SINGULAR SUMS OF SQUARES OF DEGENERATE VECTOR 6 0 FIELDS 0 2 ANTONIOBOVE,MAKHLOUFDERRIDJ,JOSEPHJ.KOHN,ANDDAVIDS.TARTAKOFF n a J Abstract. In [7], J. J. Kohn proved C∞ hypoellipticity with loss of k−1 2 derivatives in Sobolev norms (and at least that loss in L∞) for the highly 2 non-subellipticsingularsumofsquares ] Pk =LL+L|z|2kL=−L∗L−(zkL)∗zkL with L= ∂∂z +iz∂∂t. P A In this paper, we prove hypoellipticity with loss of km−1 derivatives in Sobolevnormsfortheoperator . th (0.1) PmF,k =LFmLFm+LFm|z|2kLFm with LFm= ∂∂z +iFz∂∂t, a withF(z,z)suchthat m [ (0.2) Fzz =|z|2(m−1)g, g(0)>0, sothatFz =z|z|2(m−1)h whoseprototype, whenmF(z,z)=|z|2m,is 1 v (0.3) Pm,k =LmLm+Lm|z|2kLm, Lm= ∂ +iz|z|2(m−1) ∂ , ∂z ∂t 3 forwhichtheunderlyingmanifoldisoffinitetype. 2 We give two proofs: the first using a fairlyrapid derivation of an a priori 5 estimateanalogous tothatusedbyKohnin[7]: 1 0 (0.4) kϕuk0≤Ckϕ˜PmF,kvkk−1 +Ckuk−∞ 6 m (for all u ∈C∞ with ϕ˜≡1 near supp ϕ), after deriving this estimate in the 0 0 firstpartofthepaper;thesecondusesthefarmorerapidlyderivedestimateof / h [12] and [5] (where analytic hypoellipticity for Pk and Pm,k arealso proved): t ∀v∈C∞ ofsmallsupport, a 0 m (0.5) kvk2−k−1 +kLvk20+kzkLvk20 ≤C|(PmF,kv,v)L2|+Ckvk2−N. 2m v: Wealsoprove,alongtheway,analytichypoellipticityforPmF,k.For Xi (0.6) F(z,z)=f(|z|2), weshowthattheseestimatesareoptimal. r a 1. Introduction and statement of theorems In his recent paper,[7], J.J. Kohnexhibited a sum of squaresofcomplex vector fields which satisfied the bracket condition but which was not subelliptic; nonethe- less, he showed that the operator was hypoelliptic, though with a large loss of derivatives. His example was: P =LL+L|z|2kL=−L∗L−(zkL)∗zkL with L= ∂ +iz ∂ . k ∂z ∂t Date:February2,2008. 1 2 Bove,Derridj,Kohn&Tartakoff The a priori estimate Kohn established is a strong one and in this case (since the operatorisindependent ofthe variablet,)leadsvirtuallyatonce tothe hypoel- lipticity of P : for any s, there exists a constant C such that for all smooth u and k s any pair of cut-off functions ϕ,ϕ˜ with ϕ˜≡1 near supp ϕ, (1.1) kϕuk ≤C kϕ˜P uk +C kuk s s k s+k−1 s −∞ Here the last norm stands for a norm of arbitrarily low order, with the constant preceeding it possibly depending on the order of that norm, and u assumed to be of (possibly large) compact support. Subsequently, in [5], M. Derridj and D. S. Tartakoff proved analytic hypoellip- ticity for P using rather different methods, namely they established an inequality k for functions v of small support, hence an estimate which did not require explicit cut-off functions, reservingthe necessity of localizing an actual solutionto a neigh- borhoodof a point to the proofof (analytic) hypoellipticity: for any s, there exists a constant C such that for all v ∈C∞ of small support, s 0 (1.2) kvk2 +kLvk2+kzkLvk2 ≤C |(P v,v) |+C kvk s−k−1 s s s k s s −∞ 2 which of course yields the previous estimate at once without the cut-off functions but only for u already known to have (small) compact support. Thispaperwaspartlymotivatedbythe efforttounderstandtherelationshipbe- tweentheseestimates,partlytoobtainasimpler(oratleastmoreconcise)derivation of the former, and finally to generalize these results where possible. In [12], the fourth author had already sharpened the methods of [5] to include the example of the operator ∂ ∂ (1.3) P =L L +L |z|2kL with L = +iz|z|2(m−1) m,k m m m m m ∂z ∂t based on the tangential vector fields to a domain in C2 of finite type; the technical work was heavily dependent on the methods of [4]. Both[12]and[5]includeproofsofC∞ -hypoellipticityby‘truncating’theproofs ofanalytichypoellipticity,henceusetheentiremachinerythathascometobeknown as (Tp) since [10]. ϕ In this paper, we consider the more general operator ∂ ∂ (1.4) PF =LFLF +LF |z|2kLF with LF = +iF , m,k m m m m m ∂z z∂t with (1.5) F =|z|2(m−1)g, g(0)6=0, F =z|z|2(m−1)h zz z whose prototype, when F(z,z)=|z|2m/m, is the operator P discussed above. m,k Here we establishtwo families of estimates for PF ,and provethe optimality of m,k these estimates under the additional restriction (1.6) F(z,z)=f(|z|2). We will then use one of the estimates to prove C∞ hypoellipticity with precise lossof k−1 derivativesandtheothertoproveCω hypoellipticityandtogiveanother m proof of C∞ hypoellipticity with the prescribed loss. A note onthe norms used is in order. All of our norms and derivations are done in L2(z,z)×Hs(t). There are severalreasonsfor this. First, Proposition1.1 could, fors=0,triviallyhavethenormontheleftreplacedwiththefull−k−1 norm,then 2m SingularSumsofSquaresofVectorFields 3 as mentioned below, using a cut-off in τ dual to t which tends to the identity, one can prove easily that since ∂ commutes with P, high t derivatives of the solution t belong to Hs−mm−1 (in t) provided this is true of Pu in Hs norm. But the whole classical theory of pseudo-differential operators and wave front sets allows us to microlocalize the consideration of hypoellipticity. For it is clear that if z 6= 0, the operator is elliptic and hence even analytic hypoelliptic, and gains two derivatives. For z close to zero, one must look in the cotangent space, (z,t;ζ,τ)which,inthecomplementof(z,t;0,0)wewriteastheunionofoverlapping cones: the cones Γ± contain τ = +1,ζ = 0 and τ = −1,ζ = 0 respectively, while the “elliptic” cone Γ0 contains τ = 0. In Γ0, the operator P is also elliptic, since this is true of LL. In Γ−, the operator P is maximally hypoelliptic and hence is subelliptic with loss of 1/2m derivatives (kLvk is bounded by kLvk there, hence the operator is maximally hypoelliptic, which means that the real and imaginary parts of L and L are bounded by P, and by Ho¨rmander’s condition, subelliptic and one has the estimate of Proposition 1.1 with the Hs+21m norm on the left), hence is (microlocally) hypoelliptic with a gain of 1 derivatives in that region by 2m conventionalarguments. It is only in the positive cone that all of this work is necessary, and there in addition to having estimates such as Lemma 2.3 below, we also know (as we would in Γ− as well) that |ζ| ≤ C|τ| so that estimating high derivatives in t will yield control in all directions. InthetwoPropositionswhichfollow,thenotationA.BwillmeanthatA≤CB with C uniform in v ∈ C∞ and locally so in s, and F is assumed to satisfy the 0 conditions of (2.1) above. Proposition 1.1. For v of small support, (1.7) kvk2 +kLFvk2+kzkLFvk2 .|(PF v,v) |+kvk2 , s−k−1 m s m s m,k s −∞ 2m Proposition 1.2. For any pair of cut-off functions ϕ,ϕ˜ with ϕ˜ ≡ 1 near supp ϕ, and for u of support in a fixed (not necessarily small) compact set, (1.8) kϕuk2 .kϕ˜PF uk2 +kuk2 s m,k s+k−1 −∞ m Proposition 1.3. For the case F(z,z)=f(|z|2)b(z,z),b(0)6=0 the loss in Propo- sitions 1.1 and 1.2 cannot be improved. Theorem 1. PF is locally hypoelliptic with loss of k−1 derivatives: PF u ∈ m,k m m,k Hs =⇒ u∈Hs−km−1. In the sequel, we will write L for LF,L for LF, and P for PF . m m m,k 2. Preliminary Observations and Lemmas The first observation concerns the apparent difference between the two a priori estimates in the two Propositions above and their use. The second estimate ex- plicitly introduces a second cut-off function, although, as we shall see, except for the last term, the function ϕ˜ may be replaced by certain derivatives of ϕ. That is, except for a norm of sufficiently low order, we may control the terms on the right by derivativesofthe givenlocalizing function. In fact, the same is true in the proof of (analytic) hypoellipticity using the first estimate - we proceed with a balanced localization (Tp) of high derivatives in T = ∂ and encounter errors expressed as ϕ t 4 Bove,Derridj,Kohn&Tartakoff derivatives of the localizing function we start with and then at a certain point (in this case a fraction of the derivatives we seek to estimate), we are forced to intro- duce a cut-off function with strictly larger support and to construct a whole new balanced sum (Tp˜) around this new localizing function - and for the analyticity ϕ˜ proof we need to control these supports in a very precise way. It is not at all clear how to pass from one setting to the other - neither esti- mate trivially implies the other and the proofs of hypoellipticity are not trivially comparable, but they do seem to contain the same elements. Ourfirsttechnicalobservationconcernsthedependenceoflocalizingfunctionson z.Inordertolocalizetoaneighborhoodof0,wemaytakeaproductofafunctionof z,z of small support but identically equal to one near the origin in C with another function of t only, again taken to be of small support. Whenever the first of these functions is differentiated, the resulting function is supported away from z = 0, hence in a region where the operator P is in fact subelliptic and hence far better behaved. Weshallignoresuchregionsandthustakealllocalizationstobefunctions of t only. To make the proofs of Propositions 1.1 and 1.2 flow more smoothly, we prepare some easy lemmas which will be used repeatedly in the sequel. By integration by parts andshifting powersof z fromone side of aninner productto the other, these lemmas, especially Lemma 2.4 and 2.5, which are often used, express the obvious fact that by grouping one power of z and a fractional power of Λ , effectively a t fractional power of ∂ , as a unit, say A = zΛρ, one may move powers of A from t t one side of an inner product to the other. In all of these lemmas, w will denote a smooth function of (small) compact support and the superscript ‘+’ will indicate that the function has been microlocalizedto the positive cone for the symbol of ∂ . t The estimates are locally uniform in s. Lemma 2.1. kLwk .kLwk +kzm−1wk . s−1/2 s−1/2 s Proof. Integration by parts since [L,L]=−2iF ∂ ,|F |.|z|2(m−1). (cid:3) zz t zz Lemma 2.2. kzm−1wk .kLwk +kLwk . s+1/2 s s Proof. Integration by parts since [L,L]=−2iF ∂ and F ≥c|z|2(m−1). (cid:3) zz t zz Lemma 2.3. kLw+k +kzm−1w+k .kLw+k +kwk . s s+1/2 s s Proof. The same identity where the symbol of −2i∂ has the appropriate sign. (cid:3) t Lemma 2.4. kzrwkµ ≤s.c.kzr−n1wkµ−n1ρ+l.c.kzr+n2wkµ+n2ρ, n1 ≤r. Proof. Let A=zΛρ. Then for example t kArwk2 =(Ar−n1w,Ar+n1) ≤s.c.kAr−n1wk2 +l.c.kAr+n1wk2 µ µ µ µ but then the second of these terms may be related to lower and higher powers of A, and the result follows. (cid:3) Lemma 2.5. kzrwkµ ≤l.c.kzr−n1wkµ−n1ρ+s.c.kzr+n2wkµ+n2ρ, n1 ≤r. Proof. Completely analogous. (cid:3) Lemma 2.6. kwk.kzLwk+kzLwk. SingularSumsofSquaresofVectorFields 5 Proof. This is the subelliptic multiplier argument: kwk2 =|([L,z]w,w)|≤|(w,zLw)|+|(zLw,w)| ≤s.c.kwk2+l.c.(kzLwk2+kzLwk2). (cid:3) Lemma 2.7. kϕuk .kzLϕuk +kzLϕuk . 0 0 0 Proof. This is the previous lemma with w =ϕu. (cid:3) Lemma 2.8. kϕuk .kzϕLuk +kzϕLuk +kz2mϕ′uk . 0 0 0 0 Proof. This is just the observation that |[L,ϕ]u|∼|z|2m−1|ϕ |. (cid:3) t Lemma 2.9. kzϕ′uk .kLzϕ′uk +kLzϕ′uk +kzϕ′uk 0 − 1 − 1 − 1 2m 2m 2m Proof. This is just the observation that the vector fields L and L, or rather their real and imaginary parts, satisfy the (real) bracket condition and hence form a subelliptic systeminthe usualsensewith ε=1/2m,andthenthe whole subelliptic estimate is lowered by 1/2m. (cid:3) 3. Proof of Proposition 1.1 ToproveProposition1.1,theaprioriestimateoncompactlysupportedfunctions, wesetr=−k−1 andτ = 1 .Notethatrneednotbenegative,butr−τ =− k ≤ 2m 2m 2m 0. Then we have, since r ≤τ: kvk2 =((Lz)Λrv,Λrv)=(zΛ2rv,L v)−(LΛr−τv,zΛr+τv) r t t t m t t ≤C{kLvk2+s.c.kLΛr−τvk2+l.c.kzΛτ(Λrv)k2} t t t ≤C{kLvk2+l.c.kzΛτ(Λrv)k2+s.c.kzm−1Λ21−τ(Λrv)k2} t t t t ≤C{kLvk2+l.c.kzΛτ(Λrv)k2+s.c.kzm−1Λ(m−1)τ(Λrv)k2}. t t t t When m = 1, this last term is just s.c.kvk2 but for 0 < a ≤ m−1+k, we use r Lemma 2.4 in the form kzaΛaτwk2 ≤s.c.kwk2+l.c.kzm−1+kΛ(m−1+k)τwk2 t t =s.c.kwk2+l.c.kzm−1+kΛ21(Λ(m−1+k)τ−21w)k2 t t with w=Λrv twice, once for a=1 and once for a=m−1. t Inserting this in the estimate above for kvk2, we find r kvk2 ≤C{kLvk2+kzm−1+kΛ21Λ(m−1+k)τ−12+rvk2} r t t =C{kLvk2+kzm−1+kΛ21vk2}. t since (m−1+k)τ − 1 +r =0. 2 On the other hand, we have by Lemma 2.2, kzm−1+kΛ21vk2 ≤C{kLvk2+kzkLvk}, t (whichone provesfromthe Lemma with the additionaltermkzk−1vk2 on the right and then, writing zk−1 ∼ [L,zk], absorbs this term by the other two). Thus we arrive at kvk2 ≤C{kLvk2+kzkLvk2}=C|(Pv,v)|≤l.c.kPvk2 +s.c.kvk2 r −r r 6 Bove,Derridj,Kohn&Tartakoff or kvk2 +kLvk2+kzkLvk2 .CkPvk2 ,v ∈C∞, (cid:3) −k−1 k−1 0 2m 2m 4. Proof of Proposition 1.2. The Case k =1 Fork =1wewillestablishtheestimate(foruofsmallsupportnearz =0)using only Lemmas 2.3 and 2.8: 3 (4.1) kϕuk2+kϕLuk2+kzϕLuk2 ≡ (LHS) . s s s j j=1 X N . kϕ(j)Puk2 +kϕ(N)uk2 +kuk2 . s−j/2 s−N/2 −∞ j=0 X Here and elsewhere, we will find the following definition useful: Definition 4.1. The designation “RJ” (for “Relative Junk”) will apply to any multiple of any of the terms (LHS) that we are in the process of estimating but j with lower Sobolev index and possibly a derivative on the localizing function - in other words, to a term which will be iteratively estimated at the end. For any value of k, from Lemma 2.8, (LHS) ≡kϕuk2 .kzϕLuk2+kzϕLuk2+kzmϕ′uk2 1 (Lemma 2.8 even gives z2m in place of zm). We claim that this last term is RJ. To see this, Lemma 2.3 tells us that (4.2) kzmϕ′uk2 =kzm−1zϕ′uk2 .kzLϕ′uk2 +kzϕ′uk2 =RJ 0 0 −1/2 −1/2 (provided, as we will show, that we can estimate kϕuk and kzLϕuk2). Actually, 0 0 in the next section we will see even that kzϕ′uk ∈RJ. 0 So we have, modulo RJ (LHS) .(LHS) +(LHS) ≡kzϕLuk2+kϕLuk2 1 2 3 0 0 =|−(Lϕ2|z|2Lu,u)−(Lϕ2Lu,u)|.|(ϕ2Pu,u)| +|(F ϕϕ′Lu,u)|+|(F ϕϕ′|z|2Lu,u)|+|(ϕ2zLu,u)|. z z These last two terms are easy to handle: |(F ϕϕ′Lu,u)|+|(F ϕϕ′Lu,u)| z z .s.c.kϕLuk2+l.c.kz2m−1ϕ′uk2+s.c.kzLuk2 which are absorbed modulo the term kz2m−1ϕ′uk which is RJ since 2m−1≥m. Thus in all, in the positive cone, with ϕ˜≡1 near the support of ϕ, and for any N, N kϕuk2+kϕLuk2+kzϕLuk2 . kϕ(j)Puk2 +kϕ(N)uk2 +kuk2 . s s s s−j/2 s−N/2 −∞ j=0 X Or, with ϕ˜≡1 near the support of ϕ, kϕuk2 .kϕ˜Puk2+kuk2 . s s −∞ SingularSumsofSquaresofVectorFields 7 5. Proof of Proposition 1.2. The Case k >1 ToproveProposition1.2whenk >1,whichisharder,wecannotjustusePropo- sition 1.1 with a cutoff function ϕ in front of v and then express the right hand side in terms of (ϕPv,ϕv) modulo acceptable errors, since bracket of P with ϕ introduces errorseasily absorbedonly when the basic estimate is subelliptic, which here means k =0,the well-knowncase,or at least, in Kohn’s terminology,‘no loss, no gain’, namely the case k =1 which we just considered. Instead,weproceedasfollows. Wewillestablishagaintheclassof“RelativeJunk Terms”, denoted RJ, which are of the same form as those terms being estimated but of lower Sobolev degree, and the localizing function(s) may have received a derivative. These will be treated recursively at the end, in a very simple manner, but to see that a term is RJ one may have to compare it to all eight terms below. The terms we want to estimate are eight in number, and will be referred to as (LHS) ,j =1,...8. In estimating some the others will occur, generally with small j constants,butwe setupa genericsumwith unknowncoefficients j=8C (LHS) . j=1 j j Specifically, we will establish, for suitable C to be determined relative to one an- j P other, C kϕuk+C kzϕLuk+C kϕLuk +C kϕzkLuk +C kz2k+m−2ϕuk 1 2 3 k−1 4 k−1 5 k−1 2m 2m m (*) +C kϕz2k−1Luk +C kzm−1ϕLuk +C kLϕLuk 6 k−1−1 7 k−1 8 k−1−1 m 2 m m 2 8 =E = C (LHS) ≤C kϕPuk +RJ. j j 9 k−1 m 1 X In proving (∗) we will encounter errors from microlocalization, errors which are supported in regions where the regularity is well understood. As these will be included in RJ in any case, we will omit explicit mention of terms of the form kuk . −∞ 5.1. Estimating (LHS) . Using Lemma 2.8 and then Lemmas 2.9 and 2.4: 1 (LHS) ≡kϕuk .kϕLuk +kzϕLuk +kzϕ′uk 1 0 0 0 0 .kϕLuk +l.c.kzkϕLuk +s.c.kϕLuk +kzϕ′uk . 0 k−1 − 1 0 2m 2m For third term we write, using Lemma 2.1: kϕLuk .kzm−1ϕuk +kϕLuk +kϕ′uk − 1 − 1 +1 − 1 − 1 2m 2m 2 2m 2m (the last two terms are RJ) and from Lemma 2.4, kzm−1ϕuk .s.c.kϕuk +l.c.kzm−1zkϕuk − 1 +1 0 1+k−1 2m 2 2 2m and by Lemma 2.3, kzm−1zkϕuk .kzkϕLuk +kzkϕuk +kzk+2m−1ϕ′uk . 1+k−1 k−1 k−1 k−1 2 2m 2m 2m 2m Now kzkϕuk ≤l.c.kϕuk +s.c.kzm−1zkϕuk k−1 − 1 1+k−1 2m 2m 2 2m =RJplusatermwhichcanbeabsorbedbythepreviouslefthandsideandadirect application of Lemma 2.4 yields 8 Bove,Derridj,Kohn&Tartakoff Lemma 5.1. kzk+2m−1ϕ′uk .kz2mϕ′uk +kz2k+m−2ϕ′uk =RJ. k−1 0 k−1−1 2m m 2 Putting these together, j=2 (LHS) .(LHS) +(LHS) +RJ. j 3 4 j=1 X 5.2. Estimation of (LHS) and (LHS) . Using the fact that |F |∼|z|2m−1, we 3 4 z have (LHS) +(LHS) ≡kϕLuk2 +kϕzkLuk2 3 4 k−1 k−1 2m 2m =(ϕPu,ϕu) −([L,ϕ2]|z|2kLu,u) −([ϕ2,L]Lu,u) k−1 k−1 k−1 2m 2m 2m ≤l.c.kϕPuk2 +s.c.(LHS) k−1 1 m +2|(F ϕϕ′|z|2kLu,u) |+2|(F ϕϕ′Lu,u) | z k−1 z k−1 2m 2m ≤l.c.kϕPuk2 +s.c.(LHS) +{s.c.kϕzkLuk2 +l.c.kz2m−1+kϕ′uk2 } k−1 1 k−1 k−1 m 2m 2m +{s.c.kϕzm−1Luk2 +l.c.kzmϕ′uk2}+RJ k−1 0 m .l.c.kϕPuk2 +s.c.{(LHS) +(LHS) +(LHS) }+RJ k−1 1 4 7 m using Lemma 5.1 since the right hand side (4.2) is RJ for any k. Thus, 4 C (LHS) ≤s.c.(LHS) +C kϕPuk2 +RJ. j j 7 9 k−1 m 1 X 5.3. Estimation of (LHS) and (LHS) . Settingσ = k−1−1 andusingLemma 5 6 m 2 2.3, (LHS) +(LHS) ≡kz2k+m−2ϕuk2 +kϕz2k−1Luk2 5 6 k−1 σ m .kϕz2k−1Luk2 +kϕz2k+m−3uk2 +kz2k+3m−3ϕ′uk2 +RJ σ σ σ .kϕz2k−1Luk2 +s.c.kϕz2k+2m−3uk2 +kϕuk2 +RJ σ k−1 −2m−1 m 2m .kϕz2k−1Luk2 +s.c.(LHS) +RJ σ 5 (since 2k+2m−3≥2k+m−2) with Lemma 2.4. So we have to consider (LHS) ≡kϕz2k−1Luk2 .|(Lϕ2|z|2(2k−1)Lu,u) | 6 σ σ +kϕ′z2(k+m−1)uk2 +kz2k−2ϕuk2 σ σ .|(Lϕ2|z|2(2k−1)Lu,u) |+s.c.{(LHS) +(LHS) }+RJ σ 1 5 by Lemma 2.4. Now for the inner product we have, again using Lemma 2.4, |(Lϕ2|z|2(2k−1)Lu,u) |≤|(2ϕϕ′F |z|2(2k−1)Lu,u) | σ z σ +|(ϕL|z|2kLu,z2k−2ϕu) |+|(ϕ|z|4k−3Lu,ϕu) | σ σ .s.c.kz2k−1Luk2 +l.c.kz2k+2m−2uk2 +kϕPuk2 σ σ σ +kz2k−2ϕuk2 +|(ϕLLu,z2k−2ϕu) |+s.c.kz2k−1ϕLuk2 σ σ σ so |(Lϕ2|z|2(2k−1)Lu,u) |.s.c.{(LHS) +(LHS) +(LHS) } σ 6 5 1 SingularSumsofSquaresofVectorFields 9 +kPuk2 +|(ϕLLu,z(2k−2)ϕu) |+RJ σ σ ≤s.c.{(LHS) +(LHS) +(LHS) +(LHS) }+kPuk2 +RJ. 6 5 1 8 σ Thus, so far, 6 C (LHS) ≤s.c.{(LHS) +(LHS) }+C kϕPuk2 +RJ j j 7 8 9 k−1 m 1 X but we still need to estimate both (LHS) and (LHS) since the definition of RJ 7 8 requires it. 5.4. Estimation of (LHS) and (LHS) . We proceed to estimate (a small mul- 7 8 tiple of) the following expression B, noting that kϕLLuk2 =kLϕLuk2 +RJ : σ σ B =kzm−1ϕLuk2 +kϕLLuk2 (+kϕLLuk2).kϕLLuk2 +RJ k−1 σ σ σ m .|(ϕLLLu,ϕLu) |+|(ϕLLu,ϕ′F Lu) |+RJ σ z σ =|(ϕLLLu,ϕLu) |+s.c.B+RJ σ .|(ϕLPu,ϕLu) |+|(ϕLL|z|2kLu,ϕLu) |+s.c.B+RJ σ σ =B +B +s.c.B+RJ. 1 2 Now B =|(ϕLPu,ϕLu) |≤|(ϕPu,ϕLLu) | 1 σ σ +|(F ϕϕ′Pu,Lu) |.kϕPuk2 +s.c.B+RJ z σ σ while B =|(ϕLL|z|2kLu,ϕLu) |. 2 σ .|(ϕL|z|2kLLu,ϕLu) |+|(ϕLz|z|2(k−1)Lu,ϕLu) | σ σ =B +B . 21 22 For B we have 21 B .|(ϕL|z|2kLLu,ϕLu) |+|(ϕL|z|2kF Tu,ϕLu) | 21 σ zz σ =B +B 211 212 with B .|(ϕ|z|2kLLu,LϕLu) |+|(F ϕ′|z|2kLLu,ϕLu) | 211 σ z σ ≤s.c.B+RJ. For B , the bracket [L,|z|2kF ] will enter. This will contain a factor of 212 zz z|z|2(k−1)+2(m−1), and thus we may move zm−1 to the right hand side of the inner product leavinga function g(z)z|z|2(k−1)zm−1 onthe left, for a suitable function g. We have B =|(ϕL|z|2kF Tu,ϕLu) | 212 zz σ .|(ϕLF |z|2ku,ϕLu) |+|(ϕ′LF |z|2ku,ϕLu) | zz k−1 zz σ m .|(ϕF |z|2kLu,ϕLu) |+|(ϕg(z)z|z|2(k−1)zm−1u,ϕzm−1Lu) | zz k−1 k−1 m m +|(ϕF |z|2kLu,ϕ′Lu) |+|(ϕg(z)z|z|2(k−1)zm−1u,ϕ′zm−1Lu) | zz σ σ .s.c.B+RJ +kϕz2k+m−2uk2 .s.c.B+(LHS) +RJ k−1 5 m Finally, for B we have 22 B =|(ϕLz|z|2(k−1)Lu,ϕLu) | 22 σ .|(ϕz|z|2(k−1)Lu,LϕLu) |+|(ϕF z|z|2(k−1)Lu,ϕ′Lu) | σ z σ 10 Bove,Derridj,Kohn&Tartakoff .kϕz2k−1Luk2 +s.c.B+RJ =(LHS) +s.c.B+RJ. σ 6 Thus we have s.c.B .s.c.{(LHS) +(LHS) +(LHS) +(LHS) }+kϕPuk2 +RJ 1 4 7 6 k−1 m and hence kϕuk2 .kϕPuk2 +RJ 0 k−1 m or, iterating until RJ is of arbitrarily low order, for some ϕ˜ ≡ 1 near the support of ϕ, kϕuk2 .kϕ˜Puk2 +kuk2 0 k−1 −N m 6. Proof of Proposition 1.3 (Optimality) Proof. For ϕ of compact support, we set h (z,t)=ϕv and λ λ v =exp(−λ(F −it−(F −it)2). λ If |z| is small enough, we have, from above and below, ℜ F −it−(F −it)2 =F −F2+t2 ∼|z|2m+t2, and hence that k((cid:0)λ(|z|2m +t2))Av k(cid:1) ∼ C uniformly in λ. Also, any function λ ∞ A of compact support and equal to zero in a neighborhood of the origin, such as a derivative of a localizing function identically equal to one near the origin, times v λ is of order λ−N for any N. F Now v was chosen so that L v =0, and, letting H =F −it we compute that λ λ LF v =−2λF (1+2H)v m,k λ z λ and hence that for some A, L|z|2kLv =−2λL(|z|2kF (1+2F −2it))v ∼λ|z|2k+2m−2v λ z λ λ ∼(λ|z|2m)2k+22mm−2λ1−2k+22mm−2vλ ∼λ−km−1 λ|z|2m A. Analogously, we have that as a principal term, (cid:0) (cid:1) ∂sv ∼λsv . t λ λ Hence if there is an estimate of the form kψv k .kψ˜PF v k +kv k λ 0 k,m λ r λ −∞ valid as λ → ∞, for ψ,ψ˜ ∈ C∞,ψ ≡ 1 near 0,ψ˜ ≡ 1 near supp ψ, then r ≥ k−1, 0 m and an analogousargument holds for Proposition1.1. Finally the optimality at all levels (other values of s) follows at once since the vector field ∂/∂t commutes with the differential operator PF . (cid:3) k,m