On the Fefferman-Phong Inequality and a Wiener-type Algebra of

On the Fefferman-Phong Inequality and a Wiener-type Algebra of

Pseudodifferential Operators

By

NicolasLerner∗and YoshinoriMorimoto∗∗

Abstract

We provide an extension of the Fefferman-Phong inequality to nonnegative sym- bols whose fourth derivative belongs to a Wiener-type algebra of pseudodifferential operators introduced by J. Sj¨ostrand. As a byproduct, we obtain that the number of derivatives needed to get the classical Fefferman-Phong inequality inddimensions is bounded above by 2d+ 4 +. Our method relies on some refinements of the Wick calculus, which is closely linked to Gabor wavelets. Also we use a decomposition of C^3,1nonnegative functions as a sum of squares ofC^1,1functions with sharp estimates.

In particular, we prove that aC^3,1 nonnegative functionacan be written as a finite sumP

b²_j, where eachbjisC^1,1, but also where each functionb²_jisC^3,1. A key point in our proof is to give some bounds on (bjbj) and on (bjbj).

Contents

§1. Introduction and Statement of the Results

§1.1. The Fefferman-Phong inequality and Bony’s result

§1.2. Sj¨ostrand algebra of pseudodifferential operators

§1.3. The main result

§2. The Wick Calculus of Pseudodifferential Operators

§2.1. Definitions

§2.2. Sharp estimates for the remainders

§2.3. On the composition formula for the Wick quantization

Communicated by T. Kawai. Received March 16, 2006.

2000 Mathematics Subject Classification(s): 47G30, 35S05, 42C15, 47B38, 81R30, 81S30.

∗Projet analyse fonctionnelle, Institut de Math´ematiques de Jussieu, Universit´e Pierre et Marie Curie (Paris 6), 175 rue du Chavaleret - 75013 Paris, France.

e-mail: lerner@math.jussieu.fr http://www.math.jussieu.fr//^slerner/

∗∗Graduate School of Human and Environmental. Studies, Kyoto University, Kyoto 606- 8501, Japan.

e-mail: morimoto@math.h.kyoto-u.ac.jp http://www.math.h.kyoto-u.ac.jp/^smorimoto

§3. The Proof

§3.1. Nonnegative functions as sum of squares

§3.2. Application of the Wick calculus: proof of Theorem 1.3.1

§3.3. Proof of Corollary 1.3.2 A. Appendix

§A.1. On nonnegative functions

§A.2. More properties of the algebraA

§A.3. On Leibniz formulæ

§A.4. Symmetric k-tensors as sum ofk-th powers

§A.5. From discrete sums to finite sums

§1. Introduction and Statement of the Results

§1.1. The Fefferman-Phong inequality and Bony’s result Let us consider a classical second-order symbola(x, ξ), i.e. a smooth func- tion defined onRⁿ×Rⁿ such that, for all multi-indices α, β

(1.1.1) |(∂_ξ^α∂_x^βa)(x, ξ)| ≤Cαβ(1 +|ξ|)^2−|α|.

The Fefferman-Phong inequality states that, ifasatisfies (1.1.1) and is a non- negative function, there exists Csuch that, for allu∈ S(Rⁿ),

(1.1.2) Rea(x, D)u, u_L²_(Rⁿ₎+Cu²_L2(Rⁿ)≥0, or equivalently (with an a priori different constant C)

(1.1.3) a^w+C≥0,

where a^wstands for the Weyl quantization¹ofa, (a^wu)(x) =

e^{2iπ(x−y)ξ}a x+y

2 , ξ

u(y)dydξ.

The constant C in (1.1.2–3) depends only a finite number of Cαβ in (1.1.1).

Let us ask our first question:

(1.1.4)

How many derivatives ofain (1.1.1)are needed to control C in(1.1.2)?

1The standard quantizationa(x, D) reads (a(x, D)u)(x) =R

e^2iπxξa(x, ξ)ˆu(ξ)dξ.

Looking at the proof by C. Fefferman and D. H. Phong [FP] (see also Theo- rem 18.6.8 in the third volume of [H2]), it seems clear that the number N of derivatives of aneeded to controlC should be

N = 4 +ν(n), ν depending on the dimensionn.

Since the proof is using an induction on the dimension, it is not completely obvious to answer to our question with a reasonably simple ν. On the other hand, J.-M. Bony proved in [Bo1] (Th´eor`eme 3.2) the following result: ifa(x, ξ) is a nonnegative smooth function defined on Rⁿ×Rⁿ such that

(1.1.5) |(∂_ξ^α∂_x^βa)(x, ξ)| ≤Cαβ, for|α|+|β| ≥4,

then the conclusions (1.1.2–3) hold. This result shows an interesting twofold phenomenon:

· Only derivatives with order larger than 4 are needed.

· The control of these derivatives is quite weak, of type S_0,0⁰ . In particular, the derivatives of large order do not get small (the classS_0,0⁰ does not have an asymptotic calculus).

Our answer to the question (1.1.4) is 4+2n+(for any positive). However, we shall in fact prove a much more precise result involving a Wiener-type algebra introduced by J. Sj¨ostrand in [S1]. To formulate our result, we need first to introduce that algebra.

§1.2. Sj¨ostrand algebra of pseudodifferential operators In [S1] and [S2], J. Sj¨ostrand introduced a Wiener-type algebra of pseu- dodifferential operators as follows. LetZ²ⁿ be the standard lattice in R²ⁿ_X and let 1 =

j∈Z²ⁿχ0(X −j), χ0 ∈ C_c^∞(R²ⁿ), be a partition of unity. We note χj(X) =χ0(X−j).

Proposition 1.2.1. Let abe a tempered distribution onR²ⁿ. We shall say that a belongs to the class A if ωa ∈ L¹(R²ⁿ), with ωa(Ξ) = sup_j∈Z2n

|F(χja)(Ξ)|,whereF is the Fourier transform². Moreover, we have (1.2.1) S_0,0⁰ ⊂S_0,0;2n+1⁰ ⊂ A ⊂C⁰(R²ⁿ)∩L^∞(R²ⁿ),

where S0,0;2n+1 is the set of functions defined on R²ⁿ such that

|(∂_ξ^α∂^β_xa)(x, ξ)| ≤ Cαβ for |α|+|β| ≤ 2n+ 1. A is a Banach algebra for the multiplication with the norm a_A=ω_aL1(R²ⁿ).

2(Fa)(Ξ) = R

e^−2iπXΞa(X)dX. We use also the notation D_Xj = _2iπ¹ ∂_Xj, so that F(D^αa) = Ξ^αFa.

Proof. In fact, we have the implicationsa∈ A=⇒ F(χja)∈L¹(R²ⁿ) =⇒ χja∈ C⁰∩L^∞, and, since the sum is locally finite with a fixed overlap³, we get a∈C⁰∩L^∞. Moreover, ifa∈S0,0;2n+1⁰ , i.e. is bounded as well as all its derivatives of order ≤2n+ 1, we have, with P(Ξ) = (1 +Ξ²)ⁿ the formula F(χja)(Ξ) =P(Ξ)⁻¹F

P(DX)(χja)

.We get the identity F(χja)(Ξ) =P(Ξ)⁻¹(Ξ₁+i)⁻¹F

(DX1+i)P(DX)(χja) . This entails, in the cone {Ξ∈R²ⁿ,2n|Ξ₁| ≥ Ξ}and thus everywhere⁴

|F(χja)(Ξ)| ≤P(Ξ)⁻¹(1 +Ξ)⁻¹

∈L¹(R²ⁿ)

mes(suppχ0) sup

0≤k≤2n+1a^(l)L^∞Cn, yielding the result.

Remark 1.2.2. Since 1∈ A,Ais not included inF(L¹(R²ⁿ)). Moreover AcontainsF(L¹): letabe a function inF(L¹). With the above notations, we have

|F(χja)(Ξ)|=

χˆ0(Ξ−N)ˆa(N)e^{2iπj(N−Ξ)}dN ≤

|χˆ0(Ξ−N)||ˆa(N)|dN, and thus

|ω_a(Ξ)|dΞ≤ aˆ _L1χ0_L1,which gives the inclusion. Moreover, A is a Banach commutative algebra for the multiplication.

Proposition 1.2.3. The algebra Ais stable by change of quantization, i.e. for all t real, a ∈ A ⇐⇒ J^ta = exp(2iπtDx·Dξ)a ∈ A. The bilinear map a1, a2 → a1a2 is defined onA × A and continuous valued in A, which is a (noncommutative) Banach algebra for . The maps a→ a^w, a(x, D)are continuous from AtoL(L²(Rⁿ)).

The proof is given in [S1]. A.Boulkhemair established a lot more results on this algebra in his paper [B1]. In our Appendix A.2, we give a few more properties of the algebra A, which will be useful later on in this article.

We recall that (a1a2)^w=a^w₁a^w₂ with (1.2.2) (a1a2)(X) = 2²ⁿ

R²ⁿ×R²ⁿa1(Y1)a2(Y2)e^{−4iπ[X−Y1,X−Y2]}dY1dY2, where the bracket [ , ] stands for the symplectic form: for X = (x, ξ), Y = (y, η)∈Rⁿ×Rⁿ, we have [X, Y] =ξ, y − η, x.

3If ∩j∈Jsuppχj =∅ then cardJ ≤ N0, where N0 depends only on the compact set suppχ0.

4 R²ⁿ=∪1≤k≤2n{Ξ∈R²ⁿ,2n|Ξ_k| ≥ Ξ }since the complement of that union is empty:

it is not possible to find Ξ so that max_1≤k≤2n2n|Ξ_k|< Ξ ≤2nmax_1≤k≤2n|Ξ_k|.

Comments on the Wiener Lemma. The standard Wiener’s Lemma states that ifa∈¹

Z^d

is such thatu→a ∗u=Cauis invertible as an operator on ²

Z^d

, then the inverse operator is of the formCbfor someb∈¹ Z^d

. In [S2]

the author is proving several types of Wiener lemma forA. First a commutative version, saying that if a∈ A and 1/a is a bounded function, then 1/a belongs to A. Next, Theorem 4.1 of [S2] provides a noncommutative version of the Wiener lemma for the algebra A: if an operator a^w with a ∈ Ais invertible as a continuous operator on L², then the inverse operator is b^w with b ∈ A. In the paper [GL], K. Gr¨ochenig and M. Leinert prove several versions of the noncommutative Wiener lemma, and their definition of the twisted convolution ((1.1) in [GL]) is indeed very close to (a discrete version of) the composition formula (1.2.2) above. It would be interesting to compare the methods used to prove these noncommutative versions of the Wiener lemma in the papers [GL]

and [S2].

Back to the G˚arding inequalities. Also J. Sj¨ostrand proved in Proposi- tion 5.1 of [S2] the standard G˚arding inequality with gain of one derivative for his class, in the semi-classical setting, wherehis a small parameter in (0,1]:

(1.2.3) a≥0, a∈ A=⇒a(x, hξ)^w+Ch≥0. A consequence of the result (1.1.5) of [Bo1] is that⁵

(1.2.4) a≥0, a⁽⁴⁾ ∈S_0,0⁰ =⇒a(x, hξ)^w+Ch²≥0.

Let us ask our second question. Is it possible to get an inequality with gain of 2 derivatives as in (1.2.4) and also to generalize Bony’s result by replacingS_0,0⁰ byA? That would mean that

(1.2.5) a≥0, a⁽⁴⁾∈ A=⇒a(x, hξ)^w+Ch²≥0.

From the first two inclusions in (1.2.1), we see that (1.2.5) implies (1.2.4).

Moreover the constant C in (1.2.5) will depend only on the dimension and on the norm of a⁽⁴⁾ in A, which is much more precise than the dependence of C in (1.2.4), which depends on a finite number of semi-norms of a in S_0,0⁰ . Although (1.2.5) looks stronger than (1.2.3) since h² h, it is not obvious to actually deduce (1.2.3) from (1.2.5). Anyhow we shall see that they are both true and that the proof of (1.2.3) is an immediate consequence of the

5In fact the operator h⁻²a(x, hξ)^w is unitarily equivalent toh⁻²a(h^1/2x, h^1/2ξ)^w and the function b(x, ξ) = h⁻²a(h^1/2x, h^1/2ξ) is nonnegative and satisfies b⁽⁴⁾(x, ξ) = a⁽⁴⁾(h^1/2x, h^1/2ξ) which is uniformly inS_0,0⁰ wheneverhis bounded anda⁽⁴⁾∈S⁰_0,0.

most elementary properties of the so-called Wick quantization exposed in our Section 2. Note also that a version of the H¨ormander-Melin inequality with gain of 6/5 of derivatives was given, in the semi-classical setting, by F. H´erau in [H´e]: this author used the assumption (6.4) of Theorem 6.2 of [H1], but with a limited regularity on the symbola, which is only such thata⁽³⁾∈ A.

§1.3. The main result We can state our main result.

Theorem 1.3.1. Letnbe a positive integer. There exists a constantCn

such that, for all nonnegative functions a defined on R²ⁿ satisfying a⁽⁴⁾ ∈ A, the operator a^w is semi-bounded from below and, more precisely, satisfies (1.3.1) a^w+Cna⁽⁴⁾A≥0.

The Banach algebra A is defined in Proposition 1.2.1. Note that the constant Cn depends only on the dimensionn.

The proof is given in Section 3.2.

Corollary 1.3.2. Letn be a positive integer.

(i) Let a(x, ξ) be a nonnegative function defined on Rⁿ ×Rⁿ such that (1.1.1) is satisfied for |α|+|β| ≤2n+ 5. Then(1.1.2) and(1.1.3) hold with a constant C depending only onn and onmax|α|+|β|≤2n+5Cαβ.

(ii)Leta(x, ξ, h)be a nonnegative function defined onRⁿ×Rⁿ×(0,1]such that

|(∂_ξ^α∂_x^βa)(x, ξ, h)| ≤h^|α|Cαβ, for4≤ |α|+|β| ≤2n+ 5.

Thena^w+Ch²≥0andRea(x, D)+Ch²≥0hold with a constantCdepending only onn and onmax4≤|α|+|β|≤2n+5Cαβ.

(iii)Leta(x, ξ)be a nonnegative function defined onRⁿ×Rⁿ such thata⁽⁴⁾ belong toA. Thena(x, hξ)^w+Ca⁽⁴⁾_Ah²≥0andRea(x, hD)+Ca⁽⁴⁾_Ah²≥ 0 hold with a constantC depending only on n.

(iv)Leta(x, ξ, h)be a nonnegative function defined onRⁿ×Rⁿ×(0,1]such that, for |α|+|β| = 4, the functions (x, ξ)→ (∂₁^β∂₂^αa)(xh^1/2, ξh^−1/2, h)h^−|α|

belong to A with a norm bounded above by ν0 for all h ∈ (0,1]. Then a^w+ Cν0h²≥0andRea(x, D) +Cν0h²≥0 hold with a constantC depending only on n.

That corollary is proven in Section 3.3.

Remark. It is possible to lower the requirement on the number of deriva- tives down to 2n+ 4 + (any positive ) in the statements above, by using conditions on some fractional derivatives as in Theorem 1.1 of [B2].

§2. The Wick Calculus of Pseudodifferential Operators

§2.1. Definitions

We recall here some facts on the so-called Wick quantization (see e.g. [L1]).

That tool was introduced by F. A. Berezin in [Be], and used by many authors.

In particular its role and effectiveness in proving the G˚arding inequality with gain of one derivative (once called sharp G˚arding inequality) was highlighted by the papers of A. C´ordoba and C. Fefferman [CF] and A. Unterberger [Un].

Definition 2.1.1. LetY = (y, η) be a point inRⁿ×Rⁿ. (i) The operator Σ_Y is defined as

2ⁿe^{−2π|·−Y|2}_w

. This is a rank-one orthogonal projection: Σ_Yu= (W u)(Y)τYϕwith (W u)(Y) =u, τYϕL²(Rⁿ), where ϕ(x) = 2^n/4e^−π|x|2 and (τy,ηϕ)(x) =ϕ(x−y)e^{2iπx−y2,η}.

(ii) Letabe inL^∞(R²ⁿ). The Wick quantization of ais defined as

(2.1.1) a^Wick=

R²ⁿa(Y)Σ_YdY.

(iii) Letmbe a real number. We defineS^mas the set of smooth functions p(X,Λ) defined onR²ⁿ×[1,+∞) such that, for allk∈N,

sup

Λ≥1,X∈R²ⁿ|(∂^k_Xp)(X,Λ)Λ^−m+k2|=γk(p)<∞.

The following proposition is classical and easy (see e.g. Section 5 in [L1]).

Proposition 2.1.2.

(i) Let a be in L^∞(R²ⁿ). Then a^Wick =W^∗a^µW and 1^Wick =Id_L2(Rⁿ)

where W is the isometric mapping from L²(Rⁿ)to L²(R²ⁿ) given above, and a^µ the operator of multiplication byainL²(R²ⁿ). The operatorπH=W W^∗ is the orthogonal projection on a closed proper subspaceH ofL²(R²ⁿ). Moreover, we have

(2.1.2) a^Wick

L(L²(Rⁿ))≤ a_L∞(R²ⁿ), (2.1.3) a(X)≥0 for all X impliesa^Wick≥0.

(ii) Let m be a real number and p∈S^m. Then p^Wick =p^w+r(p)^w,with r(p)∈S^m−1 so that the mappingp→r(p) is continuous. More precisely, one has

(2.1.4) r(p)(X) = ₁

0

R²ⁿ

(1−θ)p(X+θY)Y²e^−2π|Y|22ⁿdY dθ.

Note that r(p) = 0ifpis affine.

(iii) Fora∈L^∞(R²ⁿ), the Weyl symbol ofa^Wick is (2.1.5)

a∗2ⁿexp−2π| · |² which belongs toS⁰0,0 with k^th-seminormc(k)a_L∞.

(iv)With the operatorΣ_Y given in Definition 2.1.1,we have the estimate (2.1.6) Σ_YΣ_ZL(L2(Rⁿ))≤2ⁿe^{−π2|Y−Z|2}.

(v) More precisely, the Weyl symbol of Σ_YΣ_Z is, as a function of the variableX ∈R²ⁿ,

(2.1.7) e^{−π2|Y−Z|2}e−2iπ[X−Y,X−Z]2ⁿe^{−2π|X−Y+Z2 |2}.

Proposition 2.1.2 is sufficient to prove the standard G˚arding inequality with gain of one derivative, and in fact the following improvement was given by J. Sj¨ostrand in [S2].

Theorem 2.1.3. Let a be a nonnegative function defined onR²ⁿ such that the second derivatives a belongs toA. Then we have

(2.1.8) a^w+Cna_A≥0.

Proof. Although a proof of this result is given in [S2] (Proposition 5.1), it is a nice and simple introduction to our more complicated argument of Section 3. From Proposition 2.1.2, we have

a^w=a^Wick−r(a)^w≥ −r(a)^w, withr(a)(X) =₁

0

R²ⁿ(1−θ)a(X+θY)Y²e^−2π|Y|22ⁿdY dθ.SinceAis stable by translation (see Lemma A.2.1), we see that r(a) ∈ A and thus r(a)^w is bounded onL²(Rⁿ) from Proposition 1.2.3.

Remark 2.1.4. This theorem implies as well the following semi-classical version; letabe function satisfying the assumption of Theorem 2.1.3. Forh∈ (0,1], we defineAh(x, ξ) =h⁻¹a(xh^1/2, ξh^1/2).The functionAhis nonnegative with a second derivative bounded in Aby cst× a_A(see Lemma A.2.1), so that the previous theorem implies, with C depending only on the dimension, that A^w_h +Ca_A≥0.SinceA^w_h is unitarily equivalent toh⁻¹a(x, hξ)^w, this gives

(2.1.9) a(x, hξ)^w+hCa_A≥0.

§2.2. Sharp estimates for the remainders

Proposition 2.1.2 falls short of providing a proof for the Fefferman-Phong inequality, which gains two derivatives.

Lemma 2.2.1. Let abe a function defined onR²ⁿ such that the fourth derivatives a⁽⁴⁾ belong toA. Then we have

a^w=

a− 1

8πtracea _Wick

+ρ0(a⁽⁴⁾)^w, with ρ0(a⁽⁴⁾)∈ Aand more precisely ρ0(a⁽⁴⁾)A≤Cna⁽⁴⁾A.

Proof. The Weyl symbolσa ofa^Wickis σa(X) =

a(X+Y)2ⁿe^−2π|Y|2dY

=a(X) + 1

2a(X)Y²2ⁿe^−2π|Y|2dY + 1

3!

₁

0

(1−θ)³a⁽⁴⁾(X+θY)Y⁴2ⁿe^−2π|Y|2dY dθ

=a(X) + 1

8πtracea(X) + 1

3!

₁

0

(1−θ)³a⁽⁴⁾(X+θY)Y⁴2ⁿe^−2π|Y|2dY dθ.

Moreover the Weyl symbol θa of (tracea)^Wick is, from Proposition 2.1.2, θa(X) = tracea(X) +

₁

0

R²ⁿ

(1−θ)(tracea)(X+θY)Y²e^−2π|Y|22ⁿdY dθ.

As a result, the Weyl symbol of the operator

a−_8π¹ tracea_Wick is a+ 1

3!

₁

0

(1−θ)³a⁽⁴⁾(X+θY)Y⁴2ⁿe^−2π|Y|2dY dθ

− 1 8π

₁

0

R²ⁿ

(1−θ)(tracea)(X+θY)Y²e^−2π|Y|22ⁿdY dθ.

We get the equality in the lemma with (2.2.1)

ρ0(a⁽⁴⁾)(X) = 1 8π

₁

0

R²ⁿ

(1−θ)(tracea)(X+θY)Y²e^−2π|Y|22ⁿdY dθ

− 1 3!

₁

0

(1−θ)³a⁽⁴⁾(X+θY)Y⁴2ⁿe^−2π|Y|2dY dθ.

We note now thatρ0depends linearly ona⁽⁴⁾ and that (2.2.2) ρ0(a⁽⁴⁾)(X) =

₁

0

a⁽⁴⁾(X+θY)M(θ, Y)

polynomial inY,θ.

e^−2π|Y|2dY dθ.

Looking now at the formula (2.2.2) and applying Lemma A.2.1, we get ρ0(a⁽⁴⁾)_A≤

₁

0

M(θ, Y)e^−2π|Y|2dY dθC0a⁽⁴⁾_A=C1a⁽⁴⁾_A. The proof of Lemma 2.2.1 is complete.

Remark 2.2.2. We note that, from Lemma 2.2.1 and theL²boundedness of operators with symbols inA, Theorem 1.3.1 is reduced to proving

(2.2.3)

a≥0, a⁽⁴⁾∈ A=⇒ a− 1

8πtracea _Wick

is semi-bounded from below.

Naturally, one should not expect the quantitya−_8π¹ traceato be nonnegative:

this quantity will take negative values even in the simplest casea(x, ξ) =x²+ξ², so that the positivity of the quantization expressed by (2.1.3) is far from enough to get our result. We shall prove in Section 3 a stronger version of (2.2.3), but before this, we need to investigate more closely the composition formula for the Wick quantization.

§2.3. On the composition formula for the Wick quantization In this section, we prove some formulas of composition for operators with very irregular Wick symbols. The first lemma below was already proven in [L1], but we give here a complete proof for the convenience of the reader, since these (easy) computations are not completely standard.

Lemma 2.3.1. Forp, q∈L^∞(R²ⁿ)real-valued withp∈L^∞(R²ⁿ), we have

Re

p^Wickq^Wick

=

pq− 1

4π∇p· ∇q_Wick +R, R_L(L2(Rⁿ))≤C(n)p_L∞q_L∞.

The product ∇p· ∇q above makes sense(see our Appendix A.3)as a tempered distribution since∇pis a Lipschitz continuous function and∇qis the derivative of anL^∞function: in fact, we shall use as a definition (see our Appendix A.3)

∇p· ∇q=∇ ·( q

L^∞

∇p

Lip.

)− q

L^∞

∆p

L^∞

.

Proof. Using Definition 2.1.1, we see that p^Wickq^Wick=

R²ⁿ×R²ⁿp(Y)q(Z)Σ_YΣ_ZdY dZ

= p(Z) +p(Z)(Y −Z) +p2(Z, Y)(Y −Z)²

q(Z)Σ_YΣ_ZdY dZ

=

(pq)(Z)Σ_ZdZ+

p(Z)(Y −Z)Σ_YdY q(Z)Σ_ZdZ+R0, withR0=

₁

0

(1−θ)p(Z+θ(Y−Z))(Y−Z)²q(Z)Σ_YΣ_ZdY dZdθ.

Claim 2.3.2. Let ω be a measurable function defined onR²ⁿ×R²ⁿ such that

|ω(Y, Z)| ≤γ0

1 +|Y −Z|_N0 . Then the operator

ω(Y, Z)Σ_YΣ_ZdY dZis bounded onL²(Rⁿ)withL(L²(Rⁿ)) norm bounded above by a constant depending on γ0, N0. This is an immediate consequence of Cotlar’s lemma (see e.g. Lemma4.2.3in[BL]or Lemma18.6.5 in [H2])and of the formula(2.1.6).

Using that claim, we obtain that

(2.3.1) R0_L(L2(Rⁿ))≤C1(n)p_L∞(R²ⁿ)q_L∞(R²ⁿ).

We check now

(Y −Z)Σ_YdY whose Weyl symbol is, as a function ofX,

(Y −Z)2ⁿe^{−2π|X−Y|2}dY =

(X−Z)2ⁿe^{−2π|X−Y|2}dY =X−Z.

So withLZ(X) =X−Z, we have

(Y−Z)Σ_YdYΣ_Z = (X−Z)^wΣ_Z=L^w_ZΣ_Z and thus

Re

(Y −Z)Σ_YdYΣ_Z= Re(L^w_ZΣ_Z) =

(X−Z)2ⁿe^{−2π|X−Z|2}_w

= 1

4π∂Z(2ⁿe^{−2π|X−Z|2})^w, so that

(2.3.2) Re

(Y −Z)Σ_YdYΣ_Z= 1

4π∂Z(Σ_Z).

Using that pandq are real-valued, the formula for Re(p^Wickq^Wick) becomes Re

p^Wickq^Wick

=

(pq)(Z)Σ_ZdZ+

p(Z)q(Z) 1

4π∂ZΣ_ZdZ+ ReR0

= (pq)(Z)− 1

4πp(Z)·q(Z)

Σ_ZdZ

− 1

4πtracep(Z)q(Z)Σ_ZdZ+ ReR0

that is the result of the lemma, using (2.3.1) and (2.1.2) for the penultimate term on the line above.

The next lemma is more involved.

Lemma 2.3.3. For p measurable real-valued function such that p, (pp),(pp)∈L^∞, we have

(2.3.3) p^Wickp^Wick= p(Z)²− 1

4π|∇p(Z)|²

Σ_ZdZ+S,

(2.3.4) S_L(L2(Rⁿ))≤C(n)

p²_L∞+(pp)_L∞+(pp)_L∞

. Here p stands for the vector (tensor) with components (∂_X^αp)_|α|=2, whereas the components of (pp) are ∂_X^α

∂_X^β∂_X^γp

|α|=1,|β|=2,

|γ|=1 and those of (pp) are

∂_X^α p∂_X^βp

|α|=|β|=2.

Proof. We have p^Wickp^Wick=

p(Y)p(Z)Σ_YΣ_ZdY dZ

=

p(Y)

p(Y) +p(Y)(Z−Y)

Σ_YΣ_ZdY dZ +

₁

0

p(Y)(1−θ)p(Y +θ(Z−Y))dθ(Z−Y)²Σ_YΣ_ZdY dZdθ so that, using (2.3.2) for the terms pp in the double integral above, we get, noting trace(p) = ∆p,

(2.3.5) p^Wickp^Wick=

p²− 1

4π|∇p|²− 1 4πp∆p

_Wick

+ Re(Ω₀+ Ω₁+ Ω₂), with

(2.3.6) Ω₀=

₁

0

p

Y +θ(Z−Y) p

Y +θ(Z−Y)

(Z−Y)²Σ_YΣ_ZdY dZ(1−θ)dθ,

(2.3.7) Ω₁=

₁

0

p

Y +θ(Z−Y)

θ(Y −Z)

×p

Y +θ(Z−Y)

(Z−Y)²Σ_YΣ_ZdY dZ(1−θ)dθ and from the claim (2.3.2),

(2.3.8) Ω_2L(L2(Rⁿ))≤C1(n)p²_L∞. We write now Ω₀= Ω₀₀+ Ω₀₁, Ω₁= Ω₁₀+ Ω₁₁with

Ω₀₀=1 2

p(Y)p(Y)(Z−Y)²Σ_YΣ_ZdY dZ, Ω₀₁=

₁

0

(pp)

Y +θ(Z−Y)

−(pp)(Y)

(Z−Y)²

×Σ_YΣ_ZdY dZ(1−θ)dθ Ω₁₀=−1

6 ₁

0

p(Y)(Z−Y)p(Y)(Z−Y)²Σ_YΣ_ZdY dZ Ω_11L(L²_(Rⁿ))≤C2(n)(pp)_L∞.

(2.3.9)

We have also Ω₀₁= Ω₀₁₀+ Ω₀₁₁ with Ω₀₁₀= 1

6

(pp)(Y)(Z−Y)(Z−Y)²Σ_YΣ_ZdY dZ,

(2.3.10) Ω_011L(L2(Rⁿ))≤C3(n)(pp)_L∞.

From (2.3.5–10), it suffices to check that the following term is a remainder satisfying the estimate (2.3.4) to get the result of Lemma 2.3.3:

Ω = − 1 4π

p(Y) tracep(Y)Σ_YdY (2.3.11)

+1 2Re

(pp)(Y)(Z−Y)²Σ_YΣ_ZdY dZ +1

6Re

(pp)(Y)(Z−Y)(Z−Y)²Σ_YΣ_ZdY dZ

−1 6Re

₁

0

p(Y)(Z−Y)p(Y)(Z−Y)²Σ_YΣ_ZdY dZ.

The real part of the Weyl symbol of

(Zj−Yj)(Zk−Yk)(Zl−Yl)Σ_YΣ_ZdZ is (see (2.1.7))

(Zj−Yj)(Zk−Yk)(Zl−Yl)e^{−π2|Y−Z|2}

×cos(2π[X−Y, X−Z])2ⁿe^{−2π|X−Y+Z2 |2}dZ

=

TjTkTle^{−2π|T /2|2}cos(2π[X−Y, T])2ⁿe^{−2π|X−Y−T2|2}dT

=

TjTkTlcos(2π[X−Y, T])e^{−π|X−Y−T|2}dT2ⁿe^{−π|X−Y|2} =νjkl(X−Y) with

(2.3.12) νjkl(S) =

TjTkTlcos (2π[S, T])e^{−π|S−T|2}dT2ⁿe^−π|S|2

= 2ⁿe^−π|S|2

(Tj+Sj)(Tk+Sk)(Tl+Sl) cos (2π[S, T])e^−π|T|2dT

= 2ⁿe^−π|S|2

(TjTkSl+TkTlSj+TlTjSk

+SjSkSl) cos (2π[S, T])e^−π|T|2dT.

We notice that the functionS→

R²ⁿTjTkexp (2iπ[S, T])e^−π|T|2dT is a second- order derivative ofS→

R²ⁿexp (2iπ[S, T])e^−π|T|2dT =e^−π|S|2 so that 2ⁿe^−π|S|2Sl

R²ⁿTjTkcos (2π[S, T])e^−π|T|2dT =e^−2π|S|2SlPjk(S), with Pjk even, second-order and real polynomial. The function Sl1Sl2Sl3

×e^−2π|S|2 is always a linear combination of derivatives of Schwartz functions onR²ⁿ, since

ifl1< l2≤l3it is the derivative with respect toSl1ofSl2Sl3e^−2π|S|2(−4π)⁻¹, ifl1=l2< l3it is the derivative with respect toSl3ofSl1Sl2e^−2π|S|2(−4π)⁻¹, ifl1=l2=l3=l it is a linear combination of the third and first derivative with respect toSl ofe^−2π|S|2, since

(e^t2)= (12t+ 8t³)e^t2, t³e^t2 =1

8(e^t2)−3 4(e^t2).

As a result the function νjkl defined by (2.3.12) is a linear combination of derivatives with respect toSj, Sk orSlof Schwartz functions onR²ⁿ. Integrat- ing by parts in the last two terms of (2.3.11), we see that theirL(L²) norm is bounded from above byC4(n)((pp)_L∞+(pp)_L∞). Looking at (2.3.11), we see that we are left with

(2.3.13) Ω₀=− 1

4π

p(Y) tracep(Y)Σ_YdY +1 2Re

(pp)(Y)(Z−Y)²Σ_YΣ_ZdY dZ.

The real part of the operator

(Zj−Yj)(Zk−Yk)Σ_YΣ_ZdY dZ has the Weyl symbol (function ofX)

(2.3.14)

TjTke^{−π|X−Y−T|2}cos(2π[X−Y, T])dT2ⁿe^{−π|X−Y|2}

= (Xj−Yj)(Xk−Yk) +TjTk

e^−π|T|2cos(2π[X−Y, T])dT2ⁿe^{−π|X−Y|2}

= SjSk+TjTk

e^−π|T|2cos(2π[S, T])dT2ⁿe^−π|S|2, S =X−Y.

•Ifj=k, both terms in (2.3.14) are second order derivatives with respect to Y of a Schwartz function inR²ⁿ. In fact the first term is

SjSk2ⁿe^−2π|S|2 =∂Sj∂Sk

2ⁿe^−2π|S|2/16π²

=∂Yj∂Yk

2ⁿe^−2π|S|2/16π²

and the second term is equal to−SjSk2ⁿe^−2π|S|2, withj =k, also a second- order derivative. The contribution of these terms in (2.3.13) is then, after integration by parts, anL²bounded operator with norm≤C5(n)(pp)_L∞.

• Ifj=k, withj =j±n (in factj=j+nif 1≤j≤nandj =j−n if 1 +n≤j≤2n), we note that (2.3.14) is equal to

S_j²2ⁿe^−2π|S|2− 1

4π²e^−π|S|2∂_S²_j

2ⁿe^−π|S|2

= 2ⁿe^−2π|S|2

S_j²− 1

4π²(4π²S_j²−2π)

. Taking into account the contribution of these terms in (2.3.13), we see that we are left with

− 1 4π

p(Y) tracep(Y)Σ_YdY +1 2

1

2πtrace(pp)(Y)Σ_YdY = 0. The proof of Lemma 2.3.3 is complete.

§3. The Proof

§3.1. Nonnegative functions as sum of squares

Theorem 3.1.1. Let mbe a nonnegative integer. There exists an inte- ger N and a positive constant C such that the following property holds. Let a be a nonnegative C^3,1 function⁶ defined on R^m such that a⁽⁴⁾ ∈ L^∞; then we can write

(3.1.1) a=

1≤j≤N

b²_j

where the bj are C^1,1 functions such thatb_j,(b_jb_j),(bjb_j)∈L^∞. More pre- cisely, we have

(3.1.2) b_j²_L^∞+(b_jb_j)L^∞+(bjb_j)L^∞ ≤Ca⁽⁴⁾L^∞.

Note that this implies that each function bj is such that b²_j is C^3,1 and that N andC depend only on the dimensionm.

Remark 3.1.2. We shall use the following notation: letAbe a symmetric k-linear form on real normed vector spaceV. We define the norm ofAby

A= sup

T =1|AT^k|.

6AC^3,1function is aC³ function whose third-order derivatives are Lipschitz continuous.