from R n to smooth manifolds. So in some sense, we have to study how to “glue” local pseudodifferential operators to global ones. As we have seen last time, for differential operators it is easy to get global operators from local ones, because of the locality of differential operators. In general, pseudodifferential operators does not satisfy locality. For example, one can look at the Dirichlet-to-Neumann operator Λ in PSet 2, which is by definition the map that sends the Dirichlet boundary value of a Harmonic function on R n+1 + to its Neumann boundary value. It is easy to find f ∈ C ∞(R n ) with f ≡ 0 on an open subset U in R n , while Λ(f) 6= 0 on U. Fortunately, although locality fails, we do have a weaker version of locality for pseudodifferential operators, namely, the pseudolocality. Roughly speaking, locality tells us that the value Au(x) at a point x is determined by the values u(y) for y near x, while pseudolocality tells us that the value Au(x) is determined, modulo O(~ ∞) (which is negligible in semiclassical analysis), by the values u(y) for y near x. To see the pseudolocality of A = ba W , we just start with the definition of ba W , namely ba W u(x) = 1 (2π~) n Z Rn e i ~ (x−y)·ξ a( x + y 2 , ξ)u(y)dy and notice that away from the diagonal, namely in the region |x − y| > C (which can be produced by multiplying u by a cut-off function that is supported away from the point x), one can produce as many ~’s as we want via integration by parts using e i ~ (x−y)·ξ = ~ (x − y) · Dξ |x − y| 2 e i ~ (x−y)·ξ . Pseudolocality can also be explained as follows. Suppose a ∈ S(m) and suppose χ1, χ2 ∈ C ∞ 0 (R n ) are two cut-off functions such that supp(χ1) ∩ supp(χ2) = ∅. Then (c.f. PSet 3) χ1ba W χ2 = O(~ ∞). So if we take χ1 to be a cut-off function supported near x (which equals 1 in a small neighborhood of x) and take χ2 to be a cut-off function supported away, then the effect of the value of u in the support of χ2 is negligible in computing ba W u(x). 1

2 LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS ¶ Semiclassical PsDOs under coordinate change. So we still want to study the change of semiclassical pseudodifferential operator under coordinate change. Again we start with a diffeomorphism f : U ⊂ R n x → V ⊂ R n y where we always assume |∂ α f|, |∂ α f −1 | ≤ Cα, ∀α, and again we want to use f to “transplant”, now a pseudodifferential operator P defined for x-variables to a pseudodifferential operator for y-variables. However, we can’t define Pe to be the operator (f −1 ) ∗P|U f ∗ , since pseudodifferential operators are not local and thus P|U makes no sense. The solution to this issue is straightforward: instead of localize the operator, we “globalize” the functions. How do we “globalize” a locally defined function? Multiply it by a cut-off function! More precisely, suppose χ ∈ C ∞ 0 (V ), then we have maps f ∗Mχ : S (R n y ) → C ∞ 0 (U) ⊂ C ∞ 0 (R n x ) ⊂ S (R n x ) and Mχ(f −1 ) ∗ : S (R n x ) ⊂ C ∞(R n x ) → C ∞ 0 (R n y ) ⊂ S (R n y ). which extends to maps1 f ∗Mχ : S 0 (R n y ) → S 0 (R n x ) and Mχ(f −1 ) ∗ : S 0 (R n x ) → S 0 (R n y ). So given any pseudodifferential operator P on R n x , which, for simplicity, we assume its Kohn-Nirenberg symbol is a, namely P = ba KN (for a in some symbol class), and given any cut-off function χ ∈ C ∞ 0 (V ), we can define Peχ : S (R n y ) → S (R n y ) (and Peχ : S 0 (R n y ) → S 0 (R n y )) by (1) Peχ = Mχ(f −1 ) ∗P f ∗Mχ. 1Given a tempered distribution u ∈ S 0 (R n y ) and a compactly supported function χ = χ(y), we can extend Mχ to Mχ : S 0 (R n y ) → S 0 (R n y ) defined by hMχu, ϕi := hu, Mχϕi, ∀ϕ ∈ S (R n y ). The definition of pull-back of distribution is a bit complicated: Given any diffeomorphism f : U → V , one has the pull-back map on functions, f ∗ : C∞ 0 (V ) → C∞ 0 (U). By duality, we get a linear map f∗ : D(U) → D(V ), called the push-forward, defined by hf∗u, ϕi := hu, f ∗ϕi. Moreover, it can be shown that the restriction of f∗ to C∞ 0 (U) is a continuous linear map from C∞ 0 (U) to C∞ 0 (V ). By duality again, we get a pull-back map, now defined on distributions, f ∗ : D0 (V ) → D0 (U). [For pull-back by submersions, c.f. Theorem 6.1.2 in H¨ormander, Vol 1.]

LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS 5 ¶ Semiclassical PsDOs under coordinate change: invariant symbols. The theorem we just proved also holds for invariant symbols: Theorem 1.2. Suppose a ∈ S m. Then the operator Peχ defined by (1) is a pseudodifferential operator whose symbol b ∈ S m has the asymptotic expansion (3) b(y, η) ∼ a f −1 (y),(∂f) T η χ(y) 2 + X j≥1 ~ j bj (y, η), for some bj ∈ S m−j . Proof. The proof is lengthy and we will omit it. We need to apply a more complicated version of lemma of stationary phase, e.g. Theorem 7.7.7 in H¨ormander, The Analysis of Partial Differential Operators Vol I. See also Theorem 18.1.7 in H¨ormander, The Analysis of Partial Differential Operators Vol III. In fact, according to Theorem 18.1.7 in the book, one has the following somewhat different expression of b: b(y, η) ∼ X α 1 α! ∂ α ξ a(f −1 (y),(∂f) T η)(χ(y))2 (~Dz) α (e i ~ ρx(z)·η )|z=x=f−1(y) , where ρx(z) = f(x) − f(z) − ∂f(z)(x − z). It is crucial to notice that ρx(x) = 0 and ∂ρx(x) = 0, so that to produce one η-factor (and one ~ −1 -factor) from (~Dz) α (e i ~ ρx(z)·ξ )|z=x=f−1(y) you need at least two derivatives. As a result, the right hand side is an asymptotic expansion, but the α-term is NOT in ~ |α|S m−|α| , but in ~ [|α|/2]S m−[|α|/2] instead. Remark. Usually an asymptotic expansion in S m has the form a(x, ξ) ∼ X∞ j=0 ~ j aj (x, ξ), where a ∈ S m, and aj ∈ S m−j . In this setting, negligible symbols are symbols in ~ ∞S −∞. 2. Semiclassical pseudodifferential operator on manifolds ¶ Semiclassical pseudodifferential operator on manifolds. Recall that for each m, the symbol class S m consisting of symbols satisfying |∂ α x ∂ β ξ a| ≤ Cα,βhξi m−|β| for all α, β is invariant under coordinate changes in phase space, namely (x, ξ) fe(x, ξ) = y = f(x), η = (∂f −1 ) T ξ , where f : R n → R n is a diffeomorphism that has bounded derivatives (of any order), e.g. f is the identity map outside a compact set.

6 LECTURE 20: SEMI-CLASSICAL PSDOS ON MANIFOLDS Now let M be a compact smooth manifold. Then from any coordinate chart (ϕα, Uα, Vα) we get a natural diffeomorphism ϕg−1 α : Vα × R n → T ∗Uα We define S m(T ∗M) be the set of smooth functions a ∈ C ∞(T ∗M) so that for any coordinate chart (ϕα, Uα, Vα), (4) ϕg−1 α ∗ (a|T ∗U ) ∈ S m(Vα × R n ). According to the invariance property of the class S m (Theorem 1.3 in Lecture 18), the set S m(T ∗M) is well-defined. We also need to introduce the class of smoothing operators. One can prove (exercise) that for a ∈ S m, the Schwartz kernel ka of ba W is smooth off diagonal, i.e. ka = ka(x, y) ∈ C ∞(R n × R n \ {x = y}). In general, we say ba W is a smoothing operator if it has a smooth kernel ka ∈ C ∞(R n × R n ) and the kernel satisfies |∂ α x ∂ β y ka(x, y)| ≤ CαβN ( ~ hx − yi ) N for all α, β and N. One can show that any smoothing operator is the Weyl quantization of a negligible symbol a ∈ ~ ∞S −∞. On compact manifolds this condition can be a bit simpler: An operator A on a smooth manifold M is a smoothing operator if it has a smooth Schwartz kernel with |∂ α x ∂ β y ka(x, y)| ≤ CαβN ~ N . The class of smoothing operators is denoted by ~ ∞Ψ−∞(M). Now we define Definition 2.1. A linear operator A = A~ : C ∞(M) → C ∞(M) is called a (semiclassical) pseudo-differential operator of order m on M if it has the form A = X j Mχjϕ ∗ jabj W (ϕ −1 j ) ∗Mχj + ~ ∞Ψ −∞(M) for a finite collection of charts {ϕj , Uj , Vj} and cut-off functions χj ∈ C ∞ c (Uj ), where aj ∈ S m. We will denote the class of semiclassical pseudodifferential operator as define above by Ψm(M). The following equivalent characterization is left as an exercise: Proposition 2.2. A linear operator A = A~ : C ∞(M) → C ∞(M) is a (semiclassical) pseudo-differential operator of order m on M if and only if (1) for each coordinate patch (ϕα, Uα, Vα), there exists a cut-off function χ and a symbol aα,χ ∈ S m such that (ϕα) ∗MχAMχ(ϕ −1 α ) ∗ = adα,χ W (2) A is pseudolocal, i.e. for any χ1, χ2 ∈ C ∞(M) with suppχ1 ∩ suppχ2 = ∅, (5) χ1Aχ2 ∈ ~ ∞Ψ −∞(M)