Table Of ContentSUBELLIPTIC ESTIMATES FOR OVERDETERMINED SYSTEMS
OF QUADRATIC DIFFERENTIAL OPERATORS
KARELPRAVDA-STAROV
Abstract. We prove global subelliptic estimates for systems of quadratic dif-
0 ferential operators. Quadratic differential operators areoperators defined inthe
1 Weylquantizationbycomplex-valuedquadraticsymbols. Inapreviouswork,we
0 pointedouttheexistenceofaparticularlinearsubvectorspaceinthephasespace
2 intrinsicallyassociated totheirWeyl symbols,calledsingularspace, whichrules
anumber of fairlygeneral properties of non-ellipticquadratic operators. About
n thesubellipticproperties oftheseoperators, weestablishedthatquadraticoper-
a ators with zero singular spaces fulfill global subelliptic estimates with a loss of
J
derivativesdependingoncertainalgebraicpropertiesoftheHamiltonmapsasso-
2 ciatedtotheirWeylsymbols. Thepurposeofthepresentworkistoprovesimilar
1 global subelliptic estimates for overdetermined systems of quadratic operators.
We establish here a simple criterion for the subellipticity of these systems giv-
] inganexplicitmeasureofthelossofderivatives andhighlightingthenon-trivial
P
interactions playedbythedifferent operators composingthosesystems.
A
.
h
t
a 1. Introduction
m
[ 1.1. Miscellaneous facts about quadratic differential operators. In a recent
joint work with M. Hitrik, we investigatedspectral and semigroup properties of non-
1
elliptic quadratic operators. Quadratic operators are pseudodifferential operators
v
1 defined in the Weyl quantization
3 1 x+y
19 (1.1) qw(x,Dx)u(x)= (2π)n ZR2nei(x−y).ξq(cid:16) 2 ,ξ(cid:17)u(y)dydξ,
. by some symbols q(x,ξ), with (x,ξ) ∈ Rn ×Rn and n ∈ N∗, which are complex-
1
0 valued quadratic forms. Since these symbols are quadratic forms, the corresponding
0 operators in (1.1) are in fact differential operators. Indeed, the Weyl quantization
1 of the quadratic symbol xαξβ, with (α,β) ∈ N2n and |α+β| = 2, is the differential
:
v operator
i xαDβ +Dβxα
X x x , D =i−1∂ .
x x
2
r One can also notice that quadratic differential operators are a priori formally non-
a
selfadjoint since their Weyl symbols in (1.1) are complex-valued.
Considering quadratic operators whose Weyl symbols have real parts with a sign,
say here, Weyl symbols with non-negative real parts
(1.2) Re q ≥0,
we pointed out in [2] the existence of a particular linear subvector space S in the
phase space Rn × Rn intrinsically associated to their Weyl symbols q(x,ξ), called
x ξ
singular space, which seems to play a basic rôle in the understanding of a number
2000 Mathematics Subject Classification. Primary: 35B65;Secondary: 35N10.
Key words and phrases. Quadratic differential operators, overdetermined systems, subelliptic
estimates,singularspace,Wickquantization.
1
2 KARELPRAVDA-STAROV
of fairly general properties of non-elliptic quadratic operators. More specifically, we
first proved in [2] (Theorem 1.2.1) that when the singular space S has a symplectic
structure then the associated heat equation
∂u
(t,x)+qw(x,D )u(t,x)=0
(1.3) ∂t x
( u(t,·)| =u ∈L2(Rn),
t=0 0
is smoothing in every direction of the orthogonalcomplement Sσ⊥ of S with respect
to the canonical symplectic form σ on R2n,
(1.4) σ (x,ξ),(y,η) =ξ.y−x.η, (x,ξ)∈R2n,(y,η)∈R2n,
thatis,that,if(x(cid:0)′,ξ′)aresom(cid:1)elinearsymplecticcoordinatesonthe symplecticspace
Sσ⊥ then we have for all t>0, N ∈N and u∈L2(Rn),
(1.5) (1+|x′|2+|ξ′|2)N we−tqw(x,Dx)u∈L2(Rn).
We also proved in (cid:0)[2] (See Section 1.4(cid:1).1 and Theorem 1.2.2) that when the Weyl
symbol q of a quadratic operator fulfills (1.2) and an assumption of partial ellipticity
on its singular space S in the sense that
(1.6) (x,ξ)∈S, q(x,ξ)=0⇒(x,ξ)=0,
then this singular space always has a symplectic structure and the spectrum of the
operator qw(x,D ) is only composed of a countable number of eigenvalues of finite
x
multiplicity,withasimilarstructureastheoneestablishedbyJ.Sjöstrandforelliptic
quadratic operators in his classical work [18]. Elliptic quadratic operators are the
quadratic operators whose symbols satisfy the condition of global ellipticity
(x,ξ)∈R2n, q(x,ξ)=0⇒(x,ξ)=0,
onthewholephasespaceR2n. Letusrecallherethatspectralpropertiesofquadratic
operatorsareplaying a basic rôle in the analysisof partialdifferential operatorswith
double characteristics. This is particularly the case in some general results about
hypoellipticity. We refer the reader to [4], [18], as well as Chapter 22 of [5] together
with all the references given there.
In the present paper, we are interested in studying the subelliptic properties of
overdetermined systems of non-selfadjoint quadratic operators. This work can be
viewedasanaturalextensionoftheanalysisledin[17],inwhichweinvestigatedinthe
scalarcase the rôle playedby the singular space when studying subelliptic properties
of quadratic operators. We aimhere at showing how the analysisled in this previous
work can be pushed further when dealing with overdetermined systems of quadratic
operators. We shall see that the techniques introduced in [17] are sufficiently robust
to be extended to the system case and that they turn out to be sufficiently sharp
to highlight phenomena of non-trivial interactions between the different quadratic
operators composing a system. In this paper, we shall therefore be interested in
establishing some global subelliptic estimates of the type
N
(1.7) h(x,ξ)i2(1−δ) wu . kqw(x,D )uk +kuk ,
L2 j x L2 L2
j=1
(cid:13)(cid:0) (cid:1) (cid:13) X
(cid:13) (cid:13)
whereh(x,ξ)i=(1+|x|2+|ξ|2)1/2andδ >0;forsystemsoftheN quadraticoperators
qw(x,D ), with 1 ≤ j ≤ N. The positive parameter δ > 0 appearing in (1.7) will
j x
measure the loss of derivatives with respect to the elliptic case (case δ = 0). As
in the scalar case studied in [17], we aim at giving a simple criterion for systems
of quadratic operators ensuring that a global subelliptic estimate of the type (1.7)
SUBELLIPTIC ESTIMATES FOR SYSTEMS OF QUADRATIC OPERATORS 3
holds together with an explicit characterization of the associated loss of derivatives.
This loss of derivativesδ will be characterizedin terms of algebraicconditions on the
Hamiltonmapsassociatedtothe Weylsymbolsofthe quadraticoperatorscomposing
the system.
In this work, we study the subellipticity of overdetermined systems in the sense
given by P. Bolley, J. Camus and J. Nourrigat in [1] (Theorem 1.1). In this semi-
nalwork,theseauthorsstudythe microlocalsubellipticity ofoverdeterminedsystems
of pseudodifferential operators. More specifically, they establish the subellipticity of
systems composed of pseudodifferential operators with real principal symbols satis-
fying the Hörmander-Kohn condition. More generally, in the case of overdetermined
systems of non-selfadjoint pseudodifferential operators, the greatest achievements up
to now were obtained by J. Nourrigat in [8] and [9]. In these two major works,
J.Nourrigatstudiesthe microlocalsubellipticity andmaximalhypoellipticityforsys-
temsofnon-selfadjointpseudodifferentialoperatorsbythe meanofrepresentationsof
nilpotentgroups. Weshallexplaininthefollowinghowthealgebraicconditiononthe
Hamiltonmaps(1.18)inTheorem1.2.1relateswiththeseformerresults. Morespecif-
ically, we shall comment on its link with the Hörmander-Kohn condition appearing
in [1] (Theorem 1.1).
Beforegivingtheprecisestatementofourmainresult,weshallrecallmiscellaneous
notationsaboutquadraticdifferentialoperatorsandtheresultsobtainedinthescalar
case. In all the following, we consider
q :Rn×Rn → C
j x ξ
(x,ξ) 7→ q (x,ξ),
j
with 1≤j ≤N, N complex-valued quadratic forms with non-negative real parts
(1.8) Re q (x,ξ)≥0, (x,ξ)∈R2n,n∈N∗.
j
We knowfrom[6](p.425)thatthe maximalclosedrealizationofaquadraticoperator
qw(x,D ) whose Weyl symbol has a non-negative real part, i.e., the operator on
x
L2(Rn) with the domain
D(q)= u∈L2(Rn):qw(x,D )u∈L2(Rn) ,
x
coincides with the graph clo(cid:8)sure of its restriction to S(Rn), (cid:9)
qw(x,D ):S(Rn)→S(Rn).
x
Associatedtoaquadraticsymbolq is thenumericalrangeΣ(q)definedasthe closure
in the complex plane of all its values
(1.9) Σ(q)=q(Rn×Rn).
x ξ
Wealsorecallfrom[5]thattheHamiltonmapF ∈M (C)associatedtothequadratic
2n
form q is the map uniquely defined by the identity
(1.10) q (x,ξ);(y,η) =σ (x,ξ),F(y,η) , (x,ξ)∈R2n,(y,η)∈R2n,
where q ·;· sta(cid:0)nds for the(cid:1)polar(cid:0)ized form asso(cid:1)ciated to the quadratic form q. It
directly follows from the definition of the Hamilton map F that its real part and its
(cid:0) (cid:1)
imaginary part
1 1
Re F = (F +F) and Im F = (F −F),
2 2i
aretheHamiltonmapsassociatedtothequadraticformsRe q andIm q,respectively.
Onecanalsonoticefrom(1.10)thatanHamiltonmapisalwaysskew-symmetricwith
4 KARELPRAVDA-STAROV
respect to σ. This is just a consequence of the properties of skew-symmetry of the
symplectic form and symmetry of the polarized form
(1.11) ∀X,Y ∈R2n, σ(X,FY)=q(X;Y)=q(Y;X)=σ(Y,FX)=−σ(FX,Y).
Associated to the symbol q, we defined in [2] its singular space S as the following
intersection of kernels
+∞
(1.12) S = Ker Re F(Im F)j ∩R2n,
(cid:16)j\=0 (cid:2) (cid:3)(cid:17)
where the notations Re F and Im F stand respectively for the real part and the
imaginarypartoftheHamiltonmapassociatedtoq. NoticethattheCayley-Hamilton
theorem applied to Im F shows that
(Im F)kX ∈Vect X,...,(Im F)2n−1X , X ∈R2n, k ∈N,
where Vect X,...,(Im F)2n−1X(cid:0) is the vector spac(cid:1)e spanned by the vectors X, ...,
(Im F)2n−1X;andthereforethesingularspaceisactuallyequaltothefollowingfinite
(cid:0) (cid:1)
intersection of the kernels
2n−1
(1.13) S = Ker Re F(Im F)j ∩R2n.
(cid:16) j\=0 (cid:2) (cid:3)(cid:17)
Considering a quadratic operator qw(x,D ) whose Weyl symbol
x
q :Rn×Rn → C
x ξ
(x,ξ) 7→ q(x,ξ),
has a non-negative real part, Re q ≥ 0, we established in [17] (Theorem 1.2.1) that
whenitssingularspaceSisreducedto{0},theoperatorqw(x,D )fulfillsthefollowing
x
global subelliptic estimate
(1.14) ∃C >0,∀u∈D(q), h(x,ξ)i2/(2k0+1) wu ≤C kqw(x,D )uk +kuk ,
L2 x L2 L2
where k0 stands for the sma(cid:13)l(cid:0)lest non-negative(cid:1)int(cid:13)eger, 0≤(cid:0)k0 ≤2n−1, suchthat t(cid:1)he
intersectionofthefollowing(cid:13)k +1kernelswithth(cid:13)ephasespaceR2n isreducedto{0},
0
k0
(1.15) Ker Re F(Im F)j ∩R2n ={0}.
(cid:16)j\=0 (cid:2) (cid:3)(cid:17)
Notice that the loss of derivatives δ = 2k /(2k +1), appearing in the subelliptic
0 0
estimate (1.14) directly depends on the non-negative integer k characterized by the
0
algebraic condition (1.15).
More generally, considering a quadratic operator qw(x,D ) whose Weyl symbol
x
has a non-negative real part with a singular space S which may differ from {0}, but
does have a symplectic structure in the sense that the restriction of the canonical
symplectic formσ to S is non-degenerate,weprovedin[17] (Theorem1.2.2)that the
operator qw(x,D ) is subelliptic in any direction of the orthogonal complement Sσ⊥
x
ofthesingularspacewithrespecttothesymplecticformσ inthesensethat,if(x′,ξ′)
are some linear symplectic coordinates on Sσ⊥ then we have
∃C >0,∀u∈D(q), h(x′,ξ′)i2/(2k0+1) wu ≤C kqw(x,D )uk +kuk ,
L2 x L2 L2
with h(x′,ξ′)i = (1+|x(cid:13)′|(cid:0)2+|ξ′|2)1/2, wher(cid:1)e k(cid:13)stands(cid:0)for the smallest non-nega(cid:1)tive
(cid:13) (cid:13)0
integer, 0≤k ≤2n−1, such that
0
k0
(1.16) S = Ker Re F(Im F)j ∩R2n.
(cid:16)j\=0 (cid:2) (cid:3)(cid:17)
SUBELLIPTIC ESTIMATES FOR SYSTEMS OF QUADRATIC OPERATORS 5
Finally, we end these few recalls by underlining that the assumption about the sym-
plectic structure of the singular space is always fulfilled by any quadratic symbol q
which satisfies the assumption of partial ellipticity on its singular space S,
(x,ξ)∈S, q(x,ξ)=0⇒(x,ξ)=0.
We refer the reader to Section 1.4.1 in [2] for a proof of this fact.
1.2. Statementofthemainresult. ConsideringasystemofN quadraticoperators
qw(x,D ), 1≤j ≤N, whose Weyl symbols q have all non-negative real parts
j x j
(1.17) Re q (x,ξ)≥0, (x,ξ)∈R2n, n∈N∗,
j
anddenotingbyF theirassociatedHamiltonmaps,themainresultcontainedinthis
j
article is the following:
Theorem 1.2.1. Consider a system of N quadratic operators qw(x,D ), 1≤j ≤N,
j x
satisfying (1.17). If there exists k ∈N such that
0
(1.18) Ker(Re F Im F ...Im F ) ∩R2n ={0},
j l1 lk
(cid:16)0≤\k≤k0 j=1\,...,N, (cid:17)
(l1,...,lk)∈{1,...,N}k
then this overdetermined system of quadratic operators is subelliptic with a loss of
δ = 2k /(2k + 1) derivatives, that is, that there exists C > 0 such that for all
0 0
u∈D(q )∩...∩D(q ),
1 N
N
(1.19) h(x,ξ)i2/(2k0+1) wu L2 ≤C kqjw(x,Dx)ukL2 +kukL2 ,
(cid:13)(cid:0) (cid:1) (cid:13) (cid:16)Xj=1 (cid:17)
(cid:13) (cid:13)
with h(x,ξ)i=(1+|x|2+|ξ|2)1/2.
Remark. Let us make clear that the intersection of kernels
Ker(Re F Im F ...Im F ),
j l1 lk
j=1,...,N,
\
(l1,...,lk)∈{1,...,N}k
is to be understood as
Ker Re F ,
j
j=1,...,N
\
when k =0.
1.3. Example of a subelliptic system of quadratic operators. The following
example of subelliptic system of quadratic operators shows that Theorem 1.2.1 re-
allyhighlightsnewnon-trivialinteractionphenomenabetweenthedifferentoperators
composing a system, which cannot be derived from the result of subellipticity known
in the scalar case (Theorem 1.2.1 in [17]). Indeed, define the quadratic forms
q (x,ξ)=x2+ξ2+i(ξ2+x ξ ) and q˜(x,ξ)=x2+ξ2+i(ξ2+ξ ξ ),
j 1 1 1 j+1 1 j 1 1 1 j+1 1
for 1 ≤ j ≤ n−1 and (x,ξ) ∈ R2n, with n ≥ 2. A direct computation using (1.10)
and (1.13) shows that the singular space of the quadratic form
n−1
(λ q +λ˜ q˜),
j j j j
j=1
X
6 KARELPRAVDA-STAROV
for some real numbers λ ,λ˜ verifying
j j
n−1
(λ +λ˜ )>0;
j j
j=1
X
is given by
n−1
S = (x,ξ)∈R2n :x =ξ = (λ x +λ˜ ξ )=0 ,
1 1 j j+1 j j+1
n Xj=1 o
which is always a non-zero subvector space. It then follows that one cannot deduce
any result about the subellipticity of the scalar operator
n−1
(λ qw(x,D )+λ˜ q˜w(x,D )),
j j x j j x
j=1
X
inordertogetthesubellipticityoftheoverdeterminedsystemcomposedbythe2n−2
operators qw(x,D ) and q˜w(x,D ), for 1 ≤ j ≤ n−1. Nevertheless, by denoting
j x j x
respectively F and F˜ the Hamilton maps of the quadratic forms q and q˜, another
j j j j
direct computation using (1.10) shows that
Ker Re F ∩Ker(Re F Im F )∩R2n ={(x,ξ)∈R2n :x =ξ =x =0}
j j j 1 1 j+1
and
Ker Re F˜ ∩Ker(Re F˜ Im F˜ )∩R2n ={(x,ξ)∈R2n :x =ξ =ξ =0}.
j j j 1 1 j+1
One can then deduce from Theorem 1.2.1 the following global subelliptic estimate
with a loss of 2/3 derivatives
n−1
h(x,ξ)i2/3 wu . kqw(x,D )uk +kq˜w(x,D )uk +kuk .
L2 j x L2 j x L2 L2
j=1
(cid:13)(cid:0) (cid:1) (cid:13) X(cid:0) (cid:1)
(cid:13) (cid:13)
Ofcourse,Theorem1.2.1canhighlightmorecomplexinteractionsbetweenthe differ-
ent operators composing the system when we consider operators with different real
parts.
1.4. Comments on the condition for subellipticity. Theorem1.2.1gives a very
explicit and simple algebraic condition on the Hamilton maps of quadratic opera-
tors ensuring the subellipticity of the system. Let us notice that this condition is
very easy to handle and allows to directly measure the associated loss of derivatives
by a straightforward computation. We shall now explain how this is related to the
Hörmander-Kohncondition. Recallfrom[1](Theorem1.1)thattheHörmander-Kohn
condition for microlocal subellipticity of overdetermined systems of pseudodifferen-
tial operators with real principal symbols; reads as the existence of an elliptic iter-
ated commutator of the operators composing the system. In the case of a system
of non-selfadjoint quadratic operators (qw) , if we assume in addition that this
j 1≤j≤N
system is maximal hypoelliptic1, the natural condition becomes to ask the ellipticity
of an iterated commutator of the real parts ((Re q )w) and imaginary parts
j 1≤j≤N
((Im q )w) oftheoperatorscomposingthesystem. Comingbacktoourspecific
j 1≤j≤N
1. Wereferto[8]and[9]forconditionsandgeneralresultsofmaximalhypoellipticityforoverde-
terminedsystemsofnon-selfadjointpseudodifferential operators.
SUBELLIPTIC ESTIMATES FOR SYSTEMS OF QUADRATIC OPERATORS 7
condition for subellipticity (1.18), we first notice that in the scalar case, it reads as
the existence of a non-negative integer k such that
0
k0
Ker[Re F(Im F)j] ∩R2n ={0},
(cid:16)j\=0 (cid:17)
with F standing for the Hamilton map of the unique operator qw(x,D ) composing
x
the system. As recalledin [17] (Section1.2),this conditionimplies that, for anynon-
zero point in the phase space X ∈ R2n, we can find a non-negative integer k such
0
that
∀ 0≤j ≤2k−1, Hj Re q(X )=0 and H2k Re q(X )6=0,
Imq 0 Imq 0
where H stands for the Hamilton vector field of Im q,
Imq
∂Im q ∂ ∂Im q ∂
H = · − · .
Imq
∂ξ ∂x ∂x ∂ξ
This shows that the 2kth iterated commutator
[Im qw,[Im qw,[...,[Im qw,Re qw]]]...]=(−1)k(H2k Re q)w,
Imq
with exactly 2k terms Im qw in left-hand-side of the above formula; is elliptic at X ;
0
and underlines the intimate link between (1.18) and the Hörmander-Kohn condition
inthe scalarcase. In the systemcase,the situationis more complicatedandthis link
is less obvious to highlight explicitly. More specifically, we shall see in this case that
the algebraic condition (1.18) implies that the quadratic form
k0
Re q (Im F ...Im F X),
j l1 lk
k=0 j=1,...,N,
X X
(l1,...,lk)∈{1,...,N}k
is positive definite. This property implies that for any non-zero point X ∈R2n, one
0
can find k ∈N, j ∈{1,...,N} and (l ,...,l )∈{1,...,N}k such that
1 k
Re q (Im F ...Im F X )>0.
j l1 lk 0
By considering the minimal non-negative integer k with this property and using the
same argumentsas the ones developedin [2] (p.820-822),one canactually check that
any iterated commutator of order less or equal to 2k−1, that is,
[P ,[P ,[P ,[...,[P ,P ]...]]]],
1 2 3 r r+1
with r ≤ 2k−1, P = Re qw or P = Im qw; and where at least one P is equal to
l s1 l s2 l0
Re qw, for 1 ≤ s ,s ,s ≤ N; are not elliptic at X . One can also check that the
s3 1 2 3 0
non-zero term
Re q (Im F ...Im F X )>0,
j l1 lk 0
actually appears when expanding the Weyl symbol at X of the 2kth iterated com-
0
mutator
[Im qw,[Im qw,[Im qw ,[Im qw ,[...,[Im qw,[Im qw,Re qw]]]...]
lk lk lk−1 lk−1 l1 l1 j
=(−1)k(H2 ...H2 Re q )w.
Imqlk Imql1 j
However, contrary to the scalar case, there may be also other non-zero terms in this
expansion;anditisnotreallyclearifthis naturalcommutatorassociatedtothe term
Re q (Im F ...Im F X ),
j l1 lk 0
is actually elliptic at X ,
0
?
H2 ...H2 Re q (X )6=0.
Imqlk Imql1 j 0
8 KARELPRAVDA-STAROV
Thoughitmaybedifficulttodetermineexactlyateachpointwhichspecificcommuta-
toriselliptic, itisverylikelythatcondition(1.18)ensuresthatthe Hörmander-Kohn
condition is fulfilled at any non-zero point of the phase space; and that these associ-
ated elliptic commutatorsare all of order less or equal to 2k . It is actually what the
0
loss of derivatives appearing in the estimate (1.19) suggests; and this in agreement
withtheoptimallossofderivativesobtainedin[1](Theorem1.1)for2k commutators
0
1 2k
0
δ =1− = ;
2k +1 2k +1
0 0
since we measure the loss of derivatives δ with respect to the elliptic case as
N
(Λ2(1−δ))wu . kqw(x,D )uk +kuk ,
L2 j x L2 L2
j=1
(cid:13) (cid:13) X
(cid:13) (cid:13)
with Λ2 = h(x,ξ)i2, because quadratic operators have their Weyl symbols in the
symbol class S(Λ2,Λ−2dX2) whose gain is Λ2.
Becauseofthe simplicityofits assumptions,Theorem1.2.1providesaneatsetting
forprovingglobalsubelliptic estimatesforsystemsofquadraticoperators. Itispossi-
ble that some of these global subelliptic estimates for systems of quadratic operators
mayalsobederivedfromtheresultsofmicrolocalsubellipticity andmaximalhypoel-
lipticity proved in [1], [8] and [9]. However, given a particular system of quadratic
operators,one can notice that only checking the Hörmander-Kohnconditionin every
non-zero point turns out to be quite difficult to do in practice. The same comment
appliesforcheckingthemaximalhypoellipticityofthesystem. Anotherinterestofthe
approachwearedevelopingherecomesfromthe factthatthe proofofTheorem1.2.1
is purely analytic and does not require any techniques of representationsof nilpotent
groups as in [8] or [9]. Moreover, despite its length, the proof provided here only
involves fairly elementary arguments whose complexity has no degree of comparison
with the analysis led in [8] and [9].
Finally,letus end this introductionbymentioning thatthis resultofsubellipticity
forsystemsofquadraticoperatorsmaybroadennewperspectivesintheunderstanding
ofoverdeterminedsystemsofpseudodifferentialoperatorswithdoublecharacteristics;
andthattheconstructionoftheweightfunctionsinProposition2.0.1maybeoffurther
interest and direct use in future analysis of doubly characteristic problems. In the
scalar case, this construction of the weight function specific to the structure of the
doublecharacteristicsobtainedin[17](Proposition2.0.1)hasalreadyallowedtoderive
in [3] the precise asymptotics for the resolvent norm of certain class of semiclassical
pseudodifferential operators in a neighborhood of the doubly characteristic set. On
the other hand, this deeper understanding of non-trivial interactions between the
different quadratic operators composing overdetermined systems may also give hints
on how to analyze the more complex case of N by N systems of quadratic operators,
which is a topic of current interest. On that subject, we refer the reader to the
series of recent works on non-commutative harmonic oscillators by A. Parmeggiani
and M. Wakayama in [10], [11], [12], [13], [14] and [15].
2. Proof of Theorem 1.2.1
In the following, we shall use the notation S m(X)r,m(X)−2sdX2 , where Ω is
Ω
an open set in R2n, r,s∈R and m∈C∞(Ω,R∗), to stand for the class of symbols a
+ (cid:0) (cid:1)
verifying
a∈C∞(Ω), ∀α∈N2n,∃C >0, |∂αa(X)|≤C m(X)r−s|α|, X ∈Ω.
α X α
SUBELLIPTIC ESTIMATES FOR SYSTEMS OF QUADRATIC OPERATORS 9
InthecasewhereΩ=R2n,weshalldroptheindexΩforsimplicity. Weshallalsouse
the notations f .g and f ∼g, on Ω, for respectively the estimates ∃C >0, f ≤Cg
and, f .g and g .f, on Ω.
TheproofofTheorem1.2.1willrelyonthe followingkeyproposition. Considering
for 1≤j ≤N,
q :Rn×Rn → C
j x ξ
(x,ξ) 7→ q (x,ξ),
j
with n∈N∗, N complex-valued quadratic forms with non-negative real parts
(2.1) Re q (x,ξ)≥0, (x,ξ)∈R2n, 1≤j ≤N,
j
we assume that there exist a positive integer m∈N∗ andanopen set Ω in R2n such
0
that the following sum of non-negative quadratic forms satisfies
m
(2.2) ∃c >0,∀X ∈Ω , Re q (Im F ...Im F X)≥c |X|2,
0 0 j l1 lk 0
k=0 j=1,...,N,
X X
(l1,...,lk)∈{1,...,N}k
where the notation Im F stands for the imaginary part of the Hamilton map F as-
j j
sociated to the quadratic form q . Under this assumption, one can then extend the
j
construction of the bounded weight function done in the scalar case in [17] (Proposi-
tion 2.0.1) to the system case as follows:
Proposition 2.0.1. If (q ) are N complex-valued quadratic forms on R2n ver-
j 1≤j≤N
ifying (2.1) and (2.2) then there exist N real-valued weight functions
gj ∈SΩ0 1,hXi−2m2+1dX2 , 1≤j ≤N,
such that (cid:0) (cid:1)
N
2
(2.3) ∃c,c1,...,cN >0,∀X ∈Ω0, 1+ Re qj(X)+cjHImqj gj(X) ≥chXi2m+1,
j=1
X(cid:0) (cid:1)
where the notation H stands for the Hamilton vector field of the imaginary part
Imqj
of q .
j
As in [17], the constructionof these weight functions will be really the core of this
work. Thisconstructionwillbeanadaptationtothesystemcaseoftheoneperformed
in the scalar case.
To check that we can actually deduce Theorem 1.2.1 from Proposition 2.0.1, we
begin by noticing, as in [17], that the assumptions of Theorem 1.2.1 imply that the
following sum of non-negative quadratic forms
k0
(2.4) ∃c >0, r(X)= Re q (Im F ...Im F X)≥c |X|2,
0 j l1 lk 0
k=0 j=1,...,N,
X X
(l1,...,lk)∈{1,...,N}k
is actually a positive definite quadratic form. Let us indeed consider X ∈ R2n such
0
that r(X ) = 0. Then, the non-negativity of quadratic forms Re q induces that for
0 j
all 0≤k ≤k , j =1,...,N and (l ,...,l )∈{1,...,N}k,
0 1 k
(2.5) Re q (Im F ...Im F X )=0.
j l1 lk 0
10 KARELPRAVDA-STAROV
By denoting Re q (X;Y) the polar form associated to Re q , we deduce from the
j j
Cauchy-Schwarzinequality, (1.10) and (2.5) that for all Y ∈R2n,
|Re q (Y;Im F ...Im F X )|2 = |σ(Y,Re F Im F ...Im F X )|2
j l1 lk 0 j l1 lk 0
≤ Re q (Y) Re q (Im F ...Im F X )=0.
j j l1 lk 0
It follows that for all Y ∈R2n,
σ(Y,Re F Im F ...Im F X )=0,
j l1 lk 0
which implies that for all 0≤k≤k , j =1,...,N and (l ,...,l )∈{1,...,N}k,
0 1 k
(2.6) Re F Im F ...Im F X =0,
j l1 lk 0
since σ is non-degenerate. We finally deduce (2.4) from the assumption (1.18).
In the case where k =0, we notice that the quadratic form
0
q =q +...+q ,
1 N
has a positive definite real part. This implies in particular that q is elliptic on R2n.
One can therefore directly deduce from classical results about elliptic quadratic dif-
ferential operators proved in [18] (See Theorem 3.5 in [18] or comments about the
elliptic case in Theorem 1.2.1 in [17]), the natural elliptic a priori estimate
∃C >0,∀u∈D(q )∩...∩D(q ), h(x,ξ)i2 wu ≤C(kqw(x,D )uk +kuk ),
1 N L2 x L2 L2
which easily implies (1.19). (cid:13)(cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
We can therefore assume in the following that k ≥ 1 and find from Proposi-
0
tion 2.0.1 some real-valued weight functions
(2.7) gj ∈S 1,hXi−2k02+1dX2 , 1≤j ≤N,
such that (cid:0) (cid:1)
N
(2.8) ∃c,c1,...,cN >0,∀X ∈R2n, 1+ Re qj(X)+cjHImqj gj(X) ≥chXi2k02+1.
j=1
X(cid:0) (cid:1)
For0<ε≤1,weconsiderthemultipliersdefinedintheWickquantizationbysymbols
1−εc g . We recall that the definition of the Wick quantization and some elements
j j
of Wick calculus are recalled in Section 4.1. It follows from (2.7), (4.4), (4.7), (4.8)
and the Cauchy-Schwarzinequality that
N N
(2.9) Re qWicku,(1−εc g )Wicku = Re (1−εc g )WickqWick u,u
j j j j j j
j=1 j=1
X (cid:0) (cid:1) X(cid:0) (cid:0) (cid:1) (cid:1)
N N N
≤ k1−εc g k kqWickuk kuk . kqWickuk2 +kuk2 . kq˜wuk2 +kuk2 ,
j j L∞ j L2 L2 j L2 L2 j L2 L2
j=1 j=1 j=1
X X X
where
ξ
(2.10) q˜(x,ξ)=q x, ,
j j
2π
(cid:16) (cid:17)
because the operators (1−εc g )Wick whose Wick symbol are real-valued, are for-
j j
mally selfadjoint. Indeed, symbols r(q ) defined in (4.8) are here just some constants
j
since q are quadratic forms. The factor 2π in (2.10) comes from the difference of
j
normalizationschosenbetween(1.1)and(4.9)(SeeremarkinSection4.1). Sincefrom
(4.10),
ε ε Wick
(1−εc g )WickqWick = (1−εc g )q + c ∇g .∇q − c {g ,q } +S ,
j j j j j j 4π j j j 4iπ j j j j
h i
