Keywords: assi cation. P rim ary: 35B 65;Secondary: 35N 10. K ey w ords and phrases. Q uadratic di erential operators, overdeterm ined system s, subelliptic estim ates, singular space, W ick quantization

A bstract. W e prove global subelliptic estim ates for system s of quadratic differentialoperators. Q uadratic di erentialoperators are operators de ned in the W eylquantization by com plex-valued quadratic sym bols. In a previous w ork,w e pointed out the existence ofa particularlinear subvector space in the phase space intrinsically associated to their W eylsym bols,called singular space,w hich rules a num ber of fairly generalproperties of non-elliptic quadratic operators. A bout the subelliptic properties ofthese operators,w e established that quadratic operators w ith zero singular spaces ful llglobal subelliptic estim ates w ith a loss of derivatives depending on certain algebraic properties ofthe H am ilton m aps associated to their W eylsym bols. T he purpose ofthe present w ork is to prove sim ilar global subelliptic estim ates for overdeterm ined system s of quadratic operators. W e establish here a sim ple criterion for the subellipticity of these system s giving an explicit m easure ofthe loss ofderivatives and highlighting the non-trivial interactions played by the di erent operators com posing those system s.

1. Introduction 1. 1. M iscellaneous facts about quadratic di erential operators. In a recent joi nt work w i th M .H i tri k,we i nvesti gated spectraland sem i group properti es ofnonel l i pti c quadrati c operators. Q uadrati c operators are pseudodi erenti al operators de ned i n the W eylquanti zati on (1. 1) q w (x;D x )u(x)= 1 (2 ) n Z R 2n e i(x y): q x + y 2 ; u(y)dyd ;

by som e sym bol s q(x; ), w i th (x; ) 2 R n R n and n 2 N , w hi ch are com pl exval ued quadrati c form s. Si nce these sym bol s are quadrati c form s,the correspondi ng operators i n (1. 1) are i n fact di erenti aloperators. Indeed, the W eylquanti zati on ofthe quadrati c sym bolx ,w i th ( ; ) 2 N 2n and j + j= 2,i s the di erenti al operator

x D x + D x x 2 ; D x = i 1 @ x :

O ne can al so noti ce that quadrati c di erenti aloperators are a pri oriform al l y nonsel fadjoi nt si nce thei r W eylsym bol s i n (1. 1) are com pl ex-val ued.

C onsi deri ng quadrati c operators w hose W eylsym bol s have realparts w i th a si gn, say here,W eylsym bol s w i th non-negati ve realparts (1. 2) R e q 0; we poi nted out i n [START_REF] Itrik | Spectra and sem igroup sm oothing for non-ell iptic quadratic operators[END_REF] the exi stence of a parti cul ar l i near subvector space S i n the phase space R n x R n i ntri nsi cal l y associ ated to thei r W eyl sym bol s q(x; ), cal l ed si ngul ar space,w hi ch seem s to pl ay a basi c rôl e i n the understandi ng of a num ber offai rl y generalproperti es ofnon-el l i pti c quadrati c operators. M ore speci cal l y,we rst proved i n [START_REF] Itrik | Spectra and sem igroup sm oothing for non-ell iptic quadratic operators[END_REF](T heorem 1. 2. 1) that w hen the si ngul ar space S has a sym pl ecti c structure then the associ ated heat equati on (1. 3) ( @u @t (t;x)+ q w (x;D x )u(t;x)= 0 u(t; )j t= 0 = u 0 2 L 2 (R n ); i s sm oothi ng i n every di recti on ofthe orthogonalcom pl em ent S ? ofS w i th respect to the canoni calsym pl ecti c form on R 2n , (1. 4) (x; );(y; ) = : y x: ; (x; )2 R 2n ;(y; )2 R 2n ; thati s,that,i f(x 0 ; 0 )are som e l i nearsym pl ecti c coordi nateson the sym pl ecti c space S ? then we have for al lt> 0,N 2 N and u 2 L 2 (R n ),

(1. 5) (1 + j x 0 j 2 + j 0 j 2 ) N w e tq w (x;D x ) u 2 L 2 (R n ): W e al so proved i n [START_REF] Itrik | Spectra and sem igroup sm oothing for non-ell iptic quadratic operators[END_REF] (See Secti on 1. 4. 1 and T heorem 1. 2. 2) that w hen the W eyl sym bolq ofa quadrati c operatorful l l s (1. 2) and an assum pti on ofparti alel l i pti ci ty on i ts si ngul ar space S i n the sense that (1. 6) (x; )2 S; q(x; )= 0 ) (x; )= 0; then thi s si ngul ar space al ways has a sym pl ecti c structure and the spectrum ofthe operator q w (x;D x ) i s onl y com posed ofa countabl e num ber ofei genval ues of ni te m ul ti pl i ci ty,w i th a si m i l arstructure asthe one establ i shed by J.Sj ostrand forel l i pti c quadrati c operators i n hi s cl assi calwork [START_REF] Sj | P aram etrices for pseudodi erential operators w ith m ul tipl e characteristics[END_REF] . El l i pti c quadrati c operators are the quadrati c operators w hose sym bol s sati sfy the condi ti on ofgl obalel l i pti ci ty (x; )2 R 2n ; q(x; )= 0 ) (x; )= 0; on the w hol e phase space R 2n . Letusrecal lhere thatspectralproperti esofquadrati c operatorsare pl ayi ng a basi c rôl e i n the anal ysi sofparti aldi erenti aloperatorsw i th doubl e characteri sti cs. T hi s i s parti cul arl y the case i n som e general resul ts about hypoel l i pti ci ty. W e refer the reader to [START_REF]A cl ass of hypoell iptic pseudodi erential operators w ith doubl e characteristics[END_REF] , [START_REF] Sj | P aram etrices for pseudodi erential operators w ith m ul tipl e characteristics[END_REF] ,as wel las C hapter 22 of [START_REF]T he anal ysis of l inear partial di erential operators[END_REF]together w i th al lthe references gi ven there.

In the present paper, we are i nterested i n studyi ng the subel l i pti c properti es of overdeterm i ned system s of non-sel fadjoi nt quadrati c operators. T hi s work can be vi ewed asa naturalextensi on oftheanal ysi sl ed i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] ,i n w hi ch wei nvesti gated i n the scal arcase the rôl e pl ayed by the si ngul arspace w hen studyi ng subel l i pti c properti es ofquadrati c operators.W e ai m here atshow i ng how the anal ysi sl ed i n thi s previ ous work can be pushed further w hen deal i ng w i th overdeterm i ned system s ofquadrati c operators. W e shal lsee that the techni ques i ntroduced i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF]are su ci entl y robust to be extended to the system case and that they turn out to be su ci entl y sharp to hi ghl i ght phenom ena of non-tri vi al i nteracti ons between the di erent quadrati c operators com posi ng a system . In thi s paper, we shal l therefore be i nterested i n establ i shi ng som e gl obalsubel l i pti c esti m ates ofthe type (1. 7) h(x; )i 2(1 ) w u L 2 .

N X j= 1 kq w j (x;D x )uk L 2 + kuk L 2 ; w hereh(x; )i= (1+ j xj 2 + j j 2 ) 1=2 and > 0;forsystem softhe N quadrati coperators q w j (x;D x ), w i th 1 j N . T he posi ti ve param eter > 0 appeari ng i n (1. 7) w i l l m easure the l oss of deri vati ves w i th respect to the el l i pti c case (case = 0). A s i n the scal ar case studi ed i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] , we ai m at gi vi ng a si m pl e cri teri on for system s of quadrati c operators ensuri ng that a gl obalsubel l i pti c esti m ate of the type (1. 7) hol ds together w i th an expl i ci t characteri zati on ofthe associ ated l oss ofderi vati ves. T hi sl ossofderi vati ves w i l lbe characteri zed i n term sofal gebrai c condi ti onson the H am i l ton m apsassoci ated to the W eylsym bol softhe quadrati c operatorscom posi ng the system .

In thi s work,we study the subel l i pti ci ty of overdeterm i ned system s i n the sense gi ven by P.B ol l ey, J.C am us and J.N ourri gat i n [START_REF] Ourrigat | La condition de H orm ander-K ohn pour l es op erateurs pseudodi erentiel s[END_REF](T heorem 1. 1). In thi s sem inalwork,these authorsstudy the m i crol ocalsubel l i pti ci ty ofoverdeterm i ned system s ofpseudodi erenti aloperators. M ore speci cal l y,they establ i sh the subel l i pti ci ty of system s com posed ofpseudodi erenti aloperators w i th realpri nci palsym bol s sati sfyi ng the H orm ander-K ohn condi ti on. M ore general l y,i n the case ofoverdeterm i ned system s ofnon-sel fadjoi ntpseudodi erenti aloperators,the greatestachi evem ents up to now were obtai ned by J. N ourri gat i n [START_REF] Ourrigat | Subell iptic system s[END_REF] and [START_REF] Ourrigat | Syst em es sous-ell iptiques II[END_REF] . In these two m ajor works, J.N ourri gatstudi esthe m i crol ocalsubel l i pti ci ty and m axi m alhypoel l i pti ci ty forsystem sofnon-sel fadjoi ntpseudodi erenti aloperatorsby the m ean ofrepresentati onsof ni l potentgroups.W e shal lexpl ai n i n the fol l ow i ng how the al gebrai ccondi ti on on the H am i l ton m aps(1. 18)i n T heorem 1. 2. 1 rel atesw i th theseform erresul ts.M orespeci fi cal l y,we shal lcom m ent on i ts l i nk w i th the H orm ander-K ohn condi ti on appeari ng i n [START_REF] Ourrigat | La condition de H orm ander-K ohn pour l es op erateurs pseudodi erentiel s[END_REF](T heorem 1. 1).

B eforegi vi ng the preci sestatem entofourm ai n resul t,we shal lrecal lm i scel l aneous notati onsaboutquadrati c di erenti aloperatorsand the resul tsobtai ned i n the scal ar case. In al lthe fol l ow i ng,we consi der q j :R n x R n ! C (x; ) 7 ! q j (x; ); w i th 1 j N ,N com pl ex-val ued quadrati c form s w i th non-negati ve realparts (1. 8) R e q j (x; ) 0; (x; )2 R 2n ;n 2 N :

W e know from [START_REF]H orm ander,Sym pl ectic cl assi cation ofquadratic form s,and generalM ehl er form ul as[END_REF](p. 425)thatthe m axi m alcl osed real i zati on ofa quadrati c operator q w (x;D x ) w hose W eyl sym bol has a non-negati ve real part, i . e. , the operator on L 2 (R n ) w i th the dom ai n

D (q)= u 2 L 2 (R n ):q w (x;D x )u 2 L 2 (R n ) ;
coi nci des w i th the graph cl osure ofi ts restri cti on to S(R n ),

q w (x;D x ):S(R n )! S(R n ):
A ssoci ated to a quadrati c sym bolq i sthe num eri calrange (q)de ned asthe cl osure i n the com pl ex pl ane ofal li ts val ues

(1. 9) (q)= q(R n x R n ):
W eal so recal lfrom [START_REF]T he anal ysis of l inear partial di erential operators[END_REF]thattheH am i l ton m ap F 2 M 2n (C )associ ated to thequadrati c form q i s the m ap uni quel y de ned by the i denti ty (1. 10) q (x; );(y; ) = (x; );F (y; ) ; (x; )2 R 2n ;(y; )2 R 2n ; w here q ; stands for the pol ari zed form associ ated to the quadrati c form q. It di rectl y fol l ow s from the de ni ti on ofthe H am i l ton m ap F that i ts realpart and i ts i m agi nary part

R e F = 1 2 (F + F ) and Im F = 1 2i (F F );
are the H am i l ton m apsassoci ated to the quadrati c form sR e q and Im q,respecti vel y. O ne can al so noti ce from (1. 10)thatan H am i l ton m ap i sal waysskew -sym m etri cw i th respect to . T hi s i s just a consequence ofthe properti es ofskew -sym m etry ofthe sym pl ecti c form and sym m etry ofthe pol ari zed form

(1. 11) 8X ;Y 2 R 2n ; (X ;F Y )= q(X ;Y )= q(Y ;X )= (Y;F X )= (F X ;Y ):
A ssoci ated to the sym bolq,we de ned i n [START_REF] Itrik | Spectra and sem igroup sm oothing for non-ell iptic quadratic operators[END_REF]i ts si ngul ar space S as the fol l ow i ng i ntersecti on ofkernel s (1. 12)

S = + 1 \ j= 0 K er R e F (Im F ) j \ R 2n ;
w here the notati ons R e F and Im F stand respecti vel y for the real part and the i m agi nary partoftheH am i l ton m ap associ ated to q.N oti cethattheC ayl ey-H am i l ton theorem appl i ed to Im F show s that

(Im F ) k X 2 Vect X ;: : : ;(Im F ) 2n 1 X ; X 2 R 2n ; k 2 N ;
w here Vect X ;: : : ;(Im F ) 2n 1 X i s the vector space spanned by the vectors X , . . . , (Im F ) 2n 1 X ;and thereforethe si ngul arspace i sactual l y equalto the fol l ow i ng ni te i ntersecti on ofthe kernel s

(1. 13) S = 2n 1 \ j= 0 K er R e F (Im F ) j \ R 2n :
C onsi deri ng a quadrati c operatorq w (x;D x ) w hose W eylsym bol

q :R n x R n ! C (x; ) 7 ! q(x; );
has a non-negati ve realpart,R e q 0,we establ i shed i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF](T heorem 1. 2. 1) that w hen i tssi ngul arspaceS i sreduced to f0g,theoperatorq w (x;D x )ful l l sthefol l ow i ng gl obalsubel l i pti c esti m ate (1. 14) 9C > 0;8u 2 D (q); h(x; )i 2=(2k0 + 1) w u

L 2 C kq w (x;D x )uk L 2 + kuk L 2 ;
w here k 0 stands for the sm al l est non-negati ve i nteger,0 k 0 2n 1,such that the i ntersecti on ofthe fol l ow i ng k 0 + 1 kernel s w i th the phase space R 2n i s reduced to f0g,

(1. 15) k0 \ j= 0 K er R e F (Im F ) j \ R 2n = f0g:
N oti ce that the l oss of deri vati ves = 2k 0 =(2k 0 + 1), appeari ng i n the subel l i pti c esti m ate (1. 14) di rectl y depends on the non-negati ve i nteger k 0 characteri zed by the al gebrai c condi ti on (1. 15).

M ore general l y, consi deri ng a quadrati c operator q w (x;D x ) w hose W eyl sym bol has a non-negati ve realpart w i th a si ngul ar space S w hi ch m ay di er from f0g,but does have a sym pl ecti c structure i n the sense that the restri cti on of the canoni cal sym pl ecti c form to S i snon-degenerate,we proved i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF](T heorem 1. 2. 2)thatthe operator q w (x;D x ) i s subel l i pti c i n any di recti on ofthe orthogonalcom pl em ent S ? ofthe si ngul arspace w i th respectto the sym pl ecti c form i n the sense that,i f(x 0 ; 0 ) are som e l i near sym pl ecti c coordi nates on S ? then we have

9C > 0;8u 2 D (q); h(x 0 ; 0 )i 2=(2k0 + 1) w u L 2 C kq w (x;D x )uk L 2 + kuk L 2 ;
w i th h(x 0 ; 0 )i = (1 + j x 0 j 2 + j 0 j 2 ) 1=2 ,w here k 0 stands for the sm al l est non-negati ve i nteger,0 k 0 2n 1,such that

(1. 16) S = k0 \ j= 0 K er R e F (Im F ) j \ R 2n :
Fi nal l y,we end these few recal l s by underl i ni ng that the assum pti on about the sympl ecti c structure ofthe si ngul ar space i s al ways ful l l ed by any quadrati c sym bolq w hi ch sati s es the assum pti on ofparti alel l i pti ci ty on i ts si ngul ar space S, (x; )2 S; q(x; )= 0 ) (x; )= 0:

W e refer the reader to Secti on 1. 4. 1 i n [START_REF] Itrik | Spectra and sem igroup sm oothing for non-ell iptic quadratic operators[END_REF]for a proofofthi s fact.

1. 2. Statem ent ofthe m ain result. C onsi deri ng a system ofN quadrati coperators q w j (x;D x ),1 j N ,w hose W eylsym bol s q j have al lnon-negati ve realparts (1. 17) R e q j (x; ) 0; (x; )2 R 2n ; n 2 N ;

and denoti ng by F j thei rassoci ated H am i l ton m aps,the m ai n resul tcontai ned i n thi s arti cl e i s the fol l ow i ng:

T heorem 1.2.1. C onsider a system ofN quadratic operators q w j (x;D x ),1 j N , satisfying (1. 17). Ifthere exists k 0 2 N such that

Im F l k ) \ R 2n = f0g;
then this overdeterm ined system of quadratic operators is subell iptic with a l oss of = 2k 0 =(2k 0 + 1) derivatives, that is, that there exists C > 0 such that for all u 2 D (q 1 )\ : : :\ D (q N ),

(1. 19) h(x; )i 2=(2k0 + 1) w u L 2 C N X j= 1 kq w j (x;D x )uk L 2 + kuk L 2 ; with h(x; )i= (1 + j xj 2 + j j 2 ) 1=2 .
Rem ark. Let us m ake cl ear that the i ntersecti on ofkernel s \ j= 1;:::;N ; (l1 ;:::;l k )2 f1;:::;N g k K er(R e F j Im F l1 : : : Im F l k ); i s to be understood as \ j= 1;:::;N K er R e F j ;

w hen k = 0.

1. 3. E xam ple of a subelliptic system of quadratic operators. T he fol l ow i ng exam pl e of subel l i pti c system of quadrati c operators show s that T heorem 1. 2. 1 real l y hi ghl i ghtsnew non-tri vi ali nteracti on phenom ena between the di erentoperators com posi ng a system ,w hi ch cannotbe deri ved from the resul tofsubel l i pti ci ty know n i n the scal ar case (T heorem 1. 2. 1 i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] ). Indeed,de ne the quadrati c form s ( j q j + ~ j qj );

q j (x; )= x 2 1 + 2 1 + i( 2 1 + x j+ 1
for som e realnum bers j ; ~ j veri fyi ng

n 1 X j= 1 ( j + ~ j )> 0;
i s gi ven by

S = n (x; )2 R 2n :x 1 = 1 = n 1 X j= 1 ( j x j+ 1 + ~ j j+ 1 )= 0 o ;
w hi ch i s al ways a non-zero subvector space. It then fol l ow s that one cannot deduce any resul t about the subel l i pti ci ty ofthe scal ar operator n 1 X j= 1

( j q w j (x;D x )+ ~ j qw j (x;D x ));

i n orderto getthe subel l i pti ci ty ofthe overdeterm i ned system com posed by the 2n 2 operators q w j (x;D x ) and qw j (x;D x ), for 1 j n 1. N everthel ess, by denoti ng respecti vel y F j and Fj the H am i l ton m aps ofthe quadrati c form s q j and qj ,another di rect com putati on usi ng (1. 10) show s that

K er R e F j \ K er(R e F j Im F j )\ R 2n = f(x; )2 R 2n :x 1 = 1 = x j+ 1 = 0g and K er R e Fj \ K er(R e Fj Im Fj )\ R 2n = f(x; )2 R 2n :x 1 = 1 = j+ 1 = 0g:
O ne can then deduce from T heorem 1. 2. 1 the fol l ow i ng gl obalsubel l i pti c esti m ate w i th a l oss of2=3 deri vati ves

h(x; )i 2=3 w u L 2 . n 1 X j= 1 kq w j (x;D x )uk L 2 + kq w j (x;D x )uk L 2 + kuk L 2 :
O fcourse,T heorem 1. 2. 1 can hi ghl i ghtm ore com pl ex i nteracti onsbetween the di erent operators com posi ng the system w hen we consi der operators w i th di erent real parts.

1. 4. C om m ents on the condition for subellipticity. T heorem 1. 2. 1 gi vesa very expl i ci t and si m pl e al gebrai c condi ti on on the H am i l ton m aps of quadrati c operators ensuri ng the subel l i pti ci ty of the system . Let us noti ce that thi s condi ti on i s very easy to handl e and al l ow s to di rectl y m easure the associ ated l oss ofderi vati ves by a strai ghtforward com putati on. W e shal lnow expl ai n how thi s i s rel ated to the H orm ander-K ohn condi ti on.R ecal lfrom [START_REF] Ourrigat | La condition de H orm ander-K ohn pour l es op erateurs pseudodi erentiel s[END_REF](T heorem 1. 1)thattheH orm ander-K ohn condi ti on form i crol ocalsubel l i pti ci ty ofoverdeterm i ned system sofpseudodi erenti al operators w i th realpri nci palsym bol s; reads as the exi stence of an el l i pti c i terated com m utator of the operators com posi ng the system . In the case of a system of non-sel fadjoi nt quadrati c operators (q w j ) 1 j N , i f we assum e i n addi ti on that thi s system i s m axi m alhypoel l i pti c 1 ,the naturalcondi ti on becom es to ask the el l i pti ci ty of an i terated com m utator of the real parts ((R e q j ) w ) 1 j N and i m agi nary parts ((Im q j ) w ) 1 j N ofthe operatorscom posi ng the system .C om i ng back to ourspeci c condi ti on for subel l i pti ci ty (1. 18),we rst noti ce that i n the scal ar case,i t reads as the exi stence ofa non-negati ve i nteger k 0 such that

k0 \ j= 0 K er[ R e F (Im F ) j ] \ R 2n = f0g;
w i th F standi ng for the H am i l ton m ap ofthe uni que operator q w (x;D x ) com posi ng the system . A srecal l ed i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF](Secti on 1. 2),thi s condi ti on i m pl i es that,forany nonzero poi nt i n the phase space X 0 2 R 2n ,we can nd a non-negati ve i nteger k such that 8 0 j 2k 1; H j Im q R e q(X 0 )= 0 and H 2k Im q R e q(X 0 )6 = 0; w here H Im q stands for the H am i l ton vector el d ofIm q, H Im q = @Im q @ @ @x @Im q @x @ @ :

T hi s show s that the 2k th i terated com m utator [ Im q w ;[ Im q w ;[ : : : ;[ Im q w ;R e q w ] ] ] : : : ]= ( 1) k (H 2k Im q R e q) w ; w i th exactl y 2k term s Im q w i n l eft-hand-si de ofthe above form ul a;i s el l i pti c at X 0 ; and underl i nes the i nti m ate l i nk between (1. 18) and the H orm ander-K ohn condi ti on i n the scal arcase.In the system case,the si tuati on i sm ore com pl i cated and thi s l i nk i s l ess obvi ous to hi ghl i ght expl i ci tl y. M ore speci cal l y,we shal lsee i n thi s case that the al gebrai c condi ti on (1. 18) i m pl i es that the quadrati c form k0 X k= 0 X j= 1;:::;N ; (l1 ;:::;l k )2 f1;:::;N g k R e q j (Im F l1 : : : Im F l k X ); i s posi ti ve de ni te. T hi s property i m pl i es that for any non-zero poi nt X 0 2 R 2n ,one can nd k 2 N ,j 2 f1;: : : ;N g and (l 1 ;: : : ;l k )2 f1;: : : ;N g k such that R e q j (Im F l1 : : : Im F l k X 0 )> 0: B y consi deri ng the m i ni m alnon-negati ve i nteger k w i th thi s property and usi ng the sam e argum entsasthe onesdevel oped i n [START_REF] Itrik | Spectra and sem igroup sm oothing for non-ell iptic quadratic operators[END_REF](p. 820-822),one can actual l y check that any i terated com m utator oforder l ess or equalto 2k 1,that i s, [ P 1 ;[ P 2 ;[ P 3 ;[ : : : ;[ P r ;P r+ 1 ] : : : ] ] ] ] ;

1 W e refer to [START_REF] Ourrigat | Subell iptic system s[END_REF]and [START_REF] Ourrigat | Syst em es sous-ell iptiques II[END_REF]for conditions and generalresults of m axim alhypoellipticity for overdeterm ined system s ofnon-selfadjoint pseudodi erentialoperators.

w i th r 2k 1,P l = R e q w s1 or P l = Im q w s2 ;and w here at l east one P l0 i s equalto R e q w s3 ,for 1 s 1 ;s 2 ;s 3 N ; are not el l i pti c at X 0 . O ne can al so check that the non-zero term R e q j (Im F l1 : : :

Im F l k X 0 )> 0;
actual l y appears w hen expandi ng the W eylsym bolat X 0 ofthe 2k th i terated comm utator [ Im q w l k ;[ Im q w l k ;[ Im q w l k 1 ;[ Im q w l k 1 ;[ : : : ;[ Im q w l1 ;[ Im q w l1 ;R e q w j ] ] ] : : : ] = ( 1) k (H 2 Im q l k : : : H 2 Im q l 1 R e q j ) w : H owever,contrary to the scal ar case,there m ay be al so other non-zero term s i n thi s expansi on;and i ti snotreal l y cl eari fthi snaturalcom m utatorassoci ated to the term R e q j (Im F l1 : : : Im F l k X 0 ); i s actual l y el l i pti c at X 0 , H 2 Im q l k : : : H 2 Im q l 1 R e q j (X 0 ) ? 6 = 0:

T hough i tm ay bedi cul tto determ i neexactl y ateach poi ntw hi ch speci ccom m utatori sel l i pti c,i ti svery l i kel y thatcondi ti on (1. 18)ensuresthatthe H orm ander-K ohn condi ti on i s ful l l ed at any non-zero poi nt ofthe phase space;and that these associated el l i pti c com m utatorsare al loforderl ess orequalto 2k 0 . It i s actual l y w hatthe l oss ofderi vati ves appeari ng i n the esti m ate (1. 19) suggests;and thi s i n agreem ent w i th theopti m all ossofderi vati vesobtai ned i n [START_REF] Ourrigat | La condition de H orm ander-K ohn pour l es op erateurs pseudodi erentiel s[END_REF](T heorem 1. 1)for2k 0 com m utators

= 1 1 2k 0 + 1 = 2k 0 2k 0 + 1 ;
si nce we m easure the l oss ofderi vati ves w i th respect to the el l i pti c case as

( 2(1 ) ) w u L 2 . N X j= 1 kq w j (x;D x )uk L 2 + kuk L 2 ;
w i th 2 = h(x; )i 2 , because quadrati c operators have thei r W eyl sym bol s i n the sym bolcl ass S( 2 ; 2 dX 2 ) w hose gai n i s 2 . B ecause ofthe si m pl i ci ty ofi tsassum pti ons,T heorem 1. 2. 1 provi desa neatsetti ng forprovi ng gl obalsubel l i pti c esti m atesforsystem sofquadrati c operators.Iti spossibl e that som e ofthese gl obalsubel l i pti c esti m ates for system s ofquadrati c operators m ay al so be deri ved from the resul tsofm i crol ocalsubel l i pti ci ty and m axi m alhypoell i pti ci ty proved i n [START_REF] Ourrigat | La condition de H orm ander-K ohn pour l es op erateurs pseudodi erentiel s[END_REF] , [START_REF] Ourrigat | Subell iptic system s[END_REF]and [START_REF] Ourrigat | Syst em es sous-ell iptiques II[END_REF] . H owever,gi ven a parti cul ar system of quadrati c operators,one can noti ce thatonl y checki ng the H orm ander-K ohn condi ti on i n every non-zero poi nt turns out to be qui te di cul t to do i n practi ce. T he sam e com m ent appl i esforchecki ng them axi m alhypoel l i pti ci ty ofthesystem .A notheri nterestofthe approach we are devel opi ng here com esfrom the factthatthe proofofT heorem 1. 2. 1 i s purel y anal yti c and does notrequi re any techni ques ofrepresentati onsofni l potent groups as i n [START_REF] Ourrigat | Subell iptic system s[END_REF] or [START_REF] Ourrigat | Syst em es sous-ell iptiques II[END_REF] . M oreover,despi te i ts l ength, the proof provi ded here onl y i nvol ves fai rl y el em entary argum ents w hose com pl exi ty has no degree ofcom pari son w i th the anal ysi s l ed i n [START_REF] Ourrigat | Subell iptic system s[END_REF]and [START_REF] Ourrigat | Syst em es sous-ell iptiques II[END_REF] .

Fi nal l y,l etusend thi si ntroducti on by m enti oni ng thatthi s resul tofsubel l i pti ci ty forsystem sofquadrati coperatorsm ay broaden new perspecti vesi n theunderstandi ng ofoverdeterm i ned system sofpseudodi erenti aloperatorsw i th doubl e characteri sti cs; and thattheconstructi on ofthewei ghtfuncti onsi n Proposi ti on 2. 0. 1 m ay beoffurther i nterest and di rect use i n future anal ysi s of doubl y characteri sti c probl em s. In the scal ar case,thi s constructi on ofthe wei ght functi on speci c to the structure ofthe doubl echaracteri sti csobtai ned i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF](Proposi ti on 2. 0. 1)hasal ready al l owed to deri ve i n [START_REF] Itrik | D iscrete spectra and resol vent estim ates for non-sel fadjoint pseudodi erentialoperators w ith doubl e characteristics,w ork in preparation[END_REF]the preci se asym ptoti cs for the resol vent norm ofcertai n cl ass ofsem i cl assi cal pseudodi erenti aloperators i n a nei ghborhood ofthe doubl y characteri sti c set. O n the other hand, thi s deeper understandi ng of non-tri vi al i nteracti ons between the di erent quadrati c operators com posi ng overdeterm i ned system s m ay al so gi ve hi nts on how to anal yze the m ore com pl ex case ofN by N system s ofquadrati c operators, w hi ch i s a topi c of current i nterest. O n that subject, we refer the reader to the seri es of recent works on non-com m utati ve harm oni c osci l l ators by A .Parm eggi ani and M .W akayam a i n [START_REF] Parm Eggiani | W akayam a,N on-com m utative harm onic oscill ators I[END_REF] , [START_REF] Parm Eggiani | W akayam a,N on-com m utative harm onic oscill ators II[END_REF] , [START_REF] Parm | O n the spectrum and the l ow est eigenval ue of certain non-com m utative harm onic oscill ators[END_REF] , [START_REF] Parm | O n the spectrum of certain noncom m utative harm onic oscill ators,A nn[END_REF] , [START_REF] Parm Eggiani ; C Entury C O E P Rogram \d M H F | Introduction to the spectral theory of non-com m utative harm onic oscill ators C O E Lecture N ote[END_REF]and [START_REF] Parm | O n the spectrum of certain non-com m utative harm onic oscill ators and sem icl assicalanal ysis[END_REF] .

P roof of T heorem 1.2.1

In the fol l ow i ng,we shal luse the notati on S m (X ) r ;m (X ) 2s dX 2 ,w here i s an open set i n R 2n ,r;s 2 R and m 2 C 1 ( ;R + ),to stand for the cl ass ofsym bol s a veri fyi ng

a 2 C 1 ( ); 8 2 N 2n ;9C > 0; j @ X a(X )j C m (X ) r sj j ; X 2 :
In the case w here = R 2n ,we shal ldrop the i ndex forsi m pl i ci ty. W e shal lal so use the notati ons f . g and f g,on ,for respecti vel y the esti m ates 9C > 0,f C g and,f . g and g . f,on .

T he proofofT heorem 1. 2. 1 w i l lrel y on the fol l ow i ng key proposi ti on.C onsi deri ng for 1 j N ,

q j :R n x R n ! C (x; ) 7 ! q j (x; );
w i th n 2 N ,N com pl ex-val ued quadrati c form s w i th non-negati ve realparts (2. 1) R e q j (x; ) 0; (x; )2 R 2n ; 1 j N ;

we assum e thatthere exi sta posi ti ve i ntegerm 2 N and an open set 0 i n R 2n such that the fol l ow i ng sum ofnon-negati ve quadrati c form s sati s es

(2. 2) 9c 0 > 0;8X 2 0 ; m X k= 0
X j= 1;:::;N ; (l1 ;:::;l k )2 f1;:::;N g k R e q j (Im F l1 : : :

Im F l k X ) c 0 j X j 2 ;
w here the notati on Im F j stands for the i m agi nary part ofthe H am i l ton m ap F j associ ated to the quadrati c form q j . U nder thi s assum pti on,one can then extend the constructi on ofthe bounded wei ght functi on done i n the scal arcase i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF](Propositi on 2. 0. 1)to the system case as fol l ow s: P roposition 2.0.1. If(q j ) 1 j N are N com pl ex-val ued quadratic form s on R 2n verifying (2. 1) and (2. 2) then there existN real -val ued weight functions

g j 2 S 0 1;hX i 2 2m + 1 dX 2 ; 1 j N ; such that (2. 3) 9c;c 1 ;: : : ;c N > 0;8X 2 0 ; 1 + N X j= 1 R e q j (X )+ c j H Im qj g j (X ) chX i 2 2m + 1 ;
where the notation H Im qj stands for the H am il ton vector el d ofthe im aginary part ofq j .

A s i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] ,the constructi on ofthese wei ghtfuncti ons w i l lbe real l y the core ofthi s work.T hi sconstructi on w i l lbean adaptati on to thesystem caseoftheoneperform ed i n the scal ar case.

To check that we can actual l y deduce T heorem 1. 2. 1 from Proposi ti on 2. 0. 1,we begi n by noti ci ng,as i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] ,that the assum pti ons ofT heorem 1. 2. 1 i m pl y that the fol l ow i ng sum ofnon-negati ve quadrati c form s

(2. 4) 9c 0 > 0; r(X )= k0 X k= 0
X j= 1;:::;N ; (l1 ;:::;l k )2 f1;:::;N g k R e q j (Im F l1 : : :

Im F l k X ) c 0 j X j 2 ;
i s actual l y a posi ti ve de ni te quadrati c form . Let us i ndeed consi der X 0 2 R 2n such that r(X 0 ) = 0. T hen,the non-negati vi ty ofquadrati c form s R e q j i nduces that for al l0 k k 0 ,j = 1;: : : ;N and (l 1 ;: : : ;l k )2 f1;: : : ;N g k , (2. 5) R e q j (Im F l1 : : :

Im F l k X 0 )= 0:
B y denoti ng R e q j (X ;Y ) the pol ar form associ ated to R e q j , we deduce from the C auchy-Schwarz i nequal i ty,(1. 10) and (2. 5) that for al lY 2 R 2n , j R e q j (Y ;Im F l1 : : :

Im F l k X 0 )j 2 = j (Y;R e F j Im F l1 : : : Im F l k X 0 )j 2 R e q j (Y ) R e q j (Im F l1 : : : Im F l k X 0 )= 0: It fol l ow s that for al lY 2 R 2n , (Y;R e F j Im F l1 : : : Im F l k X 0 )= 0;
w hi ch i m pl i es that for al l0 k k 0 ,j = 1;: : : ;N and (l 1 ;: : : ;l k )2 f1;: : : ;N g k , (2. 6) R e F j Im F l1 : : : Im F l k X 0 = 0; si nce i s non-degenerate.W e nal l y deduce (2. 4) from the assum pti on (1. 18). In the case w here k 0 = 0,we noti ce that the quadrati c form q = q 1 + : : :+ q N ; has a posi ti ve de ni te realpart. T hi s i m pl i es i n parti cul ar that q i s el l i pti c on R 2n . O ne can therefore di rectl y deduce from cl assi calresul ts about el l i pti c quadrati c di fferenti aloperators proved i n [START_REF] Sj | P aram etrices for pseudodi erential operators w ith m ul tipl e characteristics[END_REF](See T heorem 3. 5 i n [START_REF] Sj | P aram etrices for pseudodi erential operators w ith m ul tipl e characteristics[END_REF]or com m ents about the el l i pti c case i n T heorem 1. 2. 1 i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] ),the naturalel l i pti c a pri oriesti m ate

9C > 0;8u 2 D (q 1 )\ : : :\ D (q N ); h(x; )i 2 w u L 2 C (kq w (x;D x )uk L 2 + kuk L 2 );
w hi ch easi l y i m pl i es (1. 19). W e can therefore assum e i n the fol l ow i ng that k 0 1 and nd from Propositi on 2. 0. 1 som e real -val ued wei ght functi ons (2. 7)

g j 2 S 1;hX i 2 2k 0 + 1 dX 2 ; 1 j N ; such that (2. 8) 9c;c 1 ;: : : ;c N > 0;8X 2 R 2n ; 1+ N X j= 1 R e q j (X )+ c j H Im qj g j (X ) chX i 2 2k 0 + 1 : For0 < "
N X j= 1 R e q W ick j u;(1 "c j g j ) W ick u = N X j= 1 R e (1 "c j g j ) W ick q W ick j u;u N X j= 1 k1 "c j g j k L 1 kq W ick j uk L 2 kuk L 2 . N X j= 1 kq W ick j uk 2 L 2 + kuk 2 L 2 . N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 ;
w here

(2. 10) qj (x; )= q j x; 2 ;

because the operators (1 "c j g j ) W ick w hose W i ck sym bolare real -val ued, are form al l y sel fadjoi nt. Indeed,sym bol sr(q j )de ned i n (4. 8)are here just som e constants si nce q j are quadrati c form s. T he factor 2 i n (2. 10) com es from the di erence of norm al i zati onschosen between (1. 1)and (4. 9)(See rem ark i n Secti on 4. 1).Si nce from (4. 10),

(1 "c j g j ) W ick q W ick j = h (1 "c j g j )q j + " 4 c j r g j : r q j " 4i c j fg j ;q j g i W ick + S j ;

w i th kS j k L (L 2 (R n ))
. 1,we obtai n from the factthatrealH am i l toni ansgetquanti zed i n the W i ck quanti zati on by form al l y sel fadjoi nt operators that

N X j= 1 R e (1 "c j g j ) W ick q W ick j = N X j= 1 R e S j + N X j= 1 h (1 "c j g j )R e q j + " 4 c j r g j : r R e q j + " 4 c j H Im qj g j i W ick ;
because g j are real -val ued sym bol s.Si nce R e q j 0 and g j 2 L 1 (R n ),we can choose the posi ti ve param eter " su ci entl y sm al lsuch that

8 1 j N ;8X 2 R 2n ; 1 "c j g j (X ) 1 2 ;
i n order to deduce from (2. 8),(2. 9) and (4. 3) that (2. 11) (hX i

2 2k 0 + 1 ) W ick u;u . kuk 2 L 2 + N X j= 1 kq w j uk 2 L 2 + N X j= 1
(r g j : r R e q j ) W ick u;u ;

because from (4. 1) and (4. 2),1 W ick = Id: O ne can then com pl ete the proofofT heorem 1. 2. 1 by fol l ow i ng exactl y the sam e reasoni ng as the one used i n [START_REF] Ravda-Starov | Subell iptic estim ates for quadratic di erentialoperators, accepted for publication in A m erican JournalofM athem atics[END_REF] . W e recal l thi s reasoni ng here for the sake of com pl eteness ofthi s work.

B y denoti ng X = x; =(2 ) and O p w S(1;dX 2 ) the operators obtai ned by the W eylquanti zati on ofsym bol s i n the cl ass S(1;dX 2 ),i t fol l ow s from (4. 7),(4. 8) and usualresul ts ofsym bol i c cal cul us that (2. 12) hX i

2 2k 0 + 1 W ick h X i 2 2k 0 + 1 w 2 O p w S(1;dX 2 ) and (2. 13) h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w h X i 2 2k 0 + 1 w 2 O p w S(1;dX 2 ) ; si nce k 0 0. B y usi ng that h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w u;u = h X i 1 2k 0 + 1 w u 2 L 2 ;
we therefore deduce from (2. 11) and the C al der on-Vai l l ancourttheorem that

(2. 14) h X i 1 2k 0 + 1 w u 2 L 2 . kuk 2 L 2 + N X j= 1 kq w j uk 2 L 2 + N X j= 1
(r g j : r R e q j ) W ick u;u :

T hen,we get from (2. 7) and (4. 3) that (2. 15) (r g j : r R e q j ) W ick u;u . j r R e q j j W ick u;u : R ecal l i ng now the wel l -know n i nequal i ty

(2. 16) j f 0 (x)j 2 2f(x)kf 00 k L 1 (R ) ;
ful l l ed by any non-negati ve sm ooth functi on w i th bounded second deri vati ve, we deduce from another use of(4. 3) that (2. 17) j r R e q j j W ick u;u .

((R e q j ) 1
2 ) W ick u;u . (1 + R e q j ) W ick u;u ;

si nce R e q j i s a non-negati ve quadrati c form and that 2(R e q j ) 1 2

1 + R e q j : B y usi ng the sam e argum ents as i n (2. 9),we obtai n that

(1 + R e q j ) W ick u;u = (R e q j ) W ick u;u + kuk 2 L 2 = R e(q W ick j u;u)+ kuk 2 L 2 kq W ick j uk L 2 kuk L 2 + kuk 2 L 2 . kq W ick j uk 2 L 2 + kuk 2 L 2 . kq w j uk 2 L 2 + kuk 2 L 2 :
It therefore fol l ow s from (2. 14),(2. 15) and (2. 17)that (2. 18) h X i

1 2k 0 + 1 w u 2 L 2 . kuk 2 L 2 + N X j= 1 kq w j uk 2 L 2 :
In order to i m prove the esti m ate (2. 18), we careful l y resum e our previ ous anal ysi s and noti ce that our previ ous reasoni ng has i n fact establ i shed that

h X i 1 2k 0 + 1 w u 2 L 2 . kuk 2 L 2 + N X j= 1 R e q W ick j u;(1 "c j g j ) W ick u + N X j= 1 (r g j : r R e q j ) W ick u;u . kuk 2 L 2 + N X j= 1 R e q W ick j u;(1 "c j g j ) W ick u + N X j= 1 j R e(q W ick j u;u)j . kuk 2 L 2 + N X j= 1 R e qw j u;(1 "c j g j ) W ick u + N X j= 1 j R e(q w j u;u)j ; because (1 "c j g j ) W ick i s a bounded operator on L 2 (R n ), (2. 19) k(1 "c j g j ) W ick k L (L 2 ) k1 "c j g j k L 1 (R 2n ) :
B y appl yi ng thi sesti m ate to h X i

1 2k 0 + 1 w
u,we deduce from (2. 13)and the C al der on-Vai l l ancourt theorem that

(2. 20) h X i 2 2k 0 + 1 w u 2 L 2 . N X j= 1 R e qw j h X i 1 2k 0 + 1 w u; h X i 1 2k 0 + 1 w u + N X j= 1 R e qw j h X i 1 2k 0 + 1 w u;(1 "c j g j ) W ick h X i 1 2k 0 + 1 w u + h X i 1 2k 0 + 1 w u 2 L 2 + kuk 2 L 2 :
T hen,by noti ci ng that the com m utator

(2. 21) qw j ; h X i 1 2k 0 + 1 w 2 O p w S hX i 1 2k 0 + 1 ;hX i 2 dX 2 ;
because qj i s a quadrati c form ,and that

(2. 22) h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w Id 2 O p w S(hX i 2 ;hX i 2 dX 2 ) ;
we deduce from standard resul ts ofsym bol i c cal cul us and the C al der on-Vai l l ancourt theorem that qw j ; h X i

1 2k 0 + 1 w u L 2 . qw j ; h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w u L 2 + kuk L 2 . h X i 1 2k 0 + 1 w u L 2 + kuk L 2 : (2. 23)
B y i ntroduci ng thi s com m utator, we get from the C auchy-Schwarz i nequal i ty and (2. 23) that R e qw j h X i

1 2k 0 + 1 w u; h X i 1 2k 0 + 1 w u . R e qw j u; h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w u + h X i 1 2k 0 + 1 w u 2 L 2 + kuk 2 L 2 :
A notheruseoftheC auchy-Schwarzi nequal i ty and theC al der on-Vai l l ancourttheorem w i th (2. 13) gi ves that R e qw j u; h X i

1 2k 0 + 1 w h X i 1 2k 0 + 1 w u . kq w j uk L 2 h X i 2 2k 0 + 1 w u L 2 + kq w j uk L 2 kuk L 2 :
W e then deduce from (2. 18) and the previ ous esti m ate that

N X j= 1 R e qw j h X i 1 2k 0 + 1 w u; h X i 1 2k 0 + 1 w u . h X i 2 2k 0 + 1 w u L 2 N X j= 1 kq w j uk L 2 + N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 :
B y usi ng agai n the C auchy-Schwarz i nequal i ty,(2. 18),(2. 19),(2. 20) and (2. 23),thi s esti m ate i m pl i es that

h X i 2 2k 0 + 1 w u 2 L 2 . N X j= 1 R e qw j ; h X i 1 2k 0 + 1 w u;(1 "c j g j ) W ick h X i 1 2k 0 + 1 w u (2. 24) + N X j= 1 R e qw j u; h X i 1 2k 0 + 1 w (1 "c j g j ) W ick h X i 1 2k 0 + 1 w u + N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 . N X j= 1 R e qw j u; h X i 1 2k 0 + 1 w (1 "c j g j ) W ick h X i 1 2k 0 + 1 w u + N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 . N X j= 1 kq w j uk L 2 h X i 1 2k 0 + 1 w (1 "c j g j ) W ick h X i 1 2k 0 + 1 w u L 2 + N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 ;
because we get from (2. 19) and (2. 23) that R e qw j ; h X i

1 2k 0 + 1 w u;(1 "c j g j ) W ick h X i 1 2k 0 + 1 w u . h X i 1 2k 0 + 1 w u 2 L 2 + h X i 1 2k 0 + 1 w u L 2 kuk L 2 :
N oti ce now that (2. 7),(4. 5) and (4. 6) i m pl y that

h X i 1 2k 0 + 1 w ;(1 "c j g j ) W ick 2 O p w S(1;dX 2 ) ;
si nce (1 "c j g j ) W ick = gw j ,w i th gj 2 S(1;dX 2 ) and k 0 0. B y i ntroduci ng thi s new com m utator,we deduce from the C al der on-Vai l l ancourt theorem ,(2. 13),(2. 18) and (2. 19) that h X i

1 2k 0 + 1 w (1 "c j g j ) W ick h X i 1 2k 0 + 1 w u L 2 . h X i 1 2k 0 + 1 w u L 2 + (1 "c j g j ) W ick h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w u L 2 . h X i 1 2k 0 + 1 w u L 2 + h X i 1 2k 0 + 1 w h X i 1 2k 0 + 1 w u L 2 . h X i 2 2k 0 + 1 w u L 2 + h X i 1 2k 0 + 1 w u L 2 + kuk L 2 . h X i 2 2k 0 + 1 w u L 2 + N X j= 1 kq w j uk L 2 + kuk L 2 :
R ecal l i ng (2. 24),we can then use thi s l ast esti m ate to obtai n that

(2. 25) h X i 2 2k 0 + 1 w u 2 L 2 . N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 :
B y nal l y noti ci ng from the hom ogenei ty ofdegree 2 ofq j that we have (q j T )(x; )= 1 2 q j (x; ); i fT stands for the reall i near sym pl ecti c transform ati on

T (x; )= (2 ) 1 2 x;(2 ) 1 2
;

we deduce from the sym pl ecti c i nvari ance ofthe W eylquanti zati on (T heorem 18. 5. 9 i n [START_REF]T he anal ysis of l inear partial di erential operators[END_REF] ) that hX i

2 2k 0 + 1 w u 2 L 2 . N X j= 1 kq w j uk 2 L 2 + kuk 2 L 2 ;
w hi ch proves T heorem 1. 2. 1.

3. P roof of P roposition 2.0.1

W e prove Proposi ti on 2. 0. 1 by i nducti on on the posi ti ve i nteger m 1 appeari ng i n (2. 2). Let m 1,we shal lassum e that Proposi ti on 2. 0. 1 i s ful l l ed for any open set 0 ofR 2n ,w hen the posi ti ve i nteger i n (2. 2) i s stri ctl y sm al l er than m .

In the fol l ow i ng,we denoteby , and w som eC 1 (R ;[ 0;1] )functi onsrespecti vel y sati sfyi ng M ore generi cal l y,we shal ldenote by j , j and w j ,j 2 N ,som e other C 1 (R ;[ 0;1] ) functi onssati sfyi ng si m i l arproperti esasrespecti vel y , and w w i th possi bl y di erent choi ces for the posi ti ve num eri calval ues w hi ch de ne thei r support l ocal i zati ons.

Let 0 be an open setofR 2n such that(2. 2)i sful l l ed. C onsi deri ng the quadrati c form s

(3. 4) r1;p (X )= N X j= 1
R e q j (X ;Im F p X );

(3. 5) rk;p (X )= X j= 1;:::;N (l1 ;:::;l k 1 )2 f1;:::;N g k 1 R e q j (Im F l1 : : : Im F l k 1 X ;Im F l1 : : :

Im F l k 1 Im F p X ); for any 1 p N ,2 k m ; (3. 6) r 0 (X )= N X j= 1 R e q j (X ); r k (X )=
X j= 1;:::;N (l1 ;:::;l k )2 f1;:::;N g k R e q j (Im F l1 : : :

Im F l k X );
for any 1 k m ;and de ni ng

(3. 7) gm ;p (X )= r m 1 (X )hX i 2(2m 1) 2m + 1 hX i 4m 2m + 1 rm ;p (X );
w here i s the functi on de ned i n (3. 1) and 1 p N ,we get from Lem m a 4. 2. 1 that

H Im qp gm ;p (X )= 2 r m 1 (X )hX i 2(2m 1) 2m + 1
X j= 1;:::;N (l1 ;:::;lm 1 )2 f1;:::;N g m 1 R e q j (Im F l1 : : :

Im F lm 1 Im F p X ) hX i 4m 2m + 1 (3. 8) + 2 r m 1 (X )hX i 2(2m 1) 2m + 1
X j= 1;:::;N (l1 ;:::;lm 1 )2 f1;:::;N g m 1 R e q j (Im F l1 : : : Im F lm 1 X ;Im F l1 : : : In order to veri fy thi s,we noti ce from Lem m a 4. 2. 6 that the quadrati c form s (3. 10) R e q j (Im F l1 : : : Im F lm 1 X ;Im F l1 : : :

Im F lm 1 (Im F p ) 2 X ) hX i 4m 2m + 1 + H Im qp r m 1 (X )hX i 2(2m 1) 2m + 1 rm ;p (X ) hX i 4m 2m + 1 + r m 1 (X )hX i 2( 2m
Im F lm 1 Im F p X ) and
(3. 11) R e q j (Im F l1 : : :

Im F lm 1 X ;Im F l1 : : : Im F lm 1 (Im F p ) 2 X );
bel ong to the sym bolcl ass (3. 12)

S hX i 4m 2m + 1 ;hX i 2(2m 1) 2m + 1 dX 2 ;
for any open set i n R 2n w here r m 1 (X ) . hX i

2(2m 1)
2m + 1 . To check thi s,we just use i n addi ti on to Lem m a 4. 2. 6 the obvi ous esti m ates R e q j (Im F l1 : : :

Im F lm 1 Im F p X ) 1 2 . hX i
and R e q j (Im F l1 : : : N ext,by usi ng that Im q p 2 S hX i 2 ;hX i 2 dX 2 ; si nce Im q p i s a quadrati c form ,we get from (3. 1), (3. 5) ; we deduce from (3. 8),(3. 14),(3. 15) and (3. 16) that there exi sts 0 a C 1 (R ;[ 0;1] ) functi on sati sfyi ng si m i l ar properti es as i n (3. 2),w i th possi bl y di erent posi ti ve num eri calval ues for i ts support l ocal i zati on,such that,9c 1 ;c 2 > 0,8X 2 R 2n ,

Im F lm 1 (Im F p ) 2 X ) 1
c 1 + c 2 0 r m 1 (X )hX i 2(2m 1) 2m + 1 hX i 2 2m + 1 + N X p= 1 H Im qp gm ;p (X ) (3. 17) 2 r m 1 (X )hX i 2(2m 1) 2m + 1 r m (X ) hX i 4m 2m + 1 :
R ecal l i ng (2. 2),one can nd som e posi ti ve constants c 3 ;c 4 > 0 such that

(3. 18) m 1 X k= 0 r k (X ) c 3 j X j 2 ;
on the open set

(3. 19) 1 = X 2 R 2n :r m (X )< c 4 j X j 2 \ 0 : W hen m
2,one can nd accordi ng to our i nducti on hypothesi s som e real -val ued functi ons (3. 20) gm ;p 2 S 1 1;hX i For conveni ence,we set i n the fol l ow i ng g1;p = 0 w hen m = 1. B y choosi ng sui tabl y 0 and w 0 som e C 1 (R ;[ 0;1] ) functi ons sati sfyi ng si m i l ar properti es as the functi ons respecti vel y de ned i n (3. 1)and (3. 3),w i th possi bl y di erentposi ti venum eri calval ues for thei r support l ocal i zati ons,such that (3. 22) supp 0 r m (X )j X j 2 w 0 (X ) X 2 R 2n :r m (X )< c 4 j X j 2 ;

and setti ng (3. 23) G m ;p (X )= gm ;p (X )+ 0 r m (X )j X j 2 w 0 (X )g m ;p (X ); X 2 0 ;

we deduce from a strai ghtforward adaptati on ofthe Lem m a 4.

H Im qp 0 r m (X )j X j 2 w 0 (X ) gm ;p (X )2 S 0 1;hX i 2 2m
1 dX 2 ; and then deduce from (3. 17),(3. 19),(3. 21), (3. 22) and (3. 23) that there exi st som e posi ti ve contants c 6;p ;c 7 > 0 such that for al lX 2 0 ,

N X p= 1 R e q p (X )+ c 6;p H Im qp G m ;p (X ) + 1 + c 7 0 r m 1 (X )hX i 2(2m 1) 2m + 1 hX i 2 2m + 1 & r m 1 (X )hX i 2(2m 1) 2m + 1 r m (X ) hX i 4m 2m + 1 + 0 r m (X )j X j 2 w 0 (X )hX i 2 2m 1 ; w hen m 2. Si nce hX i 2 2m
1 & hX i 2 2m + 1 and r m (X )

hX i 4m 2m + 1 & j X j 2 2m + 1 ;
w hen r m (X ) & j X j 2 , we deduce from the previ ous esti m ate by di sti ngui shi ng the regi ons i n 0 w here r m (X ). j X j 2 and r m (X )& j X j 2 ; accordi ng to the support ofthe functi on 0 r m (X )j X j 2 ; that one can nd a C 1 (R ;[ 0;1] ) functi on w 1 w i th the sam e ki nd ofsupport as the functi on de ned i n (3. 3) such that

(3. 26) 9c 8;p ;c 9 > 0;8X 2 0 ; N X p= 1 R e q p (X )+ c 8;p H Im qp G m ;p (X ) + c 9 w 1 r m 1 (X )hX i 2(2m 1) 2m + 1 hX i 2 2m + 1 + 1 & hX i 2 2m + 1 ; w hen m 2. W hen m = 1,we noti ce from (2. 2) that (3. 27) r 1 (X )& hX i 2 ;
on any set w here (3. 28) j X j c 10 and r 0 (X )= N X p= 1 R e q p (X ) hX i i fthe posi ti ve constant c 10 i s chosen su ci entl y l arge. M oreover,si nce i n thi s case G 1;p = g1;p and that R e q p 0,one can deduce from (3. 1),(3. 3),(3. 17),(3. 27) and (3. 28),by di sti ngui shi ng the regi ons i n 0 w here r 0 (X ). hX i that the esti m ate (3. 26) i s al so ful l l ed i n the case m = 1. C onti nui ng our study of the case w here m = 1,we noti ce from (3. 3) and R e q p 0,that one can esti m ate

w 1 r 0 (X )hX i 2 3 hX i 2 3 . r 0 (X )= N X p= 1
R e q p (X );

for al lX 2 R 2n . It therefore fol l ow s that one can nd c 11;p > 0 such that for al l X 2 0 ,

N X p= 1 R e q p (X )+ c 11;p H Im qp G 1;p (X ) + 1 & hX i 2 3 ;
w hi ch provesProposi ti on 2. 0. 1 i n the case w here m = 1,and ouri nducti on hypothesi s i n the basi s case.

A ssum i ng i n the fol l ow i ng that m 2,we shal lnow work on the term

w 1 r m 1 (X )hX i 2(2m 1) 2m + 1 hX i 2 2m + 1 ;
appeari ng i n (3. 26).B y consi deri ng som e constants j 1,for0 j m 2,w hose val ues w i l lbe successi vel y chosen i n the fol l ow i ng,we shal lprove that one can w ri te that for al lX 2 R 2n , (3. 29) w 1 r m 1 (X )

hX i 2(2m 1) 2m + 1 ! W 0 (X ) 0 (X ) + m 2 X j= 1 W 0 (X ) j Y l= 1 W l (X ) j (X )+ W 0 (X ) m 1 Y l= 1 W l (X ) ; w i th (3. 30) j (X )= j r m j 2 (X ) r m j 1 (X ) 2m 2j 3 2m 2j 1 ! ; 0 j m 2; (3. 31) W j (X )= w 2 j 1 r m j 1 (X ) r m j (X ) 2m 2j 1 2m 2j+ 1 ! ; 1 j m 1; (3. 32) W 0 (X )= w 1 r m 1 (X ) hX i 2(2m 1) 2m + 1 ! ;
w here i stheC 1 (R ;[ 0;1] )functi on de ned i n (3. 1),and w 2 i sa C 1 (R ;[ 0;1] )functi on sati sfyi ng si m i l ar properti es as the functi on de ned i n (3. 3),w i th possi bl y di erent posi ti ve num eri calval ues for i ts support l ocal i zati on,i n order to have that In order to check (3. 29),we begi n by noti ci ng from (3. 3),(3. 31) and (3. 32) that for 0 j m 1, (3. 34) r m j 1 (X ) W l ; i f1 j m 1; or,supp W 0 ; i fj = 0: N oti ce that the constants i n the esti m ates (3. 34) onl y depend on the val ues ofthe param eters 0 , . . . , j 1 but not on l ,w hen l j. T hi s show s that the functi ons W l ; i f1 j m 2; or,supp W 0 ; i fj = 0; we deduce the esti m ate (3. 29)from a ni te i terati on by usi ng the fol l ow i ng esti m ates

1 2m 2j 1 & r m j (X ) 1 2m 2j+ 1 & : : :& r m 1 (X )
W 0 W 0 0 + W 0 W 1 and W 0 j Y l= 1 W l W 0 j Y l= 1 W l j + W 0 j+ 1 Y l= 1 W l ;
for any 1 j m 2. O ne can al so noti ce that (3. 35) i m pl i es that

(3. 36) 1 j + m 2 X k= j+ 1 k Y l= j+ 1 W l k + m 1 Y l= j+ 1 W l ;
on the support ofthe functi on

supp W 0 j Y l= 1
W l ; i f1 j m 2; or,supp W 0 ; i fj = 0:

Si nce R e q p 0,we then get from (3. 34) that

(3. 37) 8X 2 R 2n ; W 0 (X ) m 1 Y l= 1 W l (X ) hX i 2 2m + 1 ã 0 ;:::; m 2 N X p= 1
R e q p (X ); w here ã 0 ;:::; m 2 i s a posi ti ve constant w hose val ue depends on the param eters

( l ) 0 l m 2 : W e de ne for 1 p N , (3. 38) p j;p (X )= W 0 (X ) j Y l= 1 W l (X ) j (X ) rm j 1;p (X ) r m j 1 (X ) 2m 2j 2 2m 2j 1
; for 1 j m 2,and

(3. 39) p 0;p (X )= W 0 (X ) 0 (X ) rm 1;p (X ) for any 0 j m 2.

r m 1 (X )
W e shal l now study the Poi sson brackets H Im qp p j;p . In doi ng so, we begi n by w ri ti ng that

H Im qp p j;p (X )= H Im qp W 0 (X ) j Y l= 1 W l (X ) j (X ) rm j 1;p (X ) r m j 1 (X ) 2m 2j 2 2m 2j 1 (3. 41) + W 0 (X ) j Y l= 1 W l (X ) H Im qp j (X ) rm j 1;p (X ) r m j 1 (X ) 2m 2j 2 2m 2j 1 + W 0 (X ) j Y l= 1 W l (X ) j (X )H Im qp r m j 1 (X ) 2m 2j 2 2m 2j 1 rm j 1;p (X ) + W 0 (X ) j Y l= 1 W l (X ) j (X ) H Im qp rm j 1;p (X ) r m j 1 (X ) 2m 2j 2 2m 2j 1 + j X l= 1 W 0 (X ) H Im qp W l (X ) j Y k = 1 k 6 = l W k (X ) j (X ) rm j 1;p (X ) r m j 1 (X ) 2m 2j 2 2m 2j 1
; for 1 j m 2. W e denote by respecti vel y B 1;j;p ,B 2;j;p ,B 3;j;p ,B 4;j;p and B 5;j;p the ve term s appeari ng i n the ri ght hand si de of (3. 41). W e al so w ri te i n the case w here j = 0,

H Im qp p 0;p (X )= H Im qp W 0 (X ) 0 (X ) rm 1;p (X ) r m 1 (X ) 2m 2 2m 1 (3. 42) + W 0 (X ) H Im qp 0 (X ) rm 1;p (X ) r m 1 (X ) 2m 2 2m 1 + W 0 (X ) 0 (X )H Im qp r m 1 (X ) 2m 2 2m 1 rm 1;p (X ) + W 0 (X ) 0 (X ) H Im qp rm 1;p (X ) r m 1 (X ) 2m 2 2m 1
;

and denote as before by respecti vel y B 1;0;p ,B 2;0;p ,B 3;0;p and B 4;0;p the four term s appeari ng i n the ri ght hand si de of(3. 42). Si nce the constantsi n the esti m ates(3. 34)onl y depend on the val uesofthe parameters 0 , . . . , j 1 ;but noton l ,w hen l j;we noti ce from (3. 29),(3. 34)and (3. 37) that there exi st a 0 > 0 and som e posi ti ve constants a j; 0 ;:::; j 1 ,for 1 j m 1, w hoseval uesw i th respectto the param eters( l ) 0 l m 2 onl y depend on 0 , . . . , j 1 ; but not on l ,w hen l j;such that for any constants ( j ) 1 j m 2 ,w i th j 1;

and X 2 R 2n ,

w 1 r m 1 (X ) hX i 2(2m 1) 2m + 1 ! hX i 2 2m + 1 a 0 W 0 (X ) 0 (X )r m 1 (X ) 1 2m 1 (3. 43) + m 2
X j= 1 j a j; 0 ;:::

; j 1 W 0 (X ) j Y l= 1
W l (X ) j (X )r m j 1 (X ) Forany " > 0,weshal lprovethataftera properchoi cefortheconstants( j ) 0 j m 2 and ( j ) 1 j m 2 ,w i th j 1, j 1,w hose val ues w i l ldepend on ";one can nd a posi ti ve constant c 12;" > 0 such that for al lX 2 R 2n ,

(3. 46) c 12;" N X p= 1 R e q p (X )+ H Im qp p p (X ) + "hX i 2 2m + 1 w 1 r m 1 (X ) hX i 2(2m 1) 2m + 1 ! hX i 2 2m + 1 :
O nce thi s esti m ate proved,Proposi ti on 2. 0. 1 w i l ldi rectl y fol l ow from (3. 25),(3. 26), (3. 45)and (3. 46),i fwe choose the posi ti ve param eter" su ci entl y sm al land consi der the wei ght functi ons g p = c 13;" G m ;p + c 14;" p p ; 1 p N ; after a sui tabl e choi ce for the posi ti ve constants c 13;" and c 14;" .

Let " > 0,i t therefore rem ai ns to choose properl y these constants ( j ) 0 j m 2 and ( j ) 1 j m 2 ,w i th j 1, j 1,i n order to sati sfy (3. 46). R ecal l i ng from (4. 22) that for al l1 p N and 0 s m 2,

(3. 47) H Im qp rm s 1;p (X )= 2 X j= 1;:::;N (l1 ;:::;lm s 2 )2 f1;:::;N g m s 2 R e q j (Im F l1 : : :

Im F lm s 2 Im F p X ) + 2
X j= 1;:::;N (l1 ;:::;lm s 2 )2 f1;:::;N g m s 2 R e q j (Im F l1 : : : Im F lm s 2 X ;Im F l1 : : : R e q j (Im F l1 : : : Im F lm s 2 Im F p X ); w hi l e usi ng (3. 47),gi ve exactl y two ti m es the term a 0 W 0 (X ) 0 (X )r m 1 (X )

Im F lm s 2 (Im F p ) 2 X );
1 2m 1 (3. 48) + m 2
X j= 1 j a j; 0 ;:::

; j 1 W 0 (X ) j Y l= 1
W l (X ) j (X )r m j 1 (X )

0 hX i 2 2m + 1 " m 1 hX i 2 2m + 1 :
B y noti ci ng from (3. 34) that the esti m ates

(3. 50) r m (X ). hX i 2 . r m 1 (X ) 2m + 1 2m 1 ;
areful l l ed on the supportofthe functi on W 0 ,wededucefrom (3. 1),(3. 30)and (3. 42) that the m odul us ofthe term s B 3;0;p can be esti m ated as

a 0 N X p= 1 j B 3;0;p (X )j= a 0 N X p= 1 r m 1 (X ) 2m 2 2m 1 H Im qp r m 1 (X ) 2m 2 2m 1 r m 1 (X ) 2m 2 2m 1 rm 1;p (X ) W 0 (X ) 0 (X ) . 1 2 0 W 0 (X ) 0 (X )r m 1 (X ) 1 2m 1 ;
for al lX 2 R 2n ; si nce from Lem m a 4. 2. 8 and Lem m a 4. 2. 10,we have for any p i n f1;: : : ;N g that

r m 1 (X ) 2m 2 2m 1 H Im qp r m 1 (X ) 2m 2 2m 1 . r m 1 (X ) 1 2m 1 and r m 1 (X ) 2m 2 2m
1 rm 1;p (X ) . 0 ; on the support ofthe functi on W 0 (X ) 0 (X ):B y possi bl y i ncreasi ng su ci entl y the val ue of the constant 0 w hi ch i s of course possi bl e w hi l e keepi ng (3. 49), one can controlthi s term w i th the \good" term (3. 48).

N ext,we deduce from (3. 1),(3. 30),(3. 42),(3. 50)and Lem m a 4. 2. 9 that the m odul us ofthe second term s i n B 4;0;p associ ated to 2 X j= 1;:::;N (l1 ;:::;lm 2 )2 f1;:::;N g m 2 R e q j (Im F l1 : : : Im F lm 2 X ;Im F l1 : : : Im F lm 2 (Im F p ) 2 X ); w hi l e usi ng (3. 47),denoted here B 4;0;p ,

N X p= 1 B 4;0;p (X )= W 0 (X ) 0 (X ) N X p= 1 0 B B @ H Im qp rm 1;p (X ) r m 1 (X ) 2m 2 2m 1 2 X
j= 1;:::;N (l1 ;:::;lm 2 )2 f1;:::;N g m 2 R e q j (Im F l1 : : :

Im F lm 2 Im F p X ) r m 1 (X ) 2m 2 2m 1 1 C C A ; = W 0 (X ) 0 (X ) N X p= 1 H Im qp rm 1;p (X ) r m 1 (X ) 2m 2 2m 1 2r m 1 (X ) 1 2m 1
! can be esti m ated as

a 0 N X p= 1 j B 4;0;p (X )j. 1 2 0 W 0 (X ) 0 (X )r m 1 (X ) 1 2m 1 ;
for al l X 2 R 2n . B y possi bl y i ncreasi ng su ci entl y the val ue of the constant 0 w hi ch i s ofcourse possi bl e w hi l e keepi ng (3. 49),one can al so controlthi s term w i th the \good" term (3. 48). T he val ue ofthe constant 0 i s now de ni ti vel y xed. In (3. 42),i t onl y rem ai ns to study the term s B 2;0;p .

A bout these term s,we deduce from j B 2;0;p (X )j. W 0 (X )W 1 (X )r m 1 (X )

1 2m 1 :
B y usi ng now (3. 34) and (3. 36) w i th j = 1,we obtai n that for al lX 2 R 2n ,

a 0 N X p= 1 j B 2;0;p (X )j c m 1; 0 ;:::; m 2 W 0 (X ) m 1 Y l= 1 W l (X ) N X p= 1 R e q p (X ) + m 2 X j= 1 c j; 0 ;:::; j 1 W 0 (X ) j Y l= 1 W l (X ) j (X )r m j 1 (X ) 1 2m 2j 1 ; w hi ch i m pl i es that (3. 52) a 0 N X p= 1 j B 2;0;p (X )j c m 1; 0 ;:::; m 2 N X p= 1 R e q p (X ) + m 2 X j= 1 c j; 0 ;:::; j 1 W 0 (X ) j Y l= 1 W l (X ) j (X )r m j 1 (X ) 1 2m 2j 1 ;
w here the quanti ti es c j; 0 ;:::; j 1 stand for posi ti ve constants w hose val ues depend on 0 , . . . , j 1 ,but not on ( k ) j k m 2 and ( k ) 1 k m 2 ;accordi ng to the rem ark done after (3. 34). O ne can therefore choose the constant 1 1 i n (3. 44)su ci entl y l arge i n orderto absorb the term ofthe i ndex j = 1 i n the sum appeari ng i n the ri ght hand si de ofthe esti m ate (3. 52)by the term ofsam e i ndex i n the \good" term (3. 48). T hi s i s possi bl e si nce the constants a 1; 0 and c 1; 0 are now xed after our choi ce of the param eter 0 .

T hi s ends our step i ndex j = 0 i n w hi ch we have chosen the val ues for the two constants 0 and 1 1.W eshal lnow expl ai n how to choosetherem ai ni ng constants ( j ) 1 j m 2 and ( j ) 2 j m 2 i n (3. 44) i n order to sati sfy (3. 46). T hi s choi ce w i l l al so determ i ne the val uesofthe constants(a j; 0 ;:::; j 1 ) 1 j m 2 appeari ng i n (3. 44). A fter thi s step i ndex j = 0,we have m anaged to absorb al lthe term s appeari ng i n (3. 42) i n the \good" term (3. 48) at the excepti on ofa rem ai nder com i ng from (3. 49) and (3. 52),

m 2 X j= 2 c j; 0 ;:::; j 1 W 0 (X ) j Y l= 1 W l (X ) j (X )r m j 1 (X ) 1 2m 2j 1 + " m 1 hX i 2 2m + 1 ;
w here one recal lthat the posi ti ve constants c j; 0 ;:::; j 1 onl y depend on 0 , . . . , j 1 , but not on ( k ) j k m 2 and ( k ) 1 k m 2 : W e proceed i n the fol l ow i ng by ni te i nducti on and assum e that,atthe begi nni ng ofthe step i ndex k,w i th 1 k m 2,we have al ready chosen the val ues for the constants ( j ) 0 j k 1 and ( j ) 1 j k i n (3. 44);and that these choi ces have al l owed to absorb al lthe term s appeari ng i n the ri ght hand si de of(3. 42) and (3. 41),w hen 1 j k 1,i n the \good" term (3. 48) at the excepti on ofa rem ai nder term

(3. 53) k m 1 "hX i 2 2m + 1 + m 2
X j= k+ 1 cj; 0 ;:::; j 1 ; 1 ;:::

; k 1 W 0 (X ) j Y l= 1
W l (X ) j (X )r m j 1 (X )

1 2m 2j 1 ;
w here the quanti ti es cj; 0 ;:::; j 1 ; 1 ;:::; k 1 stand for posi ti ve constants w hose val ues onl y depend on 0 , . . . , j 1 , 1 , . . . , k 1 ;but not on ( l ) j l m 2 and ( l ) k l m 2 .

W e shal lnow expl ai n how to choose the constants k and; k+ 1 ,w hen k m 3; i n thi s step i ndex k i n order to absorb the term s appeari ng i n the ri ght hand si de of (3. 41),w hen j = k,at the excepti on ofa rem ai nder term ofthe type (3. 53) w here k w i l lbe repl aced by k+ 1;i n the \good" term (3. 48).Si nce the constants( j ) 0 j k 1 and ( j ) 1 j k have al ready been chosen,we shal lonl y underl i ne i n the fol l ow i ng the dependence ofour esti m ates w i th respect to the other param eters ( j ) k j m 2 and ( j ) k+ 1 j m 2 ,w hose val ues rem ai n to be chosen.

W

:; k 1 N X p= 1 j B 1;k;p (X )j. 1 2 k hX i 2 2m + 1 " m 1 hX i 2 2m + 1 ;
si nce the constants k , 0 , . . . . , k 1 have al ready been xed.

N ext,we deduce from (3. 1),(3. 30),(3. 34)and (3. 41)thatthe m odul usofthe term s B 3;k;p can be esti m ated as k a k; 0 ;:::

; k 1 N X p= 1 j B 3;k;p (X )j = k a k; 0 ;:::; k 1 N X p= 1 r m k 1 (X ) 2m 2k 2 2m 2k 1 H Im qp r m k 1 (X ) 2m 2k 2 2m 2k 1 r m k 1 (X ) 2m 2k 2 2m 2k 1 rm k 1;p (X ) W 0 (X ) k Y l= 1 W l (X ) k (X ) . 1 2 k W 0 (X ) k Y l= 1 W l (X ) k (X )r m k 1 (X )
: ;N g that r m k 1 (X ) 2m 2k 2 2m 2k 1 H Im qp r m k 1 (X ) 2m 2k 2 2m 2k 1
. r m k 1 (X ) R e q j (Im F l1 : : : Im F l m k 2 X ;Im F l1 : : : Im F l m k 2 (Im F p ) 2 X ); w hi l e usi ng (3. 47),denoted here B 4;k;p , R e q j (Im F l1 : : :

1 2m 2k 1 and r m k 1 (X ) 2m 2k 2 2m 2k 1 rm k 1;p (X ) .
N X p= 1 B 4;k;p (X )= W 0 (X ) k Y l= 1 W l (X ) k (X ) N X p= 1 0 B B @ H Im qp rm k 1;p (X ) r m k 1 (X )
Im F l m k 2 Im F p X ) r m k 1 (X ) 2m 2k 2 2m 2k 1 1 C C A ; = W 0 (X ) k Y l= 1 W l (X ) k (X ) N X p= 1 H Im qp rm k 1;p (X ) r m k 1 (X ) 2m 2k 2 2m 2k 1 2r m k 1 (X ) 1 2m 2k 1
! can be esti m ated as k a k; 0 ;:::

; k 1 N X p= 1 j B 4;k;p (X )j. 1 2 k W 0 (X ) k Y l= 1 W l (X ) k (X )r m k 1 (X ) 1 2m 2k 1 ;
for al l X 2 R 2n . B y possi bl y i ncreasi ng su ci entl y the val ue of the constant k w hi ch i s ofcourse possi bl e w hi l e keepi ng (3. 54),one can al so controlthi s term w i th the \good" term (3. 48).

For 1 l k and 1 p N ,we shal lnow study the term j B 2;k;p (X )j. W 0 (X )

B 5;k;p;l (X )= W 0 (X ) H Im qp W l (X ) k Y j= 1 j6 = l W j (X ) k (X ) rm k 1;p (X ) r m k 1 (X )
1 2 k W 0 (X ) l 1 Y j= 1 W j (X ) l 1 (X )r m l (X )
k+ 1 Y l= 1 W l (X ) r m k 1 (X ) 1 2m 2k 1 :
B y di sti ngui shi ng two cases,we rst assum e i n the fol l ow i ng that k m 3. In thi s case,by usi ng (3. 34) and (3. 36)w i th j = k + 1,we obtai n that for al lX 2 R 2n , k a k; 0 ;:::; k 1 N X p= 1 j B 2;k;p (X )j c 0 m 1; 0 ;:::; m 2 ; 1 ;:::; k W 0 (X )

m 1 Y l= 1 W l (X ) N X p= 1 R e q p (X ) + m 2 X
j= k+ 1 c 0 j; 0 ;:::; j 1 ; 1 ;:::; k W 0 (X ) W l (X ) j (X )r m j 1 (X )

j Y l= 1 W l (X ) j (X )r m j 1 (X )
1 2m 2j 1 ;
w here the quanti ti es c 0 j; 0 ;:::; j 1 ; 1 ;:::; k stand for posi ti ve constants w hose val ues onl y depend on 0 , . . . , j 1 , 1 , . . . , k ,but not on ( l ) j l m 2 and ( l ) k+ 1 l m 2 . Indeed,we recal lthatthe constantsappeari ng i n the esti m ates(3. 34)onl y depend on the val ues ofthe param eters 0 , . . . , j 1 ;but not on ( l ) j l m 2 and ( l ) 1 l m 2 . O ne can therefore choose the constant k+ 1 1 i n (3. 44)su ci entl y l arge i n orderto absorb the term ofi ndex j = k + 1 i n the sum (3. 53);and the term ofi ndex j = k + 1 i n the sum appeari ng i n the ri ght hand si de of the esti m ate (3. 56),by the term of sam e i ndex i n the \good" term (3. 48).

W hen k = m 2 and taki ng m 2 = 1,i tfol l ow sfrom (3. 34),used w i th j = m 1, and (3. 55) that for al lX 2 R 2n , m 2 a m 2; 0 ;:::; m 3 N X p= 1 j B 2;m 2;p (X )j. W 0 (X )

m 1 Y l= 1 W l (X ) r 1 (X ) 1 3 (3. 57) . N X p= 1 R e q p (X ):
T hi s process al l ow s us to achi eve the constructi on of the wei ght functi ons p p , 1 p N ,sati sfyi ng (3. 46),w hi ch ends the proofof(3. 46). T hi s al so ends the proofof Proposi ti on 2. 0. 1.

A ppendix

4. 1. W ick calculus. T he purpose ofthi s secti on i s to recal lthe de ni ti on and basi c properti esofthe W i ck quanti zati on thatwe need forthe proofofT heorem 1. 2. 1.W e fol l ow here the presentati on ofthe W i ck quanti zati on gi ven by N .Lerner i n [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF]and refer the reader to hi s work for the proofs ofthe resul ts recal l ed bel ow .

T he m ai n property ofthe W i ck quanti zati on i s i ts property ofposi ti vi ty,i . e. ,that non-negati ve H am i l toni ans de ne non-negati ve operators a 0 ) a W ick 0: W e recal l that thi s i s not the case for the W eyl quanti zati on and refer to [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF] for an expl i ci t exam pl e of non-negati ve H am i l toni an de ni ng an operator w hi ch i s not non-negati ve.

B eforede ni ng properl y theW i ck quanti zati on,we rstneed to recal lthede ni ti on ofthe wave packets transform ofa functi on u 2 S(R n ), W u(y; )= (u;' y;

) L 2 (R n ) = 2 n =4 Z R n u(x)e (x y) 2
e 2i (x y): dx; (y; )2 R 2n : w here ' y; (x)= 2 n =4 e (x y) 2 e 2i (x y): ; x 2 R n ; and x 2 = x 2 1 + : : :+ x 2 n . W i th thi sde ni ti on,one can check (see Lem m a 2. 1 i n [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF] )that the m appi ng

u 7 ! W u i sconti nuousfrom S(R n )to S(R 2n ),i som etri c from L 2 (R n )to L 2 (R 2n
) and that we have the reconstructi on form ul a w hi ch i s a rank-one orthogonalprojecti on

Y u (x)= W u(Y )' Y (x)= (u;' Y ) L 2 (R n ) ' Y (x);
we de ne the W i ck quanti zati on ofany L 1 (R 2n ) sym bola as

(4. 2) a W ick = Z R 2n a(Y ) Y dY :
M ore general l y,one can extend thi s de ni ti on w hen the sym bola bel ongs to S 0 (R 2n ) by de ni ng the operator a W ick for any u and v i n S(R n ) by

< a W ick u;v > S 0 (R n );S (R n ) = < a(Y );( Y u;v) L 2 (R n ) > S 0 (R 2n );S (R 2n ) ;
w here < ; > S 0 (R n );S (R n ) denotes the dual i ty bracketbetween the spaces S 0 (R n ) and S(R n ). T he W i ck quanti zati on i s a posi ti ve quanti zati on (4. 3) a 0 ) a W ick 0:

In parti cul ar,realH am i l toni ans get quanti zed i n thi s quanti zati on by form al l y sel fadjoi ntoperatorsand onehas(seeProposi ti on 3. 2 i n [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF] )thatL 1 (R 2n )sym bol sde ne bounded operators on

L 2 (R n ) such that (4. 4) ka W ick k L (L 2 (R n )) kak L 1 (R 2n ) :
A ccordi ng to Proposi ti on 3. 3 i n [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF] ,the W i ck and W eylquanti zati ons ofa sym bola are l i nked by the fol l ow i ng i denti ti es (1 )a 00 (X + Y )Y 2 e 2 jY j 2 2 n dY d ; X 2 R 2n ;

i fwe use here the norm al i zati on chosen i n [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF]for the W eylquanti zati on (4. 9) (a w u)(x)= Z R 2n e 2i (x y): a x + y 2 ; u(y)dyd ;

w hi ch di ers from the one chosen i n thi s paper. B ecause of thi s di erence i n norm al i zati ons, certai n constant factors w i l l natural l y appear i n the core of the proof of T heorem 1. 2. 1 w hi l e usi ng certai n form ul as of Secti on 4. 1, but these are m i nor adaptati ons. W e al so recal lthe fol l ow i ng com posi ti on form ul a obtai ned i n the proof ofProposi ti on 3. 4 i n [START_REF] Lerner | W ick cal cul us ofpseudodi erentialoperators and som e ofits appl ications[END_REF] ,

(4. 10) a W ick b W ick = h ab 1 4 a 0 b 0 + 1 4i fa;bg i W ick + S; w i th kSk L (L 2 (R n )) d n kak L 1 2 (b);w hen a 2 L 1 (R 2n
) and b i s a sm ooth sym bol sati sfyi ng

2 (b)= sup X 2 R 2n ; T 2 R 2n ;jT j= 1 j b (2) (X )T 2 j< + 1 :
T he term d n appeari ng i n the previ ous esti m ate stands for a posi ti ve constant dependi ng onl y on the di m ensi on n,and the notati on fa;bg denotesthe Poi sson bracket fa;bg = @a @ @b @x @a @x @b @ : 4. 2. Som e technical lem m as. T hi s second part ofthe appendi x i s devoted to the proofs ofseveraltechni call em m as.

Lem m a 4.2.1. For any 1 j N , 1 p N , (l 1 ;: : : ;l k ) 2 f1;: : : ;N g k and s 1 ;s 2 2 N ,we have H Im qp R e q j Im F l1 : : :

Im F l k (Im F p ) s1 X ;Im F l1 : : : Im F l k (Im F p ) s2 X (4. 11)
= 2R e q j Im F l1 : : :

Im F l k (Im F p ) s1 + 1 X ;Im F l1 : : : Im F l k (Im F p ) s2 X + 2R e q j Im F l1 : : : Im F l k (Im F p ) s1 X ;Im F l1 : : : Im F l k (Im F p ) s2 + 1 X ;
where R e q j (X ;Y ) stands for the pol arized form associated to the quadratic form R e q j . Proof of Lem m a 4.2.1. W e begi n by noti ci ng from (1. 10) and the skew -sym m etry property ofH am i l ton m aps (1. 11) that the H am i l ton m ap ofthe quadrati c form r(X )= R e q j Im F l1 : : :

Im F l k (Im F p ) s1 X ;Im F l1 : : : Im F l k (Im F p ) s2 X ; i s gi ven by (4. 12) F = 1 2 ( 1) k+ s1 (Im F p ) s1 Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s2 + 1 2 ( 1) k+ s2 (Im F p ) s2 Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s1 ; si nce ( 1) k+ s1 X ;(Im F p ) s1 Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s2 X (4. 13) = Im F l1 : : : Im F l k (Im F p ) s1 X ;R e F j Im F l1 : : : Im F l k (Im F p ) s2 X
= R e q j Im F l1 : : :

Im F l k (Im F p ) s1 X ;Im F l1 : : : Im F l k (Im F p ) s2 X
= R e q j Im F l1 : : :

Im F l k (Im F p ) s2 X ;Im F l1 : : : Im F l k (Im F p ) s1 X = Im F l1 : : : Im F l k (Im F p ) s2 X ;R e F j Im F l1 : : : Im F l k (Im F p ) s1 X = ( 1) k+ s2 X ;(Im F p ) s2 Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s1 X :
T hen,a di rect com putati on (see Lem m a 2 i n [START_REF] Ravda-Starov | C ontraction sem igroups of ell iptic quadratic di erentialoperators[END_REF] ) show s that the H am i l ton m ap of the quadrati c form H Im qp r = Im q p ;r = @Im q p @ : @r @x @Im q p @x : @r @ ; i s gi ven by the com m utator 2[ Im F p ; F ] ,that i s,

H Im qp r(X )= 2 X ;[ Im F p ; F ] X :
A com putati on as i n (4. 13) then al l ow s to di rectl y get (4. 11).

Lem m a 4.2.2. C onsider a C 1 (R ) function f such that

f 2 L 1 (R ) and 9c 1 ;c 2 > 0; supp f 0 x 2 R :c 1 j xj c 2 ;
and r a non-negative quadratic form then for all0 < ; on \ supp W 0 j ,and that the two deri vati ves 0 and w 0 2 ofthe functi ons appeari ng i n (3. 30)and (3. 31)have si m i l artypes ofsupportas the functi on de ned i n (3. 2),we noti ce that we are exactl y i n the setti ng studi ed i n Lem m a 4. 2. 4 w i th j repl aced by j 1. W e therefore deduce the resul t ofLem m a 4. 2. 5 from our anal ysi s l ed i n the proofofLem m a 4. 2. 4.

Lem m a 4.2.6. Ifs 1 ,s 2 2 N ,1 j;p N ,(l 1 ;: : : ;l k )2 f1;: : : ;N g k then we have R e q j (Im F l1 : : : Im F l k (Im F p ) s1 X ;Im F l1 : : : Im F l k (Im F p ) s2 X ) R e q j (Im F l1 : : : Im F l k (Im F p ) s1 X ) 1 2 R e q j (Im F l1 : : :

Im F l k (Im F p ) s2 X ) 1 2
r k+ s1 (X ) 1 2 r k+ s2 (X ) 1 2 and r R e q j (Im F l1 : : : Im F l k (Im F p ) s1 X ;Im F l1 : : : Im F l k (Im F p ) s2 X )

. R e q j (Im F l1 : : : Im F l k (Im F p ) s1 X ) 1 2 + R e q j (Im F l1 : : : Im F l k (Im F p ) s2 X ) . r k+ m ax(s1 ;s2 ) (X ) ProofofLem m a 4.2.6. B y reason ofsym m etry,we can assum e i n the fol l ow i ng that s 1 s 2 . R ecal l i ng thatthe quadrati c form R e q j i snon-negati ve,the rstesti m ate i s a di rect consequence of(3. 6) and the C auchy-Schwarz i nequal i ty. A bout the second esti m ate,we recal lfrom (4. 12)that the H am i l ton m ap ofthe quadrati c form R e q j (Im F l1 : : : Im F l k (Im F p ) s1 X ;Im F l1 : : :

Im F l k (Im F p ) s2 X ); i s 1 2
( 1) k+ s1 (Im F p ) s1 Im F l k : : : Im F l1 R e F j Im F l1 : : :

Im F l k (Im F p ) s2 + 1 2
( 1) k+ s2 (Im F p ) s2 Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s1 :

A di rect com putati on as i n (3. 18) of [START_REF] Ravda-Starov | C ontraction sem igroups of ell iptic quadratic di erentialoperators[END_REF]show s that r R e q j (Im F l1 : : : Im F l k (Im F p ) s1 X ;Im F l1 : : : . r R e q j (Im F l1 : : : Im F l k (Im F p ) s X )

Im F l k (Im F p ) s2 X ) (4.
. R e q j (Im F l1 : : : Im F l k (Im F p ) s X ) and j r rm j 1;p (X )j. r m j 1 (X )

1 2 + r m j 2 (X ) 1 2
. r m j 1 (X )

1 2 ;
on ,we get that the quadrati c form rm j 1;p bel ongs to the sym bolcl ass S r m j 1 (X ) In the fol l ow i ng l em m as,we shal lcareful l y study the dependence ofthe esti m ates w i th respect to the l arge param eter j 1.

Lem m a 4.2.9. For any 0 j m 2,we have for allX 2 , Lem m a 4.2.11. C onsider the functions j and W j+ 1 de ned in (3. 30) and (3. 31) then for any 0 j m 2 and 1 p N ,we have for allX 2 , j H Im qp j (X )j. R e q s (Im F l1 : : : Im F lm j 2 X ;Im F l1 : : : Im F lm j 2 Im F p X ):

It fol l ow s from the C auchy-Schwarz i nequal i ty that for al lX 2 , R e q s (Im F l1 : : : Im F lm j 2 X ;Im F l1 : : : Im F lm j 2 Im F p X ) (4. 27) R e q s (Im F l1 : : : Im F lm j 2 X ) 1 2 R e q s (Im F l1 : : : Im F lm j 2 Im F p X ) 1 2 r m j 2 (X ) 1 2 r m j 1 (X ) T hen,by w ri ti ng that H Im qp j r m j 2 (X ) r m j 1 (X ) Lem m a 4.2.12. For m 2,consider the function W 0 de ned in (3. 32) then for all X 2 R 2n and 1 p N , j H Im qp W 0 (X )j. hX i ProofofLem m a 4.2.12. Si nce j r Im q p (X )j. hX i,because Im q p i sa quadrati cform , Lem m a 4. 2. 12 i s then a consequence of(3. 3), (3. 6)

\

  j= 1;:::;N ; (l1 ;:::;l k )2 f1;:::;N g k K er(R e F j Im F l1 : : :

fx 2 R

 2 :1 j xj 2g; supp fx 2 R :1=2 j xj 3g; and (3. 3) w = 1 on fx 2 R :j xj 2g; supp w fx 2 R :j xj 1g:

1 R

 1 e q p (X )+ c 5;p H Im qp gm ;p (X ) & hX i

  are wel l -de ned on the support of the functi on W 0 . N ow , by noti ci ng from (3. 1), (3. 3),(3. 30),(3. 31) and (3. 33) that (3. 35) 1 j + W j+ 1 ; on the support ofthe functi on supp W 0 j Y l= 1

2 N X p= 1 R e q p + N X p= 1 H 0 N X p= 1 B 2 X j= 1 j 1 N X p= 1 B

 211012111 one can noti ce by expandi ng the term 2a m 1; 0 ;:::; m Im qp p p ; by usi ng (3. 41),(3. 42) and (3. 44)that the term s i n a 4;0;p + m a j; 0 ;:::; j 4;j;p ! ; produced by the term s associ ated to X j= 1;:::;N (l1 ;:::;lm s 2 )2 f1;:::;N g m s 2

  (3. 1),(3. 30),(3. 42),(3. 50),Lem m a 4. 2. 8 and Lem m a 4. 2. 11 that for al lX 2 R 2n ,

1 W 2 X

 12 l (X ) k (X ): B y possi bl y i ncreasi ng su ci entl y the val ue of the constant k w hi ch i s of course possi bl e w hi l e keepi ng (3. 54),one can controlthi s term w i th the \good" term(3. 48).N ext,we deduce from (3. 1),(3. 30),(3. 34),(3. 41)and Lem m a 4. 2. 9 that the m odul us ofthe second term s i n B 4;k;p associ ated to j= 1;:::;N (l1 ;:::;l m k 2 )2 f1;:::;N g m k 2

2 X

 2 j= 1;:::;N (l1 ;:::;l m k 2 )2 f1;:::;N g m k 2

; 1 l r m l 1 1 ; 1 N X p= 1 j

 11111 appeari ng i n the term B 5;k;p i n (3. 41). B y noti ci ng that r m l 2 (X ) on the support ofthe functi on H Im qp W l+ 1 ,i t fol l ow s from (3. 1),(3. 3),(3. 30),(3. 31), (3. 32),(3. 34),(3. 50),Lem m a 4. 2. 8 and Lem m a 4. 2. 13 that for al lX 2 R 2n , k a k; 0 ;:::; k B 5;k;p;1 (X )j.

1 2 k 1 N X p= 1 j

 1211 W 0 (X ) 0 (X )r m 1 (X )1 2m1 and k a k; 0 ;:::; k B 5;k;p;l (X )j.

1 N X p= 1 B 5 1 N X p= 1

 11511 l 2. B y possi bl y i ncreasi ng agai n the val ue of the constant k , one can therefore controlthe term k a k; 0 ;:::; k ;k;p ; w i th the \good" term (3. 48). T he val ue ofthe constant k i s now de ni ti vel y xed.A bout the term s B 2;k;p ,we deduce from (3. 1),(3. 30),(3. 34),(3. 41),Lem m a 4. 2. 8 and Lem m a 4. 2. 11 that for al lX 2 R 2n , (3. 55) k a k; 0 ;:::; k

1 N X p= 1 j 2 X

 112 i m pl i es that (3. 56) k a k; 0 ;:::; k B 2;k;p (X )j c 0 m 1; 0 ;:::; m 2 ; 1 ;:::; k j= k+ 1 c 0 j; 0 ;:::; j 1 ; 1 ;:::; k W 0 (X ) j Y l= 1

(4. 1 )

 1 8u 2 S(R n );8x 2 R n ; u(x)= Z R 2n W u(y; )' y; (x)dyd :B y denoti ng Y the operatorde ned i n the W eylquanti zati on by the sym bol p Y (X )= 2 n e 2 jX Y j 2 ; Y = (y; )2 R 2n ;

4

 4 + Y )e 2 jY j 2 2 n dY ; X 2 R 2n ; and (4. 7) a W ick = a w + r(a) w ; w here r(a) stands for the sym bol (

  20) = ( 1) k+ s1 + 1 (Im F p ) s1 Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s2 + ( 1) k+ s2 + 1 (Im F p ) s2 Im F l k : : : Im F l1 R e F j ImF l1 : : : Im F l k (Im F p ) s1 on I n stands here for the n by n i denti ty m atri x. W e deduce from (2. 16) and (4. 20) that for any s 2 N , j (Im F p ) s Im F l k : : : Im F l1 R e F j Im F l1 : : : Im F l k (Im F p ) s X j (4. 21)

1 2 : 1 ;

 121 B y usi ng tw i ce the esti m ate (4. 21) w i th respecti vel y X and (Im F p ) s2 s1 X ,and the i ndex s = s 1 ,we deduce from (3. 6)and (4. 20)the second esti m ate i n Lem m a 4. 2. 6.Lem m a 4.2.7. C onsider the quadratic form rm j 1;p de ned in (3. 4)and (3. 5)then for any 0 j m 2 and 1 p N , rm j 1;p (X ) which im pl ies in particul ar that rm j 1;p (X ) r m j 1 (X )

2 :

 2 ProofofLem m a 4.2.7. Si nce from Lem m a 4. 2. 6, jr m j 1;p (X )j. r m j 1 (X )

2 : 1

 21 O ne can then deduce the resul t ofLem m a 4. 2. 7 from Lem m a 4. 2. 3.W hen addi ng a l arge param eter j 1 i n the descri pti on of the open set , a strai ghtforward adaptati on of the proof of the previ ous l em m a gi ves the fol l ow i ng L 1 ( ) esti m ate w i th respect to thi s param eter.Lem m a 4.2.8. C onsider the quadratic form rm j 1;p de ned in (3. 4)and (3. 5)then for any 0 j m 2 and 1 p N , r m j 1 (X ) rm j 1;p (X ) L 1 ( ) .

1 j r m j 1

 11

N X p= 1 H

 1 Im qp rm j 1;p (X ) r m j 1 (X )

1 j

 1 r m j 1 (X )

  m j (X ). r m j 1 (X )

1 W

 1 j+ 1 (X );if is any open setwhere r m j 1 (X )& hX i

1 j r m j 1 with j 1 .!. 4 X

 1114 ProofofLem m a 4.2.11. W e begi n by noti ci ng from (3. 31) and (3. 33) that (4. 25) 0 j r m j 2 (X ) r m j 1 (X ) W j+ 1 (X ); and by w ri ti ng from Lem m a 4. 2. 1 that (4. 26) H Im qp r m j 2 (X ) = s= 1;:::;N (l1 ;:::;lm j 2 )2 f1;:::;N g m j 2

j 1 : 1 j r m j 1

 111 Im qp r m j 2 (X ) r m j 1 (X ) r m j 2 (X )H Im qp r m j 1 (X ) r m j 1 (X ) 1+ 2m 2j 3 2m 2jLem m a 4. 2. 11 i sa consequenceof(3. 30),(4. 23),(4. 24),(4. 26),(4. 27)and (4. 28),si nce r m j 2 (X )

  ;hX i 2 dX 2 ; we obtai n (3. 9) from (3. 1),(3. 5),(3. 6),(3. 7),(3. 10),(3. 12) and Lem m a 4. 2. 2.D enoti ng respecti vel y A 1;p , A 2;p , A 3;p and A 4;p the four term s appeari ng i n the ri ghthand si deof(3. 8),we rstnoti cefrom (3. 1),(3. 10),(3. 12),(3. 13)and Lem m a 4. 2. 2 that

	2 . hX i: 2m + 1 dX 2 : 4m 2m + 1 (3. 14) M oreover,si nce (3. 13) hX i 4m 2m + 1 2 S hX i A 2;p 2 S 1;hX i 2(2m 1)

  ch we have the esti m ate(3. 43). To prove the esti m ate (3. 46),i t w i l ltherefore be su ci entto check that al lthe other term s appeari ng i n (3. 41)and (3. 42)can al so be al labsorbed i n the term (3. 48)aftera properchoi ceforthe constants( j ) 0 j m 2 and ( j ) 1 j m 2 ;at the excepti on ofa rem ai nder term i n Lem m a 4. 2. 8 and Lem m a 4. 2. 12 that one can choose the rst constant 0 1 such that for al lX 2 R 2n ,

					1
					2m	2j 1
			N X
		+ a m 1; 0 ;:::; m 2	R e q p (X );
			p= 1
	2 2m + 1 : for w hi "hX i
		N X			1
	(3. 49)	a 0	j B 1;0;p (X )j.	2
		p= 1

W e shal lchoose these constants i n the fol l ow i ng order 0 , 1 , 1 , 2 ,. . . . , m 2 and m 2 . W e successi vel y study the rem ai ni ng term si n (3. 41)and (3. 42),by i ncreasi ng val ue ofthe i nteger 0 j m 2. W e rst noti ce from (3. 1),(3. 3),(3. 30),(3. 32),(3. 42),

  for al lX 2 R 2n ; si nce from Lem m a 4. 2. 8 and Lem m a 4. 2. 10,we have for any p i n f1;: :

		1
	2m	2k 1 ;

  Lem m a 4.2.5. C onsider the function W j de ned in (3. 31) then for any 1 j m 1, W

	ProofofLem m a 4.2.4. W e rst noti ce from (3. 1) and (3. 30) that
						2m	2j 3
		r m j 2 (X ) r m j 1 (X ) 2m	2j 1 ;
	on \ supp 0 j . Si nce from (2. 16),
						1
	(4. 18)	j r r m j 2 (X )j. r m j 2 (X ) 2
						2m	2j 3
			. r m j 1 (X ) 2(2m	2j 1) ;
	on \ supp 0 j ,we deduce thatthe quadrati c sym bolr m j 2 (X )bel ongsto the cl ass
	(4. 19)	S \ supp 0 j r m j 1 (X ) 2m 2m	r m j 1 (X ) 2m 2j 1 ; 2j 3 dX 2 2m	2j 1 2j 3	:
	It fol l ow s from Lem m a 4. 2. 3 that
		r m j 2 (X ) r m j 1 (X ) 2m 2j 3 2m 2j 1	2 S \ supp 0 j 1; r m j 1 (X ) 2m dX 2 2m	2j 1 2j 3	;
	w hi ch i m pl i es that
					2m 2m	2j 3 2j 1 dX 2 :
	T hi s ends the proofofLem m a 4. 2. 4.
					2(2m	2j 1)
					2m + 1	;
	which im pl ies in particul ar that
		W j 2 S 1;hX i	2(2m 2m + 1 2j 1)	dX 2 :
	ProofofLem m a 4.2.5. B y noti ci ng from (3. 3) and (3. 31) that
						2m	2j 1
		r m j 1 (X ) r m j (X ) 2m	2j+ 1
	and
				2(2m	2j+ 1)
		r m j (X )& hX i	2m + 1

1,

(4. 14)

f r(X )hX i 2 2 S(1;hX i 2 dX 2 ):

j 2 S 1;r m j 1 (X ) j 2 S 1;r m j 1 (X ) 1 dX 2 ;

if is any open setwhere r m j 1 (X )& hX i

  ,(3. 32) and Lem m a 4. 2. 2. D epartm ent of M athem atics, Im perial C ollege London, H uxley B uilding, 180 Q ueen's G ate,London SW 7 2A Z, U K E -m ailaddress: k.pravda-starov@imperial.ac.uk U R L:http://www2.imperial.ac.uk/~kpravdas/index.html

ProofofLem m a 4.2.2. It i s su ci ent to check that (4. 15) r r(X )hX i 2 2 S hX i ;hX i 2 dX 2 ; w here i s a sm al l open nei ghborhood of supp f 0 r(X )hX i 2 : W e deduce from (2. 16) and the fact that r(X ) i s a non-negati ve quadrati c form that r(X ) hX i 2 and j r r(X ) j. r(X ) 1=2 . hX i ;

on . B y noti ci ng that 0 < 1,hX i r 2 S(hX i r ;hX i 2 dX 2 ),for any r 2 R ;and that the functi on r(X ) i s just a quadrati c form ,we di rectl y deduce (4. 15) from the previ ous esti m ates and the Lei bni z' s rul e,si nce r(X )2 S hX i 2 ;hX i 2 dX 2 :

In al lthe fol l ow i ng l em m as,we shal ldenote by r k the quadrati c form s de ned i n (3. 6) for 0 k m .

Lem m a 4.2.3. For alls 2 R and 0 j m 2,we have r m j 1 (X ) s 2 S r m j 1 (X ) s ;r m j 1 (X ) 1 dX 2 ;

if is any open setwhere

:

Proof of Lem m a 4.2.3. R ecal l i ng from (3. 6) that the sym bolr m j 1 (X ) i s a nonnegati ve quadrati c form and that we have from (2. 16) that (4. 16) j r r m j 1 (X )j. r m j 1 (X )

w hi ch i m pl i es that for al ls 2 R ,

17)

. r m j 1 (X )

on ,we noti ce that the resul t ofLem m a 4. 2. 3 i s therefore a strai ghtforward consequence ofthe Lei bni z' s rul e.

Lem m a 4.2.4.C onsider the function j de ned in (3. 30)then forany 0 j m 2,

if is any open setwhere

X s= 1;:::;N (l1 ;:::;lm j 2 )2 f1;:::;N g m j 2 R e q s (Im F l1 : : :

X s= 1;:::;N (l1 ;:::;lm j 2 )2 f1;:::;N g m j 2 R e q s (Im F l1 : : : Im F lm j 2 X ;Im F l1 : : :

Lem m a 4. 2. 9 i s then a consequence ofthe fol l ow i ng esti m ate R e q s (Im F l1 : : : Im F lm j 2 X ;Im F l1 : : :

. 1 2 j r m j 1 (X ); ful l l ed on that we obtai n from the C auchy-Schwarz i nequal i ty.

Lem m a 4.2.10. For any 0 j m 2 and 1 p N ,we have for allX 2 , r m j 1 (X )

. r m j 1 (X )

ProofofLem m a 4.2.10. W e begi n by w ri ti ng from (3. 6) and Lem m a 4. 2. 1 that (4. 23) H Im qp r m j 1 (X ) = 4 X s= 1;:::;N (l1 ;:::;lm j 1 )2 f1;:::;N g m j 1 R e q s (Im F l1 : : : Im F lm j 1 X ;Im F l1 : : :

Lem m a 4. 2. 10 i s then a consequence ofthe fol l ow i ng esti m ate R e q s (Im F l1 : : : Im F lm j 1 X ;Im F l1 : : : Im F lm j 1 Im F p X ) (4. 24) R e q s (Im F l1 : : : Im F lm j 1 X ) 1 2 R e q s (Im F l1 : : :

. r m j 1 (X ) 1+ ful l l ed on that we obtai n from the C auchy-Schwarz i nequal i ty.

Lem m a 4.2.13. C onsider the function W j+ 1 de ned in (3. 31) then for any 0 j m 2 and 1 p N ,we have for allX 2 , j H Im qp W j+ 1 (X )j.

!

. j (X );

and that the deri vati ves of j and W j+ 1 are exactl y the sam e types offuncti ons. It fol l ow s that Lem m a 4. 2. 13 i s just a strai ghtforward consequence ofLem m a 4. 2. 11.

R eferences