Global subelliptic estimates for Kramers-Fokker-Planck operators with some class of polynomials

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Global subelliptic estimates for Kramers-Fokker-Planck operators with some class of polynomials

Mona Ben Said

To cite this version:

Mona Ben Said. Global subelliptic estimates for Kramers-Fokker-Planck operators with some class of polynomials. 2019. �hal-01955908v2�

Global subelliptic estimates for Kramers-Fokker-Planck operators with some class of polynomials

Mona Ben Said

Laboratoire Analyse, G´ eom´ etrie et Applications Universit´ e Paris 13

99 Avenue Jean Baptiste Cl´ ement 93430 Villetaneuse, France

[email protected] May 27, 2019

Abstract

In this article we study some Kramers-Fokker-Planck operators with a polynomial potential V(q) of degree greater than two having quadratic limiting behavior. This work provides an accurate global subelliptic estimate for KFP operators under some conditions imposed on the potential.

Key words: subelliptic estimates, compact resolvent, Kramers-Fokker-Planck operator.

MSC-2010: 35Q84, 35H20, 35P05, 47A10, 14P10

Contents

1 Introduction and main results 2

2 Preliminary results 5

3 Proof of Theorem 1.1 14

4 Applications 24

Appendix A Slow metric, partition of unity 29

Appendix B Around Tarski-Seidenberg theorem 34

1 Introduction and main results

The Kramers-Fokker-Planck operator reads KV =p.∂q−∂qV(q).∂p+ 1

2(−∆p+p²) , (q, p)∈R^2d, (1.1) whereqdenotes the space variable,pdenotes the velocity variable,x.y =

d

P

j=1

x_jy_j,x² =

d

P

j=1

x²_j and the potential V(q) = P

|α|≤r

Vαq^α is a real-valued polynomial function onR^d withd^◦V =r.

There have been several works concerned with the operator K_V with diversified ap- proaches. In this article we impose some kind of assumptions on the polynomial potential V(q), so that the Kramers-Fokker-Planck operator K_V admits a global subelliptic estimate and has a compact resolvent. This problem is closely related to the return to the equilibrium for the Kramers-Fokker-Planck operator (see [HeNi][Nie][Nou]). As mentioned in [HerNi]

and [Nie], the analysis of K_V is also strongly linked to the one of the Witten Laplacian

∆⁽⁰⁾_V = −∆_q +|∇V(q)|² −∆V(q). This relation yielded to the following conjecture estab- lished by Helffer-Nier:

(1 +K_V)⁻¹ compact ⇔(1 + ∆⁽⁰⁾_V )⁻¹ compact . (1.2) This conjecture has been partially resolved in simple cases (see for example [HeNi], [HerNi]

and [Li]), whereas for the operator ∆⁽⁰⁾_V very general criteria of compactness work for poly- nomial potiential V(q) of arbitrary degree. These last criteria require an analysis of the degeneracies at infinity of the potential and rely on extremely sophisticated tools of hypoel- lipticity developed by Helffer and Nourrigat in the 1980’s (see [HeNo], [Nie]). Among the particularities of these last analysis, we mention that the compactness results obtained for degenerate potentials at infiniy were not the same for ∆⁽⁰⁾_+V as ∆⁽⁰⁾_−V. The typical example which was considered is the case V(q₁, q₂) =q²₁q₂² in dimension d= 2: The operator ∆⁽⁰⁾_−V has a compact resolvent, while ∆⁽⁰⁾_+V has not.

In the case of the Kramers-Fokker-Planck operator, there have been extensive works concerned with the case d^◦V ≤ 2 (see [Hor][HiPr][Vio][Vio1][AlVi][BNV]). Nevertheless, as far as general potential is concerned, different kind of sufficient conditions on V(q) had been examined by H´erau-Nier [HerNi], Helffer-Nier [HeNi], Villani [Vil] and Wei-Xi Li [Li]. These first results considered only variants of the elliptic situation at the infinity ( for non-degenerate potential), which did not distinguish the sign ±V(q). Lately a significant improvement of those works has been done by Wei-Xi Li [Li2] based on some multipliers methods. In [Li2], Wei-Xi Li showed that for potentials similar to V(q₁, q₂) =q₁²q₂² the results forK±V were the same as for ∆⁽⁰⁾_±V, thus comforting the idea that the conjecture (1.2) is true.

The ultimate goal would be to develop a complete recurrence with respect tod^◦V for the Kramers-Fokker-Planck operator like it is possible to do for the Witten Laplacian as recalled in [HeNi] (cf. Teorem 10.16 page 106) and [Nie] by following the general approach of Helffer- Nourrigat in [HeNo] and [Nou]. Although we are not able to write a complete induction, we

establish here subelliptic estimates for K_V for a rather general class of polynomial potentials with criteria which distinguish clearly the sign ±V(q). The asymtotic behaviour of those polynomials is governed by at most quadratic parameter dependent potentials, and the global subelliptic estimates in which arise some logarithmic weights are know to be essentially optimal in the quadratic case (see [BNV]).

Denoting

O_p = 1

2(D_p²+p²), and

X_V =p.∂_q−∂_qV(q).∂_p ,

we can rewrite the Kramers-Fokker-Planck operator K_V defined in (1.1) asK_V =X_V +O_p . Notations: Throughout the paper we use the notation

h·i=p

1 +| · |² .

For an arbitrary polynomial V(q) of degree r, we denote for allq∈R^d Tr_+,V(q) = X

ν∈Spec(HessV) ν>0

ν(q),

Tr−,V(q) =− X

ν∈Spec(HessV) ν≤0

ν(q).

Futhermore, for a polynomial P ∈E_r :={P ∈R[X₁, ..., X_d], d^◦P ≤r} and all natural num- ber n∈ {1, ..., r}, we define the functions R^≥n_P :R^d →Rand R⁼ⁿ_P :R^d →R by

R^≥n_P (q) = X

n≤|α|≤r

∂_q^αP(q)

1

|α| , (1.3)

R⁼ⁿ_P (q) = X

|α|=n

|∂_q^αP(q)|^|α|1 . (1.4)

For arbitrary real functions A and B, we make also use of the following notation AB ⇐⇒ ∃c≥1 :c⁻¹|B| ≤ |A| ≤c|B| .

This work is essentially based on the recent publication by Ben Said, Nier, and Viola [BNV], which deals with the analysis of Kramers-Fokker-Planck operators with polynomials of degree less than 3. In this case we define the constants A_V and B_V by

A_V = max{(1 + Tr_+,V)^2/3,1 + Tr−,V} , B_V = max{min

q∈R^d

|∇V(q)|^4/3, 1 + Tr−,V

log(2 + Tr−,V)²}.

As proved in [BNV], there is a constant c > 0 such that the following global subelliptic estimate with remainder

kK_Vuk²_L2(R^2d)+A_Vkuk²_L2(R^2d) ≥c

kO_puk²_L2(R^2d)+kX_Vuk²_L2(R^2d)

+kh∂_qV(q)i^2/3uk²_L2(R^2d)+khD_qi^2/3uk_L²_(R^2d₎ (1.5) holds for all u ∈ C₀^∞(R^2d). Moreover, if V does not have any local minimum, that is if Tr−,V + min

q∈R^d

|∇V(q)| 6= 0, there exists a constant c >0 such that

kK_Vuk²_L2(R^2d)≥c B_Vkuk²_L2(R^2d) , (1.6) holds for allu∈ C₀^∞(R^2d). Hence combining (1.6) and (1.5), there is a constantc >0 so that

kK_Vuk²_L2(R^2d)≥ c 1 + ^A_B^V

V

kO_puk²_L2(R^2d)+kX_Vuk²_L2(R^2d)

+kh∂_qV(q)i^2/3uk²_L2(R^2d)+khD_qi^2/3uk_L²_(R^2d₎

(1.7) is valid for allu∈ C₀^∞(R^2d).The constants appearing in (1.5), (1.6) and (1.7) are independent of the potentialV.We recall here that for a smooth potentialV ∈ C^∞(R^d), our operatorK_V is essential maximal accretive when endowed with the domainC₀^∞(R^2d) [HeNi] (cf. Proposition 5.5 page 44). As a result the domain of its closure is given by

D(K_V) =

u∈L²(R^2d), K_Vu∈L²(R^2d) .

Consequently by density of C₀^∞(R^2d) inD(K_V) all estimates stated in this paper, which are checked with C₀^∞(R^2d) functions, can be extended to the domain of K_V.

Given a polynomial V(q) with degree r greater than two, our result will require the following assumption after setting for κ >0

Σ(κ) = n

q∈R^d, |∇V(q)|⁴³ ≥κ

|HessV(q)|+R^≥3_V (q)⁴+ 1o .

Assumption 1. There exist large constantsκ0, C1 >1such that for allκ≥κ0 the polynomial V(q) satisfies the following properties

Tr−,V(q)≥ 1

C₁Tr_+,V(q), for all q∈R^d\Σ(κ)with |q| ≥C₁ , (1.8) moreover if R^d\Σ(κ) is not bounded

q→∞lim

q∈R^d\Σ(κ)

R^≥3_V (q)⁴

|HessV(q)| = 0 . (1.9)

Our main result is the following.

Theorem 1.1. LetV(q)be a polynomial of degreer greater than two verifying Assumption 1.

Then there exists a strictly positive constant C_V >1 (depending on V) such that kK_Vuk²_L2 +C_Vkuk²_L2 ≥ 1

C_V

kL(O_p)uk²_L2 +kL(h∇V(q)i²³)uk²_L2

+kL(hHessV(q)i¹²)uk²_L2 +kL(hD_qi²³)uk²_L2

, (1.10) holds for all u∈D(K_V) where L(s) = _log(s+1)^s+1 for any s≥1.

Corollary 1.2. If V(q) is polynomial of degree greater than two that satisfies Assumption 1, then the Kramers-Fokker-Planck operator K_V has a compact resolvent.

Proof. Proof of Corollary 1.2

Assume 0< δ <1.Define the functions f_δ :R^d→R by

f_δ(q) =|∇V(q)|^43(1−δ)+|HessV(q)|^1−δ . From (1.10) in Theorem 1.1 there is a constant C_V >1 such that

kK_Vuk²_L2+C_Vkuk²_L2 ≥ 1 C_V

hu, f_δui+kL(O_p)uk²_L2 +kL(hD_qi²³)uk²_L2

,

holds for all u ∈ C₀^∞(R^2d) and all δ ∈ (0,1). In order to prove that the operator K_V has a compact resolvent it is sufficient to show that lim

q→+∞fδ(q) = +∞.

To do so, assume A >0 and denoteκ=A^1−δ1 . Ifq ∈Σ(κ), one has

|∇V(q)|^43(1−δ)≥κ^1−δ=A . Else ifq∈R^d\Σ(κ) by (1.9) in Assumption 1, _q→∞lim

q∈R^d\Σ(κ)

|HessV(q)|= +∞.Hence there exists a constant η >0 such that |HessV(q)|^1−δ ≥A for all q ∈R^d\Σ(κ) with |q| ≥η.

Remark 1.3. The results of Theorem 1.1 and Corollary 1.2 can be extended in the case when V =V₁+V₂ where V₁ is polynomial satisfying Assumption 1 and V₂ is a function in S(R^d).

2 Preliminary results

This work is essentially based on two main strategies. The first one consists in the use of a partition of unity which is the most important tool that allows one to pass from local to global estimates.

In this paper, given a polynomial V(q) we make use of a locally finite partition of unity with respect to the position variable q∈R^d

X

j∈N

χ²_j(q) = X

j∈N

χe²_j

R^≥3_V (q_j)⁻¹(q−q_j)

= 1 (2.1)

where

suppχe_j ⊂B(q_j, a) and χe_j ≡1 in B(q_j, b)

for some q_j ∈ R^d with 0 < b < a independent of j ∈ N. Such a partition is described more precisely in Lemma A.7 after taking n= 3. In our study introducing this partition yields to errors to be well controlled.

The second approach lies in the decomposition of the operatorK_V onto two parts so that the first one be a Kramers-Fokker-Planck operator with polynomial potential of degree less than three. On this way, based on [BNV], we derive the result of Theorem 1.1.

In order to prove Theorem 1.1 we need the following basic lemmas.

Lemma 2.1. Assume V ∈ Er with degree r ∈ N. Consider the Kramers-Fokker-Planck operator K_V defined as in (1.1). For a locally finite partition of unity namely P

j∈N

χ²_j(q) = 1 one has

kK_Vuk²_L2(R^2d)=X

j∈N

kK_V(χ_ju)k²_L2(R^2d)− k(p∂_qχ_j)uk²_L2(R^2d) , (2.2) for all u∈ C₀^∞(R^2d).

In particular when the degree of V is larger than two and the cutoff functions χ_j have the form (2.1), there exists a constant c_d >0 (depending on the dimension d) so that

kK_Vuk²_L2(R^2d) ≥X

j∈N

kK_V(χ_ju)k²_L2(R^2d)−c_dR^≥3_V (q_j)²kpχ_juk²_L2(R^2d) , (2.3) holds for all u∈ C₀^∞(R^2d).

Proof. First letV ∈E_r with r∈Nis the degree of V.Assume thatu∈ C₀^∞(R^2d).On the one hand,

kK_Vuk²_L2 =X

j∈N

hK_Vu, χ²_jK_Vui=X

j∈N

hu, K_V^∗χ²_jK_Vui. On the other hand,

X

j∈N

kKV(χju)k²_L2 =X

j∈N

hu, χjK_V^∗KVχjui. Putting the above equalities together

kK_Vuk²_L2 −X

j∈N

kK_V(χ_ju)k²_L2 =X

j∈N

hu,(K_V^∗χ²_jK_V −χ_jK_V^∗K_Vχ_j)ui .

Using commutators, we compute

K_V^∗χ²_jK_V =K_V^∗χ_j[χ_j, K_V ] +K_V^∗χ_jK_Vχ_j

=K_V^∗χ_j[χ_j, K_V ] + [K_V^∗, χ_j]K_Vχ_j+χ_jK_V^∗K_Vχ_j

=K_V^∗χ_j[χ_j, K_V ] + [K_V^∗, χ_j]

[K_V, χ_j] +χ_jK_V

+χ_jK_V^∗K_Vχ_j . Thus

K_V^∗χ²_jK_V −χ_jK_V^∗K_Vχ_j =K_V^∗χ_j[χ_j, K_V ] + [K_V^∗, χ_j]χ_jK_V + [K_V^∗, χ_j]◦[K_V, χ_j]. Now it is easy to check the following commutation relations







[χ_j, K_V ] =−[K_V, χ_j] =−[p∂_q, χ_j(q) ] =−p∂_qχ_j [K_V^∗, χ_j] = [−p∂_q, χ_j(q) ] =−p∂_qχ_j

[K_V^∗, χ_j]◦[K_V, χ_j] =−(p∂_qχ_j)² . Collecting the terms, we obtain

X

j∈N

(K_V^∗χ²_jK_V −χ_jK_V^∗K_Vχ_j) = X

j∈N

K_V^∗χ_j(−p∂_qχ_j) + (−p∂_qχ_j)χ_jK_V −(p∂_qχ_j)²

=X

j∈N

K_V^∗

∂_q(χ²_j 2 )

−p∂_q(χ²_j

2 )K_V −(p∂_qχ_j)²

=−(p∂_qχ_j)² ,

where in the last line we make use simply P

j∈N

χ²_j(q) = 1.

From this follows immediately the identity kKVuk²_L2 =X

j∈N

kKV(χju)k²_L2− k(p∂qχj)uk²_L2

for all u∈ C₀^∞(R^2d).

Next, suppose that the degree ofV is greater than two andχ_j(q) = χe_j

R_V^≥3(q_j)⁻¹(q−q_j) for all index j and any q ∈R^d with

supp χe_j ⊂B(q_j, a) and χe_j ≡1 in B(q_j, b) . Then we can write

X

j∈N

k(p∂_qχ_j)uk² =X

j∈N

X

j⁰∈N

k(p∂_qχ_j)χ_j⁰uk²

≤c_dX

j∈N

R^≥3_V (q_j)²kpχ_juk² ,

where c_d is a constant that depends only on the dimension d. Here the last inequality is due to the fact that for each indexj there are finitely manyj⁰ such that (∂_qχ_j)χ_j⁰ is nonzero.

Before stating the following lemma, we fix and remind some notations.

Notations 2.2. Let V be a polynomial of degree r larger than two. Consider a locally finite partition of unity P

j∈N

χ²_j(q) = 1 described as in (2.1).

Set for all κ >0

J(κ) = n

j ∈N, such that suppχj ⊂Σ(κ) o

, where we recall that

Σ(κ) =n

q ∈R^d, |∇V(q)|⁴³ ≥κ

|HessV(q)|+R_V^≥3(q)⁴+ 1o .

For a given κ > 0 and all index j ∈ N, let V_j² be the polynomial of degree less than three given by

V_j²(q) = X

0≤|α|≤2

∂_q^αV(q⁰_j)

α! (q−q_j⁰)^α , (2.4)

where

( q_j⁰ =q_j ifj ∈J(κ) q_j⁰ ∈(suppχ_j)∩

R^d\Σ(κ) else.

Lemma 2.3. Assume V a polynomial of degree r larger than two. Consider a locally finite partion of unity described as in (2.1). For a multi-index α ∈ N^d of length |α| ∈ {1,2} and all j ∈N, one has

∂_q^αV(q)−∂_q^αV_j²(q)

≤c_α,d,r

R^≥3_V (q_j⁰)|α|

(2.5) for any q∈suppχ_j =B(q_j, aR^≥3_V (q_j)⁻¹), where c_α,d,r = P

3≤|β|≤r

β!a^−|β|+|α|.

As a consequence, if V satisfies Assumption 1, there exists a large constant κ₁ ≥ κ₀ so that for all κ≥κ₁

•if j ∈J(κ) 2⁻¹

∂_qV_j²(q)

≤ |∂_qV(q)| ≤2

∂_qV_j²(q)

for every q∈suppχ_j , (2.6)

•if j /∈J(κ)

2⁻¹

HessV_j²(q)

≤ |HessV(q)| ≤2

HessV_j²(q)

, (2.7)

for any q∈suppχ_j with |q| ≥C₂(κ) where C₂(κ)>0 is a large constant that depends on κ.

Proof. Let V be a polynomial of degree r greater than two. In this proof we are going to need the following equivalence

R^≥3_V (q)R^≥3_V (q⁰), (2.8)

satisfied for all q, q⁰ ∈supp χ_j and proved in Lemma A.5. That is there is a constant C >1 such that for every q, q⁰ ∈suppχj,

R^≥3_V (q) R^≥3_V (q⁰)

±1

≤C . (2.9)

Assume α ∈N^d of length|α| ∈ {1,2}. For everyj ∈N, observe that ∂_q^αV(q)−∂_q^αV_j²(q)

=| X

3≤|β|≤r β≥α

β!

(β−α)!∂_q^βV(q⁰_j)(q−q_j⁰)^β−α|

≤ X

3≤|β|≤r β≥α

β!

(β−α)!

∂_q^βV(q_j⁰)

q−q_j⁰

|β|−|α|

,

for any q ∈ R^d. Hence regarding the equivalence (2.9), there exists a constant c_α,d,r > 0 (depending as well on the multi-index α, the dimesion d and the degree r of V) so that

∂_q^αV(q)−∂_q^αV_j²(q)

≤ X

3≤|β|≤r β≥α

β!

(β−α)!

R^≥3_V (q_j⁰) |β|

aR^≥3_V (qj)

−|β|+|α|

≤ X

3≤|β|≤r β≥α

β!

(β−α)!a^−|β|+|α|

R^≥3_V (q_j⁰)|α|

≤c_α,d,r

R^≥3_V (q_j⁰)|α|

, (2.10)

holds for all q in the support ofχj, wherecα,d,r = P

3≤|β|≤r

β!a^−|β|+|α|.

In the rest of the proof, let the polynomialV(q) satisfies Assumption 1. In vue of (2.10), we get when |α|= 1

∇V(q)− ∇V_j²(q)

≤c_1,d,r R^≥3_V (q_j⁰), (2.11)

for allj ∈N and anyq∈suppχ_j,wherec_1,d,r = P

3≤|β|≤r

β!a^−|β|+1.Givenκ≥κ₀,assume first that j ∈J(κ).By virtue of the equivalence (2.9), it results from (2.11)

∇V(q)− ∇V_j²(q)

≤c_1,d,rC R^≥3_V (q), (2.12)

for every q ∈suppχ_j.Then we obtain

∇V(q)− ∇V_j²(q)

≤ c_1,d,rC

κ¹⁴ |∇V(q)|¹³

≤ c_1,d,rC

κ¹⁴ |∇V(q)| (2.13)

for all q ∈ supp χ_j. For the above second inequality we know that |∇V(q)| ≥ 1 for every q ∈suppχ_j, indeed since j ∈J(κ),

|∇V(q)| ≥κ³⁴ ≥κ

3 4

0 ≥1. Taking the constant κ₁ ≥κ₀ such that ^c1,d,rC

κ

1 4 1

≤ ¹₂, we get for every κ≥κ₁

|∇V(q)| − |∇V_j²(q)|

≤ |∇V(q)− ∇V_j²(q)| ≤ 1

2|∇V(q)| , for any q ∈suppχ_j when j ∈J(κ). Therefore

1

2|∇V_j²(q)| ≤ |∇V(q)| ≤ 3

2|∇V_j²(q)|

holds for all q ∈suppχ_j when j ∈J(κ).

On the other hand when|α|= 2,by (2.10) and (2.9) there is a constantc_2,d,r >0 so that for all j ∈N

|∂_q^αV(q)−∂_q^αV_j²(q)| ≤c_2,d,rC²R^≥3_V (q)² . (2.14) holds for every q ∈ supp χ_j, where c_2,d,r = P

3≤|β|≤r

β! a^−|β|+2. Given κ ≥ κ₀ assume now j 6∈J(κ).Using the fact that R^≥3_V (q)≥R^=r_V (0) for everyq ∈R^d, we derive from (2.14) that

|∂_q^αV(q)−∂_q^αV_j²(q)| ≤c_2,d,rC²R^≥3_V (q)⁴ R^=r_V (0)² , for all q∈suppχ_j.

Assuming κ ≥ κ₀ and j /∈ J(κ), we obtain using the previous inequality and applying Lemma B.6

X

|α|=2

|∂_q^αV(q)| − X

|α|=2

|∂_q^αV_j²(q)|

≤ X

|α|=2

|∂_q^αV(q)−∂_q^αV_j²(q)| ≤ 1

2|HessV(q)|,

for any q ∈ supp χj with |q| ≥ C2(κ) where C2(κ) is a strictly positive large constant depending on κ. In other words,

1

2|HessV_j²(q)| ≤ |HessV(q)| ≤ 3

2|HessV_j²(q)|

holds for all q ∈suppχ_j with |q| ≥C₂(κ) and j /∈J(κ).

Lemma 2.4. Given two positive operators A and B such that kuk² <hu, Aui ≤ hu, Bui for all u∈ D where D is dense in D(A^1/2), one has

hu, A^α0

(log(A^α0/2))^kui ≤ hu, B^α0

(log(B^α0/2))^kui , (2.15) for all u∈ D, any α₀ ∈[0,1] and every natural number k.

Proof. Assume that A, B are two positive operators so that

kuk² <hu, Aui ≤ hu, Bui , (2.16) holds for allu∈ D.Referring to [Sim] (see Proposition 6.7 and Example 6.8), for any positive operator C and everyα∈(0,1) we can write

C^α = 2 sin(πα) π

Z +∞

0

w^α−1(C+w)⁻¹Cdw . (2.17) From (2.16) and (2.17)

kuk² <hu, A^αui ≤ hu, B^αui , (2.18) for any u∈ D and every α∈[0,1].

Furthermore, for any positive operator C with domain D(C) we define its logarithm for all u∈D(C) by

hu,log(C)ui= lim

α→0⁺hu,C^α−1

α ui , (2.19)

where the operator C^α is given in (2.17).

Using (2.16) and (2.19)

kuk² <hu,log(A)ui ≤ hu,log(B)ui , (2.20) holds for all u∈ D.Integrating (2.18) with respect to α over [0, α₀] whereα₀ ∈[0,1] we get

hu, 1

log(A)(A^α0 −I)ui ≤ hu, 1

log(B)(B^α0 −I)ui. (2.21) Furthermore by (2.20)

hu, 1

log(B)ui ≤ hu, 1

log(A)ui<kuk² . (2.22)

Therefore from (2.21) and (2.22)

hu, A^α0

log(A)ui ≤ hu, B^α0 log(B)ui . holds for any α₀ ∈[0,1].Then by induction on k ∈N,we obtain

hu, A^α0

(log(A))^kui ≤ hu, B^α0

(log(B))^kui , for all α₀ ∈[0,1] and every natural number k.Or equivalently

hu, A^α0

(log(A^α0/2))^kui ≤ hu, B^α0

(log(B^α0/2))^kui , for every α₀ ∈[0,1], k∈N.

Lemma 2.5. AssumeV(q)a polynomial of degree greater than two. Let P

j∈N

χ²_j(q)be a locally finite partition of unity defined as in (2.1).

There is a constant c >0 such that

hu,(1 +D_q²+R^≥3_V (q)⁴)^αui ≤cX

j∈N

hu, χ_j(1 +D_q²+R^≥3_V (q_j⁰)⁴)^αχ_jui , (2.23)

is valid for all u∈ C₀^∞(R^2d) and any α ∈[0,1].

As a consequence, there exists a constant c >0 so that X

j∈N

kL

(1 +D²_q+R^≥3_V (q⁰_j)⁴)¹³

χ_juk² ≥ 1 ckL

(1 +D²_q+R^≥3_V (q)⁴)¹³

uk² , (2.24)

holds for all u∈ C₀^∞(R^2d), where L(s) = _log(s+1)^s+1 for all s≥1.

Proof. We first set E0 =L²(R^2d) andE1 = n

u∈L²(R^2d), hu,(1−∆q+R^≥3_V (q)⁴)ui<+∞o endowed respectively with the norms k · k_E0 =k · k_L2(R^2d) and k · k_E1 defined as follows for all u∈L²(R^2d)

kuk²_E1 =kuk²_L2(R^2d)+kD_quk²_L2(R^2d)+kR_V^≥3(q)²uk²_L2(R^2d)

=k(1−∆_q+R^≥3_V (q)⁴)^1/2uk²_L2(R^2d) .

By Simader theorem (which states that if W ∈ C^∞(R^d) and −∆ +W(x) is a symetric non negative operator on C₀^∞(R^d) then −∆ +W(x) is essentially self adjoint on C₀^∞(R^d)), the operator 1−∆_q+R_V^≥3(q)⁴ is essentially self adjoint onC₀^∞(R^2d) and henceE₁ corresponds to the spectrally defined subspace of L²(R^2d).

Given a partition of unity as in (2.1), define the linear map

T :E₀ →(L²(R^2d))^N, u7→(u_j)j∈N = (χ_ju)j∈N ,

and denoteF₀ := ImT.Notice thatT :E₀ →F₀ is a unitary isometry. Indeed for all u∈E₀, kT uk²_F0 =X

j∈N

kχ_juk²_L2 =kuk²_L2 =kuk²_E0 , (2.25) further the inverse map of T is well defined by

T⁻¹ :F₀ →E₀, (u_j)_j∈N7→u=X

j∈N

χ_ju_j .

Now introduce the set F₁ =

(

(u_j)j∈N ∈F₀, X

j∈N

hu_j,(1−∆_q+R^≥3_V (q⁰_j)⁴)u_ji<+∞

) ,

with its associated norm defined for all (u_j)_j∈N ∈F₁ by k(u_j)j∈Nk²_F1 =X

j∈N

ku_jk²_L2(R^2d)+kD_qu_jk²_L2(R^2d)+kR^≥3_V (q_j⁰)²u_jk²_L2(R^2d)

=X

j∈N

k(1−∆_q+R^≥3_V (q_j⁰)⁴)^1/2u_jk²_L2(R^2d) . Assume u∈E₀. For all j ∈N, letq_j⁰ ∈supp χ_j. Observe that

| kT uk²_F

1 − kuk²_E

1|=|X

j∈N

hu_j,(1−∆_q+R^≥3_V (q⁰_j)⁴)u_ji − hu,(1−∆_q+R^≥3_V (q)⁴)ui|

=|X

j∈N

hu_j,−∆_qu_ji − hu,−∆_qui+X

j∈N

hu_j,(R^≥3_V (q_j⁰)⁴−R^≥3_V (q)⁴)u_ji|

≤ |X

j∈N

hu_j,−∆_qu_ji − hu,−∆_qui|+X

j∈N

hu_j,|R^≥3_V (q_j⁰)⁴−R^≥3_V (q)⁴|u_ji. (2.26) Since we are dealing with cutoff functions satisfying P

j∈N

|∇χ_j|² ≤ c R^≥3_V (q)² and owning to the equivalence R^≥3_V (q)R^≥3_V (q_j⁰) for all q∈suppχ_j, it follows from (2.26)

| kT uk²_F1 − kuk²_E1| ≤c₁X

j∈N

hu_j, R^≥3_V (q_j⁰)⁴u_ji ≤c₁kT uk²_F1 , and

| kT uk²_F

1 − kuk²_E

1| ≤c⁰₁hu, R^≥3_V (q)⁴ui ≤c⁰₁kuk²_E

1 ,

where c₁, c⁰₁ are two strictly positive constants. As a result 1

p(c₁+ 1)kuk_E1 ≤ kT uk_F1 ≤p

(c⁰₁ + 1)kuk_E1 . (2.27) In view of (2.25) and (2.27), we conclude by interpolation that for all α ∈[0,1]

T :E_α →F_α,

verifies kTkL(E_α,Fα) ≤ c^α and kT⁻¹kL(F_α,Eα) ≤ c^α, where E_α and F_α are two interpolated spaces endowed respectively with the norms

kuk_Eα =k(1−∆_q+R^≥3_V (q)⁴)^α/2uk_L²_(R^2d₎ , and

k(v_j)j∈Nk_Fα =X

j∈N

k(1−∆_q+R_V^≥3(q⁰_j)⁴)^α/2u_jk_L²_(R^2d₎ . Hence there is a constant c > 0 so that

hu,(1 +D²_q+R^≥3_V (q)⁴)^αui ≤cX

j∈N

hu, χ_j(1 +D_q²+R^≥3_V (q_j⁰)⁴)^αχ_jui , (2.28) holds for allu∈ C₀^∞(R^2d) and anyα ∈[0,1].In order to prove (2.24), repeat the same process as in Lemma 2.4. Starting with

kuk² <hu,(1 +D²_q+R^≥3_V (q)⁴)^αui ≤cX

j∈N

hu, χ_j(1 +D_q²+R^≥3_V (q_j⁰)⁴)^αχ_jui , (2.29) for all u ∈ C₀^∞(R^2d) and anyα ∈ [0,1], remark that when integrating over α ∈ [0,²₃] we can interchange the sum and the integral in the left hand side of (2.29) since the partition of unity is locally finite.

This completes the proof.

3 Proof of Theorem 1.1

In this section we present the proof of Theorem 1.1. In the sequel for a given polynomial V(q) with degree r greater than two, we always use a locally finite partition of unity

X

j∈N

χ²_j(q) =X

j∈N

χe²_j

R_V^≥3(q_j)⁻¹(q−q_j)

= 1 , where

suppχe_j ⊂B(q_j, a) and χe_j ≡1 in B(q_j, b)

for some q_j ∈ R^d with 0 < b < a independent of the natural numbers j, defined more specifically as in Lemma A.7 with n= 3.

Proof. Let V(q) be a polynomial with degree larger than two that satisfies Assumption 1.

Assume u∈ C₀^∞(R^2d). In the whole proof we denote u_j =χ_ju for all natural number j.

From Lemma 2.1 we get

kK_Vuk²_L2(R^2d) ≥X

j∈N

kK_Vu_jk²_L2(R^2d)−c_dR^≥3_V (q_j)²kpu_jk²_L2(R^2d) . (3.1) Given κ≥κ₀, set

J(κ) = n

j ∈N, such that suppχj ⊂Σ(κ) o

.

For all index j ∈N,let V_j² be the polynomial of degree less than three given by V_j²(q) = X

0≤|α|≤2

∂_q^αV(q_j⁰)

α! (q−q_j⁰)^α , where

( q⁰_j =q_j if j ∈J(κ) q⁰_j ∈(suppχ_j)∩

R^d\Σ(κ) else.

We associate with each polynomialV_j² the Kramers-Fokker-Planck operatorK_V²

j.Observe that using the parallelogram law 2(kxk²+kyk²)− kx+yk² =kx−yk² ≥0,

X

j∈N

kK_Vu_jk²_L2(R^2d)=X

j∈N

kK_V²

ju_j+ (K_V −K_V²

j)u_jk²_L2(R^2d)

≥ 1 2

X

j∈N

kK_V²

ju_jk²_L2(R^2d)− k(∇V(q)− ∇V_j²(q))∂_pu_jk²_L2(R^2d) (3.2) On the other hand, by (2.5) in Lemma 2.3

X

j∈N

k(∇V(q)− ∇V_j²(q))∂pujk²_L2(R^2d)≤c1,d,r

X

j∈N

R^≥3_V (q_j⁰)²k∂pujk²_L2(R^2d) . (3.3)

Combining (3.1), (3.2) and (3.3) we get immediately kK_Vuk²_L2(R^2d) ≥ 1

2 X

j∈N

kK_V²

ju_jk²_L2(R^2d)−c_1,d,rR^≥3_V (q_j⁰)²k∂_pu_jk²_L2(R^2d)−c_dR^≥3_V (q_j)²kpu_jk²_L2(R^2d)

Therefore, making use of the equivalence (A.5), it follows kK_Vuk²_L2(R^2d)≥ 1

2 X

j∈N

kK_V²

ju_jk²_L2(R^2d)−c⁰_d,rR^≥3_V (q_j⁰)²hu_j, O_pu_ji_L²_(R^2d₎ , (3.4)

where c⁰_d,r= 2(c²_1,d,r+c_d).

Using respectively the Cauchy Schwarz inequality then the Cauchy inequality with epsilon (for any real numbers a, band all >0, ab≤a²+ ₄¹b²),

c⁰_d,rR^≥3_V (q⁰_j)²hu_j, O_pu_ji=c⁰_d,rR^≥3_V (q_j⁰)²Rehu_j, K_V²

ju_ji

≤c⁰_d,rR^≥3_V (q⁰_j)²ku_jk · kK_V²

ju_jk

≤

c⁰_d,rR^≥3_V (q_j⁰)²2

ku_jk²+ 1 4kK_V²

ju_jk² . Putting the above estimate and (3.4) together we obtain

kK_Vuk² ≥X

j∈N

1 4kK_V²

ju_jk²−(c⁰_d,r)²R^≥3_V (q_j⁰)⁴ku_jk² . (3.5) From now on assume κ ≥ κ₁, where κ₁ ≥ κ₀ is introduced in Lemma 2.3. Remember as well that the constants C₁, C₂(κ) are given respectively in Assumption 1 (see (1.8)) and Lemma 2.3 (see (2.7)). By introducing C(κ)≥max(C1, C2(κ)) , which will be fixed later, we set for each κ,

I(κ) = n

j ∈N, such that suppχ_j ⊂

q ∈R^d, |q| ≥C(κ) o .

The rest of the proof is divided into three steps. The first one is devoted to the control of the terms in the the left hand side of (3.5) for which j ∈ I(κ) for some large κ ≥ κ₀ to be chosen. At the end of the first step the constantsκ > κ1 and C(κ)≥max(C1, C2(κ)) will be fixed. So on, the second step is concerned with the remaining terms for which the support of the cutoff functions χ_j are included in some closed ballB(0, C⁰(κ)). We finally sum up all the terms and refer to Lemma 2.5 after some elementary optimization trick in the last step.

Step 1, j∈I(κ), κ≥κ₁ to be fixed: As proved in [BNV], there is a constant c >0 such that for all j ∈I(κ)

kK_V²

ju_jk²+A_V²

jku_jk² ≥c

kO_pu_jk²+kh∂_qV_j²(q)i^2/3u_jk²+khD_qi^2/3u_jk

, (3.6) where

A_V²

j = max{(1 + Tr_+,V²

j)^2/3,1 + Tr_−,V²

j}

= max{(1 + Tr_+,V(q⁰_j))^2/3,1 + Tr−,V(q_j⁰)} . Hence there is a constant C > 0 so that

kK_V²

ju_jk²+ (1 + 10C)t⁴_jku_jk² ≥C

kO_pu_jk²+kh∂_qV_j²(q)i^2/3u_jk²

+khD_qi^2/3u_jk+ 10Ct⁴_jku_jk²

, (3.7)