LECTURE18:SEMI-CLASSICALPsDOsWITHCLASSICAISYMBOLS1. CLASSICAL SYMBOLS Change of coordinates in T*Rn.We are aiming at extending the definition of semiclassical pseudodifferentialoperators acting on L?(Rn) to operators acting on functions on manifolds. For thispurpose we need to study how a semiclassical pseudodifferential operator will changeundera coordinate change.Let's start by studying how coordinates will change in phase space.Supposef : Rn → Rn is a diffeomorphism, which represents a coordinate changerwy=f(a)in configuration space.Then the induced coordinate change in phase space will havetheform(r,E) ~ (y=f(r),n=n(r,s)To figure out the new cotangent variables n, let's recall the definition of cotangentvariable &: Given any 1-form α at a point , the coordinate of α is determined bythe equationa=Sidci+...+EndanSimilarly in the new coordinate system (y, n), one must havea=midyi+..+nndyn.Itfollowsthatndy =der=$())dy0So the new cotangent coordinates n is given bya-iTEn=(lgr Change of symbols under coordinate change in T*RnNow we study the following problem: if we change the coordinate system in theconfiguration space (andthuschangethecoordinate system in thephase spaceasdescribed above), how will the symbol change?For this purpose, let's suppose a = a(r, s). Consider the coordinate change(,5) ~ (() (%-1)T
LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS 1. Classical symbols ¶ Change of coordinates in T ∗R n . We are aiming at extending the definition of semiclassical pseudodifferential operators acting on L 2 (R n ) to operators acting on functions on manifolds. For this purpose we need to study how a semiclassical pseudodifferential operator will change under a coordinate change. Let’s start by studying how coordinates will change in phase space. Suppose f : R n → R n is a diffeomorphism, which represents a coordinate change x y = f(x) in configuration space. Then the induced coordinate change in phase space will have the form (x, ξ) (y = f(x), η = η(x, ξ)). To figure out the new cotangent variables η, let’s recall the definition of cotangent variable ξ: Given any 1-form α at a point x, the coordinate ξ of α is determined by the equation α = ξ1dx1 + · · · + ξndxn. Similarly in the new coordinate system (y, η), one must have α = η1dy1 + · · · + ηndyn. It follows that η T dy = ξ T dx = ξ T ( ∂x ∂y )dy. So the new cotangent coordinates η is given by η = ([∂y ∂x] −1 ) T ξ. ¶ Change of symbols under coordinate change in T ∗R n . Now we study the following problem: if we change the coordinate system in the configuration space (and thus change the coordinate system in the phase space as described above), how will the symbol change? For this purpose, let’s suppose a = a(x, ξ). Consider the coordinate change (x, ξ) y(x),([∂y ∂x] −1 ) T ξ 1
2LECTURE18:SEMI-CLASSICALPSDOSWITHCLASSICALSYMBOLSLet a be the function a(y,n) = a(r,s). Suppose a E S(m). ThenOyk0[(-1)Oma(y,n)) aua(y,n)+Or,a(r,s) =arSo in general a(r, s) will not be a symbol in S(m), even if we pose a boundednessassumptionontheJacobianmatrixof thecoordinatechangeinconfigurationmspace (which is not a very restrictiveassumption sincewewill mainlyworkoncompact manifolds).The bad term is the appeared in the expression. Even if weare working on compact manifold, the cotangent variable is still unbounded, sothat O,aisno longer bounded bym.(Onthe other hand,it is easyto seethattheE-derivatives of a behaves as nice as n-derivatives of a if we assume the boundednessof the Jacobian. So the S-term is essentially the only bad term.)Classical symbols.To solve this problem, we will have to restrict ourselves to order functions ofspecial form.In view of the computation above, we are naturallylead to study thosesymbols which will lost one -order after taking each -derivative.Definition 1.1. A classical symbol of order m E R is a function a E S((s)m) sothat1oaga(r,S)/≤ Ca,B(E)m-18l(1)for all multi-indices a and B. As usual, the symbol a is allowed to be h-dependent,in which case we require the constant Ca,β to be uniform for h e (O,ho)We will denote this class sm, and denotem=[aw lae sm].We will also denoteSo=UJ sms-8=nsm,mezmezand亚=Um亚-00Om,mEZmenFor later purpose, we list several simple properties of classical symbols whose proofsare quite obvious:Proposition1.2.Wehave(1) If a e sm, then ogaga e Sm-l0l.(2) If a e Smi,be Sm2, then ab e Sm1+m2 and [a,b) e Sm1+m2-1(3) c S-00
2 LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS Let ˜a be the function ˜a(y, η) = a(x, ξ). Suppose ˜a ∈ S(m). Then ∂xja(x, ξ) = ∂yk ∂xj ∂yk a˜(y, η) + ∂ ∂xj ([∂y ∂x] −1 ) T ξ k ∂ηk a˜(y, η). So in general a(x, ξ) will not be a symbol in S(m), even if we pose a boundedness assumption on the Jacobian matrix ∂yk ∂xj of the coordinate change in configuration space (which is not a very restrictive assumption since we will mainly work on compact manifolds). The bad term is the ξ appeared in the expression. Even if we are working on compact manifold, the cotangent variable ξ is still unbounded, so that ∂xa is no longer bounded by m. (On the other hand, it is easy to see that the ξ-derivatives of a behaves as nice as η-derivatives of ˜a if we assume the boundedness of the Jacobian. So the ξ-term is essentially the only bad term.) ¶ Classical symbols. To solve this problem, we will have to restrict ourselves to order functions of special form. In view of the computation above, we are naturally lead to study those symbols which will lost one ξ-order after taking each ξ-derivative. Definition 1.1. A classical symbol of order m ∈ R is a function a ∈ S(hξi m) so that (1) |∂ α x ∂ β ξ a(x, ξ)| ≤ Cα,βhξi m−|β| for all multi-indices α and β. As usual, the symbol a is allowed to be ~-dependent, in which case we require the constant Cα,β to be uniform for ~ ∈ (0, ~0). We will denote this class S m, and denote Ψ m = {ba W | a ∈ S m}. We will also denote S −∞ = \ m∈Z S m, S∞ = [ m∈Z S m. and Ψ −∞ = \ m∈Z Ψ m, Ψ ∞ = [ m∈Z Ψ m. For later purpose, we list several simple properties of classical symbols whose proofs are quite obvious: Proposition 1.2. We have (1) If a ∈ S m, then ∂ α x ∂ β ξ a ∈ S m−|β| . (2) If a ∈ S m1 , b ∈ S m2 , then ab ∈ S m1+m2 and {a, b} ∈ S m1+m2−1 . (3) S ⊂ S −∞
3LECTURE18:SEMI-CLASSICALPSDOSWITHCLASSICALSYMBOLSTInvarianceofclassical symbolsNowweprovethattheclassical symbols is invariant undercoordinatechange(under the boundedness assumption on the derivatives of the Jacobian):Theorem1.3.Let f :Rn→Rn be a diffeomorphism with(2)[0% fl ≤Ca, 10° f-1] ≤Cαfor all multi-indices a. Then for any classical symbol a E sm, its “pull back"f-1(r),f*a(r,s) := a (is also a classical symbol in Sm.Proof. If we write c(r,$) = a(f(r),s) then f*a = c(r,(ofa)TE), and thus thefunction(f*a)hastheformZCyap(oagc)sp/+0≤al,l0/=/lIt followsZa (f*a)Capuka(O1og+-c)Ep-AI+≤al,o=[pl,//=/I,=β,v≥,p≥)So we conclude[aga (a) / ≤Ec(5)m-10l-1+/l()l-/l ≤C(5)m-1l,口Remark. In applications the diffeomorphism f can be taken to be a orientation-preserving diffeomorphism from a bounded region to anotherbounded region, whichcould be extended to a diffeomorphism on Rn that is the identity map outside acompact set, so that the condition (2)hold automatically.↑ Polyhomogeneous symbols Sphg.In literature there is another widely used symbol class, the class of polyhomoge-neous symbols, which is by definition the space of symbols a E Sm of the formT7ak0Awhere ak is a symbol which is a degree k homogeneous function with respect to s,namely,as(c, ) =a(c,s)Onecanprove that this class is also invariant under coordinate change,and establishsymboliccalculusforthisclassasbefore.Notethatpseudodifferential operatorswithpolyhomogeneous symbols already contains all differential operators
LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS 3 ¶ Invariance of classical symbols. Now we prove that the classical symbols is invariant under coordinate change (under the boundedness assumption on the derivatives of the Jacobian): Theorem 1.3. Let f : R n → R n be a diffeomorphism with (2) |∂ α f| ≤ Cα, |∂ α f −1 | ≤ Cα for all multi-indices α. Then for any classical symbol a ∈ S m, its “pull back” fe∗a(x, ξ) := a f −1 (x), ∂f ∂x f−1(x) !T ξ is also a classical symbol in S m. Proof. If we write c(x, ξ) = a(f(x), ξ) then fe∗a = c x,(∂fx) T ξ , and thus the function ∂ α x (fe∗a) has the form X |γ|+|σ|≤|α|,|σ|=|ρ| Cγσρ(∂ γ x ∂ σ ξ c)ξ ρ It follows ∂ α x ∂ β ξ fe∗ a = X |γ|+|σ|≤|α|,|σ|=|ρ|,|κ|=|λ|,|ν|=β,ν≥κ,ρ≥λ Cγσρνκλ(∂ γ x ∂ σ+ν−κ ξ c)ξ ρ−λ . So we conclude ∂ α x ∂ β ξ fe∗ a ≤ XChξi m−|σ|−|ν|+|κ| hξi |ρ|−|λ| ≤ Chξi m−|β| . Remark. In applications the diffeomorphism f can be taken to be a orientationpreserving diffeomorphism from a bounded region to another bounded region, which could be extended to a diffeomorphism on R n that is the identity map outside a compact set, so that the condition (2) hold automatically. ¶ Polyhomogeneous symbols S m phg. In literature there is another widely used symbol class, the class of polyhomogeneous symbols, which is by definition the space of symbols a ∈ S m of the form a ∼ Xm k=−∞ ak, where ak is a symbol which is a degree k homogeneous function with respect to ξ, namely, ak(x, λξ) = λ k ak(x, ξ). One can prove that this class is also invariant under coordinate change, and establish symbolic calculus for this class as before. Note that pseudodifferential operators with polyhomogeneous symbols already contains all differential operators.
LECTURE18:SEMI-CLASSICALPSDOSWITHCLASSICALSYMBOLS2.SEMICLASSICALPSEUDODIFFERENTIALOPERATORSWITHCLASSICALSYMBOLS Symbolic calculus for classical symbols.Note that sm C S(s)m), so the formula we have proven for S(s)m) can beapplied to classical symbols.Moreover,duetothethe improvementunderdifferen-tiationin.therearenewfeaturesforthesymboliccalculusforsymbolsinSmTheorem 2.1. Suppose a e sm and b e Sm2, then a*b e Smi+m2, Moreover,1 (ih)kEhN+1gm1+m2-N-1(3)a*b-(DD,-DD,)[a(r,)(u,n)k=0.n=Proof. We have already known a *b e S(($)mi+m2) anda * b(r,E) = e (De -D,-D:-D,)(a(r,E)b(y, n)|1 (ih)(De· Dy - Dr- Dn)*[a(c, )b(y, n)k!2k1-0To estimate the remainder, we need to use the remainder formula for Taylor's ex-pansion, namely the left hand side of (3) equals(1 - t)Ne(De -D,-De-D,)nN+1 (Dg - Dy - Da- D,)N+ [a(r, )b(y, n)RN=CNdtSince (Dg Dy - D-D,)N+1 encounters exactly N +1 derivatives of and n, wehave1 (inh)kEhkgmi+m2-k-(Dg·Dy- Dr-Dn)*[a(r,E)b(y, n))Ck :k!2kr.n=andN+1(De · D, - Da · D,)N+1[a(,)b(y,n) es(E)m-k(n)ma-(N+1-k)On the other hand, we have seen in Lecture 1o (Theorem 3.2)that the operatoret(Dg -Dy-Dar-D) preserve such spaces. It followsN1 (ih)*E hN+1S(S)mi+m2-N-1a*b--(Dg-Dy-Dr·Dn)[a(c, E)b(y, n))k！2kk=0.n=Now for any Q, β, we have1B|1-(ab)=aagck+o2agRial-1e sm1+m2-k-10l+lls(5)m1+m2-18l)gak=0k=0
4 LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS 2. Semiclassical Pseudodifferential operators with classical symbols ¶ Symbolic calculus for classical symbols. Note that S m ⊂ S(hξi m), so the formula we have proven for S(hξi m) can be applied to classical symbols. Moreover, due to the the improvement under differentiation in ξ, there are new features for the symbolic calculus for symbols in S m. Theorem 2.1. Suppose a ∈ S m1 and b ∈ S m2 , then a ? b ∈ S m1+m2 . Moreover, (3) a?b− X N k=0 1 k! (i~) k 2 k (Dξ ·Dy−Dx·Dη) k [a(x, ξ)b(y, η)] y=x,η=ξ ∈ ~ N+1S m1+m2−N−1 . Proof. We have already known a ? b ∈ S(hξi m1+m2 ) and a ? b(x, ξ) = e i~ 2 (Dξ ·Dy−Dx·Dη) (a(x, ξ)b(y, η)) y=x,η=ξ ∼ X∞ k=0 1 k! (i~) k 2 k (Dξ · Dy − Dx · Dη) k [a(x, ξ)b(y, η)] y=x,η=ξ . To estimate the remainder, we need to use the remainder formula for Taylor’s expansion, namely the left hand side of (3) equals RN = CN Z 1 0 (1 − t) N e i~ 2 t(Dξ ·Dy−Dx·Dη) ~ N+1 (Dξ · Dy − Dx · Dη) N+1 [a(x, ξ)b(y, η)] y=x,η=ξ dt. Since (Dξ · Dy − Dx · Dη) N+1 encounters exactly N + 1 derivatives of ξ and η, we have ck := 1 k! (i~) k 2 k (Dξ ·Dy−Dx·Dη) k [a(x, ξ)b(y, η)] y=x,η=ξ ∈ ~ kS m1+m2−k and (Dξ · Dy − Dx · Dη) N+1 [a(x, ξ)b(y, η)] ∈ N X +1 k=0 S(hξi m1−k hηi m2−(N+1−k) ) On the other hand, we have seen in Lecture 10 (Theorem 3.2) that the operator e i~ 2 t(Dξ ·Dy−Dx·Dη) preserve such spaces. It follows a?b− X N k=0 1 k! (i~) k 2 k (Dξ ·Dy−Dx·Dη) k [a(x, ξ)b(y, η)] y=x,η=ξ ∈ ~ N+1S(hξi m1+m2−N−1 ). Now for any α, β, we have ∂ α x ∂ β ξ (a ? b) = | X β|−1 k=0 ∂ α x ∂ β ξ ck + ∂ α x ∂ β ξ R|β|−1 ∈ | X β|−1 k=0 ~ kS m1+m2−k−|β| + ~ |β|S(hξi m1+m2−|β| )
LECTURE18:SEMI-CLASSICALPSDOSWITHCLASSICALSYMBOLS5which implies10ga%(a *b)/ ≤ Ca,B(E)m1+ma-10l,where Ca,β is uniform for h e (0, ho). So a*b e Sm+m2.The proof of (3) is similar: To prove [a8ag Rnl ≤ Ca,β(s)m+m2-1al, we split Rnintotwoparts,withfirst[β|-1termshandledtermbyterminSmi+m2-N-1-k-18]and theremaining terms handled together as an element in S($)mi+ma-N-1-l).Remark. Similar property holds for other t-quantizations.As a consequence, we haveCorollary 2.2. If Ae mi,B e m2, then(1) AB E m1+m2.(2) [A, B] e hm1+m2-1Mappingpropertiesforoperatorsinm.For operators A e m, it is natural to consider the action of A on Sobolev spacesHr(m) with m = (s)s. We denoteHi:= Hr((s)) = [uE" / (P)s e L?].Since Sm S((s)m), we immediately get the following version of Calderon-Vaillancourttheorem for classical symbols:Proposition 2.3. Suppose a E Sm.Then aW e C(Hr, H-m).Note that if Ae m, thenWado,A = -hDe,a"and adp,A=-Dr,So for any ii,...,iM,ji,..,jn E {l, ...,n], one hasado.*.adQmadp,..adp.wA=(-1)M+NhMooga EhMym-Mand thusIadQ. ado adp, adp All(H+m,H+M) = O(h).Conversely we also has the following Beals's characterization for operators in m:Theorem 2.4 (Beals theorem for m). A continuous linear operator A : → )isin class m if and onlyif thereerists seR and foranyi,...,im,ji,..,jnE[1,..,n], one hasIlado. .adQw adp, adpwAllc(H+",H+M) = O(h)
LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS 5 which implies |∂ α x ∂ β ξ (a ? b)| ≤ Cα,βhξi m1+m2−|β| . where Cα,β is uniform for ~ ∈ (0, ~0). So a ? b ∈ S m1+m2 . The proof of (3) is similar: To prove |∂ α x ∂ β ξ RN | ≤ Cα,βhξi m1+m2−|β| , we split RN into two parts, with first |β| − 1 terms handled term by term in S m1+m2−N−1−k−|β| , and the remaining terms handled together as an element in S(hξi m1+m2−N−1−|β| ). Remark. Similar property holds for other t-quantizations. As a consequence, we have Corollary 2.2. If A ∈ Ψm1 , B ∈ Ψm2 , then (1) AB ∈ Ψm1+m2 . (2) [A, B] ∈ ~Ψm1+m2−1 . ¶ Mapping properties for operators in Ψm. For operators A ∈ Ψm, it is natural to consider the action of A on Sobolev spaces H~(m) with m = hξi s . We denote H s ~ := H~(hξi s ) = {u ∈ S 0 | hPi s ∈ L 2 }. Since S m ⊂ S(hξi m), we immediately get the following version of Calderon-Vaillancourt theorem for classical symbols: Proposition 2.3. Suppose a ∈ S m. Then ba W ∈ L(Hs ~ , Hs−m ~ ). Note that if A ∈ Ψm, then adQjA = −~D[ξja W and adPjA = −D[xja W . So for any i1, · · · , iM, j1, · · · , jN ∈ {1, · · · , n}, one has adQi1 · · · adQiM adPj1 · · · adPiN A = (−1)M+N ~ M∂ α x ∂ β ξ a ∈ ~ MΨ m−M and thus kadQi1 · · · adQiM adPj1 · · · adPiN AkL(H s+m ~ ,Hs+M ~ ) = O(~ M). Conversely we also has the following Beals’s characterization for operators in Ψm: Theorem 2.4 (Beals theorem for Ψm). A continuous linear operator A : S → S 0 is in class Ψm if and only if there exists s ∈ R and for any i1, · · · , iM, j1, · · · , jN ∈ {1, · · · , n}, one has kadQi1 · · · adQiM adPj1 · · · adPiN AkL(H s+m ~ ,Hs+M ~ ) = O(~ M).
6LECTURE18:SEMI-CLASSICALPSDOSWITHCLASSICALSYMBOLS Sharp Garding inequality for m.Supposea E Sm then aw maps L?to H-m.And map Hm to L?We also have the following sharp Garding inequality for classical symbols:Theorem 2.5 (Sharp Garding inequality for classical symbols). Suppose a E Smis a real-valued symbol such that a ≥0. Then there erists C≥0 such that for anyue,(aWu, u) ≥-Chl[lull1)/2Proof.Wefirstobserve that it is enoughtoprovethetheorem form=O.Supposethis is done, then for any other m e R and a sm, we letb = (5)-m/2 *a*(E)-m/2.Then byTheorem 2.1, b e So, and thus(aWu, u) = (bW。(E)m/2"u,《E)m/2"u) ≥-Chll《E)m/2" ull/2=-ChllullSoinwhatfollowsweassumem=O, andproveCh(aWu, u) ≥ -Chu/-1/2 =《s)-1Fhu(S)Pds(2元h)nNote that in Lecture 15 we already proved (for nonnegative symbol a eso S(1))(aWu, u) ≥ -Chllul/22. To improve the estimate from L?-norm to H-1/2-norm, wewill use dyadic decomposition in frequency space to decompose a and thus aw, andestimateterm byterm:Lemma 2.6 (Dyadic partition of unity). There erists bo E Co(Rn)and ECo(Rn /[0]), both with alues in [0,1], such that1 = o(E) +(2-ie),j=0口Proof. Exercise.In what follows we fix such an dyadic POU and denote ;()= (2-ig). Thena(r, ) =;(5)a(r, ) =a;(c,2-i),j=0j=0where ao(r,S) = bo(s)a(r, s) and a;(r,s) = (s)a(r, 2is) for j ≥ 1. One has[0g0ga,/ ≤Cap21l0l max(2's)-10lCap.EEsupp()ie. a, E S(1) with S(1)-seminorm bounds uniform in j. It follows that《a,Wu, ) ≥ -Ch[ul/2(R"),where C is independent of j
6 LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS ¶ Sharp G˚arding inequality for Ψm. Suppose a ∈ S m then ba W maps L 2 to H −m ~ . And map Hm to L 2 We also have the following sharp G˚arding inequality for classical symbols: Theorem 2.5 (Sharp G˚arding inequality for classical symbols). Suppose a ∈ S m is a real-valued symbol such that a ≥ 0. Then there exists C ≥ 0 such that for any u ∈ S , hba W u, ui ≥ −C~kuk 2 H (m−1)/2 ~ Proof. We first observe that it is enough to prove the theorem for m = 0. Suppose this is done, then for any other m ∈ R and a ∈ S m, we let b = hξi −m/2 ? a ? hξi −m/2 . Then by Theorem 2.1, b ∈ S 0 , and thus hba W u, ui = hbb W ◦ h\ξim/2 W u,h\ξim/2 W ui ≥ −C~kh\ξim/2 W ukH −1/2 ~ = −C~kuk 2 H m−1/2 ~ . So in what follows we assume m = 0, and prove hba W u, ui ≥ −C~kuk 2 H −1/2 ~ = −C~ (2π~) n Z Rn hξi −1 |F~u(ξ)| 2 dξ. Note that in Lecture 15 we already proved (for nonnegative symbol a ∈ S 0 ⊂ S(1)) hba W u, ui ≥ −C~kuk 2 L2 . To improve the estimate from L 2 -norm to H−1/2 -norm, we will use dyadic decomposition in frequency space to decompose a and thus ba W , and estimate term by term: Lemma 2.6 (Dyadic partition of unity). There exists ψ0 ∈ C ∞ 0 (R n ) and ψ ∈ C ∞ 0 (R n \ {0}), both with values in [0, 1], such that 1 = ψ0(ξ) +X∞ j=0 ψ(2−j ξ). Proof. Exercise. In what follows we fix such an dyadic POU and denote ψj (ξ) = ψ(2−j ξ). Then a(x, ξ) = X∞ j=0 ψj (ξ)a(x, ξ) = X∞ j=0 aj (x, 2 −j ξ), where a0(x, ξ) = ψ0(ξ)a(x, ξ) and aj (x, ξ) = ψ(ξ)a(x, 2 j ξ) for j ≥ 1. One has |∂ α x ∂ β ξ aj | ≤ Cαβ2 j|β| max ξ∈supp(ψ) h2 j ξi −|β| ≤ Ceαβ. i.e. aj ∈ S(1) with S(1)-seminorm bounds uniform in j. It follows that habj W u, ui ≥ −C~kuk 2 L2(Rn) , where C is independent of j.
LECTURE18:SEMI-CLASSICALPSDOSWITHCLASSICALSYMBOLS7Now we use the decompositionaw-ah=2-hjWe choose bo e Co(IRn) and e Co(Rn /fo!), both with values in [0,1], such that6 = 1 on supp(b), o = 1 on supp(to),and denote ;(s) = b(2-i) as before. Then - 1 on supp(a,) andWWWW山aw=Wl=2-+O()aj"=2-h=2-jhIt follows that for any ue(aWu,u) =(a,"(h=2-ihu,u)W~W[=2-inb , )-O(N2-iN)lull>>W≥-Ch2-ib /2-0()l-1/2,Now we use the following fact: there exists C> O such that for any u E,(4)Z2-≤Cu1/3j=0Proof:Recall fromthedefinition (Lecture16)thatIull2《s)-1/Fnu()12de-1/2（2元h）nSinceZ2-/2=2h J2--4Since2-C(s)-1,supp()and since for any S,there are at most finitely many different j's so that b,()+ 0andthereisatleastonejwithbi()=O,weconclude2-j1b;()/≤C(E)-1,EEsupp()It follows(aw u, u) ≥ -Chll/m-/2口
LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS 7 Now we use the decomposition ba W = X j abj W |~=2−j~. We choose ψe0 ∈ C0(R n ) and ψe ∈ C ∞ 0 (R n \ {0}), both with values in [0, 1], such that ψe = 1 on supp(ψ), ψe0 = 1 on supp(ψ0), and denote ψej (ξ) = ψe(2−j ξ) as before. Then ψe = 1 on supp(aj ) and ψ cej W abj W |~=2−j~ b ψe W = b ψe W abj W b ψe W ~=2−j~ = abj W |~=2−j~ + O(~ ∞ j ). It follows that for any u ∈ S , hba W u, ui = X j habj W |~=2−j~u, ui ≥ X j habj W |~=2−j~ b ψe W u, b ψe W ui − O(~ N 2 −jN )kukH −1/2 ~ ≥ −C~ X j 2 −j k b ψe W uk 2 L2 − O(~ N )kukH −1/2 ~ , Now we use the following fact: there exists C > 0 such that for any u ∈ S , (4) X∞ j=0 2 −j kψ cej W uk 2 L2 ≤ Ckuk 2 H −1/2 ~ . Proof: Recall from the definition (Lecture 16) that kuk 2 H −1/2 ~ = 1 (2π~) n Z Rn hξi −1 |F~u(ξ)| 2 dξ. Since X j 2 −j kψ cej W uk 2 L2 = 1 (2π~) n Z Rn X j 2 −j |ψej | 2 |F~u| 2 dξ, Since 2 −j ≤ Chξi −1 , ξ ∈ supp(ψej ) and since for any ξ, there are at most finitely many different j’s so that ψej (ξ) 6= 0 and there is at least one j with ψej (ξ) = 0, we conclude X j 2 −j |ψej (ξ)| ≤ Chξi −1 , ξ ∈ supp(ψej ) It follows hba W u, ui ≥ −C~kuk 2 H −1/2 ~
0LECTURE 18:SEMI-CLASSICALPSDOS WITHCLASSICALSYMBOLSRemark.· Sharp Garding inequality also holds for other t-quantizations, inwhich case one only need to replace (awu, u) by Re(atu, u).·One also has the stronger Fefferman-Phong inequality:(aw u, ) ≥ -Ch lll/(.m-2)/2
8 LECTURE 18: SEMI-CLASSICAL PSDOS WITH CLASSICAL SYMBOLS Remark. • Sharp Garding inequality also holds for other t-quantizations, in which case one only need to replace hba W u, ui by Rehba tu, ui. • One also has the stronger Fefferman-Phong inequality: hba W u, ui ≥ −C~ 2 kuk 2 H (m−2)/2 ~