LECTURE14:L?-THEORYOFSEMICLASSICALPsDOs:ELLIPTICITYToday we study the invertibility of semiclassical pseudodifferential operators: inparticular we want to find out conditions so that Opt(a) is invertible on L(Rn),and the inverse is another semiclassical pseudodifferential operator. Here is anilluminating example: suppose we want to solve a partial differential equation aaDu = f.lol<kByapplyingtheFouriertransform,weget( aas")a= f.lal<kSo if the symbol function p(r, E) = jal<k aa is invertible, then we get= p(r,s)-1fand thus at least formally,u= F-1(p(r,E)-1Ff) =(p(r,E)-1)KN(f)In fact this argument is one of the original motivations of introducing pseudodiffer-ential operators in mathematics, and it fits into our general philosophy perfectly: theinvertibility of a (semiclassical) pseudodifferential operator is closely related to theinvertibility of its symbol function! Of course we should be a little bit more careful:not only we want p(r,) to be invertible, but also we want the inverse p(r, ) to beinsomenicesymbolclass1.ELLIPTICITYTDefinition of ellipticity.Now we study the invertibility of aw. By the above argument, we see that foraw to be invertible, we want a to be invertible as a function. Moreover, supposea E S(m), where m = m(r,s) be an order function on R2n. Then we want theinverse 1/a to be in some nice symbol class.For example, if a E s(1), then itis natural to require 1/a E S(1) (so that the quantization of the inverse is still abounded linear operator). This amounts to require the function |1/al to be bounded-
LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY Today we study the invertibility of semiclassical pseudodifferential operators: in particular we want to find out conditions so that Opt ~ (a) is invertible on L 2 (R n ), and the inverse is another semiclassical pseudodifferential operator. Here is an illuminating example: suppose we want to solve a partial differential equation X |α|≤k aαD αu = f. By applying the Fourier transform, we get ( X |α|≤k aαξ α )ˆu = ˆf. So if the symbol function p(x, ξ) = P |α|≤k aαξ α is invertible, then we get uˆ = p(x, ξ) −1 ˆf and thus at least formally, u = F −1 (p(x, ξ) −1Ff) = (p(\x, ξ) −1 ) KN (f). In fact this argument is one of the original motivations of introducing pseudodifferential operators in mathematics, and it fits into our general philosophy perfectly: the invertibility of a (semiclassical) pseudodifferential operator is closely related to the invertibility of its symbol function! Of course we should be a little bit more careful: not only we want p(x, ξ) to be invertible, but also we want the inverse p(x, ξ) to be in some nice symbol class. 1. Ellipticity ¶ Definition of ellipticity. Now we study the invertibility of ba W . By the above argument, we see that for ba W to be invertible, we want a to be invertible as a function. Moreover, suppose a ∈ S(m), where m = m(x, ξ) be an order function on R 2n . Then we want the inverse 1/a to be in some nice symbol class. For example, if a ∈ S(1), then it is natural to require 1/a ∈ S(1) (so that the quantization of the inverse is still a bounded linear operator). This amounts to require the function |1/a| to be bounded 1
2LECTURE14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITYabove, or equivalently, require [al to be bounded below:[a(r,s,h)/ ≥C >0, V(r,) R2n,h (0,1)1Similarly for a E S(m), the natural symbol class for 1/a is 1/m, and thus we wantthe function [1/al to be bounded above by Cm. As a result, it is natural to requirethatalis bounded blowbya constantmultipleoftheorderfunctionm.Itturnsoutthat this condition is also sufficient to guarantee that the inverse 1/a is in S(1/m):Lemma 1.1. Suppose there erists C > 0 such that a(r,s,h)|≥ Cm holds for all(c,s)eR2n and all he (0, 1), then 1/aE S(1/m)Proof. A tedious computation shows that (1/a) has the form1Z0LaBia)CBi..Bka.aβ1.+β&=α,|B,/21口The conclusion follows.SowedefineDefinition 1.2. We say a symbol a E S(m) is (semiclassically) elliptic in S(m) ifthere exists a constant Co > 0 (independent of h) such that(1)[a(r,s)/≥ComonR2nSo by Lemma 1.1, if a E S(m) is elliptic, then 1 E S(). Note that by theformula in the proof of Lemma 1.l, we can say more: if a e Ss(m) is elliptic inS(m), then e Ss(=).Throughout this lecture we always assume 0< <1/2.Constructingparametrix.First we prove that for an elliptic symbol a, the operator aw is “almost invert-ible":Theorem1.3.Supposethereerists 0≤<andan order functionm sothataESs(m)is elliptic in S(m).Then there eristsb,cESs()so thatawbw=Id+rw(2)woaw=Id+rwfor some ri,r2 = O(h) in S(1). Moreover, for m = 1 we can take b = c.lIn previous lectures we wrote h e (o, ho) and we used the special case h = 1. This is not aserious problem becausewe can always do rescaling.2Both equations are understood to be hold on dense subsets of L?(R") which contain 9
2 LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY above, or equivalently, require |a| to be bounded below: |a(x, ξ, ~)| ≥ C > 0, ∀(x, ξ) ∈ R 2n , ~ ∈ (0, 1]1 . Similarly for a ∈ S(m), the natural symbol class for 1/a is 1/m, and thus we want the function |1/a| to be bounded above by Cm. As a result, it is natural to require that |a| is bounded blow by a constant multiple of the order function m. It turns out that this condition is also sufficient to guarantee that the inverse 1/a is in S(1/m): Lemma 1.1. Suppose there exists C > 0 such that |a(x, ξ, ~)| ≥ Cm holds for all (x, ξ) ∈ R 2n and all ~ ∈ (0, 1], then 1/a ∈ S(1/m). Proof. A tedious computation shows that ∂ α (1/a) has the form ∂ α ( 1 a ) = 1 a X β1+···+βk=α,|βj |≥1 Cβ1···βk Y j ( 1 a ∂ βja). The conclusion follows. So we define Definition 1.2. We say a symbol a ∈ S(m) is (semiclassically) elliptic in S(m) if there exists a constant C0 > 0 (independent of ~) such that (1) |a(x, ξ)| ≥ C0m on R 2n . So by Lemma 1.1, if a ∈ S(m) is elliptic, then 1 a ∈ S( 1 m ). Note that by the formula in the proof of Lemma 1.1, we can say more: if a ∈ Sδ(m) is elliptic in S(m), then 1 a ∈ Sδ( 1 m ). Throughout this lecture we always assume 0 ≤ δ < 1/2. ¶ Constructing parametrix. First we prove that for an elliptic symbol a, the operator ba W is “almost invertible”: Theorem 1.3. Suppose there exists 0 ≤ δ < 1 2 and an order function m so that a ∈ Sδ(m) is elliptic in S(m). Then there exists b, c ∈ Sδ( 1 m ) so that2 (2) ba W ◦ bb W = Id + rb1 W , bc W ◦ ba W = Id + rb2 W for some r1, r2 = O(~ ∞) in S(1). Moreover, for m = 1 we can take b = c. 1 In previous lectures we wrote ~ ∈ (0, ~0) and we used the special case ~ = 1. This is not a serious problem because we can always do rescaling. 2Both equations are understood to be hold on dense subsets of L 2 (R n) which contain S .
3LECTURE 14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITYProof.The idea is to approximate the inverse step by step: in each step we use a"smaller"adjustment (of order O(hk))toget a better approximation (with errorterm of order O(h+1), and then we add all adjustments via Borel's lemma to getthe O(h)-approximated inverse that wewant.So of course thefirst step is to start with bi =1/a, then bi E Ss(1/m). Accordingto the composition formula (see the remark at the end of Lecture 10),(3)a*b1=1-hl-28r1for some ii E Ss(1). This is of course only the first approximation. To get betterapproximation,wemaytrytofindb,sothata (61 + b2) = 1 - t2(1-28)r2(4)for some 2E Ss(1).Plugging (3) into (4), we see that we need to find b2 so thata62=hl-28(—l-28元2)Comparing this “target equation"with the equation (3),we can easily find a candi-date for b2: if we take b2 = h1-28bi ii, then (4) is fulflled; more over the candidateisperfectbecause. we have b2 = hl-28bi *i e hl-25 Ss(1/m), which is “"smaller" than br. we have r2 = r1*ri e Ss(1), so the remainder h2(1-26)r2 E O(h2(1-20) is evensmaller.Now we repeat the process. It is easy to see that if we takebk+1=k(1-28)1i*·*1,then ba+1 E hk(1-20)Ss(1/m) anda * (6i + b2 + . + b+1) = 1 (k+1)(1-20) rk+1,where rk+1 =ii .. * ri E Ss(1). Now we apply Borel's Lemma (c.f. Lecture 10)toget b E S()which is theasymptotic sumb~b1+b2+b3+..Thenab=1+O(h).It follows thatawbw=Id+nwfor someri =O(h) in S(1).This gives thefirst identity.To prove the second identity, we first repeat the same argument above to findcE S() so thatWWoaw=Id+r2for some r = O(h) in S(1)
LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY 3 Proof. The idea is to approximate the inverse step by step: in each step we use a “smaller” adjustment (of order O(~ k ) ) to get a better approximation (with error term of order O(~ k + 1)), and then we add all adjustments via Borel’s lemma to get the O(~ ∞)-approximated inverse that we want. So of course the first step is to start with ˜b1 = 1/a, then ˜b1 ∈ Sδ(1/m). According to the composition formula (see the remark at the end of Lecture 10), (3) a ? ˜b1 = 1 − ~ 1−2δ r˜1 for some ˜r1 ∈ Sδ(1). This is of course only the first approximation. To get better approximation, we may try to find ˜b2 so that (4) a ? ( ˜b1 + ˜b2) = 1 − ~ 2(1−2δ) r˜2 for some ˜r2 ∈ Sδ(1). Plugging (3) into (4), we see that we need to find ˜b2 so that a ? ˜b2 = ~ 1−2δ (˜r1 − ~ 1−2δ r˜2). Comparing this “target equation” with the equation (3), we can easily find a candidate for ˜b2: if we take ˜b2 = ~ 1−2δ˜b1 ? r˜1, then (4) is fulfilled; more over the candidate is perfect because • we have ˜b2 = ~ 1−2δ˜b1 ? r˜1 ∈ ~ 1−2δSδ(1/m), which is “smaller” than ˜b1. • we have ˜r2 = ˜r1 ?r˜1 ∈ Sδ(1), so the remainder ~ 2(1−2δ) r˜2 ∈ O(~ 2(1−2δ) ) is even smaller. Now we repeat the process. It is easy to see that if we take ˜bk+1 = ~ k(1−2δ)˜b1 ? r˜1 ? · · · ? r˜1, then ˜bk+1 ∈ ~ k(1−2δ)Sδ(1/m) and a ? ( ˜b1 + ˜b2 + · · · + bk+1) = 1 − ~ (k+1)(1−2δ) r˜k+1, where rk+1 = ˜r1 ? · · · ? r1 ∈ Sδ(1). Now we apply Borel’s Lemma (c.f. Lecture 10) to get b ∈ S( 1 m ) which is the asymptotic sum b ∼ ˜b1 + ˜b2 + b3 + · · · . Then a ? b = 1 + O(~ ∞). It follows that ba W ◦ bb W = Id + rb1 W for some r1 = O(~ ∞) in S(1). This gives the first identity. To prove the second identity, we first repeat the same argument above to find c ∈ S( 1 m ) so that bc W ◦ ba W = Id + br 0 2 W for some r 0 2 = O(~ ∞) in S(1)
4LECTURE14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITYFinally we assume m = 1, in which case all operators involved are boundedlinear operators and thus aredefined on thewholeL?(Rn).Since(Id+")ow=Woaw。bw=W(Id+rW)we seeWw-aw+awonw-rwbw-aw+wfor some r = O(h) in S(). It followswoaw=cwoaw+rwoaw-Id+rw口for some r2 = O(h) in S(1).The operator w is called a parametrir of aw. The inverse of elliptic h-PsDO's.Since ri = O(h) in S(1), by Calderon-Vaillancourt Theorem, rW is a boundedlinear operator on L?(Rn), and moreover, for ho small enough,I/WIlc(L2(R") 0Moreover,I(Id+W)-'llc(L2(R) ≤2.It thus follows from the first equation in (2) thataw。(@W。(Id +rW)-1)= Id.Similarly we have((Id +rw)-1 bw)oaw = Id.Note that if a e S(1), then b S(1). So all operators appeared in the previous twoformulaes are bounded linear operators (thus well-defined) on L?(Rn), and in thiscasegetCorollary 1.4. Suppose a E Ss(1) is elliptic in S(1). Then aw is invertible with(aW)-1 =bW。(Id +rW)-1 = (Id +rW)-1 obWRemark. In the proof, instead of using ri,we may simply use the error of the firstapproximation, namely 1 - a + (1/a) = tl-28r1 = O(t1-25) e S(1).Another simple consequencefrom the boundedness of (Id +r2W)-1 is the fol-lowing classical-quantum correspondencephenomena:since the absolutevalue ofan elliptic symbol has a positive lower bound (w.r.t.m), the operator norm of itsquantized operator also has a positive lower bound
4 LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY Finally we assume m = 1, in which case all operators involved are bounded linear operators and thus are defined on the whole L 2 (R n ). Since (Id + br 0 2 W ) ◦ bb W = bc W ◦ ba W ◦ bb W = bc W ◦ (Id + rb1 W ) we see bb W = bc W + bc W ◦ rb1 W − br 0 2 W ◦ bb W = bc W + rb W for some r = O(~ ∞) in S( 1 m ). It follows bb W ◦ ba W = bc W ◦ ba W + rb W ◦ ba W = Id + rb2 W for some r2 = O(~ ∞) in S(1). The operator bb W is called a parametrix of ba W . ¶ The inverse of elliptic ~-PsDO’s. Since r1 = O(~ ∞) in S(1), by Calderon-Vaillancourt Theorem, rb1 W is a bounded linear operator on L 2 (R n ), and moreover, for ~0 small enough, krb1 W kL(L2(Rn)) < 1/2, ∀~ ∈ (0, ~0). It follows that for ~ ∈ (0, ~0), the operator (Id + rb1 W ) −1 exists and is the bounded linear operator given by the Neumann series (Id + rb1 W ) −1 = X k≥0 (−1)k (rb1 W ) k . Moreover, k(Id + rb1 W ) −1kL(L2(Rn)) ≤ 2. It thus follows from the first equation in (2) that ba W ◦ (bb W ◦ (Id + rb1 W ) −1 ) = Id. Similarly we have ((Id + rb2 W ) −1 ◦ bb W ) ◦ ba W = Id. Note that if a ∈ S(1), then b ∈ S(1). So all operators appeared in the previous two formulaes are bounded linear operators (thus well-defined) on L 2 (R n ), and in this case get Corollary 1.4. Suppose a ∈ Sδ(1) is elliptic in S(1). Then ba W is invertible with (ba W ) −1 = bb W ◦ (Id + rb1 W ) −1 = (Id + rb2 W ) −1 ◦ bb W . Remark. In the proof, instead of using r1, we may simply use the error of the first approximation, namely 1 − a ? (1/a) = ~ 1−2δ r˜1 = O(~ 1−2δ ) ∈ S(1). Another simple consequence from the boundedness of (Id + rb2 W ) −1 is the following classical-quantum correspondence phenomena: since the absolute value of an elliptic symbol has a positive lower bound (w.r.t. m), the operator norm of its quantized operator also has a positive lower bound.
LECTURE14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITY5Corollary1.5.Suppose a ESs(m)is elliptic in S(m). Supposem≥1. Then thereerists ho>0 and C>0 such that for all h E (0, ho),IaW IIc(L2(R") ≥ C.Proof. Since m ≥ 1, c E S(1/m) C S(1). So aW is bounded. It followsIlullL2(R") = I(Id + rW)-1 。W) oaW ulL2(R") ≤ ClaW ullz2(R"),口T The adjoint action.The invertibility of aw (for elliptic a E S(1)) on the space L?(Rn) (as a boundedlinear operator)is not that satisfying, since we don't know whether the inverse of awis still a semiclassical pseudodifferential operator. What we need is a criterion fora bounded linear operator A on L?(Rn) to be the quantization of a bounded symbola e s(1).For this purpose we introduce a notation, the adjoint action between operators:For operators A and B we denote(5)adAB =[A,B]= AB- BA.Erample. In PSet 2-6 we used the relation[A, BC] = [A, B]C + B[A, C].Using thisnewnotation, we can rewrite it asadA(BC)= (ad^B)C +B(ad^C),If B is invertible and we take C = B-1 in the above formula, we get(6)adA(B-1) = -B-1(adAB)B-1.Erample. In Lecture 9 we have seen (whose proof was left as an exercise in PSet 2)that for a linear symbol l(r,s) = cr + d .E, the quantization condition is exact,namely[w,aw] =.a"Using the adjoint action, we can rewritethis asadawaw(7)[l,a
LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY 5 Corollary 1.5. Suppose a ∈ Sδ(m) is elliptic in S(m). Suppose m ≥ 1. Then there exists ~0 > 0 and C > 0 such that for all ~ ∈ (0, ~0), kba W kL(L2(Rn)) ≥ C. Proof. Since m ≥ 1, c ∈ S(1/m) ⊂ S(1). So bc W is bounded. It follows kukL2(Rn) = k((Id + rb2 W ) −1 ◦ bc W ) ◦ ba W ukL2(Rn) ≤ Ckba W ukL2(Rn) . ¶ The adjoint action. The invertibility of ba W (for elliptic a ∈ S(1)) on the space L 2 (R n ) (as a bounded linear operator) is not that satisfying, since we don’t know whether the inverse of ba W is still a semiclassical pseudodifferential operator. What we need is a criterion for a bounded linear operator A on L 2 (R n ) to be the quantization of a bounded symbol a ∈ S(1). For this purpose we introduce a notation, the adjoint action between operators: For operators A and B we denote (5) adAB = [A, B] = AB − BA. Example. In PSet 2-6 we used the relation [A, BC] = [A, B]C + B[A, C]. Using this new notation, we can rewrite it as adA(BC) = (adAB)C + B(adAC). If B is invertible and we take C = B−1 in the above formula, we get (6) adA(B −1 ) = −B −1 (adAB)B −1 . Example. In Lecture 9 we have seen (whose proof was left as an exercise in PSet 2) that for a linear symbol l(x, ξ) = c · x + d · ξ, the quantization condition is exact, namely [bl W , ba W ] = ~ i {[l, a} W . Using the adjoint action, we can rewrite this as (7) adblW ba W = ~ i {[l, a} W .
6LECTURE14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITYSemi-classical Beals'stheorem,Now we try to search for the criterion. Suppose A =aw for some a E S(1).Then for any linear function I = l(r, $) = c. r + d ., the Poisson bracket[1,a] =(d,or,a+c,Og,a) e S(1).As a consequence, for any N and any linear functions li, ,ln on R2n, we have.0 adwAllc(L2(R") = O().JladIt turns out that this condition is not only necessary but also sufficient conditionfor an operator A to be the quantization of a bounded symbol a E S(1):Theorem 1.6 (Semiclassical Beals's theorem). Let A :→ be a continuouslinear operator. Then A = aw for a symbol a E S(1) if and only if for all N =O,1,2,... and all linear functions li,...,l on R2n, we have(8)Iladew o... adwAllc(L2(Rn) = O(h)).We postpone the proof to the end of this lecture.T The inverse of an elliptic h-PsDO as an PsDO.Now we apply semiclassical Beals' theorem to study the left and right inversesofaw,where a eSs(m)is elliptic.It is enough to consider thefollowing question:given a symbol r which is O(h) in S(1). Assume I/Wlc(L2(R") < 1/2. Is theinverse(Id +rW)-1 = (-1)*(W)kk≥0a semiclassical pseudodifferential operator? The answer is yes. In fact, since r Ehs(1) (sowe onlyneed touser=rintheproof of Theorem1.3for somek≥1/(1 -28)),Beals's theorem impliesIlad,w o... o adw(Id +W)llc(L2(R") = O(N)for any linear functions li,..:,ln on R2n,Nowweplay a magic:we apply theformula (6)recursivelytoconcludeIlad,w o..0 adw(Id +rW)-1llc(L2(R") = O(t)).So according to the semiclassical Beals's theorem, the operator (Id + hrW)-1 is asemiclassical pseudodifferential operator, ie. c e S(1) such that (id+hrW)-1 = wIn conclusion, we provedTheorem 1.7. Suppose a E Ss(1) is elliptic elliptic in S(1), then as a boundedlinear operator on L?(Rn), aw is invertible. Moreover, there erists b E Ss(1) so that(aW)-1 = 6w
6 LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY ¶ Semi-classical Beals’s theorem. Now we try to search for the criterion. Suppose A = ba W for some a ∈ S(1). Then for any linear function l = l(x, ξ) = c · x + d · ξ, the Poisson bracket {l, a} = X(dj∂xja + cj∂ξja) ∈ S(1). As a consequence, for any N and any linear functions l1, · · · , lN on R 2n , we have kadlb1 W ◦ · · · ◦ ad lcN W AkL(L2(Rn)) = O(~ N ). It turns out that this condition is not only necessary but also sufficient condition for an operator A to be the quantization of a bounded symbol a ∈ S(1): Theorem 1.6 (Semiclassical Beals’s theorem). Let A : S → S 0 be a continuous linear operator. Then A = ba W for a symbol a ∈ S(1) if and only if for all N = 0, 1, 2, · · · and all linear functions l1, · · · , lN on R 2n , we have (8) kadlb1 W ◦ · · · ◦ ad lcN W AkL(L2(Rn)) = O(~ N ). We postpone the proof to the end of this lecture. ¶ The inverse of an elliptic ~-PsDO as an PsDO. Now we apply semiclassical Beals’ theorem to study the left and right inverses of ba W , where a ∈ Sδ(m) is elliptic. It is enough to consider the following question: given a symbol r which is O(~ ∞) in S(1). Assume krb W kL(L2(Rn)) < 1/2. Is the inverse (Id + rb W ) −1 = X k≥0 (−1)k (rb W ) k a semiclassical pseudodifferential operator? The answer is yes. In fact, since r ∈ ~S(1) (so we only need to use r = rk in the proof of Theorem 1.3 for some k ≥ 1/(1 − 2δ)), Beals’s theorem implies kadlb1 W ◦ · · · ◦ ad lcN W (Id + rb W )kL(L2(Rn)) = O(~ N ) for any linear functions l1, · · · , lN on R 2n . Now we play a magic: we apply the formula (6) recursively to conclude kadlb1 W ◦ · · · ◦ ad lcN W (Id + rb W ) −1 kL(L2(Rn)) = O(~ N ). So according to the semiclassical Beals’s theorem, the operator (Id + ~rb W ) −1 is a semiclassical pseudodifferential operator, i.e. ∃c ∈ S(1) such that (Id+~rb W ) −1 = bc W . In conclusion, we proved Theorem 1.7. Suppose a ∈ Sδ(1) is elliptic elliptic in S(1), then as a bounded linear operator on L 2 (R n ), ba W is invertible. Moreover, there exists b ∈ Sδ(1) so that (ba W ) −1 = bb W
LECTURE14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITY7Similarly if a E Ss(m) is elliptic in S(m), and if m ≥ 1, then by repeating thearguments above, we can prove.aw has a left/right inverse which are bounded linear operators.: the left/right inverses of aw are semiclassical pseudodifferential operatorswith symbol in Ss(1/m).If we denote the left/right inverse of aw by bw and ew, then at least on g (we usethefact thatfora in any symbol class S(m),aw mapsto),wehavebw-bwoawew=cwBut since both w and w are continuous linear functionals on L?(Rn), and since is dense in L?(Rn), we conclude that bW = W on L?(Rn). In other words,Corollary 1.8. If a E Ss(m) is elliptic in S(m), and if m ≥ 1, then there eristsbe Ss(1/m) so that awobw- Id and bwoaw= Id.We will call this bw the inverse of aw.2.PROOFOF THE SEMICLASSICAL BEALS'STHEOREM The proof of the semiclassical Beals's theorem.Proof. We have seen that if a E S(1), then A = aw satisfies (8)Conversely, suppose the estimate (8) holds. We first assume h = 1. By theSchwartz kernel theorem that we mentioned in Lecture 6,Au(r) = / KA(r,9)u(y)dyfor some KA E '(R2n). According to the Fourier inversion formula, we have el(e--)sKa(*+, +wKA(r,y)=7)dwds(2元)n J2222So if we setww-msKA(a+ ,)dw,a(r,s) =2thene(r-)a(*,s)ds,KA(r,y) =(2元)n2i.e. A = aW|n=1. It remains to prove a e S(1). To do so, we apply our hypothesisto functions l, = &j and l, = E, and use (7) to conclude thatII(0a")h=1llc(L2) ≤ Cafor all multi-indices α.It remains to control theL-norm of a from the operatornorm of the operators ea. For this purpose we need the following “reverse" tothe Calderon-Vailancourt theorem:
LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY 7 Similarly if a ∈ Sδ(m) is elliptic in S(m), and if m ≥ 1, then by repeating the arguments above, we can prove • ba W has a left/right inverse which are bounded linear operators. • the left/right inverses of ba W are semiclassical pseudodifferential operators with symbol in Sδ(1/m). If we denote the left/right inverse of ba W by bb W and bc W , then at least on S (we use the fact that for a in any symbol class S(m), ba W maps S to S ), we have bb W = bb W ◦ ba W bc W = bc W . But since both bb W and bc W are continuous linear functionals on L 2 (R n ), and since S is dense in L 2 (R n ), we conclude that bb W = bc W on L 2 (R n ). In other words, Corollary 1.8. If a ∈ Sδ(m) is elliptic in S(m), and if m ≥ 1, then there exists b ∈ Sδ(1/m) so that ba W ◦ bb W = Id and bb W ◦ ba W = Id. We will call this bb W the inverse of ba W . 2. Proof of the semiclassical Beals’s theorem ¶ The proof of the semiclassical Beals’s theorem. Proof. We have seen that if a ∈ S(1), then A = ba W satisfies (8). Conversely, suppose the estimate (8) holds. We first assume ~ = 1. By the Schwartz kernel theorem that we mentioned in Lecture 6, Au(x) = Z Rn KA(x, y)u(y)dy for some KA ∈ S 0 (R 2n ). According to the Fourier inversion formula, we have KA(x, y) = 1 (2π) n Z Rn Z Rn e i(x−y−w)·ξKA( x + y 2 + w 2 , x + y 2 − w 2 )dwdξ. So if we set a(x, ξ) = Z Rn e −iw·ξKA(x + w 2 , x − w 2 )dw, then KA(x, y) = 1 (2π) n Z Rn e i(x−y)·ξ a( x + y 2 , ξ)dξ, i.e. A = ba W |~=1. It remains to prove a ∈ S(1). To do so, we apply our hypothesis to functions lj = xj and lj = ξj and use (7) to conclude that k(∂dαa W )~=1kL(L2) ≤ Cα for all multi-indices α. It remains to control the L ∞-norm of ∂ αa from the operator norm of the operators ∂dρa W . For this purpose we need the following “reverse” to the Calderon-Vailancourt theorem:
6LECTURE 14:L2-THEORY OFSEMICLASSICALPSDOS:ELLIPTICITY(Recall:The Calderon-Vailancout tells us that the operator norm ofawisbounded bytheLoo-norm of aforfinitelymanyQ.Sothefollowing theorem is a reverse:)Theorem 2.1. Let a e S'(R2n) and assume that for all multi-indiceswith ml ≤ 2n + 1, the operator raE (L?(R"). Then a EL(R") andIll/~ ≤c Z th1/2/1ira" ll(2(*),(9)hl<MnSo by Theorem 2.1,sup a°al ≤ Ca.R2In other words, a E S(1).Finally one can use the re-scaling trick to convert the general case to the h = 1口case. An “"reverse" to the Calderon-Vaillancourt Theorem.It remains to prove Theorem 2.1.Proof. We will only prove the theorem for h = 1 and for the Kohn-Nirenberg quanti-zation. The general case follows from re-scaling trick and the change of quantizationformula.Again we prove a local estimate and then globalize it, this time since we areestimating the Lo-norm, we don't even need to choose a partition of unity: toglobalize the estimate it is enough to choose a cut-off function that is positive locally,prove a local L estimate, then use translation to get the demanded global estimate.So we let = p(r) and = (s) be functions in which equals 1 near 0. PutX(r,E) =(r)(E)eir-5.Then x E (IR2n), and Ix(r, $)| = 1 near (0, 0)The local estimate that we need isLemma 2.2 (Local L-estimate). Let a E '(R2n) and assume that(10)aNE C(L2(R"),Vwithl<2n+1ThenIlxall≤CZ rakIlc(L2(R")-/≤2n+1We postpone the proof to the end
8 LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY (Recall: The Calderon-Vailancout tells us that the operator norm of ba W is bounded by the L ∞-norm of ∂ αa for finitely many α. So the following theorem is a reverse:) Theorem 2.1. Let a ∈ S 0 (R 2n ) and assume that for all multi-indices γ with |γ| ≤ 2n + 1, the operator ∂dγa W ∈ L(L 2 (R n )). Then a ∈ L ∞(R n ) and (9) kakL∞ ≤ C X |γ|≤Mn ~ |γ|/2 k∂dγa W kL(L2(Rn)). So by Theorem 2.1, sup R2n |∂ α a| ≤ Cα. In other words, a ∈ S(1). Finally one can use the re-scaling trick to convert the general case to the ~ = 1 case. ¶ An “reverse” to the Calderon-Vaillancourt Theorem. It remains to prove Theorem 2.1. Proof. We will only prove the theorem for ~ = 1 and for the Kohn-Nirenberg quantization. The general case follows from re-scaling trick and the change of quantization formula. Again we prove a local estimate and then globalize it, this time since we are estimating the L ∞-norm, we don’t even need to choose a partition of unity: to globalize the estimate it is enough to choose a cut-off function that is positive locally, prove a local L ∞ estimate, then use translation to get the demanded global estimate. So we let ϕ = ϕ(x) and ψˆ = ψˆ(ξ) be functions in S which equals 1 near 0. Put χ(x, ξ) = ϕ(x)ψb(ξ)e ix·ξ . Then χ ∈ S (R 2n ), and |χ(x, ξ)| = 1 near (0, 0). The local estimate that we need is Lemma 2.2 (Local L ∞-estimate). Let a ∈ S 0 (R 2n ) and assume that (10) ∂dγa KN ∈ L(L 2 (R n )), ∀γ with |γ| ≤ 2n + 1. Then kχakL∞ ≤ C X |γ|≤2n+1 k∂dγa KN kL(L2(Rn)). We postpone the proof to the end.
9LECTURE14:L2-THEORYOFSEMICLASSICALPSDOS:ELLIPTICITYWe denote ay.n(r, s) = a(r + y, E + n) be the translation of a by (y, n). SinceIx(r,$)/=1 ina neighborhood of (o,0),wehaveIllL ≤, sup, Ixay,ll ≤supCx Z Iray.nk ll(L2(R"),(y,n)ERIl<2n+1On the other hand, one can check that if we set U :L?→L? to be the operatorUu(r)=e-ir(r + y),then U is unitary anday.nKN = UaKNu".ItfollowsIIray-n Ic(L2(R) =IIra llc(L2(R),口This completes the proof. Proof of the local Lo estimate.Finallyfinally,itremainstoprovetheLestimate,namely,Lemma2.2We first observe that12(x)()()+((),I/xal L~(R2mn) ≤ Now for any (y, n), we have[y°mP F(r,t)-(y,n)(xa)(y, n)/ =F(r,)(y.n) DD(xa)≤C[F(rs)-(y,n)De,eaDrexlIpl.≤+βSince x(r,E) = (r)()eir- we have[F(r,t)(y.n)(De,eaDrex)]e-i(ms)(w)(Dga)D(r)D(E)geirsdrds(De,ea) (a D()e-ig) Fz-e(D(z + n)b(z+ n)eir:E drdsei(a=2)(De,ea)(D?(z + n)b(2 + n)dzdE) ( D(r)e-ir) dr(D(z+ n)(2 + n), D(a)e-i)Dpa)/L2(R"). ?(z + n)2b(2 + n)l2(n) Il D(a)e-i1l2(n)Ic(L2(R")
LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY 9 We denote ay,η(x, ξ) = a(x + y, ξ + η) be the translation of a by (y, η). Since |χ(x, ξ)| = 1 in a neighborhood of (0, 0), we have kakL∞ ≤ sup (y,η)∈R2n kχay,ηkL∞ ≤ sup Cχ X |γ|≤2n+1 k∂\γay,η KN kL(L2(Rn)). On the other hand, one can check that if we set U : L 2 → L 2 to be the operator Uv(x) = e −iη·x v(x + y), then U is unitary and ady,η KN = Uba KN U ∗ . It follows k∂\γay,η KN kL(L2(Rn)) = k∂dγa KN kL(L2(Rn)). This completes the proof. ¶ Proof of the local L ∞ estimate. Finally finally, it remains to prove the L ∞ estimate, namely, Lemma 2.2. We first observe that kχakL∞(R2n) ≤ 1 (2π) 2n kF(χa)kL1(R2n) ≤ Ckh(y, η)i 2n+1F(χa)kL∞(R2n) . Now for any (y, η), we have y α η βF(x,ξ)→(y,η)(χa)(y, η) = F(x,ξ)→(y,η)D α xD β ξ (χa) ≤ C X |ρ|,|γ|≤|α+β| F(x,ξ)→(y,η)D ρ x,ξaDγ x,ξχ Since χ(x, ξ) = ϕ(x)ψb(ξ)e ix·ξ we have F(x,ξ)→(y,η)(D ρ x,ξaDγ x,ξχ) = Z R2n e −i(x,ξ)·(y,η) (D ρ x,ξa)D γ1 x ϕ(x)D γ2 ξ ψb(ξ)x γ3 ξ γ4 e ix·ξ dxdξ = Z R2n (D ρ x,ξa) x γ3D γ1 x ϕ(x)e −ix·y Fz→ξ(D γ4 z (z + η) γ2ψ(z + η))e ix·ξ dxdξ = Z R Z R2n e i(x−z)·ξ (D ρ x,ξa)(D γ4 z (z + η) γ2ψ(z + η))dzdξ x γ3D γ1 x ϕ(x)e −ix·y dx = (\D ρ x,ξa) KN (D γ4 z (z + η) γ2ψ(z + η)), x γ3D γ1 x ϕ(x)e −ix·y L2(Rn) ≤ (\D ρ x,ξa) KN L(L2(Rn)) · |D γ4 z (z + η) γ2ψ(z + η)kL2(Rn) · x γ3D γ1 x ϕ(x)e −ix·y L2(Rn)
10LECTURE 14:L?-THEORY OFSEMICLASSICAL PSDOS:ELLIPTICITYSincep,bE,weget-KNIxallL≤c Z 11rakllc(L2(R"):Il≤2n+1
10 LECTURE 14: L 2 -THEORY OF SEMICLASSICAL PSDOS: ELLIPTICITY Since ϕ, ψ ∈ S , we get kχakL∞ ≤ C X |γ|≤2n+1 k∂dγa KN kL(L2(Rn))