《半经典微局部分析 Semiclassical Microlocal Analysis》课程教学资源（英文讲义）Lec16 GENERALIZED SOBOLEV SPACES

LECTURE16:GENERALIZEDSOBOLEVSPACES1.GENERALIZED SOBOLEV SPACESFor most of the previous five lectures, we are studying aw for a e S(1), sincein this case aw is a bounded linear operator on L?(IRn). A natural question is: formore general m, what can be said for aw with a e S(m)? In particular for those mwhich diverges to +oo as z-→ oo, what can we say about aw with a E S(m)? The Sobolev space.Let's start with an example. Considera(r, s) = [s12.Then wehaveaw=-h?A,which is of course one of the most important operators in geometry and analysis.Wemaytakem(r,s) = (s)2.Then it is easy to see a e S(m). Since a S(1), the operator △ is unbounded (andthus is only densely defined)onL?(Rn). However, it is a standard fact form PDEthat admits a natural domain:the Sobolev spaceH?(R") =[uE L(R") AuE L?(Rn))which isaHilbertspacewiththeSobolevnormIIul 2(R") = Z IIDul|L2(R")la≤2Here isanother way to think of the Sobolevspace H?(Rn):(I -h?△)H?(R") = L?(R")By definition we have (I -h?△)H?(Rn) L?(R").To prove the reverse inclusion,for any f EL?(R")we need to solve the PDE-h△u+u=f.We have solved such equations via Fourier transform at thebeginning of Lecture14:12(z) = Fr(1 +ISpF(0),It remains to check u e L2(R") and △u L?(R"), both of which are consequencesof the fact Fr is an isomorphism on L?(Rn).1

LECTURE 16: GENERALIZED SOBOLEV SPACES 1. Generalized Sobolev spaces For most of the previous five lectures, we are studying ba W for a ∈ S(1), since in this case ba W is a bounded linear operator on L 2 (R n ). A natural question is: for more general m, what can be said for ba W with a ∈ S(m)? In particular for those m which diverges to +∞ as z → ∞, what can we say about ba W with a ∈ S(m)? ¶ The Sobolev space. Let’s start with an example. Consider a(x, ξ) = |ξ| 2 . Then we have ba W = −~ 2∆, which is of course one of the most important operators in geometry and analysis. We may take m(x, ξ) = hξi 2 . Then it is easy to see a ∈ S(m). Since a 6∈ S(1), the operator ∆ is unbounded (and thus is only densely defined) on L 2 (R n ). However, it is a standard fact form PDE that ∆ admits a natural domain: the Sobolev space H 2 (R n ) = {u ∈ L 2 (R n ) | ∆u ∈ L 2 (R n )}, which is a Hilbert space with the Sobolev norm kukH2(Rn) = X |α|≤2 kD αukL2(Rn) . Here is another way to think of the Sobolev space H2 (R n ): (I − ~ 2∆)H 2 (R n ) = L 2 (R n ) By definition we have (I − ~ 2∆)H2 (R n ) ⊂ L 2 (R n ). To prove the reverse inclusion, for any f ∈ L 2 (R n ) we need to solve the PDE −~ 2∆u + u = f. We have solved such equations via Fourier transform at the beginning of Lecture 14: u(x) = F −1 ~ ( 1 1 + |ξ| 2 F~(f)). It remains to check u ∈ L 2 (R n ) and ∆u ∈ L 2 (R n ), both of which are consequences of the fact F~ is an isomorphism on L 2 (R n ). 1

2LECTURE16:GENERALIZEDSOBOLEVSPACESNote that I -h? is invertible since its symbol 1+[$/2 is elliptic in S(m), wherem(r,)=(s)2.So we may rewrite theabove equation asH2(R") = (I - h△)-1L?(R")Recall:althoughI-h?△is onlydenselydefined,theinverse(1-h?)-1isagloballydefined compact operator on L?(R").The Sobolev norm can also be defined via the operator . More precisely, onecan prove that the Sobolev norm alluded to above is equivalent to the Sobolev normIlull (R") :=I(I -2)u/L2(R"),In other words, the Sobolev space isnot only a space defined for the Laplace operator,butalsoaspacedefined viatheoperator△ The generalized Sobolev spaces.Observe that the most important thing in the above discussion is that the symbol1 +s2 is elliptic in S(m), where m(r,$) =($)? ≥ 1. Inspired by this observation.we may define, for any order function m ≥ 1 and any elliptic symbol g e S(m), thefollowinggeneralized SoboleunormIlulHr(m.9) := gWul/2.We know that this is well-defined at least for all u E . Let's first investigate thedependence of this norm with the elliptic symbol g. It turns out that the norm is"almost" independent of g and thus is essentially an intrinsic property of the orderfunction m:Lemma 1.1. Suppose m ≥1 and g, g' are two elliptic symbols in S(m). Then thegeneralized Sobolev norms defined via g and g' are equivalent: there erists ho > 0and C > 0 such that for all h e (o, ho),lllHn(m.g) ≤ Ilull Hn(m,g) ≤Clul H(m,g), Vu E 9.Proof. Since g is elliptic in S(m) and m ≥ 1, there exists ho > 0 and h E S(1/m)such that (gw)-1 = hw for all h e (0, ho). It follows that g' h E S(1) and thusthere exists C > 0 such thatIIgW 。hW llc(L2(R") ≤C(uniform for all h e (O,ho). It followsIul(mg) = IIg" 。hw gW ull2(R")≤CulH(m.g),口The other half can be proved by exchanging g and g' above.As a consequence, in the definition of the generalized Sobolev norm, we mayerase g and simply denote it by Il -IlHn(m):

2 LECTURE 16: GENERALIZED SOBOLEV SPACES Note that I −~ 2∆ is invertible since its symbol 1+|ξ| 2 is elliptic in S(m), where m(x, ξ) = hξi 2 . So we may rewrite the above equation as H 2 (R n ) = (I − ~ 2∆)−1L 2 (R n ). Recall: although I−~ 2∆ is only densely defined, the inverse (I−~ 2∆)−1 is a globally defined compact operator on L 2 (R n ). The Sobolev norm can also be defined via the operator ∆. More precisely, one can prove that the Sobolev norm alluded to above is equivalent to the Sobolev norm kukH2 ~ (Rn) := k(I − ~ 2∆)ukL2(Rn) . In other words, the Sobolev space is not only a space defined for the Laplace operator ∆, but also a space defined via the operator ∆. ¶ The generalized Sobolev spaces. Observe that the most important thing in the above discussion is that the symbol 1 + kξk 2 is elliptic in S(m), where m(x, ξ) = hξi 2 ≥ 1. Inspired by this observation, we may define, for any order function m ≥ 1 and any elliptic symbol g ∈ S(m), the following generalized Sobolev norm kukH~(m,g) := kgb W ukL2 . We know that this is well-defined at least for all u ∈ S . Let’s first investigate the dependence of this norm with the elliptic symbol g. It turns out that the norm is “almost” independent of g and thus is essentially an intrinsic property of the order function m: Lemma 1.1. Suppose m ≥ 1 and g, g0 are two elliptic symbols in S(m). Then the generalized Sobolev norms defined via g and g 0 are equivalent: there exists ~0 > 0 and C > 0 such that for all ~ ∈ (0, ~0), 1 C kukH~(m,g) ≤ kukH~(m,g0) ≤ CkukH~(m,g) , ∀u ∈ S . Proof. Since g is elliptic in S(m) and m ≥ 1, there exists ~0 > 0 and h ∈ S(1/m) such that (gb W ) −1 = bh W for all ~ ∈ (0, ~0). It follows that g 0 ? h ∈ S(1) and thus there exists C > 0 such that kgb0 W ◦ bh W kL(L2(Rn)) ≤ C (uniform for all ~ ∈ (0, ~0)). It follows kukH~(m,g0) = kgb0 W ◦ bh W ◦ gb W ukL2(Rn) ≤ CkukH~(m,g) . The other half can be proved by exchanging g and g 0 above. As a consequence, in the definition of the generalized Sobolev norm, we may erase g and simply denote it by k · kH~(m) :

3LECTURE16:GENERALIZEDSOBOLEVSPACESDefinition 1.2. We will define the generalized Soboleu norm associated to m to beIlulHh(m) := IIgWullL2.We will denote the completion of under the norm Il lHr(m) by Hr(m), and callit the generalized Soboleu space associated to m.Note that in the proof of the inequality IlullH(m.g") ≤ ClullHn(mg) above, weonly used the fact that g'+ h is a bounded symbol. In particular, we conclude: if gis elliptic in S(m), g is elliptic in S(m'), and if m' ≤ m, then there exists C > 0such that IlullHn(m) ≤Cllull Hh(m). In other words, we have:Corollary 1.3.Ifm≤m, thenHn(m)CHr(m')DETOUR: Choice of order function.One may ask: is there any canonical way to choose an elliptic symbol g in S(m)?For example, in the standard Sobolev space case, we used the elliptic symbol 1+Is?,which is in fact the same as m =(s)?. Note that by definition, if m E S(m), thenm isautomatically elliptic inS(m).Recall that. a continuous function m on Rd is an order function if m(z) ≤ C(z-w)N m(w).. S(m) contains those smooth functions all of whose derivatives are boundedby the function m.So in general it is not always true that m S(m): m could benon-smooth, or smoothbut quite “"oscillating"so thatits derivatives are not nicelybounded.However,ourexperience from analysis tells us that there is a big chance that these bad behaviorscould be eliminated by using convolution:Lemma 1.4.For any order function m, there erists an order function m such that(1) S(m) = S(m).(2) mES(m)Proof. Take a cut-off function n E C(Rd) with n ≥ 0 and J ndz = 1. Letm(2) =m*n(z) = / m(z -w)n(w)dwbe the convolution of m and n. According to the definition of an order function,C-(a)-~ me) ≤C(e).m(z)It followsC-1m≤m≤Cm,which implies S(m) = S(m).Moreover,for any multi-index a, by commutativity of convolution we have[2%m|=m*2l≤Cm口so mE S(m) = S(m)

LECTURE 16: GENERALIZED SOBOLEV SPACES 3 Definition 1.2. We will define the generalized Sobolev norm associated to m to be kukH~(m) := kgb W ukL2 . We will denote the completion of S under the norm k · kH~(m) by H~(m), and call it the generalized Sobolev space associated to m. Note that in the proof of the inequality kukH~(m,g0) ≤ CkukH~(m,g) above, we only used the fact that g 0 ? h is a bounded symbol. In particular, we conclude: if g is elliptic in S(m), g 0 is elliptic in S(m0 ), and if m0 ≤ m, then there exists C > 0 such that kukH~(m0) ≤ CkukH~(m) . In other words, we have: Corollary 1.3. If m0 ≤ m, then H~(m) ⊂ H~(m0 ). ¶ DETOUR: Choice of order function. One may ask: is there any canonical way to choose an elliptic symbol g in S(m)? For example, in the standard Sobolev space case, we used the elliptic symbol 1+|ξ| 2 , which is in fact the same as m = hξi 2 . Note that by definition, if m ∈ S(m), then m is automatically elliptic in S(m). Recall that • a continuous function m on R d is an order function if m(z) ≤ Chz−wi Nm(w). • S(m) contains those smooth functions all of whose derivatives are bounded by the function m. So in general it is not always true that m ∈ S(m): m could be non-smooth, or smooth but quite “oscillating” so that its derivatives are not nicely bounded. However, our experience from analysis tells us that there is a big chance that these bad behaviors could be eliminated by using convolution: Lemma 1.4. For any order function m, there exists an order function me such that (1) S(me ) = S(m). (2) me ∈ S(me ) Proof. Take a cut-off function η ∈ C ∞ 0 (R d ) with η ≥ 0 and R ηdz = 1. Let me (z) = m ∗ η(z) = Z m(z − w)η(w)dw be the convolution of m and η. According to the definition of an order function, C −1 hwi −N ≤ m(z − w) m(z) ≤ Chwi N . It follows C −1m ≤ me ≤ Cm, which implies S(m) = S(me ). Moreover, for any multi-index α, by commutativity of convolution we have |∂ αme | = |m ∗ ∂ α η| ≤ Cαm, so me ∈ S(m) = S(me ).

4LECTURE16:GENERALIZEDSOBOLEVSPACESIn what follows we will always assume m E S(m), so that in the definition ofHr(m), wecan simply takeg=m.We remark that as a direct consequence of m e S(m) and the formula for lthat we used a couple times, we have m-1 e S(m-1). More generally mt e S(mt)for any t e R. (Reason: Let lal ≥ 1. Without loss of generality, we may assumeQi≥1and denotea= (ai-1,α2,.,Qa).Then(1)(a-1)ar(0ia)loga=(a-l01a)=So if m E S(m), then a°log m is bounded for any |al ≥ 1. Since mt = etlogm, weimmediately get mt e S(mt).) As a consequence, we see (s)t e s((s)t) for any t.T The generalized Sobolev spaces: examples.Erample. If m = 1, then H(m) = L?(Rn).Erample. More generally let m = m(r) be a smooth function that depends only onr and suppose m S(m). Then mw is the “multiplication by m(r)" operator. SoHr(m) = L?(Rn,m?(r)dr): The Sobolev norm isIul|H(m) = [lul/2(R控,m2(r)dr)Erample. On the other hand, suppose m=m(s) depends only on and m E S(m).Then we have mWu= F-'[m($)Fru($)]. SomWuE L?(R")m(S)FhuE L(R").Moreover, the Sobolev norm is given byIlul(m)= (2h)-"Fnul/2(R,m2()de)Erample. In particular if m(r,)=(s)s. Then[uE S"/ / (5)2]Fru(3)PdE<+00Hr := Hh(s)) = 3and the Sobolev norm is explicitly given byIlull? = / 《5)2]Fu(E)PdENote that in the case s = k is an nonnegative integer, Hk is the usual Sobolev spacethat we are familiar with:(2)Hk = [ue gl I lull := IIDaull2 < +o0].lal<k

4 LECTURE 16: GENERALIZED SOBOLEV SPACES In what follows we will always assume m ∈ S(m), so that in the definition of H~(m), we can simply take g = m. We remark that as a direct consequence of m ∈ S(m) and the formula for ∂ α 1 a that we used a couple times, we have m−1 ∈ S(m−1 ). More generally mt ∈ S(mt ) for any t ∈ R. (Reason: Let |α| ≥ 1. Without loss of generality, we may assume α1 ≥ 1 and denote ˜α = (α1 − 1, α2, · · · , αd). Then (1) ∂ α log a = ∂ α˜ (a −1 ∂1a) = X β+γ=˜α  α˜ β  ∂ β (a −1 )∂ γ (∂1a). So if m ∈ S(m), then ∂ α log m is bounded for any |α| ≥ 1. Since mt = e tlog m, we immediately get mt ∈ S(mt ).) As a consequence, we see hξi t ∈ S(hξi t ) for any t. ¶ The generalized Sobolev spaces: examples. Example. If m = 1, then H~(m) = L 2 (R n ). Example. More generally let m = m(x) be a smooth function that depends only on x and suppose m ∈ S(m). Then mb W is the “multiplication by m(x)” operator. So H~(m) = L 2 (R n , m2 (x)dx). The Sobolev norm is kukH~(m) = kukL2(Rn x,m2(x)dx) . Example. On the other hand, suppose m = m(ξ) depends only on ξ and m ∈ S(m). Then we have mb W u = F −1 ~ [m(ξ)F~u(ξ)]. So mb W u ∈ L 2 (R n ) ⇐⇒ m(ξ)F~u ∈ L 2 (R n ). Moreover, the Sobolev norm is given by kuk 2 H~(m) = (2π~) −n kF~uk 2 L2(Rn ξ ,m2(ξ)dξ) . Example. In particular if m(x, ξ) = hξi s . Then H s ~ := H~(hξi s ) =  u ∈ S 0 | Z Rn hξi 2s |F~u(ξ)| 2 dξ < +∞  and the Sobolev norm is explicitly given by kuk 2 s = Z Rn hξi 2s |F~u(ξ)| 2 dξ. Note that in the case s = k is an nonnegative integer, Hk is the usual Sobolev space that we are familiar with: (2) H k = {u ∈ S 0 | kuk 2 k := X |α|≤k kD αuk 2 L2 < +∞}

LECTURE16:GENERALIZEDSOBOLEVSPACES52. SEMICLASSICAL PSDOACTING ON THE GENERALIZED SOBOLEVSPACES Semiclassical PsDO acting on the generalized Sobolev spaces.Just as in the previous example, △ can be defined on H?(IR"), we can proveProposition 2.1. Suppose m ≥ 1. For any a E S(m), there erists ho > 0 suchthat for any h e (o, ho), the map aw : g -→ g can be ertended to a bounded linearoperatoraw : H(m) → L?(R").Proof. The proof is almost the same as above: We take g = m. by definition,uEHr(m)mWuEL(R").As before,aw (mW)-1 is semiclassical pseudodifferential operator with symbol inS(1) and thus is a bounded linear operator on L?(Rn). It followsawullL2 = law。(mW)-1 mWullL≤ClullHn(m)口As a consequence,Corollary 2.2. Assume m ≥ 1, a E S(m) is real-valued. If a+i is elliptic in S(m),then aW : Hn(m) C L? -→ L? is self-adjoint.Proof. Since a is real-valued, aw is symmetric.By ellipticity of a +i, the operatoraW ±i : Hr(m) → L? has an inverse (which is a bounded linear operator on L?(Rn))and thus is bijective. The conclusion follows.口Another very important consequence isCorollary 2.3. Suppose m ≤1 and suppose a E S(m) is elliptic. ThenaW : L?(Rn) -→ Hr(1/m)and there eaists b e S(1/m) so that 6W : H(1/m) -→ L?(R") is the inverse ofaW.Proof. The conclusion Image(aw) c Hr(1/m) follows from the fact that (1/m)" oawhas bounded symbol and thus is a bounded linear operator on L?, so thatuEL?(Rn)(1/m)"oaWuEL?(Rn)aWuEHn(1/m)By Lecture 14, aw admits a left inverse and a right inverse, namely, there existsb, c e s(1/m) so that at least on , we haveawobw-Id,cwoaw-Id.Since each operator maps to , we get w - w on y. Since m ≤ 1, we have1/m ≥ 1. So both w and w can be extended to continuous linear operators fromHh(1/m) to L2(IR"). Since is dense in H(1/m), we conclude bw = aw. So bw is口the inverse of aw.This also implies a is bijective onto Hr(1/m)

LECTURE 16: GENERALIZED SOBOLEV SPACES 5 2. Semiclassical PsDO acting on the generalized Sobolev spaces ¶ Semiclassical PsDO acting on the generalized Sobolev spaces. Just as in the previous example, ∆ can be defined on H2 (R n ), we can prove Proposition 2.1. Suppose m ≥ 1. For any a ∈ S(m), there exists h0 > 0 such that for any ~ ∈ (0, ~0), the map ba W : S → S can be extended to a bounded linear operator ba W : H~(m) → L 2 (R n ). Proof. The proof is almost the same as above: We take g = m. by definition, u ∈ H~(m) ⇐⇒ mb W u ∈ L 2 (R n ). As before, ba W ◦ (mb W ) −1 is semiclassical pseudodifferential operator with symbol in S(1) and thus is a bounded linear operator on L 2 (R n ). It follows kba W ukL2 = kba W ◦ (mb W ) −1 ◦ mb W ukL2 ≤ CkukH~(m) . As a consequence, Corollary 2.2. Assume m ≥ 1, a ∈ S(m) is real-valued. If a+i is elliptic in S(m), then ba W : H~(m) ⊂ L 2 → L 2 is self-adjoint. Proof. Since a is real-valued, ba W is symmetric. By ellipticity of a + i, the operator ba W ±i : H~(m) → L 2 has an inverse (which is a bounded linear operator on L 2 (R n )) and thus is bijective. The conclusion follows. Another very important consequence is Corollary 2.3. Suppose m ≤ 1 and suppose a ∈ S(m) is elliptic. Then ba W : L 2 (R n ) → H~(1/m), and there exists b ∈ S(1/m) so that bb W : H~(1/m) → L 2 (R n ) is the inverse of ba W . Proof. The conclusion Image(ba W ) ⊂ H~(1/m) follows from the fact that (1\/m) W ◦ba W has bounded symbol and thus is a bounded linear operator on L 2 , so that u ∈ L 2 (R n ) =⇒ (1\/m) W ◦ ba W u ∈ L 2 (R n ) =⇒ ba W u ∈ H~(1/m). By Lecture 14, ba W admits a left inverse and a right inverse, namely, there exists b, c ∈ S(1/m) so that at least on S , we have ba W ◦ bb W = Id, bc W ◦ ba W = Id. Since each operator maps S to S , we get bb W = bc W on S . Since m ≤ 1, we have 1/m ≥ 1. So both bb W and bc W can be extended to continuous linear operators from H~(1/m) to L 2 (R n ). Since S is dense in H~(1/m), we conclude bb W = bc W . So bb W is the inverse of ba W . This also implies ba is bijective onto H~(1/m).

6LECTURE16:GENERALIZEDSOBOLEVSPACES The generalized Sobolev spaces Hr(m) for any m (with m E S(m))For anorderfunctionm with meS(m),wedefineDefinition 2.4. The generalized Soboleu space associated with m isHh(m) := (uES"I mWuE L?(R"))(3)with the Sobolev norm(4)Iul|Hn(m) = [mW ulL2(Rn).Note that this coincides with our earlier definition. Also note that for any mand any a e S(m), we have aw : " → '. So all the following expressions makesense as tempered distributions.By definition, if m ≥ 1, one may think of Hn(m) as a function space whoseelements have more regularity than those in Hr(1)-L?, while if m< 1, one maythink of Hr(m)as a“function space"whose elements have less regularity than thosein Hr(1) = L?. We have just seeing how an operator of the form aw with a e S(m)will increase or decrease the regularity according to whether m ≥ 1 or m ≤ 1. Itturns our that this is true for any two order functions:Proposition 2.5. Suppose m andm' are order functions on IR2n.For anya E S(m),wehaveaW e C(Hr(m), Hn(m'/m),with the operator norm bound uniform inh.Proof. Since m' is elliptic in S(m'), we can find b e S(1/m) such thatwomw=Idon.Since (m'/m)* a*b E S(1), we conclude that for any u E,Iawullh(m /m)=lm /maw。Wom"ul/a≤Clll;(m),where the constant C= m'/m oaw willc(L2) is uniform in h. Since S is dense ineachHn(m)(provethis!),aw extendstoabounded linearoperator fromHr(m'/m)口to Hr(m' with operatornorm bounded by the same constant C.As a consequence, we seeCorollary 2.6. If a E S(m) is elliptic, then there erists b S(1/m) such that forany m',6W= (aW)-1 E C(Hn(m'/m),H(m).Proof. We have see that there exist b,c e S(1/m) such thatwoaw=Id=awocwong.It follows bw = w on J. But bw,cw e C(Hr(m'/m), Hr(m'), so they must口coincideon L(Hr(m'/m)since is dense

6 LECTURE 16: GENERALIZED SOBOLEV SPACES ¶ The generalized Sobolev spaces H~(m) for any m (with m ∈ S(m)). For an order function m with m ∈ S(m), we define Definition 2.4. The generalized Sobolev space associated with m is (3) H~(m) := {u ∈ S 0 | mb W u ∈ L 2 (R n )} with the Sobolev norm (4) kukH~(m) = kmb W ukL2(Rn) . Note that this coincides with our earlier definition. Also note that for any m and any a ∈ S(m), we have ba W : S 0 → S 0 . So all the following expressions make sense as tempered distributions. By definition, if m ≥ 1, one may think of H~(m) as a function space whose elements have more regularity than those in H~(1) = L 2 , while if m ≤ 1, one may think of H~(m) as a “function space” whose elements have less regularity than those in H~(1) = L 2 . We have just seeing how an operator of the form ba W with a ∈ S(m) will increase or decrease the regularity according to whether m ≥ 1 or m ≤ 1. It turns our that this is true for any two order functions: Proposition 2.5. Suppose m and m0 are order functions on R 2n . For any a ∈ S(m), we have ba W ∈ L(H~(m0 ), H~(m0 /m)), with the operator norm bound uniform in ~. Proof. Since m0 is elliptic in S(m0 ), we can find b ∈ S(1/m0 ) such that bb W ◦ mc0 W = Id on S . Since (m0/m) ? a ? b ∈ S(1), we conclude that for any u ∈ S , kba W ukH~(m0/m) = km\0/m W ◦ ba W ◦ bb W ◦ mc0 W ukL2 ≤ CkukH~(m0) , where the constant C = km\0/m W ◦ba W ◦bb W kL(L2) is uniform in ~. Since S is dense in each H~(m) (prove this!), ba W extends to a bounded linear operator from H~(m0/m) to H~(m0 with operator norm bounded by the same constant C. As a consequence, we see Corollary 2.6. If a ∈ S(m) is elliptic, then there exists b ∈ S(1/m) such that for any m0 , bb W = (ba W ) −1 ∈ L(H~(m0 /m), H~(m0 )). Proof. We have see that there exist b, c ∈ S(1/m) such that bb W ◦ ba W = Id = ba W ◦ bc W on S . It follows bb W = bc W on S . But bb W , bc W ∈ L(H~(m0/m), H~(m0 )), so they must coincide on L(H~(m0/m) since S is dense.

1LECTURE16:GENERALIZEDSOBOLEVSPACESWith the“well-defined inverse"on suitable space, many earlier computationsextendsto allm.Forexample,Corollary1.3nowholdsfor anym,m'.AlsowecansayHr(m)= (mW)-1L?(R"),generalizing the similar formula at the top of page 2 for the ordinary Sobolev spaceH-(Rn).We list several otherresults,and leave theproofs as an happy exercise:. The L? dual of Hr(m) is Hr(1/m).. g =nmHr(m) and l=UmHr(m).Compactness of awfor aE S(m).We have the following result which generalize our earlier results for L?(R"):Proposition 2.7. Suppose m and m' are order functions on R2n. Iflim m(z) = 0,→then for any a E S(m), the operatoraW : Hr(m') → Hr(m)is compact.Proof. The condition implies m'/m ≥ Cm' and thusaW : H(m) → Hr(m' /m) C Hn(m').The conclusion is equivalent to the fact that the mapm" 。aW 。(m")-1 : L2(R") -→ L2(R")口is compact, which is true because m' a *b e S(m).Finally we remark that there exists Sobolev space version of Beals's theorem andSharp Garding inequality,for “classical symbols", a subset of S((s)) that consistsof symbols a = a(r,s) in S((s)) so that for all multi-indices α and β, there existsa constant Ca,β such that[ogoga(r,E)/≤ Ca,B(s)m-10l.(5)Such symbols has the nice property that they are invariant under coordinate changeand thus can be defined on manifolds

LECTURE 16: GENERALIZED SOBOLEV SPACES 7 With the “well-defined inverse” on suitable space, many earlier computations extends to all m. For example, Corollary 1.3 now holds for any m, m0 . Also we can say H~(m) = (mb W ) −1L 2 (R n ), generalizing the similar formula at the top of page 2 for the ordinary Sobolev space H2 (R n ). We list several other results, and leave the proofs as an happy exercise: • The L 2 dual of H~(m) is H~(1/m). • S = ∩mH~(m) and S 0 = ∪mH~(m). ¶ Compactness of ba W for a ∈ S(m). We have the following result which generalize our earlier results for L 2 (R n ): Proposition 2.7. Suppose m and m0 are order functions on R 2n . If limz→∞ m(z) = 0, then for any a ∈ S(m), the operator ba W : H~(m0 ) → H~(m0 ) is compact. Proof. The condition implies m0/m ≥ Cm0 and thus ba W : H~(m0 ) → H~(m0 /m) ⊂ H~(m0 ). The conclusion is equivalent to the fact that the map mc0 W ◦ a W ◦ (mc0 W ) −1 : L 2 (R n ) → L 2 (R n ) is compact, which is true because m0 ? a ? b ∈ S(m). Finally we remark that there exists Sobolev space version of Beals’s theorem and Sharp Garding inequality, for “classical symbols”, a subset of S(hξi k ) that consists of symbols a = a(x, ξ) in S(hξi k ) so that for all multi-indices α and β, there exists a constant Cα,β such that (5) |∂ α x ∂ β ξ a(x, ξ)| ≤ Cα,βhξi m−|β| . Such symbols has the nice property that they are invariant under coordinate change and thus can be defined on manifolds.