A CLASS OF COUNTEREXAMPLES TO
THE FEFFERMAN-PHONG INEQUALITY
FOR SYSTEMS
ALBERTO PARMEGGIANI
Abstract. We give here a class of counterexamples to the FeffermanPhong inequality for systems of pseudodifferential operators, which contains Brummelhuis’s one as a particular case. The main ingredient in
the proof is the use of “localized operators” associated with the system,
and Hörmander’s example of a positive-semidefinite matrix whose Weyl
quantization is not non-negative. For the considered class the Sharp
Gårding inequality cannot be improved.
1. Introduction
The Fefferman-Phong inequality ([5]) states that if a = a(x, ξ) ≥ 0 is a
scalar symbol in the class S 2 (Rn × Rn ), then the relative pseudodifferential
operator (ψdo in the sequel) a(x, Dx ) is bounded from below in L2 , that is,
there exists a constant C > 0 such that
(F P )
Re (a(x, Dx )u, u) ≥ −C ||u||20 , ∀u ∈ C0∞ (Rn ).
On the other hand, when A = A∗ = a(x, Dx ) is a properly supported
classical ψdo of order m, that is, its (total) symbol a(x, ξ) has an expansion
a(x, ξ) ∼ am (x, ξ) + am−1 (x, ξ) + . . ., where am−j is positively homogeneous
of degree m − j in the fiber variable ξ, one looks for conditions on the
principal symbol am and on the lower order term asm−1 (x, ξ) := am−1 (x, ξ) +
n
iX 2
∂xj ξj am (x, ξ), the sub-principal symbol of A, that ensure lower bound
2
j=1
estimates for the L2 -quadratic form (Au, u), u ∈ C0∞ , associated with A. The
reason why one looks at am and asm−1 is that am is invariantly defined
on T ∗ Rn \ 0, and asm−1 (x, ξ) is invariantly defined where am vanishes to
second order (which is always the case when am ≥ 0).
When am ≥ 0 on T ∗ Rn \ 0, one has the celebrated Sharp Gårding inequality (proved by L.Hörmander) which states: For every compact K ⊂ Rn there
exists CK > 0 such that
(SG)
(Au, u) ≥ −CK ||u||2(m−1)/2 , ∀u ∈ C0∞ (K).
I wish to thank the Institut Mittag-Leffler for the kind hospitality. I wish also to thank
N.Lerner and D.Tataru for useful discussions.
1
2
ALBERTO PARMEGGIANI
The next step was obtained by A.Melin (see [11]), who proved the following equivalence:
For any given ε > 0 and any given compact K ⊂ Rn there exists Cε,K > 0
such that
(M )
(Au, u) ≥ −ε ||u||2(m−1)/2 − Cε,K ||u||2(m−2)/2 , ∀u ∈ C0∞ (K)
(the so-called Melin inequality);

 am (x, ξ) ≥ 0, ∀(x, ξ) ∈ T ∗ Rn \ 0,
(1)

am (x, ξ) = 0 =⇒ asm−1 (x, ξ) + Tr+ (F (x, ξ)) ≥ 0,
where Tr+ (F (x, ξ)) is the positive trace of the fundamental matrix F (x, ξ)
defined by the identity
hQ(x,ξ) v, wiT(x,ξ) T ∗ Rn = σ(v, F (x, ξ)w), v, w ∈ T(x,ξ) T ∗ Rn ,
Pn
Q(x,ξ) being the Hessian of am /2 at (x, ξ) and with σ =
j ∧ dxj
j=1 dξP
+
∗
n
the canonical symplectic form on T R . Explicitly, Tr (F (x, ξ)) = µ>0 µ
with iµ ∈ Spec(F (x, ξ)).
It is important to remark here that the Hessian Q(x,ξ) and the positive
trace Tr+ (F (x, ξ)) are invariantly defined on the set where am vanishes
to second order. Observe also that in conditions (1) above, no assumption
on the geometry of the characteristic set
Σ := {(x, ξ) ∈ T ∗ Rn \ 0; am (x, ξ) = 0}
is made. In fact, supposing that:
(i) Σ is a smooth sub-manifold of T ∗ Rn \ 0,
(ii)
rank on the connected components of Σ, i.e. Σ 3 ρ 7→
¡ σ has constant
¢
dim Tρ Σ∩Tρ Σσ is locally constant, where Tρ Σσ is the symplectic orthogonal
of Tρ Σ,
(iii) am (x, ξ) vanishes exactly to second order on Σ,
Hörmander proved in [7] the following result: Conditions (1) above is equivalent to
For any given compact K ⊂ Rn there exists CK > 0 such that
(H)
(Au, u) ≥ −CK ||u||2(m−2)/2 , ∀u ∈ C0∞ (K).
Condition (iii) above can also be rewritten as am (x, ξ) ≈ |ξ|m distΣ (x, ξ/|ξ|)2 ,
where distΣ (x, ξ/|ξ|) denotes the distance of (x, ξ/|ξ|) to Σ.
It is important to remark the following fact, that we state already for
matrix-valued ψdo’s.
Remark 1.1. If A(x, Dx ) ∈ OPSm (Rn ; CN ) (the set of classical, properly supported, N × N matrices of classical ψdo’s of order m) has symbol
A(x, ξ) ∼ Am (x, ξ)+Am−1 (x, ξ)+. . . , where Am−j is positively homogeneous
of degree m − j in ξ, and A(x, ξ) = A(x, ξ)∗ (that is, A(x, ξ) is a Hermitian matrix of classical symbols), then (A(x, Dx ) + A(x, Dx )∗ )/2 =:
Re (A)(x, Dx ) = Aw (x, Dx ) (the Weyl-quantization of A(x, ξ)), with the total
A CLASS OF COUNTEREXAMPLES TO THE . . .
3
symbol Ã(x, ξ) of Re (A)(x, Dx ) computed by the formula (see [10], Vol.III)
Ã(x, ξ) = eihDx ,Dξ i/2 A(x, ξ). Hence, at the level of the L2 -quadratic forms,
one always has Re (A(x, Dx )u, u) = (Aw (x, Dx )u, u). We shall hence work
with the Weyl-quantization of a classical symbol. In particular, for the subprincipal symbol Ãs1 of Re (A)(x, Dx ), one has Ãs1 (x, ξ) = A1 (x, ξ).
As regards the vector-valued case, the Fefferman-Phong inequality (F P )
is in general false. In fact, in [2] R.Brummelhuis exhibited an example
(homogeneous of degree 2) for which (F P ) fails. There was already a clue
that problems might arise in vector-valued situations. L.Hörmander, in [8],
showed that the Weyl-quantization of positive-semidefinite matrices may be
not non-negative. The purpose of this paper is to introduce (building upon
the examples of Hörmander and Brummelhuis) a class of counterexamples
to the Fefferman-Phong inequality (F P ) that is geometrically characterized
(see Section 5 below). Indeed, for such a class, only the Sharp Gårding
inequality holds (and no improvement of the kind (W H) below, as suggested
by the result of [12], is possible).
We recall that conditions for Melin’s inequality (M ) to hold in the case of
systems were obtained by Brummelhuis [3], Brummelhuis and Nourrigat [4],
and by Parenti and Parmeggiani [12] (under conditions on the symplectic
type of the characteristic manifold Σ).
As regards Hörmander’s inequality (H) for N ≥ 2, conditions in the
case of systems with double characteristics (that is, Σ = {(x, ξ) ∈ T ∗ Rn \
0; det(am (x, ξ)) = 0} is a manifold on which the symplectic form has constant rank, det(am (x, ξ)) ≈ |ξ|N m distΣ (x, ξ/|ξ)2` , and dimKer am (ρ) = `,
for all ρ ∈ Σ) were given by Hörmander [7] (case of ` = 1), and by Parenti
and Parmeggiani [12] (any ` with 1 ≤ ` ≤ N and Σ symplectic). It is worth
noting that, as shown in [12], another inequality that can be obtained by
relaxing a bit the hypotheses is the following (Weak-Hörmander): For any
given compact K ⊂ X there exists a constant CK > 0 such that
(W H)
(Au, u) ≥ −CK ||u||2(m−3/2)/2 , ∀u ∈ C0∞ (K; CN ).
Inequality (W H) is already interesting in its own right.
Hence, inequality (W H) shows that when (F P ) (equivalently (H)) fails,
there might be room for intermediate inequalities of the following kind.
Definition 1.2. Let Aw (x, Dx ) = Aw (x, Dx )∗ ∈ OPSm (Rn ; CN ), with principal symbol Am (x, ξ) ≥ 0 in the sense of Hermitian matrices in CN . Let
s ∈ [0, (m − 1)/2). We will say that Aw (x, Dx ) satisfies inequality (Is ) if for
any given compact K ⊂ Rn there exists a constant CK,s > 0 such that
(Is )
(Aw (x, Dx )u, u) ≥ −CK,s ||u||2s , ∀u ∈ C0∞ (K; CN ).
The reason why we take s ∈ [0, (m − 1)/2) is that inequality (I(m−1)/2 ) is the
Sharp Gårding inequality, which already holds true for such an Aw .
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ALBERTO PARMEGGIANI
Remark 1.3. When considering inequalities such as Fefferman-Phong’s inequality for A(x, Dx ) ∈ OPS2 (Rn ; CN ) (that we also called Hörmander’s inequality, because of the presence of homogeneity), one is concerned only with
the behavior of A2 (x, ξ) + A1 (x, ξ), for the remainder gives rise to a term
which is already O(||u||20 ).
From now on, for definiteness and to relate more clearly the various inequalities to the Fefferman-Phong one, we shall consider m = 2.
In Section 2 we will describe the fundamental example by Hörmander of
a non-negative matrix whose Weyl-quantization is not non-negative, which
is a model for the class we will construct.
In Section 3 we will recall Brummelhuis’s counterexample AB to the
Fefferman-Phong inequality. It will be clear in the sequel that the failure of AB to fulfill inequality (F P ) is caused by the fact that the “localized
operator” of Brummelhuis’s system AB at the point (0, x02 , 0, 1) is unitarily
equivalent in L2 (but also equivalent in S and S 0 ) to Hörmander’s operator
AH , and therefore it cannot be non-negative. Since the non-negativity of
the localized operator is a necessary condition for all the inequalities (Is ),
s ∈ [0, 1/2) (see Section 4 below), it follows that inequality (F P ) (as well as
any of the (Is ), s ∈ [0, 1/2)) cannot hold for AB .
In Section 4 we will recall what a localized operator associated with the
system A(x, Dx ) at a characteristic point is. As recalled earlier on, its spectral properties give necessary, and in some cases sufficient, conditions for
the inequalities we are concerned with to hold.
We will describe the class of pseudodifferential systems that fail to satisfy
the Fefferman-Phong inequality in Section 5 and Section 6 below. The class
is roughly described as the Weyl-quantization of the quadratic form
A2 (x, ξ) = p(x, ξ)2 v̄ ⊗ v + 2p(x, ξ)q(x, ξ)Re(v̄ ⊗ w) + q(x, ξ)2 w̄ ⊗ w,
where v, w ∈ CN are orthonormal vectors, and p and q are real and vanish
in a “finite-type” fashion on a symplectic (conic) manifold. For this class,
it will also be seen at once that the Sharp Gårding inequality cannot be
improved, not even to the weaker (W H) inequality above (and in fact, to
none of inequalities (Is ) for all s ∈ [0, 1/2)).
We close this introduction by giving some examples (or, better, counterexamples).
Example 1.4. The first one is a 2 × 2 example in R3 . For k ∈ Z+ and
δ ∈ [0, 1), let
"
#
ξ12 + x22 ξ32 − 2x2 ξ1 ξ3
(ξ1 − x2 ξ3 )(ξ2 + x2k+1
ξ3 )
1
A2 (x, ξ) =
,
2 + x4k+2 ξ 2 + 2x2k+1 ξ ξ
(ξ1 − x2 ξ3 )(ξ2 + x2k+1
ξ
)
ξ
3
2
3
2
3
1
1
1
#
"
´
0 −i
δ³
A1 (x, ξ) =
1 + (2k + 1)x2k
.
1 ξ3
2
i 0
A CLASS OF COUNTEREXAMPLES TO THE . . .
5
Then for Aw (x, Dx ), A(x, ξ) = A2 (x, ξ) + A1 (x, ξ), the Fefferman-Phong
inequality does not hold, and the Sharp Gårding inequality cannot be improved.
Example 1.5. The second one is a 3 × 3 example in R2 . Let
√


(ξ1 + x1 ξ2 )2
−i(ξ1 + x1 ξ2 )2
2 (ξ12 − x21 ξ22 )

√
1
A(x, ξ) = 
i(ξ1 + x1 ξ2 )2
(ξ1 + x1 ξ2 )2
i 2 (ξ12 − x21 ξ22 )

.
4 √
√
2 (ξ12 − x21 ξ22 ) −i 2 (ξ12 − x21 ξ22 ) 2(ξ1 − x1 ξ2 )2
Example 1.6. The third one is a 4 × 4 example in R2 . It is very close to
Brummelhuis’s example. Let


ξ12
−ix1 ξ1 ξ2
ξ1 (ξ1 − ξ2 )
ξ1 (ξ1 + ξ2 )


2ξ2

ix
ξ
ξ
x
ix
ξ
(ξ
−
ξ
)
ix
ξ
(ξ
+
ξ
)
1
1
2
1
2
1
2
1
2
1
2
1
1
2


A(x, ξ) = 

2
2
2

3 ξ1 (ξ1 − ξ2 ) −ix1 ξ2 (ξ1 − ξ2 )
(ξ
−
ξ
)
ξ
−
ξ
1
2
1
2


ξ1 (ξ1 + ξ2 ) −ix1 ξ2 (ξ1 + ξ2 )
ξ12 − ξ22
(ξ1 + ξ2 )2 .
Example 1.7. The last one is a 2 × 2 example in R2 . It is another kind of
generalization of Brummelhuis’s example. Let k ≥ 1 and let
"
#
ξ12
−ixk1 ξ1 ξ2
A(x, ξ) =
.
2
ixk1 ξ1 ξ2
x2k
ξ
1 2
Then in all the above examples, the Fefferman-Phong inequality does not
hold for Aw (x, Dx ), and the Sharp Gårding inequality cannot be improved.
2. Hörmander’s non-positivity example
Consider the system
(2)
AH (x, ξ) =
"
x2 xξ
xξ
ξ2
#
, (x, ξ) ∈ R × R.
Then hAH (x, ξ)v, vi ≥ 0 for all (x, ξ) ∈ R2 and v ∈ C2 . In spite of this,
Hörmander proved that the the Weyl-quantization Aw
H is not non-negative.
· ¸
u1
In fact, one has, with u =
∈ S(R; C2 ) and Dx = −i∂x ,
u2
2
(Aw
H (x, Dx )u, u) = ||xu1 + Dx u2 ||0 − Im (u1 , u2 ).
If one chooses, following Hörmander, u1 = v 00 , u2 = i(v − xv 0 ), for v ∈
C0∞ (R; R), then the term ||xu1 + Dx u2 ||20 = 0 and the right-hand-side becomes
1 ¯¯ ¯¯2
−Im (u1 , u2 ) = (v 00 , v − xv 0 ) = − ¯¯v 0 ¯¯0 < 0.
2
It is important to notice that det(AH (x, ξ)) = 0 on the whole T ∗ R.
6
ALBERTO PARMEGGIANI
3. Brummelhuis’s counterexample to (F P )
Consider the system
(3)
AB (x, ξ) =
"
x21 ξ22
ix1 ξ1 ξ2
−ix1 ξ1 ξ2
ξ12
#
, (x, ξ) ∈ R2 × R2 .
One has that the operator AB (x, Dx ), in spite of having non-negative symbol, does not satisfy the Fefferman-Phong inequality (F P ). In fact, following
Brummelhuis, one considers χ, ψ ∈ C0∞ (|x1 |, |x2 | ≤ 1) such that
RR
χ(x1 , x2 )ψ(0, x2 )dx1 dx2 = 1. Take then µ > 0 a parameter to be picked
sufficiently large, and define
√
u(x1 , x2 ) := µe−iµx2 χ(µx1 , x2 ), v(x1 , x2 ) := e−iµx2 ψ(x1 , x2 ).
Then ||u||20 + ||v||20 = ||χ||20 + ||ψ||20 . Writing (recall that Dxj = i−1 ∂xj )
µ
· ¸ · ¸¶
u
u
Re AB (x, Dx )
,
= ||iDx1 v + x1 Dx2 u||20 + Re (Dx2 u, v) ,
v
v
it easily follows that there exists C > 0 and µ0 ≥ 1 such that for all µ ≥ µ0
µ
· ¸ · ¸¶
´
u
u
√ ³
Re AB (x, Dx )
,
≤ −C µ ||u||20 + ||v||20 .
v
v
It is interesting to notice that also in this case det(AB (x, ξ)) = 0 on the
whole T ∗ R2 .
4. The localized operator
In this section, we shall define the localized operator of a pseudodifferential
system in a simple case tailored to our purposes (see [1], [3], [4], [12] for more
general situations). Suppose we are given A(x, ξ) = A(x, ξ)∗ ∼ A2 (x, ξ) +
A1 (x, ξ) + . . ., an Hermitian N × N matrix of second-order classical symbols.
Suppose that det(A2 (x, ξ)) = 0 for all (x, ξ) ∈ T ∗ Rn \0, and let Σ be a closed
conic symplectic submanifold of T ∗ Rn \ 0 of codimension 2ν ≥ 2 such that
Ker (A2 (ρ)) = CN for all ρ ∈ Σ. Then, for w, w0 ∈ CN , define
0
0
0
,w
,w
aw ,w (x, ξ) := aw
(x, ξ) + aw
(x, ξ) =
2
1
= hA2 (x, ξ)w, w0 i + hA1 (x, ξ)w, w0 i, (x, ξ) ∈ T ∗ Rn \ 0.
0
,w
Since aw,w
≥ 0 on T ∗ Rn \ 0 (w = w0 ), it follows by polarization that aw
2
2
vanishes to second order on Σ for all w, w0 ∈ CN . Hence we may consider,
for it is invariantly defined then, with ρ0 = (x0 , ξ0 ) ∈ Σ, |ξ0 | = 1 and ζ =
(y, η) ∈ Tρ0 T ∗ Rn , the N × N Hermitian-valued quadratic form QA2 (ρ0 ; ζ) =
hHess(A2 /2)(ρ0 )ζ, ζiTρ0 T ∗ Rn .
Definition 4.1. (See, e.g., [3]) In this framework, the localized operator
w
Lw
A (ρ0 ) of A at ρ0 ∈ Σ is the Weyl quantization
w
Lw
A (ρ0 ; y, Dy ) := QA2 (ρ0 ; y, Dy ) + A1 (ρ0 ),
thought of as a formally self-adjoint unbounded operator in L2 (Rn ; CN ).
A CLASS OF COUNTEREXAMPLES TO THE . . .
7
We then have the following proposition, which provides a necessary condition in order for inequality (Is ) to hold for some s ∈ [0, 1/2).
Proposition 4.2. Let A(x, ξ) = A(x, ξ)∗ ∼ A2 (x, ξ) + A1 (x, ξ) + . . . be an
Hermitian N × N matrix of classical symbols of order 2. Suppose that at
some ρ0 = (x0 , ξ0 ) ∈ T ∗ Rn \ 0, |ξ0 | = 1, one has Ker (A2 (ρ0 )) = CN . If
Aw (x, Dx ) = Aw (x, Dx )∗ ∈ OPS2 (Rn ; CN ) satisfies inequality (Is ) for some
s ∈ [0, 1/2), then
³
´
w
(4)
LA (ρ0 ; y, Dy )f, f ≥ 0, ∀f ∈ S(Rn ; CN ).
Proof. The proof is classical. One just tests inequality (Is ) as t → +∞
2
on a cut-off function of the kind eit hx,ξ0 i ψ(t(x − x0 )), where t ≥ 1, ψ ∈
C0∞ (Rn ; CN ). A Taylor-expansion finishes the proof.
¤
On the other hand, since Tρ0 T ∗ Rn = Tρ0 Σ ⊕ Tρ0 Σσ (symplectic orthogonal direct sum) and ζ ∈ Tρ0 Σ =⇒ QA2 (ρ0 ; ζ) = 0, one may reduce
matters to considering only ζ ∈ Tρ0 Σσ . In fact, one may use a homogeneous
symplectomorphism χ to locally near ρ0 flatten Σ into {(y 0 , y 00 , η 0 , η 00 ) ∈
Rν × Rn−ν × Rν × (Rn−ν \ {0}); y 0 = η 0 = 0, η 00 6= 0}, whose Jacobian matrix hence coordinatizes Tρ0 Σσ as Rνt × (Rν )∗τ (such a symplectomorphism
exists by Thm. 21.2.4 of [10]). Then, by using the invariance of the WeylCalculus under linear symplectomorphisms it easy to see that (4) above
implies, with
w
Lw
A (ρ0 ) := QA2 ◦χ−1 (ρ0 ; t, Dt ) + A1 (ρ0 ),
that
ν
N
(Lw
A (ρ0 )f, f ) ≥ 0, ∀f ∈ S(R ; C ).
(5)
This latter necessary condition will be used in the proof of Theorem 5.1
below.
5. The class of counterexamples
Let ν ≥ 1, and consider real classical symbols on T ∗ Rn \ 0, αj , βj homogeneous of degree 0, and pj , qj homogeneous of degree 1, respectively,
j = 1, . . . , ν. Set
p(x, ξ) :=
ν
X
j=1
αj (x, ξ)pj (x, ξ), q(x, ξ) :=
ν
X
βj (x, ξ)qj (x, ξ).
j=1
We suppose that
Σ1 = {(x, ξ) ∈ T ∗ Rn \ 0; p1 (x, ξ) = . . . = pν (x, ξ) = 0},
and
Σ2 = {(x, ξ) ∈ T ∗ Rn \ 0; q1 (x, ξ) = . . . = qν (x, ξ) = 0},
8
ALBERTO PARMEGGIANI
are closed conic manifolds of codimension ν with transversal intersection
Σ = Σ1 ∩ Σ2 , that has therefore codimension 2ν. Suppose

{dpj (ρ)}1≤j≤ν are linearly independent ∀ρ ∈ Σ1 ,





{dqj (ρ)}1≤j≤ν are linearly independent ∀ρ ∈ Σ2 ,
(H1)




 {p , p }¯¯ = {q , q }¯¯ = 0, ∀j, k = 1, . . . , ν,
j k Σ1
j k Σ2
and
(H2)
the matrix
³
´
{pj , qk }(ρ)
1≤j,k≤ν
is invertible for all ρ ∈ Σ.
Hence, by (H1), Σ1 and Σ2 are involutive manifolds, and by (H2), Σ is
symplectic. Moreover, (H2) yields that the differentials dpj , dqk , 1 ≤ j, k ≤ ν
are linearly independent on Σ.
Next, with h·, ·i the canonical Hermitian product in CN and | · | the relative
norm, fix orthonormal vectors v1 , v2 ∈ CN , and complete {v1 , v2 } into an
Hermitian basis of CN . Let {vj∗ }j=1,...,N be the relative dual vectors, so
that vj∗ (w) = hw, vj i, j = 1, . . . , N. (Hence, when thinking of vj as column
vectors, the vj∗ are the complex-conjugate row vectors.) Let then, for (x, ξ) ∈
T ∗ Rn \ 0,
³
´
(6) L(x, ξ) : C −→ CN , L(x, ξ) : z 7−→ z p(x, ξ)v1 + q(x, ξ)v2 , z ∈ C,
and the related linear form
(7)
L(x, ξ)∗ : CN −→ C, L(x, ξ)∗ : w 7−→ hw, L(x, ξ)i, w ∈ CN .
Define hence the matrix-valued second-order classical symbol A2 (x, ξ) by
(8)
A2 (x, ξ)w = hw, L(x, ξ)iL(x, ξ), ∀w ∈ CN .
Hence A2 (x, ξ) = A2 (x, ξ)∗ ≥ 0, for
hA2 (x, ξ)w, wi = |hL(x, ξ), wi|2 , ∀w ∈ CN .
Moreover, since (CN )∗ ⊗ CN ' HomC (CN , CN ) (⊗ = ⊗C )³by the isomorphism that acts on decomposable elements as v ∗ ⊗ w 7−→ u 7→ v ∗ (u)w =
´
hu, viw , we shall write
A2 (x, ξ) = L(x, ξ)∗ ⊗ L(x, ξ) ≡ L(x, ξ)L(x, ξ)∗ ,
where the last equality is understood when thinking of L as a column vector
and L∗ as the relative complex-conjugate row vector.
It is clear that

 det A2 (x, ξ) = 0, ∀(x, ξ) ∈ T ∗ Rn \ 0,
(9)

Ker A2 (ρ) = CN , ∀ρ ∈ Σ.
One has the following theorem.
A CLASS OF COUNTEREXAMPLES TO THE . . .
9
Theorem 5.1. Suppose hypotheses (H1), (H2) are satisfied, and consider
the classical symbol A(x, ξ) = A(x, ξ)∗ ∼ A2 (x, ξ) + A1 (x, ξ) + . . . with A2
defined in (8). Suppose that there exists ρ0 = (x0 , ξ0 ) ∈ Σ, |ξ0 | = 1, such
that
ν
X
{p, q}(ρ0 ) =
αj (ρ0 )βj (ρ0 ){pj , qk }(ρ0 ) 6= 0,
j,k=1
and that there exists δ ∈ [0, 1) such that (notice that A1 (x, ξ) = A1 (x, ξ)∗
for all (x, ξ) ∈ T ∗ Rn \ 0)
D
E
D
E
A1 (ρ0 )w, w ≤ δ {p, q}(ρ0 ) Im (v2∗ ⊗ v1 )w, w , ∀w ∈ CN .
Then Aw (x, Dx ) does not satisfy the Fefferman-Phong inequality (F P ), and
the Sharp Gårding inequality cannot be improved, that is, none of inequalities
(Is ) above, with s ∈ [0, 1/2), may hold.
Proof. To prove the theorem, we just compute the localized operator (actually, a unitary equivalent of it) Lw
A at ρ0 and show that it is not nonnegative. This latter point is based upon Hörmander’s example recalled
earlier in Section 2. As it is well-known, hypotheses (H1) and (H2) ensure
that there exists a conic neighborhood Γ0 of ρ0 in T ∗ Rn \ 0, a conic neighborhood Γ̃ of (0, εn = (0, 0, . . . , 0, 1)) in Rνy0 × Ryn−ν
× Rνη0 × (Rηn−ν
\ {0}) '
00
00
∗
ν
∗
n−ν
T R × (T R
\ 0), and a smooth symplectomorphism, homogeneous of
degree 1 in the fibers, χ : Γ0 −→ Γ̃ such that χ(ρ0 ) = (0, εn ), and
χ(Γ0 ∩ Σ1 ) = {(y, η) ∈ Γ̃; η 0 = 0, η 00 6= 0},
χ(Γ0 ∩ Σ2 ) = {(y, η) ∈ Γ̃; y 0 = 0, η 00 6= 0},
χ(Γ0 ∩ Σ) = {(y, η) ∈ Γ̃; y 0 = η 0 = 0, η 00 6= 0}.
As a consequence, on Γ̃ we have
(pj ◦ χ−1 )(y, η) =
ν
X
ajk (y, η)ηk , ajk homogeneous of degree 0,
k=1
(qj ◦ χ−1 )(y, η) = |η|
ν
X
bjk (y, η)yk , bjk homogeneous of degree 0,
k=1
and for (x, ξ) ∈ Γ0 ∩ Σ, with (y, η) = χ(x, ξ),
{pj , qk }(x, ξ) = |η 00 |
ν
X
ajr (y, η)bkr (y, η).
r=1
Localizing L(x, ξ) at ρ0 by means of the coordinates (t, τ ) gives
Lρ0 (t, τ ) :=
ν
³X
j=1
ν
´
³X
´
α̃j τj v1 +
β̃j tj v2 ,
j=1
10
ALBERTO PARMEGGIANI
where
α̃j :=
ν
X
k=1

αk (ρ0 )akj (χ(ρ0 )), β̃j :=



ν
X
βk (ρ0 )bkj (χ(ρ0 )).
k=1
β̃1
α̃1
.
 . 
ν

 
Hence, with α̃ = 
 ..  and β̃ =  ..  ∈ R , we have
α̃ν
β̃ν
ν
X
(10)
α̃j β̃j =: hα̃, β̃i(ν) = {p, q}(ρ0 ) 6= 0.
j=1
Consider now a rotation R of Rν such that Rα̃ = |α̃|(ν) e1 , where e1 =
t (1, 0, . . . , 0)
1/2
∈ Rν and |α̃|(ν) = hα̃, α̃i(ν) . Then we get a symplectomorphism
−1/2
1/2
ϕ : Rν × Rν → Rν × Rν , ϕ : (t, τ ) 7−→ (|α̃|(ν) Rt, |α̃|(ν) Rτ ) = (z, ζ),
such that
³
´
1/2
1/2
|α̃|(ν) L̃ρ0 (z, ζ) := (Lρ0 ◦ ϕ−1 )(z, ζ) = |α̃|(ν) ζ1 v1 + hγ, zi(ν) v2 ,
where we have put γ := Rβ̃. Hence we obtain that in these coordinates the
localized symbol of A2 at ρ0 ∈ Σ is (writing again (t, τ ) for (z, ζ))
(11)
Aρ0 (t, τ ) = |α̃|(ν) L̃ρ0 (t, τ )∗ ⊗ L̃ρ0 (t, τ ) = |α̃|(ν) L̃ρ0 (t, τ )L̃ρ0 (t, τ )∗ .
We may hence compute
´
i X³
∂τj L̃ρ0 ∂tj L̃∗ρ0 − ∂tj L̃ρ0 ∂τj L̃∗ρ0 =
2
ν
(12)
L̃ρ0 ]L̃∗ρ0 = L̃ρ0 L̃∗ρ0 −
j=1
´
i
i ³
= L̃ρ0 L̃∗ρ0 − {L̃ρ0 , L̃∗ρ0 } = L̃∗ρ0 ⊗ L̃ρ0 − γ1 v2∗ ⊗ v1 − v1∗ ⊗ v2 .
2
2
Remark 5.2. Notice that
´
i³ ∗
v2 ⊗ v1 − v1∗ ⊗ v2 = −Im (v2∗ ⊗ v1 ) = Re i(v2∗ ⊗ v1 ),
2
is the quadratic form introduced by Hörmander in [6] (see also [9]).
Upon Weyl-quantization, since to linear symplectomorphisms of Rν × Rν
there correspond (up to factors of modulus 1) unitary transformations in
L2 (Rν ) that are also automorphisms of S(Rν ) and S 0 (Rν ), we obtain, using
(11) and (12), that the localized operator of A2 at ρ0 ∈ Σ is unitarily equivalent, through a unitary transformation U ⊗ IdCN of L2 (Rν ; CN ) ' L2 (Rν ) ⊗
CN , which is also an automorphism of S(Rν ; CN ) and of S 0 (Rν ; CN ), to
´i
h
i ³ ∗
∗
w
w
∗
Aw
γ
v
⊗
v
−
v
⊗
v
,
(t,
D
)
=
|α̃|
L̃
(t,
D
)
L̃
(t,
D
)
+
1
1
2
t
t
t
(ν)
ρ0
2
1
ρ0
ρ0
2
A CLASS OF COUNTEREXAMPLES TO THE . . .
and hence, with f =
N
X
11
fj vj ∈ S(Rν ; CN ),
j=1
¯¯2
³
´
³¯¯
´
¯¯ w
∗ ¯¯
Aw
ρ0 (t, Dt )f, f = |α̃|(ν) ¯¯L̃ρ0 (t, Dt ) f ¯¯ + γ1 Im (f1 , f2 ) =
0
´
³¯¯
¯¯2
= |α̃|(ν) ¯¯Dt1 f1 + hγ, ti(ν) f2 ¯¯0 + γ1 Im (f1 , f2 ) .
Now set t = (t1 , t0 ). Following Hörmander’s example, we find f1 , f2 ∈
C0∞ (Rν ) such that
Dt1 f1 + hγ, ti(ν) f2 = 0,
and γ1 Im (f1 , f2 ) < 0.
We choose f2 ∈ C0∞ (Rν ) of the form f2 (t1 , t0 ) = g100 (t1 )g2 (t0 ), for real valued
g1 ∈ C0∞ (R) and g2 ∈ C0∞ (Rν−1 ), so that
Z t1
f1 (t1 , t0 ) = −i
hγ, (s, t0 )i(ν) f2 (s, t0 )ds =
−∞
³
´
= −i hγ, ti(ν) g10 (t1 )g2 (t0 ) − γ1 g1 (t1 )g2 (t0 ) ∈ C0∞ (Rν ).
A computation then gives
i ¯¯ ¯¯2
(f1 , f2 ) = − γ1 ¯¯g10 ¯¯0 ||g2 ||20 .
2
Hence, for f = f1 v1 + f2 v2 , f1 , f2 chosen as above,
³
´
³
´
∗
Aw
ρ0 (t, Dt )f, f = −|α̃|(ν) γ1 Im (v2 ⊗ v1 )f, f = |α̃|(ν) γ1 Im (f1 , f2 ) =
¯¯ ¯¯2
1
= − |α̃|(ν) γ12 ¯¯g10 ¯¯0 ||g2 ||20 < 0
2
provided γ1 =
6 0, which is true by hypothesis, for we have
|α̃|(ν) γ1 = h|α̃|(ν) e1 , Rβ̃i(ν) = hRα̃, Rβ̃i(ν) = {p, q}(ρ0 ) 6= 0.
Finally, since (U ⊗ IdCN )∗ A1 (ρ0 )(U ⊗ IdCN ) = A1 (ρ0 ), the localized operator
of Aw (x, Dx ) at ρ0 is unitarily equivalent to Aw
ρ0 (t, Dt )+A1 (ρ0 ), so that using
again f constructed above, one has
³h
i
´ ³
´
³
´
∗
Aw
(t,
D
)
+
A
(ρ
)
f,
f
=
A
(ρ
)f,
f
−
{p,
q}(ρ
)
Im
(v
⊗
v
)f,
f
t
1 0
1 0
0
1
ρ0
2
³
´
≤ −(1 − δ){p, q}(ρ0 ) Im (v2∗ ⊗ v1 )f, f < 0.
Hence the necessary condition (5) is not fulfilled and this concludes the proof
of the theorem.
¤
12
ALBERTO PARMEGGIANI
6. A slight generalization
When ν = 1, we may handle cases with “finite-type” degeneracy (but
with no subprincipal term). Let p, q be real homogeneous symbols of order
1 in T ∗ Rn \ 0. Let Σ1 be the simple-characteristic set of p, that is
Σ1 = {(x, ξ) ∈ T ∗ Rn \ 0; p(x, ξ) = 0}
with dp(ρ) 6= 0 for all ρ ∈ Σ1 , and let Σ2 be another involutive closed conic
smooth manifold of T ∗ Rn \0 of codimension 1, that intersects transversally
Σ1 into a symplectic submanifold Σ = Σ1 ∩ Σ2 . Suppose that for a fixed
k ≥ 2,
¯
(H3)
Hpj q ¯Σ2 = 0, ∀j = 0, . . . , k − 1 and Hpk q(ρ) 6= 0, ∀ρ ∈ Σ.
Consider then, for L and L∗ defined as in (6) and (7) above, the second-order
homogeneous symbol
A(x, ξ) = A2 (x, ξ) = L(x, ξ)∗ ⊗ L(x, ξ) =
= p(x, ξ)2 v1∗ ⊗ v1 + 2p(x, ξ)q(x, ξ)Re (v1∗ ⊗ v2 ) + q(x, ξ)2 v2∗ ⊗ v2 .
Theorem 6.1. Suppose hypothesis (H3) be satisfied. Take ρ0 = (x0 , ξ0 ) ∈
Σ, |ξ0 | = 1. Then Aw (x, Dx ) does not satisfy the Fefferman-Phong inequality
(F P ), and the Sharp Gårding inequality cannot be improved, that is, none
of inequalities (Is ) above, with s ∈ [0, 1/2), may hold.
∗
Proof. We immediately recall that Aw
2 (x, Dx ) = (A2 (x, Dx )+A2 (x, Dx ) )/2,
so that no subprincipal term is present. Let now χ : (x, ξ) 7−→ (y, η) =
(y1 , y 0 , η1 , η 0 ) ∈ R × Rn−1 × R × (Rn−1 \ {0}) be a symplectomorphism that
locally flattens Σ1 , Σ2 and Σ near ρ0 ∈ Σ. We may then suppose χ(ρ0 ) =
(0, 0, 0, η00 ), and
(p ◦ χ−1 )(y, η) = η1 , (q ◦ χ−1 )(y, η) = α(y, η)y1k |η|,
where α is homogeneous of degree 0 in η, and α(0, y 0 , 0, η 0 ) 6= 0. Let F be
a properly supported unitary, scalar Fourier integral operator associated
with the graph of χ−1 . Consider ϕ̃γ (x) := F ϕγ (y), where γ ≥ 1 and
ϕγ (y1 , y 0 ) := eiγ
k+1 y 0 ·η 0
0
ψ2 (γy 0 )ψ1 (γy1 ),
∞
N
∞
n
N
with ψ2 ∈ C0∞ (Rn−1
y 0 ; C) and ψ1 ∈ C0 (Ry1 ; C ). Then ϕ̃γ ∈ C0 (R ; C ).
Supposing (Is ) holds for some s ∈ [0, 1/2), testing (Is ) on ϕ̃γ and taking the
limit as γ → +∞ gives
³
´
Ãw
(t,
D
)f,
f
≥ 0, ∀f ∈ S(R; CN ),
t
ρ0
where this time Ãw
ρ0 (t, Dt ) denotes the Weyl-quantization of the quadratic
form
L̃ρ0 (t, τ )∗ ⊗ L̃ρ0 (t, τ ), where L̃ρ0 (t, τ ) = τ v1 + α(0, 0, 0, η00 )tk v2 ,
A CLASS OF COUNTEREXAMPLES TO THE . . .
13
(t, τ ) ∈ R × R. In other words, we consider
X
1
Ãρ0 (t, τ ) :=
(∂ j ∂ h Ã2 )(0, 0, 0, η00 )tj τ h ,
j!h! y1 η1
j/k+h=2
where Ã2 (y, η) is the principal symbol of F ∗ A2 (x, Dx )F . This because no
subprincipal term is present in A2 (x, Dx ) at Σ (since that is the case for
(p2 )w (x, Dx ), (q 2 )w (x, Dx ) and (pq)w (x, Dx )). We hence obtain
Ãρ0 (t, τ ) = τ 2 v1∗ ⊗ v1 + 2α̃tk τ Re (v1∗ ⊗ v2 ) + α̃2 t2k v2∗ ⊗ v2 ,
P
where we have set α̃ := α(0, 0, 0, η00 ). As before, with f =
j fj vj ∈
S(R; CN ), we have
¯¯2
³
´ ¯¯
¯¯
k ¯¯
w
Ãρ0 (t, Dt )f, f = ¯¯Dt f1 + α̃t f2 ¯¯ + k α̃ Im (tk−1 f1 , f2 ).
0
Ãw
ρ0 (t, Dt )
We now prove that
cannot be non-negative in L2 . In fact, we
∞
choose ϕ ∈ C0 (R; R) and consider
Z t
(k+1)
f2 (t) := ϕ
(t), f1 (t) = −iα̃
sk ϕ(k+1) (s)ds.
−∞
C0∞
It follows that f1 , f2 ∈
and that for f = f1 v1 + f2 v2 ∈ C0∞ (R; CN ) we
have
³
´
Ãw
(t,
D
)f,
f
= k α̃ Im (tk−1 f1 , f2 ).
t
ρ0
A computation yields
¯¯2 k(k − 1) ¯¯
¯¯
h 1 ¯¯
¯¯
¯¯ k−2 (k−1) ¯¯2
¯¯
(tk−1 f1 , f2 ) = −iα̃ ¯¯tk−1 ϕ(k) ¯¯ +
¯¯t ϕ
¯¯ +
2
2
0
0
µZ t
¶ i
Z
+k(k − 1)2 (k − 2) tk−3 ϕ(k−1) (t)
sk−2 ϕ(k−1) (s)ds dt .
Z
−∞
t
Since t 7→
−∞
sk−3 ϕ(k−1) (s)ds ∈ C0∞ (R; R), integrating by parts gives
µZ
Z
t
k−3 (k−1)
ϕ
t
s
(t)
−∞
k−2 (k−1)
ϕ
¯¯Z
¯¯2
¶
¯¯
1 ¯¯¯¯ t k−3 (k−1)
(s)ds dt = ¯¯
s ϕ
(s)ds¯¯¯¯ .
2 −∞
0
Hence, recalling that (Hpk q)(ρ0 ) = k!α̃, we obtain
¯¯
¯¯
¯¯2
h¯¯
³
´
k
¯¯ k−2 (k−1) ¯¯2
k
2 ¯¯ k−1 (k) ¯¯
t
ϕ
+
k(k
−
1)
t
ϕ
(H
q)(ρ
)
Ãw
(t,
D
)f,
f
=
−
¯
¯¯
¯
¯
¯
¯
¯
0
t
ρ0
2(k!)2 p
0
0
¯¯2 i
¯¯Z t
¯¯
¯¯
k−3 (k−1)
2
¯
¯
s ϕ
(s)ds¯¯¯¯ < 0.
+k(k − 1) (k − 2) ¯¯
−∞
0
Therefore the necessary condition does not hold and this completes the proof
of the theorem.
¤
14
ALBERTO PARMEGGIANI
7. The examples of the Introduction
We now show how the examples mentioned in the Introduction fit in the
framework we developed. We let v1 = e1 , v2 = e2 be the canonical vectors
of C2 , and consider, for a fixed k ∈ Z+ , the symbols
p(x, ξ) = ξ1 − x2 ξ3 , q(x, ξ) = ξ2 + x2k+1
ξ3 , (x, ξ) ∈ T ∗ R3 .
1
Notice that for k = 0, p+iq is the symbol of −iL0 , where L0 is the celebrated
Lewy operator (which is non-solvable)
L0 =
∂
∂
∂
+i
+ i(x1 + ix2 )
.
∂x1
∂x2
∂x3
Then
Σ = {(x, ξ) ∈ T ∗ R3 \ 0; ξ1 = x2 ξ3 , ξ2 = −x2k+1
ξ3 },
1
so that (x, ξ) ∈ Σ =⇒ ξ3 6= 0, and hence
³
´
(x, ξ) ∈ Σ =⇒ {p, q}(x, ξ) = (2k + 1)x2k
+
1
ξ3 6= 0.
1
In the case of Example 1.4
L(x, ξ) = p(x, ξ)e1 + q(x, ξ)e2 =
"
ξ1 − x2 ξ3
ξ2 + x2k+1
ξ3
1
#
, (x, ξ) ∈ T ∗ R3 ,
and A(x, ξ) = A2 (x, ξ) + A1 (x, ξ), where
A2 (x, ξ) = L(x, ξ)L(x, ξ)∗ =
"
#
(ξ1 − x2 ξ3 )2
(ξ1 − x2 ξ3 )(ξ2 + x2k+1
ξ3 )
1
=
,
2k+1
2
(ξ1 − x2 ξ3 )(ξ2 + x2k+1
ξ
)
(ξ
+
x
ξ
)
3
2
3
1
1
and, for some fixed δ ∈ [0, 1),
"
#
´
0 −i
δ³
A1 (x, ξ) = δ{p, q}(x, ξ)Im(e2 ⊗ e1 ) =
.
(2k + 1)x2k
1 + 1 ξ3
2
i 0
In the case of Example 1.5, (x, ξ) ∈ T ∗ R2 , p(x, ξ) = ξ1 , q(x, ξ) = x1 ξ2 , Σ is
defined by x1 = ξ1 = 0, ξ2 6= 0,
 




1
1
ξ1 + x1 ξ2
1   1

 1

L(x, ξ) = ξ1  i  + x1 ξ2  i  =  i(ξ1 + x1 ξ2 )  ,
√
√
2
2
2 √
2
− 2
2(ξ1 − x1 ξ2 )
and A(x, ξ) = A2 (x, ξ) = L(x, ξ)L(x, ξ)∗ .
In the case of Example 1.6, (x, ξ) ∈ T ∗ R2 , p(x, ξ) = ξ1 , q(x, ξ) = x1 ξ2 , Σ is
again defined by x1 = ξ1 = 0, ξ2 6= 0,

 
 

1
0
ξ1

0
 i 
1
1 
1
 
 
 ix1 ξ2 
L(x, ξ) = √ ξ1   + √ x1 ξ2   = √ 
,
−1
3 1
3
3 ξ1 − ξ2 
ξ1 + ξ2
1
1
A CLASS OF COUNTEREXAMPLES TO THE . . .
15
and A(x, ξ) = A2 (x, ξ) = L(x, ξ)L(x, ξ)∗ .
In the case of Example 1.7, (x, ξ) ∈ T ∗ R2 , p(x, ξ) = ξ1 , q(x, ξ) = xk1 ξ2 , k ≥ 1,
Σ is again defined by x1 = ξ1 = 0, ξ2 6= 0,
#
· ¸
· ¸ "
ξ1
1
0
k
L(x, ξ) = ξ1
+ x1 ξ 2
=
,
0
i
ixk1 ξ2
and A(x, ξ) = A2 (x, ξ) = L(x, ξ)L(x, ξ)∗ .
Theorems 5.1 and 6.1 yield that all the above Aw (x, Dx ) do not satisfy the
Fefferman-Phong inequality and that the Sharp Gårding inequality cannot
be improved in these cases.
We leave it to the reader to write down other possible examples.
References
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[2] R.Brummelhuis. Sur les Inegalites de Gårding Pour les Systemes d’Operateurs PseudoDifferentiels. C.R.Acad.Sci. Paris, t.315, Serie I (1992), 149-152.
[3] R.Brummelhuis. On Melin’s inequality for systems. Comm. Partial Differential Equations 26 (2001), no. 9-10, 1559-1606.
[4] R.Brummelhuis and J.Nourrigat. A necessary and sufficient condition for Melin’s inequality for a class of systems. Journal D’Analyse Mathématique 85 (2001), 195-211.
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[6] L.Hörmander. Pseudo-differential Operators and Nonelliptic Boundary Value Problems. Annals of Mathematics 83 (1966), 129-209.
[7] L.Hörmander. The Cauchy Problem for Differential Equations with Double Characteristics. Journal D’Analyse Mathématique 32 (1977), 118-196.
[8] L.Hörmander. The Weyl Calculus of Pseudodifferential Operators. Comm. on Pure and
Applied Mathematics XXXII (1979), 359-443.
[9] L.Hörmander. On the Subelliptic Test Estimates. Comm. on Pure and Applied Mathematics XXXIII (1980), 339-363.
[10] L.Hörmander. The Analysis of Linear Partial Differential Operators, Vol.I-IV,
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(1971), 117-140.
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Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40127 Bologna, ITALY
E-mail address: [email protected]