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The Kato Smoothing Effect for Regularized Schr\"odinger Equations in Exterior Domains

We prove, under the exterior geometric control condition, the Kato smoothing effect for solutions of an inhomogenous and damped Schr\"odinger equation on exterior domains.


Introduction and results
This paper is devoted to the study of a smoothing effect for a damped Schrödinger equation on exterior domain. In order to formulate the results, we shall begin by recalling some results for Schrödinger equation linking the regularity of solutions and the geometry of domain where these equations are posed. It is well known that the free Schrödinger equation enjoys the property of the C ∞ smoothing effect, which can be described as follows: For any distribution u 0 of compact support, the solution of the Cauchy problem (i∂ t + ∆)u = 0 in R × R d u |t=0 = u 0 , is infinitely differentiable with respect to t and x when t = 0 and x ∈ R d .
Another type of smoothing effect says that if u 0 ∈ L 2 (R d ) then the solution of the Schrödinger equation satisfies the Kato 1 2 -smoothing effect (H 1/2 -smoothing effect): This property of gain of regularity has been first observed in the case of R d in the works of Constantin-Saut [12], Sjölin [31] and Vega [33] and it has been extended locally in time to variable coefficient operators with non trapping metric by Doi ( [13,15])).
In the case of domains with boundary Burq, Gérard and Tzvetkov [11] proved a local smoothing estimate for exp(it∆) in the exterior domains with non-trapping assumption. Using the T T ⋆ argument, the proof of the smoothing effect with respect to initial data in [11] is reduced to the non-homogeneous bound which, by performing Fourier transform in time, can be deduced from the bounds on the cut-off resolvent: The resolvent bound, for which the non-trapping assumption plays a crucial role, is proven for |λ| >> 1 in greater generality by Lax-Phillips [21], Melrose-Sjostrand [24,25], Vainberg [32] and Vazy-Zworski [34] The Kato-effect has been extended by Robbiano and Zuily in [30] to variable coefficients operators with unbounded potential in exterior domains with non trapping metric. The proof of their result is reduced to an estimate localized in frequency which has been established by contradiction using in a crucial way the semiclassical defect measure introduced by P. Gerard [17] (see also [22]). The use of the microlocal defect measure to prove an estimate by contradiction method (Wilcox [35]) go back to Lebeau [22]. This idea has been followed with success by several authors (see Burq [8,9,10] Aloui and Khenissi [3,4,20]).
In [10], Burq proved that the non trapping condition is necessary for the H 1/2 smoothing effect and showed, in the case of several convex obstacles satisfying certain assumptions, the smoothing effect with an ε > 0 loss: where χ is compactly supported. On the other hand, the non-trapping assumption is also equivalent to the uniform decay of the local energy for the wave equation (see [21,28,23]). For the trapping domains, when no such decay is hoped, the idea of stabilization for the wave equation is to add a dissipative term to the equation to force the energy of the solution to decrease uniformly. There is a large literature on the problem of stabilization of wave equation. In the case of bounded domains, we quote essentially the work of J. Rauch and M. Taylor [29] and the one of C. Bardos, G. Lebeau and J. Rauch [6] whose introduced and developed the geometric control condition (GCC). This condition that asserts, roughly speaking, that every ray of geometric optics enters the region where the damping term is effective in a uniform time, turns out to be almost necessary and sufficient for the uniform exponential decay of waves. In [3], Aloui and Khenissi introduced the Exterior Geometric control condition (see below Definition 1.1) and hence extended the result of [6] to the case of exterior domains (see also [4] ).
Recently, by analogy with the stabilization problem the first author [1,2] has introduced the forced smoothing effect for Schrödinger equation in bounded domains; it consists to act on the equation to produce some smoothing effects. More precisely he considered the following equation in Ω, u| R + ×∂Ω = 0, (1.1) where Ω is a bounded domain and ∆ D is the Dirichlet-Laplace operator on Ω.
Using the strategy of [11], Aloui [2] proved a weak Kato -Smoothing effect: where 0 < ε < T < ∞ and v 0 ∈ H s D (Ω), (See [2] for the definition of H s D ). By iteration of the last result, Aloui deduced also a C ∞ -smoothing effect for the regularized Schrödinger equation (1.1). Recently, Aloui, Khenissi and Vodev [5] have proved that the Geometric control condition is not necessary to obtain the forced C ∞ -smoothing effect.
On the other hand, using the arguments of [11], we can prove, for the equation (1.1) in exterior domains, the cut-off resolvent bound, which is sufficient to deduce the non-homogenous bound. But, unfortunately, the generator operator ∆ D − ia(x)(−∆ D ) 1 2 a(x) is not self-adjoint and then the T T ⋆ argument fails. For this reason, we can not prove (with this strategy) the weak Kato-smoothing effect (1.2) for exterior domains.
The question now is the following: Can we establish the Kato-smoothing effect for the regularized Schrödinger equation (1.1) for which the Geometric Control Condition is necessary? and if so, does this result still hold for exterior problems?
In this paper, we give an affirmative answer. Indeed, under the Exterior Geometric Control condition, we prove the Kato-smoothing effect and the non homogenous bound for the regularized Schrödinger equation in exterior domains. Notice that the case of bounded domains can be treated by the same method.
In order to state our results, we give several notations and assumptions. Let K be a compact obstacle in R d whose complement Ω an open set with C ∞ boundary ∂Ω andP be a second-order differential operator of the form where coefficients b jk and V are assumed to be in C ∞ (R d ), real valued, and b jk = b kj , 1 ≤ j, k ≤ d.
Throughout this paper, x := (1 + |x| 2 ) 1 2 and we denote by S Ω (M, g) the Hörmander's class of symbols if M is a weight and the metric We shall denote by p the principal symbol ofP , namely and we assume that Under the assumptions (1.4) and (1.5), the operatorP is essentially self-adjoint on C ∞ 0 (Ω) and we denote by P its self-adjoint extension. Now we set Λ = ((1 + C 0 )Id + P ) 1/2 , which is well defined by functional calculus of self-adjoint positive operators. We consider the following regularized Schrödinger equation where (u 0 , f ) ∈ C ∞ 0 (Ω) × C ∞ 0 (]0, +∞) × Ω) and a ∈ C ∞ 0 (Ω). Let's recall the Exterior Geometric Control (E.G.C.) condition [3] Definition 1.1 (E.G.C.). Let R > 0 be such that K ⊂ B R = {|x| < R} and ω be a subset of Ω. We say that ω verifies the Exterior Geometric Control condition on B R (E.G.C.) if there exists T R > 0 such that every generalized bicharacteristic γ starting from B R at time t = 0, is such that: • γ leaves R + × B R before the time T R , or • γ meets R + × ω between the times 0 and T R .
We assume also that the bicaracteristics have no contact of infinite order with the boundary (see, for a precise statement, Definition 2.11).
where u denotes the solution of (1.6). Working withũ = e i(1+C0)t u, one may assume V ≥ 1 in (1.5) and Λ = P 1/2 , which will be assumed in the sequel. It turns into the following equation where P ≥ 1.

Remarks 1.3.
1. When the obstacle is nontrapping, we obtain the result of Robbiano Zuily [30] by taking a(x) = 0 and moreover, we improve their result to non homogenous bound.
2. If we consider the equation in a bounded domain Ω of R d , and replace the exterior geometric condition (E.G.C) by the classical microlocal condition of Bardos-Lebeau-Rauch [6], we can still prove the Kato-effect and then we improve the result of Aloui [2].
3. If there is a trapped ray which does not intersect the regularized region, due to Burq [10], the Kato-effect does not hold. In this context, our result is thus optimal.
The rest of the paper is organized as follows: Section 2 is devoted to the proof of Theorem 1.2 while in the Section A we shall prove some Lemmata used in Section 2.

Proofs
Let's describe the strategy of the proof of theorem 1.2. In a first step, we reduce the estimate (1.7) to an analogue one localized in frequencies. By following a contradiction argument, we can construct an adapted microlocal defect measure. Our aim in the rest of the proof is to obtain a contradiction on this measure. First, we prove that this measure is not identically null. Next, we show that it is null on incoming set and on {a 2 > 0}. Finaly, using the geometrical assumption (E.G.C.) and that the support of this measure is propagated along the generalized flow, we conclude that the measure is identically null. This gives the contradiction.

Reduction to an estimate localized in frequency
We recall the Paley-Littlewood decomposition. Let Φ ∈ C ∞ 0 ([0, +∞)) be a decreasing function such that ψ(4 −n s), and using P ≥ 1, we have For support reason thus there exists C > 0 such that for all u ∈ L 2 (Ω), In the sequel we denote by h n = 2 −n and u n = u hn = ψ(h 2 n P )u.

Construction of microlocal defect measure
In this section we will prove the localized frequency estimate (2.4) by a contradiction argument and using microlocal defect measure. More precisely, let u h solution of We will prove by contradiction the following estimate, Assuming it is false. Taking C = k ∈ N, we deduce sequences h k → k→+∞ 0, u 0 k = u h k (0) ∈ L 2 (Ω) and g k = g h k ∈ L 2 (Ω) such that, (2.14) We normalize by the left term in (2.13), thus where, for simplicity, we have denoted u h k = u k . By the Lemma A.1 we have The sequence (u k ) is bounded in L 2 loc (R t , L 2 loc (Ω)). Indeed, for R > 0 , there exists c > 0 such that x −2s ≥ c, ∀x ∈ B(0, R) and then we have It follows from (2.17) that the sequence (W k ) is bounded in L 2 (R t , L 2 loc (R d )). We associate to a symbol b = b(x, t, ξ, τ ) ∈ C ∞ 0 (T * R d+1 ) the semiclassical pseudo-differential operator (pdo) by the formula where ϕ ∈ C ∞ 0 (R d ) is equal to one on a neighborhood of the x-projection of the support of b. As in [30] we can associate to (W k ) a semi-classical measure µ. More precisely, Proposition 2.2. There exists a subsequence (W σ(k) ) and a Radon measure µ on T * R d+1 such that for every b ∈ C ∞ 0 (T * R d+1 ) one has We prove first that the measure µ satisfies the following property.
Proposition 2.3. The support of µ is contained in the characteristic set of the operator D t + P Proof. According to (2.18), it is obvious that Therefore it remains to show that if m 0 = (x 0 , t 0 , ξ 0 , τ 0 ) with x 0 ∈ Ω, t 0 ∈ [0, T ], and τ 0 +p(x 0 , ξ 0 ) = 0 then m 0 / ∈ supp µ. For simplicity, we shall denote the sequence W σ(k) by W k .
. Recall that we have W k = 1 [0,T ] 1 Ω u k and that (u k ) is bounded sequence in L 2 ([0, T ], L 2 loc (Ω)). We set As in [30] we have On the other hand, since we have and ϕ ∈ C ∞ 0 (Ω), Then I k is a sum of four terms, For the first term I 1 k , we use the Lemma A.6, we have, and we deduce, Then we obtain, that I 1 k goes to zero by (2.15). For the second term I 2 k , .
We would like to show that one can find a neighborhood U x0 of Indeed this will imply that the point m 0 (x 0 , t 0 , ξ 0 , τ 0 ) (with τ 0 + (x 0 , ξ 0 ) = 0) does not belong to the support of µ as claimed. Formula (2.27) will be implied, by where R is a second-order differential operator, D ′ = (D 2 , ..., D d ) and L(x, D) a first order differential operator. Let us set Making the change of variable x = F −1 (y) on the right-hand side of the second line of (2.28), we see that , and ψ ∈ C ∞ 0 (U 0 ), ψ = 1 on π y suppb. To prove (2.28) it is sufficient to prove that, , ψ 0 ψ 1 = 1 on π y supp θΦχ; According to (2.31) we have, Therefore (2.28) will be proved if we can prove that To estimate the term A 2 k we need a Lemma. With U 0 introduced in (2.29), we set U + 0 = {y ∈ U 0 : y 1 > 0}. We consider a smooth solution of the problem: There exists C > 0 such that for any solution u of (2.34) and all h in ]0, 1], we have Proof of the Lemma. It is analogue to the proof of [30, Lemma A.6]. We replace in the previous Lemma g by , we obtain easily the following corollary.
where v k has been defined in (2.30) andχ ∈ C ∞ 0 (R d ). Let us go back to the estimate of A 2 k defined in (2.32). We have dt.

The microlocal defect measure does not vanish identically
First let us prove that the sequence (u k ) have mass in a compact domain.
Proof of Lemma. We prove the Lemma by contradiction. Assume that where R 0 is large enough such that supp a ⊂ {|x| ≤ R 0 /2}. Let χ ∈ C ∞ (R d ) such that χ = 1 for |x| > 2 and χ = 0 for |x| < 1. We set χ R (x) = χ(x/R) and by the choice of R 0 we have aχ R = χ R a = 0 . The function v k := χ R u k satisfies For the first term of the right hand side of (2.37) we have, where (·, ·) means the scalar product in L 2 (Ω), Using that χ R a = 0, we have And then and It follows that the symbol of [30,Lemma 4.2]) and we have (Ω) and we get To estimate we have with ψ 2 (s) = s −1 ψ 1 (s) andχ a smooth function such that,χ = 1 for |x| ≥ 1 andχ = 0 for Then the second term of (2.40) is bounded on L 2 (Ω) by Ch k . Finally, we yield by Lemma A.6, According to (2.14) and (2.15), we conclude that the second term of the right hand side of (2.37) goes to zeros when k tend to +∞ where we have used, first that E −1 ∂ x is bounded on L 2 , x s is estimate by CR s on support of χ 1 and ∂ x χ R is the product of a bounded function by R −1 , second, the symbol of where δ k → 0 when k → +∞, C is independent of R and C R may depend of R. Then we have This with (2.16) implies a contradiction with (2.36) and proves the Lemma.
In the sequel, for simplicity, we shall denote the sequence u kν found in Lemma 2.6 by u k . Thus there exist R 0 > 0, k 0 > 0 such that when R > R 0 and k > k 0 . We consider We recall that w k (t) = 1 Ω u k (t).
Proposition 2.7. There exist positive constants A 0 , R 0 , k 0 such that Corollary 2.8. The measure µ does not vanish identically.

Proof of Proposition . Set
It is easy to see that ψ ∈ L ∞ (R) and | ψ(τ )| ≤ 1 A for all τ ∈ R. We have From [30, See the proof of Proposition 6.1] we know that ψ A (h 2 k D t )δ t=a L 2 (R) ≤ Ch −1 k , then we deduce that Using (2.22) and (2.15), we can prove easily that From (2.14) we can see that Now, for B 2 k we argue as in [30, See the proof of Proposition 6.1]. Letθ ∈ C ∞ 0 (0, +∞) suchθ = 1 on the support of ψ and letθ 1 (s) = sθ(s). We have [30] and the fact that Taking k and A sufficiently large we obtain where C R1 depends on R 1 and The proof does not depend on the equation, so it remains valid in our case. Nevertheless we recall the proof in the sequel for the convenience of the reader. Before we give the end of the proof of proposition 2.7. Taking R sufficiently large and using (2.44), we obtain Return to the proof of (2.45). We have where θ ∈ C ∞ 0 (R) satisfying θ(t) = 1 if t ∈ supp(θ 1 ) and θθ 1 = θ 1 . We have by Lemma 6.3 [30] where χ 1 ∈ C ∞ 0 (Ω), χ 1 = 1 on supp(χ 1 ). Combining (2.46), (2.47) and (2.48), we obtain (2.45).

The microlocal defect measure vanishes in the incoming set
In this section we prove that the microlocal defect measure µ vanishes in the incoming set.
First remind some notation introduced in [30] section 7. We keep the same notation when it is possible.
We denote by We remind the results proved in [30] in section 7, Lemma 7.5 and Corollary 7.6. A part of the proof is in Doi [15]. We use the Weyl quantification of symbol which is denoted by Op w .
There exist a symbol Φ ∈ S(1, g) such that 0 ≤ Φ ≤ 1 and a symbol λ 1 ∈ S(1, g) such that, there exist two positive constants C, C ′ such that, Let M large enough such that, Here and in the sequel (·|·) and · denote the L 2 (Ω) inner product and norm respectively. The cutoff make sense with this L 2 product. We set, (2.54) For support reasons, we have a(1 − ϕ 1 ) = 0 thus we deduce, We have, for a constant C 1 > 0 Following (2.49) and (2.52), the support of λ 1 and ϕ 1 are disjoint, thus, taking account of (2.53), we have , and d(x 0 , ξ 0 ) = 1. According to (2.50), (2.51) and Gårding inequality, we get, where we have set Integrating (2.61) between 0 and t for t ∈ [0, T ] we obtain, By Gronwall's inequality we have for t ∈ [0, T ], Using (2.63) in (2.62), we get Following (2.14) and (2.16) we obtain supported in a neighborhood sufficiently small around (t 0 , τ 0 ) and taking account that d is supported in a neighborhood of (x 0 , ξ 0 ), we have

The microlocal defect measure vanishes on {a 2 > 0}
The goal of this section is to prove that the microlocal defect measure vanishes on {a 2 > 0}. More precisely we have the following proposition.
Proof. Taking the imaginary part of the L 2 ([0, T ] × Ω) inner product of (2.64) with u k /h k , we obtain, Using that P is self-adjoint, we get Let θ ∈ C ∞ 0 ((0, +∞)) with θ = 1 on the support of ψ. Thus we have θ(h 2 k P )u k = u k . Let θ(t) = t −1/4 θ(t), we haveθ ∈ C ∞ 0 ((0, +∞)) and, We have also, , we have by standard symbolic semi-classical calculus Thus from (2.65), we have, From (2.73) and using aW k which is the identity on T * R d+1 M \{0} (see [30] for details). Consider, near a point of the boundary z = (x 1 , x ′ , t) ∈ ∂M a geodesic system of coordinates given by the diffeomorphism F in (2.29), for which z = (0, 0, t), M = {(x 1 , x ′ , t), x 1 > 0)} and the operator D t + P has the form (near z) with R a second order tangential operator and S a first order operator. Denoting r(x 1 , x ′ , ξ ′ ) the principal symbol of R and r 0 = r| x1=0 , the cotangent bundle to the boundary T ⋆ ∂M \{0} can be decomposed (in this coordinate system) as the disjoint union of the following regions: • and the glancing region For the purpose of the proofs, it is important to consider the following subsets of the glancing region: Definition 2.11. We say that the bicaracteristics have no contact of infinite order with the boundary Now, we recall the definition of ν the measure on the boundary. By the Lemma 2.4, we see that the sequence (1 [0,T ] h k ( ∂w k ∂n )) is bounded in L 2 (R t × L 2 (∂Ω)). Therefore with the notations in (2.18) and Proposition 2.2, we have the following Lemma. Lemma 2.12. There exists a subsequence (W σ1(k) ) of (W σ(k) ) and a Radon measure ν on T ⋆ (∂Ω×R t ) such that for every b ∈ C ∞ 0 (T * (∂Ω × R t )) we have We give now two results on propagation of support of microlocal defect measure. The first, Proposition 2.13 for point inside T ⋆ M and the second, Proposition 2.15 at the boundary of M . Proof. It is enough to prove (2.77) when b(x, t, ξ, τ ) = Φ(x, ξ)χ(t, τ ) with π x supp Φ ⊂ V x0 ⊂ Ω. Let ϕ ∈ C ∞ 0 (Ω) be such that ϕ = 1 on V x0 . We introduce We claim that we have We have (1) by pseudo-differential calculus. It was proved in [30, proof of Proposition A.9] that the first and the second terms tend to zero when k → +∞. Since g k → 0 in L 2 loc , the third term tends also to zero when k → +∞. For the fourth term, according to (2.74) and (2.76), it is easy to see that it tends to zero. Thus (2.78) is proved.
In another side, it was shown in the Proposition A.9 [30] that It follows from (2.78), (2.77) that µ, H p b = 0 if b = Φχ, which implies our proposition.
We consider now the case of point m 0 = (x 0 , ξ 0 , t 0 , τ 0 ) ∈ T ⋆ R d+1 with x 0 ∈ ∂Ω. We take, as in [30], a neighborhood U x0 so small that we can perform the diffeomorphism F described in (2.29).
We first recall the Lemma A.10 established in [30].
Proposition 2.15. With the notations of Lemma 2.14 for every b ∈ C ∞ 0 (T ⋆ (U 0 × R t )), we have Proof. This proof is similar to the one of Proposition A.12 [30]. We recall some results from [30] used to prove Proposition A.12.
Lemma 2.17 (Lemma A.15 [30]). Let for j = 0, 1, The previous Lemmas still hold in our case, since they are independent of the equation.
Then lim Proof. The proof is similar to the one of Lemma A.14 [30]. We have, [30], the first and the second terms of the RHS in the previous identity tend to zero. Using that g k L 2 → 0, we can prove that the third term tends also to zero.
Following Lemma A.6 and (2.73) the forth term tends to zero. We conclude that I j k tends to zero. For J j k we argue as for I j k . Proof of Proposition 2.15. From Proposition 2.3 (τ + p)µ = 0, so we have Let consider the identity (2.79), by Lemma 2.18, the LHS tends to zero when k → +∞. By the semiclassical symbolic calculus, we have where c j ∈ C ∞ 0 (U 0 ×R d+1 ), ϕ 1 = 1 on supp ϕ, and {p, b 0 +b 1 η 1 } = 2 j=0 c j η j 1 . Hence, using Lemma 2.17 and Lemma 2.12, the RHS of (2.79) tends to We conclude that µ, which proves the Proposition 2.15.
Proposition 2.19. With the notations of [30], we have Proof. The proof is the same as of Lemma A.17 in [30]. By measure theory methods (see [8], [9] and [30]), the propagation of the measure µ along the generalized bicharacteristic flow is equivalent to Propositions 2.13, 2.15 and 2.19.

A Appendix
In this appendix, we prove some Lemmas used above. We recall the Helffer-Sjöstrand formula (see [14]) used extensively in this section. To introduce it we recall some notations. Let θ ∈ C ∞ 0 (R) and let ϕ ∈ C ∞ 0 (R) such that ϕ(t) = 1 if |t| ≤ 1 and ϕ(t) = 0 if |t| ≥ 2. Let N ≥ 2, we setθ thenθ ∈ C ∞ 0 (R 2 ) and satisfies We callθ an almost analytic extension of θ. Let P a self adjoint operator. We have the following Helffer-Sjöstrand formula The formula does not depend of N and ϕ. We recall the estimates proved in [30], Lemma A.22, we have for f = (z − h 2 P ) −1 u and ℑmz = 0, Let h n a sequence such that h n > 0 and h n → 0 when n → +∞. In the sequel, for simplicity we denote such a sequence by h. We say h → 0 instead of h n → 0 when n → +∞.
Lemma A.1. Let u h and g h satisfying and we assume that Using Thus The assumptions and the definition of k imply the Lemma. Let ψ : R → R such that ψ(t) = 0 if t ≤ α or t ≥ β where 0 < α < β.
Proof. We prove (A.4) for u ∈ C ∞ 0 (Ω). Taking the adjoint, (A.4) is equivalent to Thus it is enough to prove and Now we prove (A.5). Following the Helffer-Sjöstrand formula, whereψ is an almost analytic extension of ψ, we have where α j and c are compact supported. Following (A.7), we have two types of terms to control. First we remark that where β j and d are compact supported, following (A.7) and estimates (A.3) (with N = 3) we obtain Thus following (A.1), we have where |γ k (x)| + |γ(x)| ≤ C x s−1 ≤ C ′ , with the above notations, we have following (A.3), thus, following the proof of (A.10), we prove (A.5).
To prove (A.6), following the Helffer-Sjöstrand formula we have, With the notation above, it is enough to prove x s ], the first term is estimated following the proof of (A.9). To estimate the second term, we follow the proof of (A.11). Thus we obtain (A.12) which achieve the proof of Lemma.
Lemma A.3. Let s ∈ [0, 1] and χ a smooth function such that χ = 1 for |x| ≥ 1. We set χ R (x) = χ(x/R). There exists C > 0 such that for all u ∈ L 2 (Ω), Proof. The proof is very close to the one of Lemma A.2. By the same argument it is sufficient to prove From the Helffer-Sjöstrand formula, we obtain (as in (A.7)) Modulo negative power of ℑmz, in the first term of (A.15) h∂ xj (z − h 2 P ) −1 is bounded on L 2 (Ω) and, because x s /R is bounded on the support of χ ′ (x/R), we can write x s [(z − h 2 P ), χ R ] as a sum of term α(x)h 2 ∂ xj . This yields that x s [(z − h 2 P ), χ R ](z − h 2 P ) −1 is bounded on L 2 (Ω) by Ch modulo negative power of ℑmz. This gives the result for the first term in (A.15). Writing and arguing as for the first term, we obtain (A.13). By the same arguments and using that hV 1/2 (z − h 2 P ) −1 is bounded on L 2 (Ω) modulo negative power of ℑmz (see [30,Lemma A.22]), we obtain (A.14).
Proof. By Helffer-Sjöstrand formula, we have, with the notation of Lemma A.2, thus, following the proof of (A.10), we achieve the proof of Lemma in this case. If s < 0, we write Putting this in (A.16), we obtain two terms. The first gives The second gives 19) and the Helffer-Sjöstrand formula, we obtain the Lemma.
Remarks A.5. In the Lemma A.4, we can remove the assumption |s| ≤ 1, by commuting x s with (z − h 2 P ) −1 several times, but Lemma A.4 is sufficient for us in the sequel.
Proof. We have To estimate A, we can write By support properties of ψ and by the Lemma A.7, we have By Lemma A.4 we see easily that Summing with respect n, we obtain We can consider the right hand side of (A.21) as a convolution ℓ 1 * ℓ 2 and we obtain the estimation of this term by C Summing with respect n, we obtain We have h −α n h 1+α k = 2 −(1+α)(k−n) 2 −n ≤ 2 −(1+α)(k−n) and (2 −(1+α)j ) ∈ ℓ 1 since 1 + α > 0. We can conclude as for the term A above. We have proved the right inequality of the Lemma.
We prove the other inequality. We have, We have by properties of support of ψ, The estimate of the first term is clear, for the second using Lemma A.4, we get We have h −α+1 n h α k ≤ 2 −(1−α)(n−k) and we can conclude as above by convolution argument. For E, it follows from the support properties of ψ, Lemma A.7 and Lemma A.4, Summing with respect n, we obtain, We have h −α n h 1+α k ≤ 2 −(n−k)(1+α) and we can conclude by convolution argument.
A 1 k = {(j, n) ∈ N 2 , k ≥ j − 2 or k ≥ n − 2, and j ≥ n − 2}, In the sequel, for each set A p k we will prove (A.22). Let ψ 2 ∈ C ∞ 0 (0, +∞) such that ψ 2 = 1 on the support of ψ. We have, For B, we can see that In the last sum |j − n| ≤ 1, then we can estimate this term as the term A.
We have (by Lemma A.4 and Lemma A.10).
In by Cauchy-Schwarz inequality and as all the sums converge if α ∈ (−1, 3/2), we obtain (A.22). Now we will estimate the sum over A 2 k . We have with the function ψ 2 defined above, as ψ 0 (h 2 j P )ψ 2 (h 2 n P ) = 0, because j ≤ n − 2, h −1 j ≤ Ch −1 n , we can end the proof as for the term D above.
Finally we treat the sum over A 3 k . We have, as ψ(h 2 k P )ψ 0 (h 2 j P ) = 0. If (j, n) ∈ A 3 k , we have j ≥ k + 3 then the sum over j is less than Ch k . We obtain, h 5 k u 2 L 2 (Ω) ≤ C u 2 L 2 (Ω) . Which achieve the proof of Lemma.
Proof. We give only a sketch of proof, we use the same technic than before. By the Helffer-Sjöstrand formula, we have where z = (z 1 , z 2 ) and z j = t j + iσ j . First, we can write where the γ's are compactly supported. Second, as Lemma A.11. Let s ∈ [−1, 1], α < 3/2, there exists C > 0 such that for all u ∈ L 2 (Ω), we have Proof. We follow the same strategy than the one for the proof of Lemma A.9. We have to prove, If [j − k| ≥ 2 and |n − k| ≥ 2, the corresponding term in the sum is null. If |j − k| ≤ 1 (the case |n − k| ≤ 1 is symmetric and let to the reader). We consider two cases, the first if n ≥ k + 2, term A in the sequel, and the second if k ≥ n + 2 term B in the sequel.