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SUBELLIPTIC ESTIMATES FOR
OVERDETERMINED SYSTEMS OF QUADRATIC
DIFFERENTIAL OPERATORS
Karel Pravda-Starov
To cite this version:
Karel Pravda-Starov.
SUBELLIPTIC ESTIMATES FOR OVERDETERMINED SYSTEMS OF
O F Q U A D R A T IC D IF F E R E N T IA L O P E R A T O R S
K A R E L P R A V D A -ST A R O V
A bstract. W e prove global subelliptic estim ates for system s of quadratic dif-ferentialoperators. 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 particular linear subvector space in the phase space intrinsically associated to their W eylsym bols,called singular space,w hich rules a num ber of fairly general properties of non-elliptic quadratic operators. A bout the subelliptic properties ofthese operators,w e established that quadratic oper-ators w ith zero singular spaces ful ll global subelliptic estim ates w ith a loss of derivatives depending on certain algebraic properties ofthe H am ilton m aps asso-ciated 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 giv-ing an explicit m easure ofthe loss ofderivatives and highlightgiv-ing 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 joint w ork w ith M .H itrik,w e investigated spectraland sem igroup properties ofnon-elliptic quadratic operators. Q uadratic operators are pseudodi erential operators de ned in the W eylquantization
(1.1) qw(x;Dx)u(x)= 1 (2 )n Z R2n ei(x y):q x + y 2 ; u(y)dyd ;
by som e sym bols q(x; ), w ith (x; ) 2 Rn Rn and n 2 N , w hich are com
plex-valued quadratic form s. Since these sym bols are quadratic form s,the corresponding operators in (1.1) are in fact di erential operators. Indeed, the W eyl quantization ofthe quadratic sym bolx ,w ith ( ; ) 2 N2n and j + j= 2,is the di erential
operator
x Dx + Dxx
2 ; Dx= i
1@ x:
O ne can also notice that quadratic di erential operators are a prioriform ally non-selfadjoint since their W eylsym bols in (1.1) are com plex-valued.
C onsidering quadratic operators w hose W eylsym bols have realparts w ith a sign, say here,W eylsym bols w ith non-negative realparts
(1.2) R e q 0;
w e pointed out in [2] the existence of a particular linear subvector space S in the phase space Rn
x Rn intrinsically associated to their W eyl sym bols q(x; ), called
singular space, w hich seem s to play a basic r^ole in the understanding of a num ber
2000 M athem atics Subject C lassi 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.
offairly generalproperties ofnon-elliptic quadratic operators. M ore speci cally,w e rst proved in [2](T heorem 1.2.1) that w hen the singular space S has a sym plectic structure then the associated heat equation
(1.3) ( @u @t(t;x)+ q w(x;D x)u(t;x)= 0 u(t; )jt= 0= u02 L2(Rn);
is sm oothing in every direction ofthe orthogonalcom plem ent S ? ofS w ith respect
to the canonicalsym plectic form on R2n,
(1.4) (x; );(y; ) = :y x: ; (x; )2 R2n;(y; )2 R2n;
thatis,that,if(x0;0)are som e linearsym plectic coordinateson the sym plectic space S ? then w e have for allt> 0,N 2 N and u 2 L2(Rn),
(1.5) (1 + jx0j2+ j0j2)N we tqw(x;Dx)u 2 L2(Rn):
W e also proved in [2] (See Section 1.4.1 and T heorem 1.2.2) that w hen the W eyl sym bolq ofa quadratic operator ful lls (1.2) and an assum ption ofpartialellipticity on its singular space S in the sense that
(1.6) (x; )2 S; q(x; )= 0 ) (x; )= 0;
then this singular space alw ays has a sym plectic structure and the spectrum of the operator qw(x;D
x) is only com posed of a countable num ber of eigenvalues of nite
m ultiplicity,w ith a sim ilarstructure asthe one established by J.Sjostrand forelliptic quadratic operators in his classical w ork [18]. E lliptic quadratic operators are the quadratic operators w hose sym bols satisfy the condition ofglobalellipticity
(x; )2 R2n; q(x; )= 0 ) (x; )= 0;
on the w hole phase space R2n. Letusrecallhere thatspectralpropertiesofquadratic
operators are playing a basic r^ole in the analysis ofpartialdi erentialoperatorsw ith double characteristics. T his is particularly the case in som e general results about hypoellipticity. W e refer the reader to [4],[18],as w ellas C hapter 22 of[5]together w ith allthe references given there.
In the present paper, w e are interested in studying the subelliptic properties of overdeterm ined system s of non-selfadjoint quadratic operators. T his w ork can be view ed asa naturalextension ofthe analysisled in [17],in w hich w einvestigated in the scalar case the r^ole played by the singular space w hen studying subelliptic properties ofquadratic operators. W e aim here at show ing how the analysis led in this previous w ork can be pushed further w hen dealing w ith overdeterm ined system s ofquadratic operators. W e shallsee that the techniques introduced in [17]are su ciently robust to be extended to the system case and that they turn out to be su ciently sharp to highlight phenom ena of non-trivial interactions betw een the di erent quadratic operators com posing a system . In this paper, w e shall therefore be interested in establishing som e globalsubelliptic estim ates ofthe type
(1.7) h(x; )i2(1 ) wu L2 . N X j= 1 kqwj(x;Dx)ukL2 + kukL2;
w here h(x; )i = (1+ jxj2+ j j2)1=2and > 0;forsystem softhe N quadratic operators
qjw(x;Dx), w ith 1 j N . T he positive param eter > 0 appearing in (1.7) w ill
holds together w ith an explicit characterization ofthe associated loss ofderivatives. T his loss ofderivatives w illbe characterized in term s ofalgebraic conditions on the H am ilton m aps associated to the W eylsym bols ofthe quadratic operatorscom posing the system .
In this w ork, w e study the subellipticity of overdeterm ined system s in the sense given by P. B olley, J. C am us and J. N ourrigat in [1] (T heorem 1.1). In this sem i-nalw ork,these authorsstudy the m icrolocalsubellipticity ofoverdeterm ined system s ofpseudodi erentialoperators. M ore speci cally,they establish the subellipticity of system s com posed of pseudodi erentialoperators w ith realprincipalsym bols satis-fying the H orm ander-K ohn condition. M ore generally,in the case ofoverdeterm ined system s ofnon-selfadjoint pseudodi erentialoperators,the greatest achievem ents up to now w ere obtained by J. N ourrigat in [8] and [9]. In these tw o m ajor w orks, J.N ourrigatstudies the m icrolocalsubellipticity and m axim alhypoellipticity for sys-tem s ofnon-selfadjointpseudodi erentialoperatorsby the m ean ofrepresentationsof nilpotentgroups. W e shallexplain in the follow ing how the algebraic condition on the H am ilton m aps(1.18)in T heorem 1.2.1 relatesw ith these form erresults. M ore specif-ically,w e shallcom m ent on its link w ith the H orm ander-K ohn condition appearing in [1](T heorem 1.1).
B efore giving the precise statem entofourm ain result,w e shallrecallm iscellaneous notationsaboutquadratic di erentialoperatorsand the resultsobtained in the scalar case. In allthe follow ing,w e consider
qj:Rnx R
n ! C
(x; ) 7! qj(x; );
w ith 1 j N ,N com plex-valued quadratic form s w ith non-negative realparts (1.8) R e qj(x; ) 0; (x; )2 R2n;n 2 N :
W e know from [6](p.425)that the m axim alclosed realization ofa quadratic operator qw(x;Dx) w hose W eyl sym bol has a non-negative real part, i.e., the operator on
L2(Rn) w ith the dom ain
D (q)= u 2 L2(Rn):qw(x;Dx)u 2 L2(Rn) ;
coincides w ith the graph closure ofits restriction to S(Rn),
qw(x;Dx):S(Rn)! S(Rn):
A ssociated to a quadratic sym bolq is the num ericalrange (q) de ned asthe closure in the com plex plane ofallits values
(1.9) (q)= q(Rn
x Rn):
W e also recallfrom [5]thatthe H am ilton m ap F 2 M 2n(C )associated to the quadratic
form q is the m ap uniquely de ned by the identity
(1.10) q (x; );(y; ) = (x; );F (y; ); (x; )2 R2n;(y; )2 R2n;
w here q ; stands for the polarized form associated to the quadratic form q. It directly follow s from the de nition ofthe H am ilton m ap F that its realpart and its im aginary part
R e F = 1
2(F + F ) and Im F = 1
2i(F F );
respect to . T his is just a consequence of the properties of skew -sym m etry of the sym plectic form and sym m etry ofthe polarized form
(1.11) 8X ;Y 2 R2n; (X ;F Y )= q(X ;Y )= q(Y ;X )= (Y;F X )= (F X ;Y ): A ssociated to the sym bolq,w e de ned in [2]its singular space S as the follow ing intersection ofkernels
(1.12) S =
+ 1\ j= 0
K er R e F (Im F )j \ R2n;
w here the notations R e F and Im F stand respectively for the real part and the im aginary partofthe H am ilton m ap associated to q. N otice thatthe C ayley-H am ilton theorem applied to Im F show s that
(Im F )kX 2 Vect X ;:::;(Im F )2n 1X ; X 2 R2n; k 2 N ;
w here Vect X ;:::;(Im F )2n 1X is the vector space spanned by the vectors X , ...,
(Im F )2n 1X ;and therefore the singularspace isactually equalto the follow ing nite
intersection ofthe kernels
(1.13) S =
2n\ 1 j= 0
K er R e F (Im F )j \ R2n:
C onsidering a quadratic operator qw(x;D
x) w hose W eylsym bol
q :Rnx Rn ! C (x; ) 7! q(x; );
has a non-negative realpart,R e q 0,w e established in [17](T heorem 1.2.1) that w hen itssingularspaceS isreduced to f0g,theoperatorqw(x;D
x)ful llsthefollow ing
globalsubelliptic estim ate
(1.14) 9C > 0;8u 2 D (q); h(x; )i2=(2k0+ 1) wu
L2 C kq
w(x;D
x)ukL2+ kukL2 ; w here k0 stands for the sm allest non-negative integer, 0 k0 2n 1, such that
the intersection ofthe follow ing k0+ 1 kernels w ith the phase space R2n is reduced
to f0g, (1.15) k0 \ j= 0 K er R e F (Im F )j \ R2n = f0g:
N otice that the loss of derivatives = 2k0=(2k0+ 1), appearing in the subelliptic
estim ate (1.14) directly depends on the non-negative integer k0characterized by the
algebraic condition (1.15).
M ore generally, considering a quadratic operator qw(x;D
x) w hose W eyl sym bol
has a non-negative realpart w ith a singular space S w hich m ay di er from f0g,but does have a sym plectic structure in the sense that the restriction of the canonical sym plectic form to S is non-degenerate,w e proved in [17](T heorem 1.2.2)that the operator qw(x;D
x) is subelliptic in any direction ofthe orthogonalcom plem ent S ?
ofthe singularspace w ith respectto the sym plectic form in the sense that,if(x0;0) are som e linear sym plectic coordinates on S ? then w e have
9C > 0;8u 2 D (q); h(x0;0)i2=(2k0+ 1) wu
L2 C kq
w(x;D
w ith h(x0;0)i = (1 + jx0j2+ j0j2)1=2,w here k
0 stands for the sm allest non-negative
integer,0 k0 2n 1,such that
(1.16) S =
k0 \
j= 0
K er R e F (Im F )j \ R2n:
Finally,w e end these few recalls by underlining that the assum ption about the sym -plectic structure of the singular space is alw ays ful lled by any quadratic sym bolq w hich satis es the assum ption ofpartialellipticity on its singular space S,
(x; )2 S; q(x; )= 0 ) (x; )= 0: W e refer the reader to Section 1.4.1 in [2]for a proofofthis fact.
1.2. Statem ent ofthe m ain result. C onsidering a system ofN quadraticoperators qw
j(x;Dx),1 j N ,w hose W eylsym bols qjhave allnon-negative realparts
(1.17) R e qj(x; ) 0; (x; )2 R2n; n 2 N ;
and denoting by Fjtheir associated H am ilton m aps,the m ain resultcontained in this
article is the follow ing:
T heorem 1.2.1. C onsider a system ofN quadratic operators qjw(x;Dx),1 j N ,
satisfying (1.17). Ifthere exists k02 N such that
(1.18) \ 0 k k0 \ j= 1;:::;N ; (l1;:::;lk)2 f1;:::;N gk K er(R e FjIm Fl1:::Im Flk) \ R 2n = f0g;
then this overdeterm ined system of quadratic operators is subelliptic with a loss of = 2k0=(2k0 + 1) derivatives, that is, that there exists C > 0 such that for all
u 2 D (q1)\ :::\ D (qN), (1.19) h(x; )i2=(2k0+ 1) wu L2 C N X j= 1 kqjw(x;Dx)ukL2+ kukL2 ; with h(x; )i = (1 + jxj2+ j j2)1=2.
1.3. E xam ple of a subelliptic system of quadratic operators. T he follow ing exam ple of subelliptic system of quadratic operators show s that T heorem 1.2.1 re-ally highlightsnew non-trivialinteraction phenom ena betw een the di erentoperators com posing a system ,w hich cannot be derived from the result ofsubellipticity know n in the scalar case (T heorem 1.2.1 in [17]). Indeed,de ne the quadratic form s
qj(x; )= x21+ 2 1+ i( 2 1+ xj+ 1 1) and ~qj(x; )= x21+ 2 1+ i( 2 1+ j+ 1 1);
for 1 j n 1 and (x; ) 2 R2n,w ith n 2. A direct com putation using (1.10)
and (1.13) show s that the singular space ofthe quadratic form
nX 1 j= 1
( jqj+ ~jq~j);
for som e realnum bers j;~j verifying n 1 X j= 1 ( j+ ~j)> 0; is given by S = n (x; )2 R2n :x1= 1= n 1 X j= 1 ( jxj+ 1+ ~j j+ 1)= 0 o ;
w hich is alw ays a non-zero subvector space. It then follow s that one cannot deduce any result about the subellipticity ofthe scalar operator
n 1
X
j= 1
(jqjw(x;Dx)+ ~jq~wj(x;Dx));
in orderto getthe subellipticity ofthe overdeterm ined system com posed by the 2n 2 operators qw
j(x;Dx) and ~qjw(x;Dx), for 1 j n 1. N evertheless, by denoting
respectively Fjand ~Fj the H am ilton m aps ofthe quadratic form s qjand ~qj,another
direct com putation using (1.10) show s that
K er R e Fj\ K er(R e FjIm Fj)\ R2n = f(x; )2 R2n :x1= 1= xj+ 1= 0g
and
K er R e ~Fj\ K er(R e ~FjIm ~Fj)\ R2n = f(x; )2 R2n :x1= 1= j+ 1= 0g:
O ne can then deduce from T heorem 1.2.1 the follow ing global subelliptic estim ate w ith a loss of2=3 derivatives
h(x; )i2=3 wu
L2 .
n 1
X
j= 1
kqjw(x;Dx)ukL2 + k~qjw(x;Dx)ukL2 + kukL2:
1.4. C om m ents on the condition for subellipticity. T heorem 1.2.1 gives a very explicit and sim ple algebraic condition on the H am ilton m aps of quadratic opera-tors ensuring the subellipticity of the system . Let us notice that this condition is very easy to handle and allow s to directly m easure the associated loss ofderivatives by a straightforw ard com putation. W e shallnow explain how this is related to the H orm ander-K ohn condition. R ecallfrom [1](T heorem 1.1)thatthe H orm ander-K ohn condition form icrolocalsubellipticity ofoverdeterm ined system sofpseudodi erential operators w ith realprincipal sym bols; reads as the existence of an elliptic iterated com m utator of the operators com posing the system . In the case of a system of non-selfadjoint quadratic operators (qw
j)1 j N, if w e assum e in addition that this
system is m axim alhypoelliptic1,the naturalcondition becom es to ask the ellipticity
of an iterated com m utator of the real parts ((R e qj)w)1 j N and im aginary parts
((Im qj)w)1 j N ofthe operatorscom posing the system . C om ing back to ourspeci c
condition for subellipticity (1.18),w e rst notice that in the scalar case,it reads as the existence ofa non-negative integer k0such that
k0 \
j= 0
K er[R e F (Im F )j] \ R2n = f0g;
w ith F standing for the H am ilton m ap ofthe unique operator qw(x;D
x) com posing
the system . A s recalled in [17](Section 1.2),this condition im plies that,for any non-zero point in the phase space X0 2 R2n,w e can nd a non-negative integer k such
that
8 0 j 2k 1; HIm qj R e q(X0)= 0 and HIm q2k R e q(X0)6= 0;
w here HIm q stands for the H am ilton vector eld ofIm q,
HIm q= @Im q @ @ @x @Im q @x @ @ : T his show s that the 2kthiterated com m utator
[Im qw;[Im qw;[:::;[Im qw;R e qw]]]:::]= ( 1)k(HIm q2k R e q)w;
w ith exactly 2k term s Im qw in left-hand-side ofthe above form ula;is elliptic at X 0;
and underlines the intim ate link betw een (1.18) and the H orm ander-K ohn condition in the scalarcase. In the system case,the situation is m ore com plicated and this link is less obvious to highlight explicitly. M ore speci cally,w e shallsee in this case that the algebraic condition (1.18) im plies that the quadratic form
k0 X k= 0 X j= 1;:::;N ; (l1;:::;lk)2 f1;:::;N g k R e qj(Im Fl1:::Im FlkX );
is positive de nite. T his property im plies that for any non-zero point X02 R2n,one
can nd k 2 N ,j 2 f1;:::;N g and (l1;:::;lk)2 f1;:::;N gk such that
R e qj(Im Fl1:::Im FlkX0)> 0:
B y considering the m inim alnon-negative integer k w ith this property and using the sam e argum ents as the ones developed in [2](p.820-822),one can actually check that any iterated com m utator oforder less or equalto 2k 1,that is,
[P1;[P2;[P3;[:::;[Pr;Pr+ 1]:::]]]];
1W e refer to [8]and [9]for conditions and generalresults of m axim alhypoellipticity for
w ith r 2k 1,Pl= R e qsw1 or Pl= Im q
w
s2;and w here at least one Pl0 is equalto R e qw
s3, for 1 s1;s2;s3 N ; are not elliptic at X0. O ne can also check that the non-zero term
R e qj(Im Fl1:::Im FlkX0)> 0;
actually appears w hen expanding the W eylsym bolat X0 of the 2kth iterated com
-m utator [Im qlw k;[Im q w lk;[Im q w lk 1;[Im q w lk 1;[:::;[Im q w l1;[Im q w l1;R e q w j]]]:::] = ( 1)k(HIm q2 lk:::H 2 Im ql1R e qj)w:
H ow ever,contrary to the scalar case,there m ay be also other non-zero term s in this expansion;and it is notreally clearifthis naturalcom m utatorassociated to the term
R e qj(Im Fl1:::Im FlkX0); is actually elliptic at X0, HIm q2 lk:::H 2 Im ql1R e qj(X0) ? 6 = 0:
T hough itm ay be di cultto determ ine exactly ateach pointw hich speci c com m uta-tor is elliptic,it is very likely thatcondition (1.18)ensuresthat the H orm ander-K ohn condition is ful lled at any non-zero point ofthe phase space;and that these associ-ated elliptic com m utators are alloforder less or equalto 2k0. It is actually w hat the
loss of derivatives appearing in the estim ate (1.19) suggests; and this in agreem ent w ith the optim allossofderivativesobtained in [1](T heorem 1.1)for2k0com m utators
= 1 1 2k0+ 1
= 2k0 2k0+ 1
;
since w e m easure the loss ofderivatives w ith respect to the elliptic case as
( 2(1 ))wu L2 . N X j= 1 kqw j(x;Dx)ukL2+ kukL2;
w ith 2 = h(x; )i2, because quadratic operators have their W eyl sym bols in the
sym bolclass S( 2; 2dX2) w hose gain is 2.
B ecause ofthe sim plicity ofits assum ptions,T heorem 1.2.1 providesa neatsetting forproving globalsubelliptic estim atesforsystem sofquadratic operators.Itis possi-ble that som e ofthese globalsubelliptic estim ates for system s ofquadratic operators m ay also be derived from the results ofm icrolocalsubellipticity and m axim alhypoel-lipticity proved in [1], [8] and [9]. H ow ever,given a particular system of quadratic operators,one can notice that only checking the H orm ander-K ohn condition in every non-zero point turns out to be quite di cult to do in practice. T he sam e com m ent appliesforchecking the m axim alhypoellipticity ofthe system . A notherinterestofthe approach w e are developing here com es from the fact that the proofofT heorem 1.2.1 is purely analytic and does not require any techniques ofrepresentations ofnilpotent groups as in [8] or [9]. M oreover, despite its length, the proof provided here only involves fairly elem entary argum ents w hose com plexity has no degree ofcom parison w ith the analysis led in [8]and [9].
doublecharacteristicsobtained in [17](P roposition 2.0.1)hasalready allow ed to derive in [3]the precise asym ptotics for the resolvent norm ofcertain class ofsem iclassical pseudodi erentialoperators in a neighborhood of the doubly characteristic set. O n the other hand, this deeper understanding of non-trivial interactions betw een the di erent quadratic operators com posing overdeterm ined system s m ay also give hints on how to analyze the m ore com plex case ofN by N system s ofquadratic operators, w hich is a topic of current interest. O n that subject, w e refer the reader to the series of recent w orks on non-com m utative harm onic oscillators by A . Parm eggiani and M .W akayam a in [10],[11],[12],[13],[14]and [15].
2. P roof of T heorem 1.2.1
In the follow ing,w e shalluse the notation S m (X )r;m (X ) 2sdX2 ,w here is an open set in R2n,r;s 2 R and m 2 C1 ( ;R
+),to stand for the class ofsym bols a
verifying
a 2 C1 ( ); 8 2 N2n;9C > 0; j@
Xa(X )j C m (X )r sj j; X 2 :
In the case w here = R2n,w e shalldrop the index forsim plicity. W e shallalso use
the notations f . g and f g,on ,for respectively the estim ates 9C > 0,f C g and,f . g and g . f,on .
T he proofofT heorem 1.2.1 w illrely on the follow ing key proposition. C onsidering for 1 j N ,
qj:Rnx Rn ! C
(x; ) 7! qj(x; );
w ith n 2 N ,N com plex-valued quadratic form s w ith non-negative realparts (2.1) R e qj(x; ) 0; (x; ) 2 R2n; 1 j N ;
w e assum e that there exist a positive integer m 2 N and an open set 0in R2n such
that the follow ing sum ofnon-negative quadratic form s satis es
(2.2) 9c0> 0;8X 2 0; m X k= 0 X j= 1;:::;N ; (l1;:::;lk)2 f1;:::;N gk R e qj(Im Fl1:::Im FlkX ) c0jX j 2;
w here the notation Im Fjstands for the im aginary part ofthe H am ilton m ap Fj
as-sociated to the quadratic form qj. U nder this assum ption,one can then extend the
construction ofthe bounded w eight function done in the scalar case in [17](P roposi-tion 2.0.1) to the system case as follow s:
P roposition 2.0.1. If(qj)1 j N are N com plex-valued quadratic form s on R2n
ver-ifying (2.1) and (2.2) then there exist N real-valued weight functions gj2 S 0 1;hX i 2 2m + 1dX2 ; 1 j N ; such that (2.3) 9c;c1;:::;cN > 0;8X 2 0; 1 + N X j= 1 R e qj(X )+ cjHIm qj gj(X ) chX i 2 2m + 1;
A s in [17],the construction ofthese w eight functions w illbe really the core ofthis w ork.T hisconstruction w illbe an adaptation to the system case ofthe one perform ed in the scalar case.
To check that w e can actually deduce T heorem 1.2.1 from P roposition 2.0.1,w e begin by noticing,as in [17],that the assum ptions ofT heorem 1.2.1 im ply that the follow ing sum ofnon-negative quadratic form s
(2.4) 9c0> 0; r(X )= k0 X k= 0 X j= 1;:::;N ; (l1;:::;lk)2 f1;:::;N gk R e qj(Im Fl1:::Im FlkX ) c0jX j 2 ;
is actually a positive de nite quadratic form . Let us indeed consider X02 R2n such
that r(X0) = 0. T hen,the non-negativity ofquadratic form s R e qj induces that for
all0 k k0,j = 1;:::;N and (l1;:::;lk)2 f1;:::;N gk,
(2.5) R e qj(Im Fl1:::Im FlkX0)= 0:
B y denoting R e qj(X ;Y ) the polar form associated to R e qj, w e deduce from the
C auchy-Schw arz inequality,(1.10) and (2.5) that for allY 2 R2n,
jR e qj(Y ;Im Fl1:::Im FlkX0)j
2= j (Y;R e F
jIm Fl1:::Im FlkX0)j
2
R e qj(Y ) R e qj(Im Fl1:::Im FlkX0)= 0: It follow s that for allY 2 R2n,
(Y;R e FjIm Fl1:::Im FlkX0)= 0;
w hich im plies that for all0 k k0,j = 1;:::;N and (l1;:::;lk)2 f1;:::;N gk,
(2.6) R e FjIm Fl1:::Im FlkX0= 0;
since is non-degenerate. W e nally deduce (2.4) from the assum ption (1.18). In the case w here k0= 0,w e notice that the quadratic form
q = q1+ :::+ qN;
has a positive de nite realpart. T his im plies in particular that q is elliptic on R2n.
O ne can therefore directly deduce from classicalresults about elliptic quadratic dif-ferential operators proved in [18] (See T heorem 3.5 in [18] or com m ents about the elliptic case in T heorem 1.2.1 in [17]),the naturalelliptic a prioriestim ate
9C > 0;8u 2 D (q1)\ :::\ D (qN); h(x; )i2 wu L2 C (kq
w(x;D
x)ukL2 + kukL2); w hich easily im plies (1.19).
W e can therefore assum e in the follow ing that k0 1 and nd from P
roposi-tion 2.0.1 som e real-valued w eight funcroposi-tions (2.7) gj2 S 1;hX i 2 2k 0 + 1dX2 ; 1 j N ; such that (2.8) 9c;c1;:::;cN > 0;8X 2 R2n; 1+ N X j= 1 R e qj(X )+ cjHIm qj gj(X ) chX i 2 2k 0 + 1:
ofW ick calculus are recalled in Section 4.1. It follow s from (2.7),(4.4),(4.7),(4.8) and the C auchy-Schw arz inequality that
(2.9) N X j= 1 R e qW ick j u;(1 "cjgj)W icku = N X j= 1
R e (1 "cjgj)W ickqjW ick u;u
N X j= 1 k1 "cjgjkL1 kqW ick j ukL2kukL2 . N X j= 1 kqW ickj uk2L2+ kuk2L2 . N X j= 1 k~qwjuk2L2+ kuk2L2; w here (2.10) q~j(x; )= qj x; 2 ;
because the operators (1 "cjgj)W ick w hose W ick sym bol are real-valued, are
for-m ally selfadjoint. Indeed,syfor-m bols r(qj) de ned in (4.8) are here just som e constants
since qj are quadratic form s. T he factor 2 in (2.10) com es from the di erence of
norm alizationschosen betw een (1.1)and (4.9)(See rem ark in Section 4.1). Since from (4.10), (1 "cjgj)W ickqjW ick= h (1 "cjgj)qj+ " 4 cjr gj:r qj " 4i cjfgj;qjg iW ick + Sj;
w ith kSjkL (L2(Rn)). 1,w e obtain from the fact that realH am iltonians getquantized in the W ick quantization by form ally selfadjoint operators that
N X j= 1 R e (1 "cjgj)W ickqjW ick = N X j= 1 R e Sj + N X j= 1 h (1 "cjgj)R e qj+ " 4 cjr gj:r R e qj+ " 4 cjHIm qj gj iW ick ;
because gjare real-valued sym bols. Since R e qj 0 and gj2 L1 (Rn),w e can choose
the positive param eter " su ciently sm allsuch that 8 1 j N ;8X 2 R2n; 1 "cjgj(X )
1 2; in order to deduce from (2.8),(2.9) and (4.3) that
(2.11) (hX i2k 0 + 12 )W icku;u . kuk2
L2+ N X j= 1 k~qjwuk2L2+ N X j= 1 (r gj:r R e qj)W icku;u ;
because from (4.1) and (4.2),1W ick= Id:
O ne can then com plete the proofofT heorem 1.2.1 by follow ing exactly the sam e reasoning as the one used in [17]. W e recall this reasoning here for the sake of com pleteness ofthis w ork.
B y denoting ~X = x; =(2 ) and O pw S(1;dX2) the operators obtained by the W eylquantization ofsym bols in the class S(1;dX 2),it follow s from (4.7),(4.8) and
usualresults ofsym bolic calculus that (2.12) hX i2k 0 + 12 W ick h~X i 2 2k 0 + 1 w 2 O pw S(1;dX2) 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 pw S(1;dX2) ; since k0 0. B y using that
w e therefore deduce from (2.11) and the C alderon-Vaillancourt theorem that (2.14) h~X i 1 2k 0 + 1 wu 2 L2 . kuk 2 L2+ N X j= 1 k~qjwuk2L2 + N X j= 1 (r gj:r R e qj)W icku;u :
T hen,w e get from (2.7) and (4.3) that
(2.15) (r gj:r R e qj)W icku;u . jr R e qjjW icku;u :
R ecalling now the w ell-know n inequality
(2.16) jf0(x)j2 2f(x)kf00kL1
(R );
ful lled by any non-negative sm ooth function w ith bounded second derivative, w e deduce from another use of(4.3) that
(2.17) jr R e qjjW icku;u . ((R e qj)
1
2)W icku;u . (1 + R e q
j)W icku;u ;
since R e qj is a non-negative quadratic form and that
2(R e qj)
1
2 1 + R e qj: B y using the sam e argum ents as in (2.9),w e obtain that
(1 + R e qj)W icku;u = (R e qj)W icku;u + kuk2L2 = R e(qW ickj u;u)+ kuk2L2 kqW ickj ukL2kukL2 + kuk2
L2 . kq W ick j uk 2 L2+ kuk 2 L2 . k~q w juk 2 L2 + kuk 2 L2: It therefore follow s from (2.14),(2.15) and (2.17) that
(2.18) h~X i 1 2k 0 + 1 wu 2 L2 . kuk 2 L2+ N X j= 1 k~qwjuk2L2:
In order to im prove the estim ate (2.18), w e carefully resum e our previous analysis and notice that our previous reasoning has in fact established that
h~X i 1 2k 0 + 1 wu 2 L2 . kuk2 L2+ N X j= 1
R e qjW icku;(1 "cjgj)W icku + N X j= 1 (r gj:r R e qj)W icku;u . kuk2 L2+ N X j= 1 R e qW ick j u;(1 "cjgj)W icku + N X j= 1 jR e(qW ick j u;u)j . kuk2 L2+ N X j= 1 R e ~qjwu;(1 "cjgj)W icku + N X j= 1 jR e(~qwju;u)j;
because (1 "cjgj)W ick is a bounded operator on L2(Rn),
(2.19) k(1 "cjgj)W ickkL (L2) k1 "cjgjkL1 (R2n): B y applying this estim ate to h ~X i
1
2k 0 + 1 wu,w e deduce from (2.13)and the C alderon-Vaillancourt theorem that
T hen,by noticing that the com m utator
(2.21) q~wj; h~X i2k 0 + 11 w 2 O pw S hX i 1
2k 0 + 1;hX i 2dX2 ; because ~qj is a quadratic form ,and that
(2.22) h~X i 1
2k 0 + 1 w h~X i 1
2k 0 + 1 w Id 2 O pw S(hX i 2;hX i 2dX2) ; w e deduce from standard results ofsym bolic calculus and the C alderon-Vaillancourt theorem that ~ qwj; h~X i 1 2k 0 + 1 w u L2 . q~ w j; h~X i 1 2k 0 + 1 w h~X i 1 2k 0 + 1 w h~X i 1 2k 0 + 1 wu L2 + kukL2 . h~X i 1 2k 0 + 1 wu L2 + kukL2: (2.23)
B y introducing this com m utator, w e get from the C auchy-Schw arz inequality and (2.23) that R e ~qwj h~X i2k 0 + 11 wu; h~X i 1 2k 0 + 1 wu . R e ~qjwu; h~X i2k 0 + 11 w h~X i 1 2k 0 + 1 wu + h~X i 1 2k 0 + 1 wu 2 L2 + kuk 2 L2: A notheruse ofthe C auchy-Schw arzinequality and the C alderon-Vaillancourttheorem w ith (2.13) gives that
R e ~qwju; h ~X i 1 2k 0 + 1 w h~X i 1 2k 0 + 1 wu . k~qw jukL2 h~X i 2 2k 0 + 1 wu L2+ k~q w jukL2kukL2: W e then deduce from (2.18) and the previous estim ate that
N X j= 1 R e ~qwj h~X i 1 2k 0 + 1 wu; h~X i 1 2k 0 + 1 wu . h~X i2k 0 + 12 wu L2 N X j= 1 k~qjwukL2+ N X j= 1 k~qwjuk2L2 + kuk2L2:
B y using again the C auchy-Schw arz inequality,(2.18),(2.19),(2.20) and (2.23),this estim ate im plies that
h~X i 2 2k 0 + 1 wu 2 L2 . N X j= 1 R e q~wj; h~X i 1 2k 0 + 1 w u;(1 "cjgj)W ick h~X i 1 2k 0 + 1 wu (2.24) + N X j= 1 R e ~qwju; h~X i2k 0 + 11 w(1 "cjgj)W ick h~X i 1 2k 0 + 1 wu + N X j= 1 k~qwjuk2L2 + kuk2L2 . N X j= 1 R e ~qwju; h~X i 1 2k 0 + 1 w(1 "cjgj)W ick h~X i 1 2k 0 + 1 wu + N X j= 1 k~qwjuk2L2 + kuk2L2 . N X j= 1 k~qwjukL2 h~X i 1 2k 0 + 1 w(1 "cjgj)W ick h~X i 1 2k 0 + 1 wu L2+ N X j= 1 k~qwjuk2L2 + kuk2L2;
N otice now that (2.7),(4.5) and (4.6) im ply that
h~X i2k 0 + 11 w;(1 "cjgj)W ick 2 O pw S(1;dX2) ;
since (1 "cjgj)W ick= ~gjw,w ith ~gj2 S(1;dX2) and k0 0. B y introducing this new
com m utator,w e deduce from the C alderon-Vaillancourt theorem ,(2.13),(2.18) and (2.19) that h~X i 1 2k 0 + 1 w(1 "cjgj)W ick h~X i 1 2k 0 + 1 wu L2 . h~X i 1 2k 0 + 1 wu L2 + (1 "cjgj)W ick h~X i 1 2k 0 + 1 w h~X i 1 2k 0 + 1 wu L2 . h~X i 1 2k 0 + 1 wu L2 + h~X i 1 2k 0 + 1 w h~X i 1 2k 0 + 1 wu L2 . h~X i2k 0 + 12 wu L2 + h~X i 1 2k 0 + 1 wu L2 + kukL2 . h~X i 2 2k 0 + 1 wu L2 + N X j= 1 k~qjwukL2 + kukL2:
R ecalling (2.24),w e can then use this last estim ate to obtain that
(2.25) h~X i2k 0 + 12 wu 2 L2 . N X j= 1 k~qjwuk2L2+ kuk2L2:
B y nally noticing from the hom ogeneity ofdegree 2 of ~qj that w e have
(~qj T )(x; )=
1
2 qj(x; ); ifT stands for the reallinear sym plectic transform ation
T (x; )= (2 ) 12x;(2 ) 1 2 ;
w e deduce from the sym plectic invariance ofthe W eylquantization (T heorem 18.5.9 in [5]) that hX i2k 0 + 12 wu 2 L2 . N X j= 1 kqjwuk2L2+ kuk2L2;
w hich proves T heorem 1.2.1.
3. P roof of P roposition 2.0.1
W e prove P roposition 2.0.1 by induction on the positive integer m 1 appearing in (2.2). Let m 1,w e shallassum e that P roposition 2.0.1 is ful lled for any open set 0ofR2n,w hen the positive integer in (2.2) is strictly sm aller than m .
In the follow ing,w e denote by , and w som e C1 (R ;[0;1])functionsrespectively
satisfying
(3.1) = 1 on [ 1;1]; supp [ 2;2];
(3.2) = 1 on fx 2 R :1 jxj 2g; supp fx 2 R :1=2 jxj 3g; and
(3.3) w = 1 on fx 2 R :jxj 2g; supp w fx 2 R :jxj 1g:
M ore generically,w e shalldenote by j, jand wj,j 2 N ,som e other C1 (R ;[0;1])
Let 0 be an open setofR2n such that(2.2) is ful lled. C onsidering the quadratic form s (3.4) r~1;p(X )= N X j= 1 R e qj(X ;Im FpX ); (3.5) r~k;p(X )= X j= 1;:::;N (l1;:::;lk 1)2 f1;:::;N gk 1
R e qj(Im Fl1:::Im Flk 1X ;Im Fl1:::Im Flk 1Im FpX );
for any 1 p N ,2 k m ; (3.6) r0(X )= N X j= 1 R e qj(X ); rk(X )= X j= 1;:::;N (l1;:::;lk)2 f1;:::;N gk R e qj(Im Fl1:::Im FlkX );
for any 1 k m ;and de ning
(3.7) ~gm ;p(X )= rm 1(X )hX i 2 (2m 1 ) 2m + 1 hX i 4m 2m + 1r~ m ;p(X );
w here is the function de ned in (3.1) and 1 p N ,w e get from Lem m a 4.2.1 that HIm qp g~m ;p(X )= 2 rm 1(X )hX i 2 (2m 1 ) 2m + 1 X j= 1;:::;N (l1;:::;lm 1)2 f1;:::;N gm 1 R e qj(Im Fl1:::Im Flm 1Im FpX ) hX i2m + 14m (3.8) + 2 rm 1(X )hX i 2 (2m 1 ) 2m + 1 X j= 1;:::;N (l1;:::;lm 1)2 f1;:::;N gm 1
R e qj(Im Fl1:::Im Flm 1X ;Im Fl1:::Im Flm 1(Im Fp)
2X ) hX i2m + 14m + HIm qp rm 1(X )hX i 2 (2m 1 ) 2m + 1 ~rm ;p(X ) hX i2m + 14m + rm 1(X )hX i 2 (2m 1 ) 2m + 1 H Im qp hX i 4m 2m + 1 ~r m ;p(X ): W e rst check that (3.9) ~gm ;p2 S 1;hX i 2 (2m 1 ) 2m + 1 dX2 :
In order to verify this,w e notice from Lem m a 4.2.6 that the quadratic form s (3.10) R e qj(Im Fl1:::Im Flm 1X ;Im Fl1:::Im Flm 1Im FpX ) and
(3.11) R e qj(Im Fl1:::Im Flm 1X ;Im Fl1:::Im Flm 1(Im Fp)
2X );
belong to the sym bolclass
(3.12) S hX i2m + 14m ;hX i 2 (2m 1 )
2m + 1 dX2 ; for any open set in R2n w here r
m 1(X ) . hX i
2 (2m 1 )
2m + 1 . To check this,w e just use in addition to Lem m a 4.2.6 the obvious estim ates
R e qj(Im Fl1:::Im Flm 1Im FpX ) 1
2 . hX i and
R e qj(Im Fl1:::Im Flm 1(Im Fp)
w e obtain (3.9) from (3.1),(3.5),(3.6),(3.7),(3.10),(3.12) and Lem m a 4.2.2. D enoting respectively A1;p, A2;p, A3;p and A4;p the four term s appearing in the
righthand sideof(3.8),w e rstnoticefrom (3.1),(3.10),(3.12),(3.13)and Lem m a 4.2.2 that
(3.14) A2;p2 S 1;hX i
2 (2m 1 ) 2m + 1 dX2 : N ext,by using that
Im qp2 S hX i2;hX i 2dX2 ;
since Im qp is a quadratic form ,w e get from (3.1),(3.5),(3.6),(3.10),(3.12),(3.13)
and Lem m a 4.2.2 that
(3.15) A3;p2 S hX i 2 2m + 1;hX i 2 (2m 1 ) 2m + 1 dX2 ; since HIm qp rm 1(X )hX i 2 (2m 1 ) 2m + 1 2 S hX i 2 2m + 1;hX i 2 (2m 1 ) 2m + 1 dX2 : B y using now that
HIm qp hX i 4m
2m + 1 2 S hX i 4m
2m + 1;hX i 2dX2 ;
w e nally obtain from anotheruse of(3.1),(3.5),(3.6),(3.10),(3.12)and Lem m a 4.2.2 that
(3.16) A4;p2 S 1;hX i
2 (2m 1 ) 2m + 1 dX2 : Since the term A3;p is supported in
supp 0rm 1(X )hX i
2 (2m 1 ) 2m + 1 ;
w e deduce from (3.8), (3.14), (3.15) and (3.16) that there exists 0 a C1 (R ;[0;1])
function satisfying sim ilar properties as in (3.2),w ith possibly di erent positive nu-m ericalvalues for its support localization,such that,9c1;c2> 0,8X 2 R2n,
c1+ c2 0 rm 1(X )hX i 2 (2m 1 ) 2m + 1 hX i 2 2m + 1 + N X p= 1 HIm qp g~m ;p(X ) (3.17) 2 rm 1(X )hX i 2 (2m 1 ) 2m + 1 rm(X ) hX i2m + 14m :
R ecalling (2.2),one can nd som e positive constants c3;c4> 0 such that
(3.18)
mX 1 k= 0
rk(X ) c3jX j2;
on the open set
(3.19) 1= X 2 R2n :rm (X )< c4jX j2 \ 0:
W hen m 2, one can nd according to our induction hypothesis som e real-valued functions (3.20) ~gm ;p2 S 1 1;hX i 2 2m 1dX2 ; 1 p N ; such that (3.21) 9c5;p> 0;8X 2 1; 1 + N X p= 1 R e qp(X )+ c5;pHIm qp ~gm ;p(X ) & hX i 2 2m 1: For convenience,w e set in the follow ing ~g1;p= 0 w hen m = 1. B y choosing suitably
respectively de ned in (3.1)and (3.3),w ith possibly di erentpositivenum ericalvalues for their support localizations,such that
(3.22) supp 0 rm(X )jX j2 w0(X ) X 2 R2n :rm (X )< c4jX j2 ;
and setting
(3.23) Gm ;p(X )= ~gm ;p(X )+ 0 rm (X )jX j2 w0(X )~gm ;p(X ); X 2 0;
w e deduce from a straightforw ard adaptation of the Lem m a 4.2.2 by recalling (3.1) and (3.3) that
(3.24) 0 rm (X )jX j2 w0(X )2 S 1;hX i 2dX2 :
A ccording to (3.9) and (3.20),this im plies that (3.25) G1;p2 S 0 1;hX i 2 3dX2 and G m ;p2 S 0 1;hX i 2 2m 1dX2 ; w hen m 2. Since from (3.24),
HIm qp 0 rm(X )jX j
2 w
0(X ) 2 S 1;hX i 2dX2 ;
because Im qp is a quadratic form ,w e rst notice from (3.19),(3.20) and (3.22) that
HIm qp 0 rm (X )jX j
2 w
0(X ) ~gm ;p(X )2 S 0 1;hX i 2
2m 1dX2 ;
and then deduce from (3.17),(3.19),(3.21),(3.22) and (3.23) that there exist som e positive contants c6;p;c7> 0 such that for allX 2 0,
N X p= 1 R e qp(X )+ c6;pHIm qp Gm ;p(X ) + 1 + c7 0 rm 1(X )hX i 2 (2m 1 ) 2m + 1 hX i 2 2m + 1 & rm 1(X )hX i 2 (2m 1 ) 2m + 1 rm (X ) hX i2m + 14m + 0 rm (X )jX j2 w0(X )hX i 2 2m 1; w hen m 2. Since hX i2m2 1 & hX i 2 2m + 1 and rm(X ) hX i2m + 14m & jX j 2 2m + 1;
w hen rm(X ) & jX j2, w e deduce from the previous estim ate by distinguishing the
regions in 0w here
rm(X ). jX j2 and rm (X )& jX j2;
according to the support ofthe function
0 rm (X )jX j2 ;
that one can nd a C1 (R ;[0;1]) function w1 w ith the sam e kind ofsupport as the
function de ned in (3.3) such that
(3.26) 9c8;p;c9> 0;8X 2 0; N X p= 1 R e qp(X )+ c8;pHIm qp Gm ;p(X ) + c9w1 rm 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,w e notice from (2.2) that
(3.27) r1(X )& hX i2;
on any set w here
if the positive constant c10 is chosen su ciently large. M oreover,since in this case
G1;p= ~g1;p and that R e qp 0,one can deduce from (3.1),(3.3),(3.17),(3.27) and
(3.28),by distinguishing the regions in 0 w here
r0(X ). hX i 2 3 and r 0(X )& hX i 2 3; according to the support ofthe function
r0(X )hX i
2 3 ;
that the estim ate (3.26) is also ful lled in the case m = 1. C ontinuing our study of the case w here m = 1,w e notice from (3.3) and R e qp 0,that one can estim ate
w1 r0(X )hX i 2 3 hX i 2 3 . r0(X )= N X p= 1 R e qp(X );
for all X 2 R2n. It therefore follow s that one can nd c
11;p > 0 such that for all
X 2 0, N X p= 1 R e qp(X )+ c11;pHIm qpG1;p(X ) + 1 & hX i 2 3;
w hich provesP roposition 2.0.1 in the case w here m = 1,and ourinduction hypothesis in the basis case.
A ssum ing in the follow ing that m 2,w e shallnow w ork on the term w1 rm 1(X )hX i
2 (2m 1 ) 2m + 1 hX i
2 2m + 1;
appearing in (3.26). B y considering som e constants j 1,for 0 j m 2,w hose
values w illbe successively chosen in the follow ing,w e shallprove that one can w rite that for allX 2 R2n,
(3.29) w1 rm 1(X ) hX i 2 (2m 1 ) 2m + 1 ! ~ W0(X ) 0(X ) + mX 2 j= 1 ~ W0(X ) j Y l= 1 Wl(X ) j(X )+ ~W0(X ) mY 1 l= 1 Wl(X ) ; w ith (3.30) j(X )= jrm j 2(X ) rm j 1(X ) 2m 2j 3 2m 2j 1 ! ; 0 j m 2; (3.31) Wj(X )= w2 j 1rm j 1(X ) rm j(X ) 2m 2j 1 2m 2j+ 1 ! ; 1 j m 1; (3.32) W~0(X )= w1 rm 1(X ) hX i2 (2m2m + 11 ) ! ; w here istheC1 (R ;[0;1])function de ned in (3.1),and w
2isa C1 (R ;[0;1])function
satisfying sim ilar properties as the function de ned in (3.3), w ith possibly di erent positive num ericalvalues for its support localization,in order to have that
(3.33) supp 0 w2= 1 and supp w20 = 1 :
on the support ofthe function
supp ~W0 j
Y
l= 1
Wl ; if1 j m 1; or,supp ~W0; ifj = 0:
N otice that the constants in the estim ates (3.34) only depend on the values of the param eters 0,...,j 1 but not on l,w hen l j. T his show s that the functions
0; j Y l= 1 Wl j; for 1 j m 2; and mY 1 l= 1 Wl;
are w ell-de ned on the support of the function ~W0. N ow , by noticing from (3.1),
(3.3),(3.30),(3.31) and (3.33) that
(3.35) 1 j+ Wj+ 1;
on the support ofthe function
supp ~W0 j
Y
l= 1
Wl ; if1 j m 2; or,supp ~W0; ifj = 0;
w e deduce the estim ate (3.29) from a nite iteration by using the follow ing estim ates ~ W0 W~0 0+ ~W0W1 and ~ W0 j Y l= 1 Wl W~0 j Y l= 1 Wl j+ ~W0 j+ 1Y l= 1 Wl ;
for any 1 j m 2. O ne can also notice that (3.35) im plies that
(3.36) 1 j+ mX 2 k= j+ 1 k Y l= j+ 1 Wl k+ mY 1 l= j+ 1 Wl;
on the support ofthe function
supp ~W0 j
Y
l= 1
Wl ; if1 j m 2; or,supp ~W0; ifj = 0:
Since R e qp 0,w e then get from (3.34) that
(3.37) 8X 2 R2n; ~W 0(X ) mY 1 l= 1 Wl(X ) hX i 2 2m + 1 ~a 0;:::;m 2 N X p= 1 R e qp(X );
w here the quadratic form s ~rk;p are de ned in (3.4) and (3.5). W e get from (3.1),
(3.3), (3.30), (3.31), (3.32), (3.34), Lem m a 4.2.2, Lem m a 4.2.4, Lem m a 4.2.5 and Lem m a 4.2.7 that
(3.40) pj;p2 S 1;hX i
2 (2m 2j 3 ) 2m + 1 dX2 :
for any 0 j m 2.
W e shall now study the Poisson brackets HIm qppj;p. In doing so, w e begin by w riting that HIm qppj;p(X )= HIm qpW~0 (X ) j Y l= 1 W l(X ) j(X ) ~ rm j 1;p(X ) rm j 1(X ) 2m 2j 2 2m 2j 1 (3.41) + ~W0(X ) j Y l= 1 Wl(X ) HIm qp j (X ) ~ rm j 1;p(X ) rm j 1(X ) 2m 2j 2 2m 2j 1 + ~W0(X ) j Y l= 1 Wl(X ) j(X )HIm qp rm j 1(X ) 2m 2j 2 2m 2j 1 r~ m j 1;p(X ) + ~W0(X ) j Y l= 1 Wl(X ) j(X ) HIm qpr~m j 1;p(X ) rm j 1(X ) 2m 2j 2 2m 2j 1 + j X l= 1 ~ W0(X ) HIm qpWl (X ) j Y k = 1 k 6=l Wk(X ) j(X ) ~ rm j 1;p(X ) rm j 1(X ) 2m 2j 2 2m 2j 1 ;
for 1 j m 2. W e denote by respectively B1;j;p,B2;j;p,B3;j;p,B4;j;p and B5;j;p
the ve term s appearing in the right hand side of (3.41). W e also w rite in the case w here j = 0, HIm qpp0;p(X )= HIm qpW~0 (X ) 0(X ) ~ rm 1;p(X ) rm 1(X ) 2m 2 2m 1 (3.42) + ~W0(X ) HIm qp 0 (X ) ~ rm 1;p(X ) rm 1(X ) 2m 2 2m 1 + ~W0(X ) 0(X )HIm qp rm 1(X ) 2m 2 2m 1 r~ m 1;p(X ) + ~W0(X ) 0(X ) HIm qpr~m 1;p(X ) rm 1(X ) 2m 2 2m 1 ;
and denote as before by respectively B1;0;p,B2;0;p,B3;0;p and B4;0;p the four term s
appearing in the right hand side of(3.42).
Since the constantsin the estim ates(3.34)only depend on the valuesofthe param -eters 0,..., j 1;but not on l,w hen l j;w e notice from (3.29),(3.34) and (3.37)
that there exist a0 > 0 and som e positive constants aj;0;:::;j 1,for 1 j m 1, w hose valuesw ith respectto the param eters( l)0 l m 2only depend on 0,..., j 1;
and X 2 R2n, w1 rm 1(X ) hX i2 (2m2m + 11 ) ! hX i2m + 12 a 0W~0(X ) 0(X )rm 1(X ) 1 2m 1 (3.43) + mX 2 j= 1 jaj;0;:::;j 1W~0(X ) j Y l= 1 W l(X ) j(X )rm j 1(X ) 1 2m 2j 1 + am 1; 0;:::;m 2 N X p= 1 R e qp(X ):
T hepositiveconstanta0isindependentofany ofthe param eters( l)0 l m 2. Setting
(3.44) pp= a0p0;p+ mX 2
j= 1
jaj; 0;:::; j 1pj;p; w e know from (3.40) that
(3.45) pp2 S 1;hX i
2
2m + 1dX2 :
Forany " > 0,w e shallprove thataftera properchoice forthe constants( j)0 j m 2
and ( j)1 j m 2,w ith j 1, j 1,w hose values w illdepend on ";one can nd
a positive constant c12;"> 0 such that for allX 2 R2n,
(3.46) c12;" N X p= 1 R e qp(X )+ HIm qppp(X ) + "hX i 2 2m + 1 w 1 rm 1(X ) hX i 2 (2m 1 ) 2m + 1 ! hX i2m + 12 :
O nce this estim ate proved,P roposition 2.0.1 w illdirectly follow from (3.25),(3.26), (3.45)and (3.46),ifw e choose the positive param eter" su ciently sm alland consider the w eight functions
gp= c13;"Gm ;p+ c14;"pp; 1 p N ;
after a suitable choice for the positive constants c13;"and c14;".
Let " > 0, it therefore rem ains to choose properly these constants ( j)0 j m 2
and ( j)1 j m 2,w ith j 1, j 1,in order to satisfy (3.46).
R ecalling from (4.22) that for all1 p N and 0 s m 2, (3.47) HIm qp~rm s 1;p(X )= 2 X j= 1;:::;N (l1;:::;lm s 2)2 f1;:::;N gm s 2 R e qj(Im Fl1:::Im Flm s 2Im FpX ) + 2 X j= 1;:::;N (l1;:::;lm s 2)2 f1;:::;N gm s 2
R e qj(Im Fl1:::Im Flm s 2X ;Im Fl1:::Im Flm s 2(Im Fp)
2
X );
one can notice by expanding the term
2am 1;0;:::;m 2 N X p= 1 R e qp+ N X p= 1 HIm qppp;
by using (3.41),(3.42) and (3.44) that the term s in
produced by the term s associated to X
j= 1;:::;N
(l1;:::;lm s 2)2 f1;:::;N gm s 2
R e qj(Im Fl1:::Im Flm s 2Im FpX );
w hile using (3.47),give exactly tw o tim es the term a0W~0(X ) 0(X )rm 1(X ) 1 2m 1 (3.48) + mX 2 j= 1 jaj;0;:::;j 1W~0(X ) j Y l= 1 Wl(X ) j(X )rm j 1(X ) 1 2m 2j 1 + am 1; 0;:::; m 2 N X p= 1 R e qp(X );
for w hich w e have the estim ate (3.43). To prove the estim ate (3.46),it w illtherefore be su cient to check that allthe other term s appearing in (3.41) and (3.42) can also be allabsorbed in the term (3.48)aftera properchoice forthe constants( j)0 j m 2
and ( j)1 j m 2;at the exception ofa rem ainder term in
"hX i2m + 12 :
W e shallchoose these constants in the follow ing order 0, 1, 1, 2,...., m 2 and m 2.
W e successively study the rem aining term sin (3.41)and (3.42),by increasing value ofthe integer 0 j m 2. W e rst notice from (3.1),(3.3),(3.30),(3.32),(3.42), 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 allX 2 R2n,
(3.49) a0 N X p= 1 jB1;0;p(X )j. 1 2 0 hX i 2 2m + 1 " m 1hX i 2 2m + 1:
B y noticing from (3.34) that the estim ates (3.50) rm(X ). hX i2. rm 1(X )
2m + 1 2m 1;
are ful lled on the supportofthe function ~W0,w e deduce from (3.1),(3.30)and (3.42)
that the m odulus ofthe term s B3;0;p can be estim ated as
a0 N X p= 1 jB3;0;p(X )j= a0 N X p= 1 rm 1(X ) 2m 2 2m 1H Im qp rm 1(X ) 2m 2 2m 1 rm 1(X ) 2m 2 2m 1r~ m 1;p(X ) ~W0(X ) 0(X ) . 1 2 0 W~0(X ) 0(X )rm 1(X ) 1 2m 1;
for allX 2 R2n; since from Lem m a 4.2.8 and Lem m a 4.2.10,w e have for any p in
f1;:::;N g that rm 1(X ) 2m 2 2m 1H Im qp rm 1(X ) 2m 2 2m 1 . rm 1(X ) 1 2m 1 and rm 1(X ) 2m 2 2m 1r~ m 1;p(X ) . 1 2 0 ;
on the support ofthe function ~W0(X ) 0(X ):B y possibly increasing su ciently the
value of the constant 0 w hich is of course possible w hile keeping (3.49), one can
N ext,w e deduce from (3.1),(3.30),(3.42),(3.50) and Lem m a 4.2.9 that the m od-ulus ofthe second term s in B4;0;p associated to
2 X
j= 1;:::;N (l1;:::;lm 2)2 f1;:::;N gm 2
R e qj(Im Fl1:::Im Flm 2X ;Im Fl1:::Im Flm 2(Im Fp)
2
X );
w hile using (3.47),denoted here ~B4;0;p, N X p= 1 ~ B4;0;p(X )= ~W 0(X ) 0(X ) N X p= 1 0 B B @ HIm qpr~m 1;p(X ) rm 1(X ) 2m 2 2m 1 2 X j= 1;:::;N (l1;:::;lm 2)2 f1;:::;N gm 2 R e qj(Im Fl1:::Im Flm 2Im FpX ) rm 1(X ) 2m 2 2m 1 1 C C A ; = ~W0(X ) 0(X ) N X p= 1 HIm qp~rm 1;p(X ) rm 1(X ) 2m 2 2m 1 2rm 1(X ) 1 2m 1 !
can be estim ated as a0 N X p= 1 j~B4;0;p(X )j. 1 2 0 W~0(X ) 0(X )rm 1(X ) 1 2m 1;
for all X 2 R2n. B y possibly increasing su ciently the value of the constant 0
w hich is ofcourse possible w hile keeping (3.49),one can also controlthis term w ith the \good" term (3.48). T he value of the constant 0 is now de nitively xed. In
(3.42),it only rem ains to study the term s B2;0;p.
A bout these term s,w e deduce from (3.1),(3.30),(3.42),(3.50),Lem m a 4.2.8 and Lem m a 4.2.11 that for allX 2 R2n,
(3.51) a0 N X p= 1 jB2;0;p(X )j. ~W0(X )W 1(X )rm 1(X ) 1 2m 1:
B y using now (3.34) and (3.36) w ith j = 1,w e obtain that for allX 2 R2n,
a0 N X p= 1 jB2;0;p(X )j cm 1;0;:::;m 2W~0(X ) mY 1 l= 1 Wl(X ) N X p= 1 R e qp(X ) + mX 2 j= 1 cj; 0;:::;j 1W~0(X ) j Y l= 1 Wl(X ) j(X )rm j 1(X ) 1 2m 2j 1; w hich im plies that
(3.52) a0 N X p= 1 jB2;0;p(X )j cm 1;0;:::;m 2 N X p= 1 R e qp(X ) + mX 2 j= 1 cj; 0;:::;j 1W~0(X ) j Y l= 1 Wl(X ) j(X )rm j 1(X ) 1 2m 2j 1; w here the quantities cj; 0;:::;j 1 stand for positive constants w hose values depend on 0,..., j 1,but not on ( k)j k m 2 and ( k)1 k m 2;according to the rem ark
done after (3.34). O ne can therefore choose the constant 1 1 in (3.44) su ciently
hand side ofthe estim ate (3.52)by the term ofsam e index in the \good" term (3.48). T his is possible since the constants a1;0 and c1; 0 are now xed after our choice of the param eter 0.
T his ends our step index j = 0 in w hich w e have chosen the values for the tw o constants 0and 1 1. W e shallnow explain how to choosethe rem aining constants
( j)1 j m 2 and ( j)2 j m 2 in (3.44) in order to satisfy (3.46). T his choice w ill
also determ ine the values ofthe constants (aj;0;:::; j 1)1 j m 2appearing in (3.44). A fter this step index j = 0,w e have m anaged to absorb allthe term s appearing in (3.42) in the \good" term (3.48) at the exception ofa rem ainder com ing from (3.49) and (3.52), mX 2 j= 2 cj;0;:::;j 1W~0(X ) j Y l= 1 Wl(X ) j(X )rm j 1(X ) 1 2m 2j 1 + " m 1hX i 2 2m + 1;
w here one recallthat the positive constants cj; 0;:::;j 1 only depend on 0,..., j 1, but not on ( k)j k m 2 and ( k)1 k m 2:
W e proceed in the follow ing by nite induction and assum e that,at the beginning ofthe step index k,w ith 1 k m 2,w e have already chosen the values for the constants ( j)0 j k 1 and ( j)1 j k in (3.44);and that these choices have allow ed
to absorb allthe term s appearing in the right hand side of (3.42) and (3.41),w hen 1 j k 1,in the \good" term (3.48) at the exception ofa rem ainder term
(3.53) k m 1"hX i 2 2m + 1+ mX 2 j= k+ 1 ~ cj;0;:::;j 1;1;:::;k 1W~0(X ) j Y l= 1 Wl(X ) j(X )rm j 1(X ) 1 2m 2j 1;
w here the quantities ~cj; 0;:::;j 1; 1;:::;k 1 stand for positive constants w hose values only depend on 0,..., j 1, 1,..., k 1;but not on ( l)j l m 2 and ( l)k l m 2.
W e shallnow explain how to choose the constants k and; k+ 1,w hen k m 3;
in this step index k in order to absorb the term s appearing in the right hand side of (3.41),w hen j = k,at the exception ofa rem ainder term ofthe type (3.53) w here k w illbe replaced by k + 1;in the \good" term (3.48). Since the constants( j)0 j k 1
and ( j)1 j k have already been chosen,w e shallonly underline in the follow ing the
dependence ofour estim ates w ith respect to the other param eters ( j)k j m 2 and
( j)k+ 1 j m 2,w hose values rem ain to be chosen.
W e notice from (3.1),(3.30),(3.31),(3.32),(3.34),(3.41),Lem m a 4.2.8 and Lem m a 4.2.12 that one can assum e by choosing the constant k 1 su ciently large that
for allX 2 R2n, (3.54) kak;0;:::;k 1 N X p= 1 jB1;k;p(X )j. 1 2 k hX i 2 2m + 1 " m 1hX i 2 2m + 1;
N ext,w e deduce from (3.1),(3.30),(3.34)and (3.41)thatthe m odulusofthe term s B3;k;pcan be estim ated as
kak;0;:::;k 1 N X p= 1 jB3;k;p(X )j = kak;0;:::;k 1 N X p= 1 rm k 1(X ) 2m 2k 2 2m 2k 1H Im qp rm k 1(X ) 2m 2k 2 2m 2k 1 rm k 1(X ) 2m 2k 2 2m 2k 1r~ m k 1;p(X ) ~W0(X ) k Y l= 1 Wl(X ) k(X ) . 1 2 k W~0(X ) k Y l= 1 Wl(X ) k(X )rm k 1(X ) 1 2m 2k 1;
for allX 2 R2n; since from Lem m a 4.2.8 and Lem m a 4.2.10,w e have for any p in
f1;:::;N g that rm k 1(X ) 2m 2k 2 2m 2k 1H Im qp rm k 1(X ) 2m 2k 2 2m 2k 1 . rm k 1(X ) 1 2m 2k 1 and rm k 1(X ) 2m 2k 2 2m 2k 1r~ m k 1;p(X ) . 1 2 k ;
on the support ofthe function
~ W0(X ) k Y l= 1 Wl(X ) k(X ):
B y possibly increasing su ciently the value of the constant k w hich is of course
possible w hile keeping (3.54),one can controlthis term w ith the \good" term (3.48). N ext,w e deduce from (3.1),(3.30),(3.34),(3.41) and Lem m a 4.2.9 that the m od-ulus ofthe second term s in B4;k;p associated to
2 X
j= 1;:::;N (l1;:::;lm k 2)2 f1;:::;N g
m k 2
R e qj(Im Fl1:::Im Flm k 2X ;Im Fl1:::Im Flm k 2(Im Fp)
2X );
w hile using (3.47),denoted here ~B4;k;p, N X p= 1 ~ B4;k;p(X )= ~W0(X ) k Y l= 1 Wl(X ) k(X ) N X p= 1 0 B B @ HIm qpr~m k 1;p(X ) rm k 1(X ) 2m 2k 2 2m 2k 1 2 X j= 1;:::;N (l1;:::;lm k 2)2 f1;:::;N gm k 2 R e qj(Im Fl1:::Im Flm k 2Im FpX ) rm k 1(X ) 2m 2k 2 2m 2k 1 1 C C A ; = ~W0(X ) k Y l= 1 Wl(X ) k(X ) N X p= 1 HIm qpr~m k 1;p(X ) rm k 1(X ) 2m 2k 2 2m 2k 1 2rm k 1(X ) 1 2m 2k 1 !
can be estim ated as
for all X 2 R2n. B y possibly increasing su ciently the value of the constant k
w hich is ofcourse possible w hile keeping (3.54),one can also controlthis term w ith the \good" term (3.48).
For 1 l k and 1 p N ,w e shallnow study the term
B5;k;p;l(X )= ~W0(X ) HIm qpWl (X ) k Y j= 1 j6=l Wj(X ) k(X ) ~ rm k 1;p(X ) rm k 1(X ) 2m 2k 2 2m 2k 1 ;
appearing in the term B5;k;p in (3.41). B y noticing that
rm l 2(X ) l1rm l 1(X )
2m 2l 3 2m 2l 1;
on the support ofthe function HIm qpWl+ 1,it follow 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 allX 2 R2n,
kak;0;:::;k 1 N X p= 1 jB5;k;p;1(X )j. 1 2 k W~0(X ) 0(X )rm 1(X ) 1 2m 1 and kak;0;:::;k 1 N X p= 1 jB5;k;p;l(X )j. 1 2 k W~0(X ) l 1 Y j= 1 Wj(X ) l 1(X )rm l(X ) 1 2m 2l+ 1; w hen l 2. B y possibly increasing again the value of the constant k, one can
therefore controlthe term
kak;0;:::;k 1
N
X
p= 1
B5;k;p;
w ith the \good" term (3.48). T he value ofthe constant k is now de nitively xed.
A bout the term s B2;k;p,w e 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 allX 2 R2n,
(3.55) kak;0;:::;k 1 N X p= 1 jB2;k;p(X )j. ~W0(X ) k+ 1Y l= 1 Wl(X ) rm k 1(X ) 1 2m 2k 1: B y distinguishing tw o cases,w e rst assum e in the follow ing that k m 3. In this case,by using (3.34) and (3.36) w ith j = k + 1,w e obtain that for allX 2 R2n,
kak;0;:::;k 1 N X p= 1 jB2;k;p(X )j c0m 1; 0;:::;m 2;1;:::;k ~ W0(X ) mY 1 l= 1 Wl(X ) N X p= 1 R e qp(X ) + mX 2 j= k+ 1 c0j; 0;:::;j 1; 1;:::;k ~ W0(X ) j Y l= 1 Wl(X ) j(X )rm j 1(X ) 1 2m 2j 1; w hich im plies that
w here the quantities c0j;
0;:::;j 1;1;:::;k stand for positive constants w hose values only depend on 0,..., j 1, 1,..., k,but not on ( l)j l m 2 and ( l)k+ 1 l m 2.
Indeed,w e recallthatthe constantsappearing in the estim ates(3.34)only depend on the values 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 in (3.44)su ciently large in orderto
absorb the term ofindex j = k + 1 in the sum (3.53);and the term ofindex j = k + 1 in the sum appearing in the right hand side of the estim ate (3.56), by the term of sam e index in the \good" term (3.48).
W hen k = m 2 and taking m 2= 1,itfollow sfrom (3.34),used w ith j = m 1,
and (3.55) that for allX 2 R2n,
m 2am 2; 0;:::;m 3 N X p= 1 jB2;m 2;p(X )j. ~W0(X ) mY 1 l= 1 Wl(X ) r1(X ) 1 3 (3.57) . N X p= 1 R e qp(X ):
T his process allow s us to achieve the construction of the w eight functions pp, 1
p N ,satisfying (3.46),w hich ends the proofof(3.46). T his also ends the proofof P roposition 2.0.1.
4. A ppendix
4.1. W ick calculus. T he purpose ofthis section is to recallthe de nition and basic properties ofthe W ick quantization that w e need for the proofofT heorem 1.2.1. W e follow here the presentation ofthe W ick quantization given by N .Lerner in [7]and refer the reader to his w ork for the proofs ofthe results recalled below .
T he m ain property ofthe W ick quantization is its property ofpositivity,i.e.,that non-negative H am iltonians de ne non-negative operators
a 0 ) aW ick 0:
W e recall that this is not the case for the W eyl quantization and refer to [7] for an explicit exam ple of non-negative H am iltonian de ning an operator w hich is not non-negative.
B eforede ning properly the W ick quantization,w e rstneed to recallthe de nition ofthe w ave packets transform ofa function u 2 S(Rn),
W u(y; )= (u;’y;)L2(Rn)= 2n =4 Z
Rn
u(x)e (x y)2e 2i (x y):dx; (y; )2 R2n: w here
’y;(x)= 2n =4e (x y)
2
e2i (x y):; x 2 Rn; and x2= x2
1+ :::+ x2n. W ith this de nition,one can check (see Lem m a 2.1 in [7])that
the m apping u 7! W u is continuous from S(Rn) to S(R2n),isom etric from L2(Rn) to
L2(R2n) and that w e have the reconstruction form ula
(4.1) 8u 2 S(Rn);8x 2 Rn; u(x)=
Z
R2n
W u(y; )’y;(x)dyd :
B y denoting Y the operator de ned in the W eylquantization by the sym bol
pY(X )= 2ne 2 jX Y j
2
; Y = (y; )2 R2n; w hich is a rank-one orthogonalprojection
w e de ne the W ick quantization ofany L1 (R2n) sym bola as
(4.2) aW ick= Z
R2n
a(Y ) YdY :
M ore generally,one can extend this de nition w hen the sym bola belongs to S0(R2n)
by de ning the operator aW ickfor any u and v in S(Rn) by
< aW icku;v >S0(Rn);S (Rn)= < a(Y );( Yu;v)L2(Rn)>S0(R2n);S (R2n);
w here < ; S>0(Rn);S (Rn)denotes the duality bracket betw een the spaces S0(Rn) and S(Rn). T he W ick quantization is a positive quantization
(4.3) a 0 ) aW ick 0:
In particular,realH am iltonians get quantized in this quantization by form ally self-adjointoperatorsand onehas(see P roposition 3.2 in [7])thatL1 (R2n)sym bolsde ne
bounded operators on L2(Rn) such that
(4.4) kaW ickkL (L2(Rn)) kakL1 (R2n):
A ccording to P roposition 3.3 in [7],the W ick and W eylquantizations ofa sym bola are linked by the follow ing identities
(4.5) aW ick= ~aw; w ith (4.6) ~a(X )= Z R2n a(X + Y )e 2 jY j22ndY ; X 2 R2n; and (4.7) aW ick= aw + r(a)w; w here r(a) stands for the sym bol
(4.8) r(a)(X ) = Z 1 0 Z R2n (1 )a00(X + Y )Y2e 2 jY j22ndY d ; X 2 R2n;
ifw e use here the norm alization chosen in [7]for the W eylquantization (4.9) (awu)(x)=
Z
R2n
e2i (x y):a x + y
2 ; u(y)dyd ;
w hich di ers from the one chosen in this paper. B ecause of this di erence in nor-m alizations, certain constant factors w ill naturally appear in the core of the proof of T heorem 1.2.1 w hile using certain form ulas of Section 4.1, but these are m inor adaptations. W e also recallthe follow ing com position form ula obtained in the proof ofP roposition 3.4 in [7], (4.10) aW ickbW ick= h ab 1 4 a 0 b 0 + 1 4i fa;bg iW ick + S;
w ith kSkL (L2(Rn)) dnkakL1 2(b); w hen a 2 L1 (R2n) and b is a sm ooth sym bol satisfying
2(b)= sup
X 2 R2n ; T 2 R2n ;jT j= 1
jb(2)(X )T2j< + 1 :
T he term dn appearing in the previous estim ate stands for a positive constant
4.2. Som e technical lem m as. T his second part ofthe appendix is devoted to the proofs ofseveraltechnicallem m as.
L em m a 4.2.1. For any 1 j N , 1 p N , (l1;:::;lk) 2 f1;:::;N gk and
s1;s22 N ,we have HIm qp R e qj Im Fl1:::Im Flk(Im Fp) s1X ;Im F l1:::Im Flk(Im Fp) s2X (4.11) = 2R e qj Im Fl1:::Im Flk(Im Fp) s1+ 1X ;Im F l1:::Im Flk(Im Fp) s2X + 2R e qj Im Fl1:::Im Flk(Im Fp) s1X ;Im F l1:::Im Flk(Im Fp) s2+ 1X ; where R e qj(X ;Y ) stands for the polarized form associated to the quadratic form
R e qj.
Proof of Lem m a 4.2.1. W e begin by noticing from (1.10) and the skew -sym m etry property ofH am ilton m aps (1.11) that the H am ilton m ap ofthe quadratic form
~ r(X )= R e qj Im Fl1:::Im Flk(Im Fp) s1X ;Im F l1:::Im Flk(Im Fp) s2X ; is given by (4.12) F =~ 1 2( 1) k+ s1(Im F
p)s1Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s2
+ 1 2( 1)
k+ s2(Im F
p)s2Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s1; since
( 1)k+ s1 X ;(Im F
p)s1Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s2X (4.13) = Im Fl1:::Im Flk(Im Fp) s1X ;R e F jIm Fl1:::Im Flk(Im Fp) s2X = R e qj Im Fl1:::Im Flk(Im Fp) s1X ;Im F l1:::Im Flk(Im Fp) s2X = R e qj Im Fl1:::Im Flk(Im Fp) s2X ;Im F l1:::Im Flk(Im Fp) s1X = Im Fl1:::Im Flk(Im Fp) s2X ;R e F jIm Fl1:::Im Flk(Im Fp) s1X = ( 1)k+ s2 X ;(Im F
p)s2Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s1X : T hen,a direct com putation (see Lem m a 2 in [16]) show s that the H am ilton m ap of the quadratic form
HIm qp~r = Im qp;~r = @Im qp @ : @~r @x @Im qp @x : @~r @ ; is given by the com m utator 2[Im Fp;~F ],that is,
HIm qp~r(X )= 2 X ;[Im Fp;~F ]X : A com putation as in (4.13) then allow s to directly get (4.11).
L em m a 4.2.2. C onsider a C1 (R ) function f such that
f 2 L1 (R ) and 9c1;c2> 0; supp f0 x 2 R :c1 jxj c2 ;
ProofofLem m a 4.2.2. It is su cient to check that
(4.15) r r(X )hX i 2 2 S hX i ;hX i 2 dX2 ;
w here is a sm all open neighborhood of supp f0r(X )hX i 2 : W e deduce from
(2.16) and the fact that r(X ) is a non-negative quadratic form that r(X ) hX i2
and
jr r(X ) j. r(X )1=2. hX i ;
on . B y noticing that 0 < 1,hX ir 2 S(hX ir;hX i 2dX2),for any r 2 R ;and
that the function r(X ) is just a quadratic form ,w e directly deduce (4.15) from the previous estim ates and the Leibniz’s rule,since
r(X )2 S hX i2 ;hX i 2 dX2 :
In allthe follow ing lem m as,w e shalldenote by rk the quadratic form s de ned in
(3.6) for 0 k m .
L em m a 4.2.3. For alls 2 R and 0 j m 2,we have
rm j 1(X )s2 S rm j 1(X )s;rm j 1(X ) 1dX2 ;
if is any open set where
rm j 1(X )& hX i
2 (2m 2j 1 ) 2m + 1 :
Proof of Lem m a 4.2.3. R ecalling from (3.6) that the sym bol rm j 1(X ) is a
non-negative quadratic form and that w e have from (2.16) that (4.16) jr rm j 1(X )j. rm j 1(X )
1 2; w hich im plies that for alls 2 R ,
r rm j 1(X )s rm j 1(X )s . r rm j 1(X ) rm j 1(X ) (4.17) . rm j 1(X ) 1 2;
on ,w e notice that the result ofLem m a 4.2.3 is therefore a straightforw ard conse-quence ofthe Leibniz’s rule.
L em m a 4.2.4. C onsider the function jde ned in (3.30)then for any 0 j m 2, j2 S 1;rm j 1(X )
2m 2j 3 2m 2j 1dX2 ; if is any open set where
rm j 1(X )& hX i
2 (2m 2j 1 ) 2m + 1 ; which im plies in particular that
j2 S 1;hX i
ProofofLem m a 4.2.4. W e rst notice from (3.1) and (3.30) that rm j 2(X ) rm j 1(X ) 2m 2j 3 2m 2j 1; on \ supp 0 j. Since from (2.16), jr rm j 2(X )j. rm j 2(X ) 1 2 (4.18) . rm j 1(X ) 2m 2j 3 2 (2m 2j 1 );
on \ supp 0j,w e deduce thatthe quadratic sym bolrm j 2(X )belongsto the class
(4.19) S \ supp 0 j rm j 1(X ) 2m 2j 3 2m 2j 1; dX 2 rm j 1(X ) 2m 2j 3 2m 2j 1 :
It follow s from Lem m a 4.2.3 that rm j 2(X ) rm j 1(X ) 2m 2j 3 2m 2j 1 2 S \ supp 0 j 1; dX2 rm j 1(X ) 2m 2j 3 2m 2j 1 ;
w hich im plies that
j2 S 1;rm j 1(X )
2m 2j 3 2m 2j 1dX2 : T his ends the proofofLem m a 4.2.4.
L em m a 4.2.5. C onsider the function Wj de ned in (3.31) then for any 1 j
m 1,
Wj2 S 1;rm j 1(X ) 1dX2 ;
if is any open set where
rm j 1(X )& hX i
2 (2m 2j 1 ) 2m + 1 ; which im plies in particular that
Wj2 S 1;hX i
2 (2m 2j 1 ) 2m + 1 dX2 :
ProofofLem m a 4.2.5. B y noticing from (3.3) and (3.31) that rm j 1(X ) rm j(X ) 2m 2j 1 2m 2j+ 1 and rm j(X )& hX i 2 (2m 2j+ 1) 2m + 1 ; on \ supp W 0
j,and that the tw o derivatives
0and w0
2 ofthe functions appearing
in (3.30) and (3.31) have sim ilar types ofsupport as the function de ned in (3.2),w e notice that w e are exactly in the setting studied in Lem m a 4.2.4 w ith j replaced by j 1. W e therefore deduce the result of Lem m a 4.2.5 from our analysis led in the proofofLem m a 4.2.4.
L em m a 4.2.6. Ifs1,s22 N ,1 j;p N ,(l1;:::;lk)2 f1;:::;N gk then we have
R e qj(Im Fl1:::Im Flk(Im Fp)
s1X ;Im F
l1:::Im Flk(Im Fp)
s2X ) R e qj(Im Fl1:::Im Flk(Im Fp)
s1X )12R e q
j(Im Fl1:::Im Flk(Im Fp)
and
r R e qj(Im Fl1:::Im Flk(Im Fp)
s1X ;Im F l1:::Im Flk(Im Fp) s2X ) . R e qj(Im Fl 1:::Im Flk(Im Fp) s1X )12 + R e qj(Im Fl 1:::Im Flk(Im Fp) s2X )12 . rk+ m ax(s 1;s2)(X ) 1 2:
Proof of Lem m a 4.2.6. B y reason ofsym m etry,w e can assum e in the follow ing that s1 s2. R ecalling that the quadratic form R e qjis non-negative,the rst estim ate is
a direct consequence of(3.6) and the C auchy-Schw arz inequality. A bout the second estim ate,w e recallfrom (4.12) that the H am ilton m ap ofthe quadratic form
R e qj(Im Fl1:::Im Flk(Im Fp)
s1X ;Im F l1:::Im Flk(Im Fp) s2X ); is 1 2( 1) k+ s1(Im F
p)s1Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s2
+ 1 2( 1)
k+ s2(Im F
p)s2Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s1: A direct com putation as in (3.18) of[16]show s that
r R e qj(Im Fl1:::Im Flk(Im Fp)
s1X ;Im F
l1:::Im Flk(Im Fp)
s2X ) (4.20)
= ( 1)k+ s1+ 1 (Im F
p)s1Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s2
+ ( 1)k+ s2+ 1 (Im F
p)s2Im Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s1
w here
= 0 In In 0
:
T he notation In stands here for the n by n identity m atrix. W e deduce from (2.16)
and (4.20) that for any s 2 N ,
j(Im Fp)sIm Flk:::Im Fl1R e FjIm Fl1:::Im Flk(Im Fp)
s X j (4.21) . r R e qj(Im Fl 1:::Im Flk(Im Fp) sX ) . R e qj(Im Fl 1:::Im Flk(Im Fp) s X )21:
B y using tw ice the estim ate (4.21) w ith respectively X and (Im Fp)s2 s1X ,and the
index s = s1,w e deduce from (3.6)and (4.20)the second estim ate in Lem m a 4.2.6.
L em m a 4.2.7. C onsider the quadratic form ~rm j 1;pde ned in (3.4) and (3.5) then
for any 0 j m 2 and 1 p N , ~ rm j 1;p(X ) rm j 1(X ) 2m 2j 2 2m 2j 1 2 S 1;rm j 1(X ) 2m 2j 3 2m 2j 1dX2 ; if is any open set where
rm j 1(X )& hX i 2 (2m 2j 1 ) 2m + 1 and rm j 2(X ). rm j 1(X ) 2m 2j 3 2m 2j 1; which im plies in particular that
ProofofLem m a 4.2.7. Since from Lem m a 4.2.6, j~rm j 1;p(X )j. rm j 1(X ) 2m 2j 2 2m 2j 1 and jr ~rm j 1;p(X )j. rm j 1(X ) 1 2 + r m j 2(X ) 1 2 . rm j 1(X ) 1 2;
on ,w e get that the quadratic form ~rm j 1;pbelongs to the sym bolclass
S rm j 1(X ) 2m 2j 2 2m 2j 1;r m j 1(X ) 2m 2j 3 2m 2j 1dX2 : O ne can then deduce the result ofLem m a 4.2.7 from Lem m a 4.2.3.
W hen adding a large param eter j 1 in the description of the open set , a
straightforw ard adaptation of the proof of the previous lem m a gives the follow ing L1 ( ) estim ate w ith respect to this param eter.
L em m a 4.2.8. C onsider the quadratic form ~rm j 1;pde ned in (3.4) and (3.5) then
for any 0 j m 2 and 1 p N , rm j 1(X ) 2m 2j 2 2m 2j 1r~ m j 1;p(X ) L1 ( ). 1 2 j ;
if is any open set where
rm j 1(X )& hX i 2 (2m 2j 1 ) 2m + 1 and rm j 2(X ). j1rm j 1(X ) 2m 2j 3 2m 2j 1; with j 1.
In the follow ing lem m as,w e shallcarefully study the dependence ofthe estim ates w ith respect to the large param eter j 1.
L em m a 4.2.9. For any 0 j m 2,we have for allX 2 ,
N X p= 1 HIm qpr~m j 1;p(X ) rm j 1(X ) 2m 2j 2 2m 2j 1 2rm j 1(X ) 1 2m 2j 1 . 1 2 j rm j 1(X ) 1 2m 2j 1;
if is any open set where
