Spectral inequality and resolvent estimate for the bi-Laplace operator
SSPECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THEBI-LAPLACE OPERATOR
J ´ER ˆOME LE ROUSSEAU AND LUC ROBBIANOA bstract . On a compact Riemannian manifold with boundary, we prove a spectral in-equality for the bi-Laplace operator in the case of so-called “clamped” boundary condi-tions, that is, homogeneous Dirichlet and Neumann conditions simultaneously. We alsoprove a resolvent estimate for the generator of the damped plate semigroup associatedwith these boundary conditions. The spectral inequality allows one to observe finite sumsof eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset ofthe manifold. Moreover, the constant that appears in the inequality grows as exp( C µ / )where µ is the largest eigenvalue associated with the eigenfunctions appearing in the sum.This type of inequality is known for the Laplace operator. As an application, we obtaina null-controllability result for a higher-order parabolic equation. The resolvent estimateprovides the spectral behavior of the plate semigroup generator on the imaginary axis.This type of estimate is known in the case of the damped wave semigroup. As an ap-plication, we deduce a stabilization result for the damped plate equation, with a log-typedecay.The proofs of both the spectral inequality and the resolvent estimate are based onthe derivation of di ff erent types of Carleman estimates for an elliptic operator related tothe bi-Laplace operator: in the interior and at some boundaries. One of these estimatesexhibits a loss of one full derivative. Its proof requires the introduction of an appropriatesemi-classical calculus and a delicate microlocal argument.K eywords : high-order operators; boundary value problem; spectral inequality; resolventestimate; interpolation inequality; controllability; stabilization; Carleman estimate; semi-classical calculus.AMS 2010 subject classification : 35B45; 35J30; 35J40; 35K25; 35S15; 74K20; 93B05;93B07; 93D15. C ontents
1. Introduction 31.1. On Carleman estimates 51.2. A method to prove the spectral inequality for the Laplace operator 61.3. Outline of the proof of the spectral inequality for the bi-Laplace operator 81.4. On Carleman estimates for higher-order elliptic operators 111.5. Some perspectives 121.6. Notation 131.7. Some basic properties of the bi-Laplace operator 13
Date : October 5, 2018.Part of this article was written when both authors were on a leave at Insitut Henri-Poincar´e, Paris. Theywish to thank the institute for its hospitality. Both authors acknowledge support from Agence Nationale dela Recherche (grant ANR-13-JS01-0006 - iproblems - Probl`emes Inverses). The authors also wish to thanktwo anonymous referees who have contributed to improve the presentation of this article by their remarks. a r X i v : . [ m a t h . A P ] N ov J ´ER ˆOME LE ROUSSEAU AND LUC ROBBIANO
2. Estimate away from boundaries 142.1. Simple-characteristic property of second-order factors 142.2. Local Carleman estimates away from boundaries 153. Estimate at the boundary { s = } { s = } , S ) × ∂ Ω E − E \ F F , S ) × ∂ Ω Q k PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 3
1. I ntroduction
Let A be the positive Laplace operator on a compact Riemannian manifold ( Ω , g ), of dimension d ≥
1, with nonempty boundary ∂ Ω . In local coordinates, it reads A = − ∆ = | g | − / (cid:80) ≤ i , j ≤ d D i (cid:0) | g | / g i j D j (cid:1) , where D = − i ∂ . For boundary conditions, say of homogeneous Dirichlet type , we denote by0 < ω ≤ · · · ≤ ω j ≤ · · · , the eigenvalues of the operator A , associated with a family ( φ j ) j ∈ N of eigenfunctions that form a Hilbert basis for L ( Ω ). We refer to this selfadjoint operator as theDirichlet Laplace operator. The following spectral inequality originates from [LR95, LZ98, JL99]. Theorem 1.1.
Let O be an open subset of Ω . There exists C > such that (cid:107) u (cid:107) L ( Ω ) ≤ Ce C ω / (cid:107) u (cid:107) L ( O ) , ω > , u ∈ Span { φ j ; ω j ≤ ω } . (1.1)It provides an observation estimate of finite sums of eigenfunctions. The constant Ce C ω / inthe inequality is in fact optimal if O (cid:98) Ω [JL99, LL12], and can be seen as a measure of theloss of orthogonality of the eigenfunctions φ j when restricted to O . This inequality has variousapplications. It can be used to prove the null-controllability of the heat equation [LR95] (seealso the review article [LL12]), the null-controllability of the thermoelasticity system [LZ98], thenull-controllability of the thermoelastic plate system [BN02, Mil07], and the null-controllabilityof some systems of parabolic PDEs [L´ea10]. It can also be used to estimate the ( d − ff measure of the nodal set of finite sums of eigenfunctions of A , in the case of an analytic Rie-mannian manifold [JL99], recovering the result of [Lin91], that generalizes a result of [DF88] foreigenfunctions.Consider now the unbounded operator acting on H ( Ω ) × L ( Ω ) A = (cid:32) − A α (cid:33) , with domain D ( A ) = ( H ( Ω ) ∩ H ( Ω )) × H ( Ω ), where α ( x ) is a nonnegative function. One canprove the following resolvent estimate [Leb96]. Theorem 1.2.
Let O be an open subset of Ω and α be such that α ( x ) ≥ δ > on O . Then, theunbounded operator i σ Id −A is invertible on H = H ( Ω ) × L ( Ω ) for all σ ∈ R and there existK > and σ > such that (1.2) (cid:107) ( i σ Id −A ) − (cid:107) L ( H , H ) ≤ Ke K | σ | , σ ∈ R , | σ | ≥ σ . This resolvent estimate allows one to deduce a logarithmic type stabilization result for thedamped wave equation ∂ t y + Ay + α∂ t y = , y | t = = y , ∂ t y | t = = y , y | [0 , + ∞ ) × ∂ Ω = , for y and y chosen su ffi ciently regular, e.g. ( y , y ) ∈ D ( A ) [Leb96, Bur98, BD08].It is quite natural to wish to obtain similar inequalities for higher-order elliptic operators on Ω ,along with appropriate boundary conditions. The bi-Laplace operator, that can be encountered inmodels originating from elasticity for example, appears as a natural candidate for such a study. Tounderstand some of the issues associated with the boundary conditions one may wish to imposelet us consider the case of a spectral inequality of the form of (1.1). If the boundary conditions What we describe is yet valid for more general boundary conditions of Lopatinskii type for the Laplaceoperator.
J ´ER ˆOME LE ROUSSEAU AND LUC ROBBIANO used for the bi-Laplace operator precisely make it the square of the Laplace operator A (with itsboundary conditions) then the spectral inequality is obvious as the eigenfunctions are the same forthe two operators and λ j ≥ (cid:112) λ j is onefor the Laplace operator. To be clearer, let us consider the positive Dirichlet Laplace operator A .If A is the bi-Laplace operator on Ω along with the boundary conditions u | ∂ Ω = ∆ u | ∂ Ω = φ j ) j ∈ N introduced above, is in fact composed of eigenfunctions for A associatedwith the eigenvalues λ j = ω j . This set of boundary conditions is known as the “hinged” boundaryconditions. We refer to this operator as the “hinged” bi-Laplace operator, and, for this operator,with Theorem 1.1, we directly have the following spectral inequality, for O ⊂ Ω , (cid:107) u (cid:107) L ( Ω ) ≤ Ce C λ / (cid:107) u (cid:107) L ( O ) , λ > , u ∈ Span { φ j ; λ j ≤ λ } . (1.3)One is naturally inclined to consider another set of boundary conditions, the so-called “clamped”boundary conditions, u | ∂ Ω = ∂ ν u | ∂ Ω =
0, where ν is the outward normal to ∂ Ω . We refer tothis operator as the “clamped” bi-Laplace operator. It is sometimes referred to as the Dirichlet-Neumann bi-Laplace operator. Eigenfunctions of the “clamped” bi-Laplace operator are not re-lated to eigenfunctions of the Dirichlet Laplace operator. In fact, observe that an eigenfunctionof the “clamped” bi-Laplace operator cannot be an eigenfunction for the Laplace operator A , in-dependently of the boundary conditions used for A . Indeed, from unique continuation arguments,if a H -function φ is such that A φ = λφ on Ω and φ | ∂ Ω = ∂ ν φ | ∂ Ω =
0, then φ vanishes identi-cally. Thus, a spectral inequality for the “clamped” bi-Laplace cannot be deduced from a similarinequality for the Laplace operator A with some well chosen boundary conditions. Yet, such aninequality is valuable to have at hand, in particular as the “clamped” bi-Laplace operator appearsnaturally in models. It is however often disregarded in the mathematical literature and replacedby the “hinged” bi-Laplace operator for which analysis can be more direct, in particular for thereasons we put forward above. A resolvent estimate of the form of (1.2) is also of interest towardsstabilization results.The main purpose of the present article is to show that a spectral inequality of the form of (1.1)and a resolvent estimate of the form (1.2) hold for the “clamped” bi-Laplace operator and, moregenerally, to provide some analysis tools to carefully study fourth-order operators that have a prod-uct structure. Carleman estimates will be central in the analysis here and we shall show how theirderivation is feasible when the so-called sub-ellipticity condition does not hold, which is typicalfor product operators. If B is the “clamped” bi-Laplace operator, that is, the unbounded operator B = ∆ on L ( Ω ), with domain D ( B ) = H ( Ω ) ∩ H ( Ω ), which turn B into a selfadjoint operator,let ( ϕ j ) j ∈ N be a family of eigenfunctions of B that form a Hilbert basis for L ( Ω ), associated withthe eigenvalues 0 < µ ≤ · · · ≤ µ j ≤ · · · (the selfadjointness of B and the existence of such afamily are recalled in Section 1.7 below). We shall prove the following spectral inequality. Theorem 1.3 (Spectral inequality for the “clamped” bi-Laplace operator) . Let O be an open subsetof Ω . There exists C > such that (cid:107) u (cid:107) L ( Ω ) ≤ Ce C µ / (cid:107) u (cid:107) L ( O ) , µ > , u ∈ Span { ϕ j ; µ j ≤ µ } . Note that the spectral inequality of Theorem 1.3 was recently proven in [AE13] and [Gao16].In [AE13] the coe ffi cients and the domain are assumed to be analytic (the techniques used for theproof are then very di ff erent and exploit the analytic properties of the eigenfunctions). In [Gao16],the result is obtained in one space dimension; yet , therein, the factor e C µ / is replaced by e C µ / ,yielding a weaker form of the spectral inequality.We shall present a null controllability result for the parabolic equation associated with B whichis a consequence of this spectral inequality. Such a result can be found in [AE13, EMZ15] inthe case of analytic coe ffi cients and domain. Here, coe ffi cients are only assumed smooth. We PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 5 conjecture that regularity could be lowered as low as W , ∞ for the coe ffi cients in the principal partof the operator. This would require further developments in the line of what is done in [DCFL + H ( Ω ) × L ( Ω ),(1.4) B = (cid:32) − B α (cid:33) , with domain D ( B ) = ( H ( Ω ) ∩ H ( Ω )) × H ( Ω ), where α ( x ) is a nonnegative function. Theorem 1.4.
Let O be an open subset of Ω and α be such that α ( x ) ≥ δ > on O . Then, theunbounded operator i σ Id −B is invertible on H = H ( Ω ) × L ( Ω ) for all σ ∈ R and there existK > and σ > such that (cid:107) ( i σ Id −B ) − (cid:107) L ( H , H ) ≤ Ke K | σ | / , σ ∈ R , | σ | ≥ σ . We shall present a log type stabilization result that is a consequence of Theorem 1.4 for thefollowing damped plate equation ∂ t y + ∆ y + α∂ t y = , y | t = = y , ∂ t y | t = = y , y | [0 , + ∞ ) × ∂ Ω = ∂ ν y | [0 , + ∞ ) × ∂ Ω = . Both the proofs of the spectral inequality and the resolvent estimate are based on Carlemanestimates for the fourth-order operator P = D s + B .The subject of the present article is connected to that of unique continuation, in particularthrough the use of Carleman estimates. Moreover, the spectral inequality of Theorem 1.3 is aquantified version of the unique continuation property for finite sums of eigenfunctions. Thereis an extensive literature on unique continuation for di ff erential operators; yet, positive resultsrequire assumptions on the operator or on the hypersurface across which unique continuation ispursued. For instance, a simple-root assumption is often made following the work of A. Calder´on[Cal58] or the celebrated strong pseudo-convexity condition is assumed following the work ofL. H¨ormander [H¨or58, H¨or63]. For second-order elliptic operators (with smooth complex co-e ffi cients) these assumptions are fulfilled. However, for higher-order operators they may not besatisfied. Counterexamples for the non uniqueness of fourth-order and higher-order operators withsmooth coe ffi cients can be found in [Pli61] and [H¨or75]. See also the monograph [Zui83] for man-ifold positive and negative results. The question of strong unique continuation is also of interest forhigher-order operators; see for instance [AB80] for a positive result and [Ali80] for a large class ofnegative results. Note that the above literature concerns unique continuation properties away fromboundaries . For the results of Theorems 1.3 and 1.4 the analysis we use mainly focuses on theneighborhood of the boundary of the open set Ω . There are few results on unique continuation neara boundary. Under the strong pseudo-convexity condition the unique continuation property can beobtained, even for higher-order operators; see [Tat96] and [BL15]. For the operator P = D s + B that we consider here, the strong pseudo-convexity property fails to hold near the boundary andalso away from it. General approaches as developed in [Tat96, BL15] cannot be used. This is oneof the interests of the present article.1.1. On Carleman estimates.
Carleman estimates are weighted a priori inequalities for the solu-tions of a partial di ff erential equation (PDE), where the weight is of exponential type. For a partialdi ff erential operator Q away from boundaries , it takes the form: (cid:107) e τϕ w (cid:107) L (cid:46) (cid:107) e τϕ Qw (cid:107) L , w ∈ C ∞ c ( Ω ) , τ ≥ τ . J ´ER ˆOME LE ROUSSEAU AND LUC ROBBIANO
The exponential weight involves a parameter τ that can be taken as large as desired. Additionalterms in the l.h.s., involving derivatives of u , can be obtained depending on the order of Q and onthe joint properties of Q and ϕ . For instance for a second-order operator Q , such an estimate cantake the form τ / (cid:107) e τϕ u (cid:107) L + τ / (cid:107) e τϕ D x u (cid:107) L (cid:46) (cid:107) e τϕ Qu (cid:107) L , τ ≥ τ , u ∈ C ∞ c ( Ω ) . (1.5)One says that this estimate is characterized by the loss of a half derivative . This terminologyoriginates from the underlying semi-classical calculus where one gives the same strengths to theparameter τ and to D . Whereas Q is a second-order operator, the l.h.s. only exhibits derivatives orpowers of τ of order 3 /
2. For most operators, this cannot be improved [H¨or63, H¨or85a]. In theproof of a Carleman estimate one introduces the so-called conjugated operator Q ϕ = e τϕ Qe − τϕ ,and estimate (1.5) reads τ / (cid:107) v (cid:107) L + τ / (cid:107) D x v (cid:107) L (cid:46) (cid:107) Q ϕ v (cid:107) L , τ ≥ τ , v = e τϕ u , u ∈ C ∞ c ( Ω ) . This type of estimate was used for the first time by T. Carleman [Car39] to achieve uniquenessproperties for the Cauchy problem of an elliptic operator. Later, A.-P. Calder´on and L. H¨ormanderfurther developed Carleman’s method [Cal58, H¨or58]. To this day, the method based on Carlemanestimates remains essential to prove unique continuation properties; see for instance [Zui83] foran overview. On such questions, more recent advances have been concerned with di ff erentialoperators with singular potentials, starting with the contribution of D. Jerison and C. Kenig [JK85].There, Carleman estimates rely on L p -norms rather than L -norms as in the estimates above. Theproof of such L p Carleman estimates is very delicate. The reader is also referred to [Sog89, KT01,KT02, DSF05, KT05]. In more recent years, the field of applications of Carleman estimates hasgone beyond the original domain; they are also used in the study of: • Inverse problems, where Carleman estimates are used to obtain stability estimates for theunknown sought quantity (e.g. coe ffi cient, source term) with respect to norms on mea-surements performed on the solution of the PDE, see e.g. [BK81, Isa98, Kub00, IIY03];Carleman estimates are also fundamental in the construction of complex geometrical op-tic solutions that lead to the resolution of inverse problems such as the Calder´on problemwith partial data [KSU07, DSFKSU09]. • Control theory for PDEs; Carleman estimates yield the null controllability of linear par-abolic equations [LR95] and the null controllability of classes of semi-linear parabolicequations [FI96, Bar00, FCZ00]. They can also be used to prove unique continuationproperties, that in turn are crucial for the treatment of low frequencies for exact control-lability results for hyperbolic equations as in [BLR92].To indicate how the spectral inequality of Theorem 1.3 for the bi-Laplace operator B can beproven by means of Carleman estimates, we first review a method, that yields the spectral inequal-ity of Theorem 1.1 for the Laplace operator A . In this introductory section, we have chosen tomainly focus on the method of proof of the spectral inequality; a comprehensive presentation in-cluding a presentation of the proof of the resolvent estimates of Theorems 1.2 and 1.4 would notbring any further insight to the reader as the line of arguments is quite similar.1.2. A method to prove the spectral inequality for the Laplace operator.
The method wedescribe here originates from [LR95]. We consider the elliptic operator P A = D s + A on Z = (0 , S ) × Ω , for some S > < α < S /
2. Three di ff erenttypes of Carleman estimates are proven for the operator P A : (i) in the interior of (0 , S ) × Ω ; (ii)at the boundary { s = } × Ω ; (iii) at the boundary ( α, S − α ) × ∂ Ω . The three regions where theseCarleman estimates are derived are illustrated in Figure 1. It is simpler to first describe Case (i), PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 7 S − α Ω V V S α z (3) O z (2) V z (1) F igure
1. Location and geometry of the three types of estimates. Dashedare level sets for the weight functions ϕ used in the Carleman estimates.Arrows represent the directions of the (non vanishing) gradient of ϕ . that is, the estimate in the interior. In Figure 1, this corresponds to the neighborhood V of somepoint z (1) ∈ Z . There, the Carleman estimate for this operator P A is of the form described above,that is, τ / (cid:107) e τϕ w (cid:107) L ( Z ) + τ / (cid:107) e τϕ D z w (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ P A w (cid:107) L ( Z ) , (1.6)where the weight function ϕ = ϕ ( z ) is real-valued with a non-vanishing gradient, τ is a largepositive parameter, and w is any smooth function compactly supported in V . In fact, this estimateholds if the so-called sub-ellipticity condition is fulfilled by P A and ϕ . If p A ( z , ζ ) is the principalsymbol of P A , the sub-ellipticity condition in V reads p A ( z , ζ + i τ d ϕ ( z )) = ⇒ i { p A ( z , ζ + i τ d ϕ ( z )) , p A ( z , ζ + i τ d ϕ ( z )) } > , (1.7)for z ∈ V , ζ ∈ R d + , and τ ≥
0. It is in fact equivalent to a Carleman estimate of the form (1.6) for P A (see [H¨or63] or [LL12]). Observe that p A ( z , ζ + i τ d ϕ ( z )) is the semi-classical principal symbolof the conjugated operator P A ,ϕ = e τϕ P A e − τϕ .The function ϕ is chosen of the form ϕ ( z ) = exp( − γ | z − z (1) | ) and V is an annulus around z (1) ,thus avoiding where the gradient of ϕ vanishes (see Figure 1). For γ > ffi ciently large,one can prove that the sub-ellipticity condition (1.7) holds and thus estimate (1.6) is achieved (seee.g. [LR95] or [LL12]).From estimate (1.6), one can deduce the following local interpolation inequality, for all r > ffi ciently small, for some δ ∈ (0 ,
1) (see e.g. [LR95]), (cid:107) v (cid:107) H ( B ( z (1) , r )) (cid:46) (cid:107) v (cid:107) − δ H ( Z ) (cid:16) (cid:107) P A v (cid:107) L ( Z ) + (cid:107) v (cid:107) H ( B ( z (1) , r )) (cid:17) δ , v ∈ H ( Z ) . (1.8)We now consider Case (ii). In a neighborhood V of a point z (2) ∈ { } × O , one can derive anestimate of the same form as (1.6), yet, with two trace terms in the r.h.s., that is, (cid:80) | β |≤ τ / −| β | (cid:107) e τϕ D β w (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ P A w (cid:107) L ( Z ) + τ / (cid:16) | e τϕ w | s = + | H ( O ) + | e τϕ ∂ s w | s = + | L ( O ) (cid:17) , (1.9) J ´ER ˆOME LE ROUSSEAU AND LUC ROBBIANO for τ ≥ τ ≥ w smooth up to the boundary { s = } , with supp( w ) ∩ Z ⊂ V , with V asrepresented in Figure 1. This can be obtained by locally choosing a weight function of the form ϕ ( z ) = exp( γψ ( z )) with ψ ( z ) such that ∂ s ψ ( z ) ≤ − C < V and choosing the parameter γ > ffi ciently large (see e.g. [LZ98]). We use the notation (cid:107) . (cid:107) for functions in the interior of thedomain and | . | for functions on the boundaries.From estimate (1.9) one deduces the following local interpolation inequality: there exist V ⊂ V and δ ∈ (0 ,
1) such that (cid:107) v (cid:107) H ( V ∩ Z ) (cid:46) (cid:107) v (cid:107) − δ H ( Z ) (cid:16) (cid:107) P A v (cid:107) L ( Z ) + | v | s = + | H ( O ) + | ∂ s v | s = + | L ( O ) (cid:17) δ , v ∈ H ( Z ) . (1.10)We finally consider Case (iii). In a neighborhood of a point z (3) ∈ ( α, S − α ) × ∂ Ω , one canderive an estimate of the same form as (1.6), yet, with a single trace term in the r.h.s., that is,(1.11) (cid:80) | β |≤ τ / −| β | (cid:107) e τϕ D β w (cid:107) L ( Z ) + τ / | e τϕ ∂ ν w | ∂ Z | L (( α, S − α ) × ∂ Ω ) (cid:46) (cid:107) e τϕ P A w (cid:107) L ( Z ) + (cid:80) | β (cid:48) |≤ τ / −| β (cid:48) | | e τϕ D β (cid:48) T w | ∂ Z | L (( α, S − α ) × ∂ Ω ) , for τ ≥ τ ≥ w smooth up to the boundary ( α, S − α ) × ∂ Ω , with supp( w ) ∩ Z ⊂ V , with V as represented in Figure 1. This can be obtained by locally choosing a weight function of theform ϕ ( z ) = exp( γψ ( z )) with ψ ( z ) such that ∂ ν ψ ( z ) ≤ − C < V , where ν is the outward normalto ∂ Ω , and choosing the parameter γ > ffi ciently large (see e.g. [LR95]). Here, for | β (cid:48) | ≥ D β (cid:48) T stand as di ff erentiations in the tangential directions only, along vector fields that form a localframe.From estimate (1.11) one deduces the following local interpolation inequality: there exist V ⊂ V , with V neighborhood of z (3) in Z , some open subset Q ⊂ V with positive distance to theboundary, and δ ∈ (0 ,
1) such that (cid:107) v (cid:107) H ( V ∩ Z ) (cid:46) (cid:107) v (cid:107) − δ H ( Z ) (cid:16) (cid:107) P A v (cid:107) L ( Z ) + (cid:107) v (cid:107) H ( Q ) (cid:17) δ , v ∈ H ( Z ) , v | (0 , S ) × ∂ Ω = . (1.12)The three interpolation inequalities (1.8), (1.10), and (1.12) can be used to form a global inter-polation inequality, by means of compactness arguments. In particular, the interior inequality (1.8)permits the “propagation” of the estimate. Then, there exists δ ∈ (0 , (cid:107) v (cid:107) H (( α, S − α ) × Ω ) (cid:46) (cid:107) v (cid:107) − δ H ( Z ) (cid:16) (cid:107) P A v (cid:107) L ( Z ) + | v | s = + | H ( O ) + | ∂ s v | s = + | L ( O ) (cid:17) δ , (1.13)for v ∈ H ( Z ) satisfying v | (0 , S ) × ∂ Ω =
0. This inequality then implies the spectral property for theLaplace operator for u = (cid:80) ω j ≤ ω u j φ j ∈ Span { φ j ; ω j ≤ ω } , if applied to a well chosen function v ( s , x ), namely, v ( s , x ) = (cid:80) ω j ≤ ω u j ω − / j sinh( ω / j s ) φ j ( x ) . Details can for instance be found in [LL12]. In the present paper, we shall apply this approach forthe bi-Laplace operator, the argument is provided in details in Section 5.2.1.3.
Outline of the proof of the spectral inequality for the bi-Laplace operator.
Above wedescribed how Carleman estimates can be used to prove a spectral inequality of the form givenin Theorem 1.1 for the Laplace operator. To prove the spectral inequality of Theorem 1.3 forthe “clamped” bi-Laplace operator, we shall prove several Carleman estimates for the followingfourth-order elliptic operator P = D s + ∆ on Z = , S ) × Ω . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 9
As for P A above, we shall prove such estimates at three di ff erent locations: (i) in the interiorof (0 , S ) × Ω , in Section 2; (ii) at the boundary { s = } × Ω , in Section 3; (iii) at the boundary( α, S − α ) × ∂ Ω , in Section 4. In Section 5, these three types of estimations are then used to achievelocal interpolation inequalities that can be used to prove, first, a global interpolation inequality and,second, the spectral inequality of Theorem 1.3. Note that for the proof of the resolvent estimate ofTheorem 1.4 only steps (ii) and (iii) are needed. Cases (i) and (ii).
The weight functions that we shall use will be the same as that used for theoperator P A for Cases (i) and (ii). In Case (ii), the estimate we obtain for P takes the form τ − / (cid:80) | α |≤ (cid:107) τ −| α | e τϕ D α s , x u (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ Pu (cid:107) L ( Z ) + (cid:80) j = (cid:16) τ / − j | e τϕ D js u | s = + | L ( Ω ) + | e τϕ D js u | s = + | H / − j ( Ω ) (cid:17) , for functions localized near a point z (2) ∈ { } × O , with O ⊂ Ω . We have observation terms at theboundary { s = } . We use the notation (cid:107) . (cid:107) for functions in the interior of the domain and | . | forfunctions on the boundaries.Note that this estimate is characterized by the loss of half-derivative, similarly to the estimateone can derive for P A . In fact, the sub-ellipticity condition holds in V despite the fact that P ϕ = e τϕ Pe − τϕ can be written as a product of two operators, P ϕ = Q Q , as, here, char( Q ) ∩ char( Q ) = ∅ . In Case (i), however, the estimate we obtain is characterized by the loss of one full derivative,taking the form (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α u (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ Pu (cid:107) L ( Z ) , for functions compactly supported away from boundaries. In fact, this loss cannot be improved asexplained in Section 1.4. Here also, the operator P ϕ can be written as a product of two operators, P ϕ = Q Q , and here, as opposed to Case (ii), we have char( Q ) ∩ char( Q ) (cid:44) ∅ .We provide fairly short proofs of the Carleman estimates in Cases (i) and (ii) in Sections 2 and3. Note, however, that the loss of a full derivative in Case (i) does not create any obstruction to thederivation of a local interpolation inequality in Section 5. Remark 1.5.
Sub-ellipticity does not hold in V . The reader should note that the failure of thesub-ellipticity property does not automatically imply a loss of one full derivative. The phenomenathat can occur require a fine analysis to be understood. This is carried out in [Ler88]. Roughlyspeaking, if sub-ellipticity does not hold, and if some iterated Poisson brackets vanish up to order k and an iterated Poisson bracket of order k + k / ( k +
1) derivative. In the present case, as we can prove that the loss of one fullderivative cannot be improved, we then know that all the iterated Poisson brackets used in [Ler88]vanish. The essential problem is that the conjugated operator P ϕ can be written as a product oftwo operators Q Q , and in the case char( Q ) ∩ char( Q ) (cid:44) ∅ , not only does sub-ellipticity nothold, but we see that the iterated Poisson brackets also vanish. Case (iii).
This case is delicate and the derivation of the Carleman estimate at the boundary ( α, S − α ) × ∂ Ω is one of the main results of the present article. This case is also precisely where we haveto take into account the boundary conditions for the bi-Laplace operator B . The estimate we obtainin Case (iii) in Section 4 is characterized by the loss of one full derivative and, as for case (i), thiscannot be improved as explained in Section 1.4. This is a source of major complications for theproof of the Carleman estimate itself. As in Case (i) this, however, does not create any obstruction in the derivation of the local interpolation inequality in Section 5. In fact, the proof of the localCarleman estimate in V , a neighborhood of a point of the boundary ( α, S − α ) × ∂ Ω , requiresmicrolocal arguments. This implies the introduction of microlocalization operators that realizesome partition of unity in phase space over V . For each induced microlocal region, a Carlemanestimate is derived. One region is less favorable: there, the fourth-order conjugated operator P ϕ can we written as a product of four first-order factors, and two of them fail to be elliptic. Moreover,their characteristic sets intersect; sub-ellipticity does not hold there and, in fact, this generates aloss of a full derivative in the estimation. There, the a priori estimate one derives permits to onlyestimate the semi-classical H -norm, viz., (cid:107) w (cid:107) ,τ (cid:16) τ (cid:107) w (cid:107) L + (cid:107) w (cid:107) H . In other microlocal regionsover V , the conjugated operator P ϕ exhibits at most a non elliptic first-order factor only yieldinga half derivative loss as sub-ellipticity holds. If one does not proceed carefully, the derivation inthe least favorable region yields error terms that can be of the same strength as the norm (cid:107) w (cid:107) ,τ ,preventing to conclude positively to the Carleman estimate.We define the weight function in the form ϕ ( z ) = e γψ ( z ) and keep track of the parameter γ that ismeant to be large. The function ψ is chosen such that ∂ ν ψ ≤ − C < γ for general classes of operators. That analysis is carried out away fromboundaries. Here, we use that approach by means of a tangential Weyl-H¨ormander calculus. Theintroduction of the second large parameter γ allows us to handle some error terms in the derivationof the Carleman estimate in V . This is however not su ffi cient to have control over all the errorterms that appear in the microlocal region within V where sub-ellipticity does not hold, since theoperator under study is a product of two second-order operators (see above).Yet, when one attempts to derive the estimate, one realizes that the derivation is possible in thecase ϕ , and thus ψ , only depend on the normal variable to the boundary. Yet, for the interpolationinequality we wish to derive at the boundary ( α, S − α ) × ∂ Ω , some convexity of the level setsof the weight function ϕ is needed: ϕ cannot be constant along the boundary. This is illustratedin Figure 1 (in the neighborhood V ). We thus introduce the function ψ ε ( z ) = ψ ( ε z (cid:48) , z N ), where z (cid:48) denotes the tangential variables and z N denotes the normal variable (in local coordinates where theboundary is given by { z N = } ), and we set ϕ ( z ) = e γψ ε ( z ) . Here, ε is a small parameter, ε ∈ (0 , τ , the second large parameter γ , and this new parameter ε ∈ (0 ,
1) that controlsthe convexity of the level sets of the weight function. Note that even in the case ψ = ψ ( z n ), theproof of the Carleman estimate relies on taking the second parameter γ su ffi ciently large (see theend of Proposition 4.25 below). The introduction of the parameter ε alone would not be su ffi cient.Only the joint introduction of the two parameters allows us to conclude positively to the Carlemanestimate in the microlocal region where a full derivative is lost. PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 11
All the di ff erent microlocal estimates need to be derived within the refined semi-classical calcu-lus with three parameters. Arguments are based on the ellipticity or sub-ellipticity of the di ff erentfactors building the fourth-order operator P ϕ , and the position of theirs roots in the complex plane.This analysis follows in part from the di ff erent works [Bel03, LR10, LR11, LL13, CR14].Eventually, the various microlocal estimates we obtain need to be patched together. This proce-dure generates commutators of the fourth operator P ϕ and the microlocal cut-o ff s, leading to somethird-order error terms that can be handled thanks to the better microlocal estimates obtained awayfrom the least favorable region.Near a point of the boundary ∂ Z = (0 , S ) × ∂ Ω locally written in the form { x d = } with Z = (0 , S ) × Ω = { x d > } , the estimate we obtain, for τ and γ large and ε small, is of the form γ (cid:80) | α |≤ (cid:107) ˜ τ −| α | e τϕ D α s , x u (cid:107) L ( Z ) + (cid:80) ≤ j ≤ | e τϕ D rx d u | ∂ Z | / − j , ˜ τ (cid:46) (cid:107) e τϕ Pu (cid:107) L ( Z ) + (cid:80) j = , | e τϕ D jx d u | ∂ Z | / − j , ˜ τ . On the l.h.s. we find norms of all traces; on the r.h.s. we only have observation with the traces u | ∂ Z and D x d u | ∂ Z associated with the clamped boundary conditions. Here ˜ τ = τγϕ .1.4. On Carleman estimates for higher-order elliptic operators. If Q is an elliptic operator ofeven order m , and ϕ is a weight function such that the couple ( P , ϕ ) satisfies the sub-ellipticitycondition (as stated above), then a Carleman estimate can be obtained, even at a boundary, forinstance with the results of [BL15]. We use those results in Section 3 for the proof of the Carlemanestimate at the boundary { s = } .If m ≥
4, it is however quite natural to not have the sub-ellipticity condition, in particular if theoperator Q is in the form of a product of two operators, say Q = Q Q . Denote by q , q , and q theprincipal symbols of Q , Q , and Q respectively. The conjugated operator Q ϕ = e τϕ Qe − τϕ reads Q ϕ = Q ,ϕ Q ,ϕ , with Q k ,ϕ = e τϕ Q k e − τϕ , k = ,
2. If we have char( Q ,ϕ ) ∩ char( Q ,ϕ ) (cid:44) ∅ then thesub-ellipticity condition fails to hold. In fact, if q ϕ , q ,ϕ , and q ,ϕ are the semi-classical principalsymbols of Q ϕ , Q ,ϕ , and Q ,ϕ , that is, q ϕ = q ( z , ζ + i τ d ϕ ( z )) and q k ,ϕ = q k ( z , ζ + i τ d ϕ ( z )), k = , { q ϕ , q ϕ } = | q ,ϕ | { q ,ϕ , q ,ϕ } + | q ,ϕ | { q ,ϕ , q ,ϕ } + f | q ,ϕ | | q ,ϕ | , for some function f . Thus { q ϕ , q ϕ } vanishes if q ,ϕ = q ,ϕ =
0. Then, the sub-ellipticity propertyof (1.7) cannot hold for Q .Observe that in the above example we have d z ,ζ q ( z , ζ + i τ d ϕ ( z )) = q ( z , ζ + i τ d ϕ ( z )) = q ( z , ζ + i τ d ϕ ( z )) =
0. The following proposition (that applies to operators that need not beelliptic) shows that in such case of symbol “flatness”, the Carleman estimate we can derive for Q exhibits at least a loss of one full derivative. Proposition 1.6.
Let Q = Q ( z , D z ) be a smooth operator of order m ≥ in Z, an open subset of R N . Assume further that there exist a smooth weight function ϕ ( z ) , C > , τ > , a multi-index α with ≤ | α | ≤ m, and δ ≥ such that τ m − −| α | + δ (cid:107) e τϕ D α z u (cid:107) L ≤ C (cid:107) e τϕ Qu (cid:107) L , (1.14) for τ ≥ τ and for u ∈ C ∞ ( R N ) with supp( u ) ⊂ Z. Let q ( z , ζ ) be the principal symbol of Q. If thereexist z ∈ Z, ζ ∈ R N and τ > such that θ α (cid:44) , with θ = ζ + i τ d ϕ ( z ) , andq ( z , θ ) = q ϕ ( z , ζ , τ ) = , d z ,ζ q ( z , θ ) = , then δ = . In other words, if there is a point ( x , ξ , τ ) where the symbol q ϕ vanishes at second order, thenif a Carleman estimate holds it exhibits at least the loss of a full derivative.We refer to Section A.1 for a proof. Remark 1.7.
This loss of at least one full derivative shows that the analysis of [Ler88] cannot beapplied here, as it concerns Carleman estimate with losses of less that one derivative. In particular,one can check that iterated Poisson brackets used in [Ler88] all vanish at points where q ϕ vanishesat second order.In dimension greater than 1, this proposition applies to the bi-Laplace operator B introducedabove on the manifold Ω . If a ( x , ξ ) is the principal symbol of the Laplace operator in a local chart V , for all x ∈ V , there exists ξ and τ > a ( x , ξ + i τ d x ϕ ( x )) =
0. Then, the symbol b = a vanishes at second order at ( x , ξ + i τ d x ϕ ( x ). Hence, we cannot hope for a Carlemanestimate for B with a loss of less than one full derivative. In fact, such an estimate can be obtainedby using twice in cascade the Carleman estimate for the Laplace operator. This is consistent, asthe estimate for the Laplace operator exhibits a loss a half derivative in dimension greater than 1(if ϕ is chosen such that sub-ellipticity holds – see [LL12]).In dimension one, however, B = D x and the conjugated operator ( D x + i τ d ϕ ( x )) is elliptic (inthe sense of semi-classical operators) if d ϕ ( x ) (cid:44) Ω . Then, the resulting Carleman estimate ischaracterized by no derivative loss.Concerning the operator P = D s + B in Z = (0 , S ) × Ω , that is central in the present article, wewrite P = P P with P k = ( − k iD s + A . Setting P k ,ϕ = e τϕ P k e − τϕ , with semi-classical principalsymbols given by p k ,ϕ ( z , ζ, τ ) = ( − k i ( σ + i τ∂ s ϕ ( z )) + a ( x , ξ + i τ d x ϕ ( z )) , k = , , where z = ( s , x ) ∈ Z and ζ = ( σ, ξ ) ∈ R + d = R N . Let d ≥
2. If, for some z ∈ Z , we have ∂ s ϕ ( z ) =
0, if we choose ξ ∈ R d and τ > a ( x , ξ + i τ d x ϕ ( z )) =
0, then for σ = ζ = (0 , ξ ) ∈ R N and θ = (0 , ξ ) + i τ (0 , d x ϕ ( z )) and p k ,ϕ ( z , ζ , τ ) = p k ( z , θ ) = d z ,ζ p ( z , θ ) =
0, where p and p k are the principal symbols of P and P k , k = ,
2. Hence, in aneighborhood of z , Proposition 1.6 applies.This situation occurs in Cases (i) and (iii) described in Section 1.3 and Figure 1. In the neigh-borhoods V and V introduced there, we have points where ∂ s ϕ vanishes (as can observed by theshapes of the level sets of ϕ in Figure 1). This explains why we can only obtain estimates with aloss of one full derivative for those cases. In case (ii), however, this does not occur, and there weobtain an estimation with only a loss of a half derivative.1.5. Some perspectives.
The present article deals with the natural “clamped” boundary condi-tions, that is, homogeneous Dirichlet and Neumann conditions simultaneously. In the light ofthe results obtained here and those that can be obtained for very general boundary conditions ofLopatinskii type in [Tat96, BL15], for instance for unique continuation through the derivationof Carleman estimates at the boundary for general elliptic operators with complex coe ffi cient incases where the sub-ellipticity property hold, one is inclined to attempt to prove estimates similarto those proven in the present article, in the case of an operator, such as the operator P = D s + B studied here, for which the sub-ellipticity condition cannot hold everywhere and for general bound-ary conditions of Lopatinskii type.Here, we considerer the bi-Laplace operator B = ∆ . It would be of interest to consider moregeneral polyharmonic operators such as ∆ k , k ∈ N , on Ω along with natural boundary conditions,e.g., u | ∂ Ω = , . . . , ∂ k − ν u | ∂ Ω = , or more general Lopatinskii type conditions. PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 13
Notation.
We shall use some spaces of smooth functions in the closed half space. We set S ( R N + ) = (cid:8) u | R N + ; u ∈ S ( R N ) (cid:9) . The reader needs to be warned that in some sections z ∈ R N will denote ( x , s ), with x ∈ R d = R N − and s ∈ R , and thus, there, z N = s . This is the case in Section 3. In other sections, z willdenote ( s , x ), and thus there z N = x d . This is the case of Section 4 and Appendices A.2 and B.Some specific notation for semi-classical tangential operators will be introduced in Section 3.1,and they allow us to derive the Carleman estimate for D s + B at the boundary { } × Ω (Cases(i) above). Semi-classical calculus is characterized by the presence of a large parameter denotedby τ here, that is precisely the large parameter that appears in the Carleman estimates (for readersfamiliar with semi-classical analysis this is done by taking τ = / h where h is the Planck constant.)A special class of semi-classical calculus is also introduced in Section 4.1 and is characterizedby three parameters. This calculus is essential in the proof of the Carleman estimate for D s + B atthe boundary (0 , S ) × ∂ Ω (Case (iii) above).In this article, when the constant C is used, it refers to a constant that is independent of thesemi-classical parameters, e.g. τ , γ , ε . Its value may however change from one line to another. Ifwe want to keep track of the value of a constant we shall use another letter.For concision, we use the notation (cid:46) for ≤ C , with a constant C >
0. We also write a (cid:16) b todenote a (cid:46) b (cid:46) a . As done above, we shall use the notation (cid:107) . (cid:107) for functions in the interior of thedomain and | . | for functions on the boundaries.We finish this introductory section by stating some basic properties of the “clamped” bi-Laplaceoperator that will be used at places in this article (some were implicitly used above).1.7. Some basic properties of the bi-Laplace operator.
We recall here some facts on the “clamped”bi-Laplace operator. We define the operator B = ∆ on L ( Ω ) with domain D ( B ) = H ( Ω ) ∩ H ( Ω ). Proposition 1.8.
The operator ( B , D ( B )) is selfadjoint on L ( Ω ) and maximal monotone. In particular, if µ ≥
0, there exists C > f ∈ L ( Ω ), there exists a unique u ∈ D ( B ) such that ∆ u + µ u = f , and (cid:107) u (cid:107) H ( Ω ) ≤ C (cid:107) f (cid:107) L ( Ω ) . (1.15)This can be proven by first finding a unique solution in H ( Ω ) with the Lax-Milgram theoremand then applying Theorem 20.1.2 in [H¨or85b, Section 20.1]. Note in particular that (cid:107) ∆ u (cid:107) L is aequivalent norm on H ( Ω ) ∩ H ( Ω ) by (1.15).As a consequence of Proposition 1.8 we have the existence of a Hilbert basis for L ( Ω ) madeof eigenfunctions. Corollary 1.9.
There exist ( µ j ) j ∈ N ⊂ R , and ( ϕ j ) j ∈ N ⊂ D ( B ) such that < µ ≤ µ ≤ · · · ≤ µ j ≤ · · · , lim j →∞ µ j = + ∞ , B ϕ j = µ j ϕ j , and the family ( ϕ j ) j forms a Hilbert basis for L ( Ω ) . Corollary 1.10.
The operator ( B , D ( B )) generates an analytic C -semigroup S ( t ) on L ( Ω ) .For T > , y ∈ L ( Ω ) , and f ∈ L (0 , T ; H − ( Ω )) , there exists a uniquey ∈ L ([0 , T ]; H ( Ω )) ∩ C ([0 , T ]; L ( Ω )) ∩ H (0 , T ; H − ( Ω )) , given by y ( t ) = S ( t ) y + ∫ t S ( t − s ) f ( s ) ds, such that ∂ t y + ∆ y = f for t ∈ (0 , T ) a.e. , y | t = = y For semigroup theory we refer the reader to [Paz83].For the operator B defined in (1.4) we have the following property. Proposition 1.11.
The spectrum of B is contained in { z ∈ C ; Re( z ) > } . Moreover, for z ∈ C suchthat Re z < , we have (cid:107) ( z Id H −B ) U (cid:107) H ≥ | Re z | (cid:107) U (cid:107) H , U ∈ D ( B ) , with H = H ( Ω ) × L ( Ω ) . With the Hille-Yoshida theorem [Paz83, Theorem 3.1, Chapter 1] we then have the followingresults.
Corollary 1.12.
The unbounded operator ( B , D ( B )) generates a C -semigroup of contraction Σ ( t ) on H . Corollary 1.13.
For ( y , y ) ∈ D ( B ) there exists a uniquey ∈ C ([0 , + ∞ ); L ( Ω )) ∩ C ([0 , + ∞ ); H ( Ω )) ∩ C ([0 , + ∞ ); D ( B )) , such that ∂ t y + ∆ y + α∂ t y = in L ∞ ((0 , + ∞ ); L ( Ω )) , y | t = = y , ∂ t y | t = = y . The solution is given by the first component of Σ ( t ) Y with Y = ( y , y ) . The energy t (cid:55)→ E ( y )( t ) with E ( y )( t ) = (cid:107) ∂ t y ( t ) (cid:107) L ( Ω ) + (cid:107) ∆ y ( t ) (cid:107) L ( Ω ) , (1.16) is nonincreasing: for ≤ t ≤ t we have E ( y )( t ) − E ( y )( t ) = − ∫ t t (cid:107) α / ∂ t y ( t ) (cid:107) L ( Ω ) dt .
2. E stimate away from boundaries
For operators exhibiting at most double (complex) roots, estimates can be found in the proof ofTheorem 28.1.8 in [H¨or85a]. Here, the structure of the operator P is explicit which allows one toexpose the argumentation in a self contained yet short presentation.2.1. Simple-characteristic property of second-order factors.
We consider the augmented op-erator P = D s + B in Z = (0 , S ) × Ω , remaining away from boundaries here. We write P = P P , with P k = ( − k iD s + A . (2.1)Here, we show that P and P both satisfy the so-called simple characteristic property in the caseof a weight function whose di ff erential does not vanish.Let (cid:96) ( z , ζ ), with ( z , ζ ) ∈ R N × R N , be polynomial of degree m in ζ , with smooth coe ffi cient in z .For z (cid:55)→ M ( z ) ∈ R N \ { } , we introduce the map ρ z ,ζ, M : R + → C ,θ (cid:55)→ (cid:96) ( z , ζ + i θ M ( z )) . (2.2) Definition 2.1.
Let W be an open set of R N . We say that (cid:96) satisfies the simple-characteristicproperty in direction M in W if, for all z ∈ W , we have ζ = θ = ρ z ,ζ, M has adouble root.We can formulate this condition as follows (cid:96) ( z , ζ + i θ M ( z )) = d ζ (cid:96) ( z , ζ + i θ M ( z ))( M ( z )) = ⇒ ζ = , θ = . (2.3) PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 15
Lemma 2.2.
Let W be an open set of R N . If N ≥ and (cid:96) ( z , ζ ) is of order two (with complexcoe ffi cients) and elliptic for z ∈ W, then for any map z (cid:55)→ M ( z ) ∈ R N \ { } , (cid:96) satisfies the simple-characteristic property in direction M in W. Proof . The proof can be adapted from classical ideas (see [LM68, proof of Proposition 1.1, Chap-ter 2] or [H¨or83]). We consider the polynomial f z ,ζ, M ( t ) = (cid:96) ( z , ζ + tM ( z )) where t is a complexvariable, for z ∈ W , ζ ∈ R N .If ζ is colinear to M ( z ), e.g. ζ = α M ( z ) then f z ,ζ, M ( t ) = ( α + t ) (cid:96) ( z , M ( z )). Because of theellipticity of (cid:96) , (cid:96) ( z , M ( z )) (cid:44)
0, and we only have t = − α as a double real root for f .We set J = R N \ Span( M ( z )). Note that z is fixed here and Span( M ( z )) is a vector line. The set J is connected as N ≥
3. Let now ζ ∈ J , that is, ζ is not colinear to M ( z ). As (cid:96) is elliptic, the rootsof f z ,ζ, M cannot be real numbers. We denote by m + ( ζ ) and m − ( ζ ) the number of roots with positiveand negative imaginary parts, respectively. We have 2 = m + ( ζ ) + m − ( ζ ). Since roots are continuousw.r.t. ζ and cannot be real, they remain in the upper- or lower-half complex plane as ζ varies in J ,as J is connected, meaning that m + and m − are then invariant. In particular, m + ( ζ ) = m + ( − ζ ) and m − ( ζ ) = m − ( − ζ ). Observing however, that if t is a root of t (cid:55)→ (cid:96) ( z , ζ + tM ( z )) then − t is a rootof t (cid:55)→ (cid:96) ( z , − ζ + tM ( z )), we find that m + ( ζ ) = m − ( − ζ ). This gives m + ( ζ ) = m − ( ζ ) =
1. Hence,complex roots are simple.In any case, we see that if the map θ (cid:55)→ ρ z ,ζ, M = f z ,ζ, M ( i θ ) has a double real root θ then θ = ζ =
0. The simple-characteristic property is thus fulfilled. (cid:4)
If we consider a weight function ψ = ψ ( s , x ), for the operators P k , k = ,
2, introduced in (2.1),we have the following proposition.
Proposition 2.3.
Let k = or . Assume that d ψ (cid:44) in (0 , S ) × Ω . Then, P k satisfies thesimple-characteristic property in direction d ψ in (0 , S ) × Ω . Proof . Here, the dimension is N = d +
1. The case d ≥ d =
1. Then, the principal symbol of A reads a ( x , ξ ) = α ( x ) ξ , with α ( x ) ≥ C >
0. We set M ( z ) = ( M σ ( z ) , M ξ ( z )) = d ψ ( z ) ∈ R N \ { } . We write ρ in place of ρ ( z ,ζ , M ) for concision.With ζ = ( σ, ξ ), we have ρ ( θ ) = p k (cid:0) z , ζ + i θ M (cid:1) = ( − k i (cid:0) σ + i θ M σ (cid:1) + α ( x ) (cid:0) ξ + i θ M ξ (cid:1) = α ( x ) ξ − α ( x )( θ M ξ ) − − k θσ M σ + i (cid:0) ( − k σ − ( − k ( θ M σ ) + θα ( x ) ξ M ξ (cid:1) . We thus have ∂ θ ρ ( θ ) = − α ( x ) θ M ξ − ( − k σ M σ + i (cid:0) α ( x ) ξ M ξ − ( − k θ M σ (cid:1) . Assuming that M σ (cid:44)
0, if ∂ θ ρ = θ = ( − k α ( x ) ξ M ξ M σ , and σ = − α ( x ) ξ M ξ M σ . This yields ρ = α ( x ) ξ (cid:16) + ( − k i α ( x ) M ξ / M σ (cid:17)(cid:16) + α ( x ) M ξ / M σ (cid:17) . In this case, we thus have ρ = ∂ θ ρ = θ = ζ = ( σ, ξ ) = (0 , M σ =
0. Since M (cid:44)
0, we find that ∂ θ ρ = θ = ξ =
0. Then ρ = σ =
0. Hence, in any case, the simple characteristic property is fulfilled. (cid:4)
Local Carleman estimates away from boundaries.
Let V be an open subset of Z = (0 , S ) × Ω . We set z = ( s , x ). Let L = L ( z , D z ) be a di ff erential operator of order m , with smooth principalsymbol, (cid:96) ( z , ζ ). Definition 2.4.
Let ϕ ( z ) be defined and smooth in V and such that | d ϕ | ≥ C >
0. We say that thecouple ( L , ϕ ) satisfies the sub-ellipticity condition in V if we have (cid:96) ( z , ζ + i τϕ ( z )) = ⇒ i { (cid:96) ( z , ζ + i τ d ϕ ( z )) , (cid:96) ( z , ζ + i τ d ϕ ( z )) } = { Re (cid:96) ( z , ζ + i τ d ϕ ( z )) , Im (cid:96) ( z , ζ + i τ d ϕ ( z )) } > , for all z ∈ V and ζ ∈ R N and τ > ψ ( z ) be smooth in V and such that | d ψ | ≥ C > V . We define ϕ ( z ) = exp( γψ ( z )).Sub-ellipticity for the couple ( P k , ϕ ) can be easily achieved by the following lemma. Lemma 2.5.
The couple ( P k , ϕ ) satisfies the sub-ellipticity condition in V for γ > chosen su ffi -ciently large. Proof . By Proposition 2.3 we see that P k satisfies the simple-characteristic property in direction d ψ in V . This implies that ψ is strongly pseudo-convex with respect to P k in the sense given in[H¨or85a, Section 28.3] at every point in V . We then obtain that the couple ( P k , ϕ ) satisfies the sub-ellipticity condition in V for γ > ffi ciently large by Proposition 28.3.3 in [H¨or85a]. (cid:4) A consequence of the sub-ellipticity property is the following Carleman estimate for P k in V ,that is, away from boundaries. Proposition 2.6.
Let k = or . Let ϕ = exp( γψ ) with | d ψ | ≥ C > in V. For γ > chosensu ffi ciently large, there exist C > and τ such that (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ D α z u (cid:107) L ( Z ) ≤ C (cid:107) e τϕ P k u (cid:107) L ( Z ) , for τ ≥ τ and u ∈ C ∞ c ( V ) . We refer to [H¨or85a, Theorem 28.2.3] for a proof. In fact, to incorporate the term associatedwith | α | = P k , k = ,
2, we deduce the following estimate for the operator P = P P . Proposition 2.7.
Let ϕ = exp( γψ ) with | d ψ | ≥ C > in V. For γ > chosen su ffi ciently large,there exist C > and τ such that (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α z u (cid:107) L ( Z ) ≤ C (cid:107) e τϕ Pu (cid:107) L ( Z ) , for τ ≥ τ and u ∈ C ∞ c ( V ) . This estimate is characterized by the loss of a full derivative.
Proof . With the estimate of Proposition 2.6 for the operator P applied to P u ∈ C ∞ ( V ) we have (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ D α z P u (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ Pu (cid:107) L ( Z ) . Observing that [ D α z , P ] is a di ff erential operator of order 1 + | α | we obtain (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ P D α z u (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ Pu (cid:107) L ( Z ) + (cid:80) | β |≤ τ / −| β | (cid:107) e τϕ D β z u (cid:107) L ( Z ) . (2.4)Applying now the estimate of Proposition 2.6 for the operator P to D α z u ∈ C ∞ ( V ) we obtain (cid:80) | δ |≤ τ / −| δ | (cid:107) e τϕ D δ + α z u (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ P D α z u (cid:107) L ( Z ) . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 17
With (2.4) we then obtain (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α z u (cid:107) L ( Z ) (cid:16) (cid:80) | δ |≤ (cid:80) | α |≤ τ −| δ |−| α | (cid:107) e τϕ D δ + α z u (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ Pu (cid:107) L ( Z ) + (cid:80) | β |≤ τ / −| β | (cid:107) e τϕ D β z u (cid:107) L ( Z ) . We then conclude by choosing τ > ffi ciently large. (cid:4)
3. E stimate at the boundary { s = } Tangential semi-classical calculus and associated Sobolev norms.
Considering boundaryproblems, we shall locally use coordinates so that the geometry is that of the half space R N + = { z ∈ R N , z N > } , z = ( z (cid:48) , z N ) with z (cid:48) ∈ R N − , z N ∈ R . We shall use the notation (cid:37) = ( z , ζ, τ ) and (cid:37) (cid:48) = ( z , ζ (cid:48) , τ ) in this section. (This notation is not tobe confused with that introduced and used in Section 4 and Appendix B.)Let a ( (cid:37) (cid:48) ) ∈ C ∞ ( R N + × R N − ), with τ as a parameter in [1 , + ∞ ) and m ∈ R , be such that, for allmulti-indices α, β , we have | ∂ α z ∂ βζ (cid:48) a ( (cid:37) (cid:48) ) | ≤ C α,β λ m −| β | T ,τ , z ∈ R N + , ζ (cid:48) ∈ R N − , τ ∈ [1 , + ∞ ) , where λ T ,τ = | ζ (cid:48) | + τ . We write a ∈ S m T ,τ . We also define S −∞ T ,τ = ∩ r ∈ R S r T ,τ . For a ∈ S m T ,τ we callprincipal symbol, σ ( a ), the equivalence class of a in S m T ,τ / S m − T ,τ . Note that we have λ m T ,τ ∈ S m T ,τ .If a ( (cid:37) (cid:48) ) ∈ S m T ,τ , we setOp T ( a ) u ( z ) : = (2 π ) − ( N − ∫ R N − e i ( z (cid:48) ,ζ (cid:48) ) a ( (cid:37) (cid:48) ) ˆ u ( ζ (cid:48) , z N ) d ζ (cid:48) , for u ∈ S ( R N + ), where ˆ u is the partial Fourier transform of u with respect to the tangential variables z (cid:48) . We denote by Ψ m T ,τ the set of these pseudo-di ff erential operators. For A ∈ Ψ m T ,τ , σ ( A ) = σ ( a )will be its principal symbol in S m T ,τ / S m − T ,τ . We also set Λ m T ,τ = Op T ( λ m T ,τ ), m ∈ R .Let m ∈ N and m (cid:48) ∈ R . If we consider a of the form a ( (cid:37) ) = m (cid:80) j = a j ( (cid:37) (cid:48) ) ζ jN , a j ∈ S m + m (cid:48) − j T ,τ , we define Op( a ) : = (cid:80) mj = Op T ( a j ) D jz N . We write a ∈ S m , m (cid:48) τ and Op( a ) ∈ Ψ m , m (cid:48) τ .We define the following norm, for m ∈ N and m (cid:48) ∈ R , (cid:107) u (cid:107) m , m (cid:48) ,τ (cid:16) m (cid:80) j = (cid:107) Λ m + m (cid:48) − j T ,τ D jz N u (cid:107) + (cid:107) u (cid:107) m ,τ = (cid:107) u (cid:107) m , ,τ (cid:16) m (cid:80) j = (cid:107) Λ m − j T ,τ D jz N u (cid:107) + , u ∈ S ( R N + ) , where (cid:107) . (cid:107) + : = (cid:107) . (cid:107) L ( R N + ) . We have (cid:107) u (cid:107) m ,τ (cid:16) (cid:80) | α |≤ m α ∈ N N τ m −| α | (cid:107) D α u (cid:107) + , and in the case m (cid:48) ∈ N we have (cid:107) u (cid:107) m , m (cid:48) ,τ (cid:16) (cid:80) α N ≤ m (cid:80) | α |≤ m + m (cid:48) α = ( α (cid:48) ,α N ) ∈ N N τ m + m (cid:48) −| α | (cid:107) D α u (cid:107) + . If m , m (cid:48) ∈ N and m (cid:48)(cid:48) , m (cid:48)(cid:48)(cid:48) ∈ R , and if a ∈ S m (cid:48)(cid:48) T ,τ , then we have (cid:107) Op T ( a ) u (cid:107) m (cid:48) , m (cid:48)(cid:48)(cid:48) ,τ ≤ C (cid:107) u (cid:107) m (cid:48) , m (cid:48)(cid:48) + m (cid:48)(cid:48)(cid:48) ,τ , u ∈ S ( R N + ) . If a ∈ S m , m (cid:48)(cid:48) T ,τ , then we have (cid:107) Op T ( a ) u (cid:107) m (cid:48) , m (cid:48)(cid:48)(cid:48) ,τ ≤ C (cid:107) u (cid:107) m + m (cid:48) , m (cid:48)(cid:48) + m (cid:48)(cid:48)(cid:48) ,τ , u ∈ S ( R N + ) . The following argument will be used on many occasions in what follows, for m ∈ N , m (cid:48) , (cid:96) ∈ R ,with (cid:96) > (cid:107) w (cid:107) m , m (cid:48) ,τ (cid:28) (cid:107) w (cid:107) m , m (cid:48) + (cid:96),τ . (3.1)At the boundary { z N = } we define the following norms, for m ∈ N and m (cid:48) ∈ R , | tr( u ) | m , m (cid:48) ,τ = m (cid:80) j = | Λ m − j + m (cid:48) T ,τ D jz N u | z N = + | L ( R N − ) , u ∈ S ( R N + ) . Statement of the Carleman estimate.
In this section, we consider z = ( x , s ) ∈ R N with x ∈ R d and s ∈ R . We also set Z = Ω × (0 , S ). We write x = z (cid:48) and s = z N , in connexion with thenotation introduced for the tangential calculus in Section 3.1.Let z = ( x ,
0) with x ∈ Ω . We consider a function ψ ∈ C ∞ ( R N ) such that ∂ s ψ ( z ) ≤ − C < V of z in R × Ω . We then set ϕ ( z ) = e γψ ( z ) .Using the notation introduced in Section 3.1 for semi-classical norms, we have the followingCarleman estimate at the boundary Ω × { } for functions defined in { s ≥ } ∩ V . Theorem 3.1.
Let P = D s + B = D s + ∆ on Z = Ω × (0 , S ) . Let W be an open set of R N suchthat W (cid:98) V. For γ > chosen su ffi ciently large, there exist τ ≥ and C > such that (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ D α s , x u (cid:107) L ( Z ) ≤ C (cid:16) (cid:107) e τϕ Pu (cid:107) L ( Z ) + (cid:80) j = | tr( e τϕ D js u ) | , / − j ,τ (cid:17) , for τ ≥ τ and for u = w | Z , with w ∈ C ∞ c ( R d × R ) and supp( w ) ⊂ W. This Carleman estimate is characterized by the loss of a half derivative.
Corollary 3.2.
Let P = D s + B = D s + ∆ on Z = Ω × (0 , S ) . Let W be an open set of R N suchthat W (cid:98) V. For γ > chosen su ffi ciently large, there exist τ ≥ and C > such that (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ D α s , x u (cid:107) L ( Z ) ≤ C (cid:16) (cid:107) e τϕ Pu (cid:107) L ( Z ) + τ / (cid:80) j = | tr( e τϕ D js u ) | , − j ,τ (cid:17) , for τ ≥ τ and for u = w | Z , with w ∈ C ∞ c ( R d × R ) and supp( w ) ⊂ W. Proofs are given below.3.3.
Sub-ellipticity property.
As in Section 2.1, we write P = P P with P k = ( − k iD s + A ,and P ϕ = e τϕ Pe − τϕ = Q Q with Q k = e τϕ P k e − τϕ . The principal symbol of q k , in the sense ofsemi-classical operators, is given by q k ( z , ζ, τ ) = ( − k i ( σ + i ˆ τ σ ) + a ( x , ξ + i ˆ τ ξ ) , ˆ τ ( z , τ ) = (ˆ τ ξ , ˆ τ σ ) = τ d ϕ ∈ R N , where a ( x , ξ ) denotes the principal symbol of the Laplace operator A .Recalling the definition of the semi-classical characteristic set of a (pseudo-)di ff erential opera-tor A , with principal symbol a ( (cid:37) ),char( A ) = { (cid:37) = ( z , ζ, τ ) ∈ V × R N × R + ; ( ζ, τ ) (cid:44) (0 , , and a ( (cid:37) ) = } , we have the following results for the characteristic sets of Q k , k = , PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 19
Lemma 3.3.
In V, we have char( Q ) ∩ char( Q ) = ∅ . Proof . Let (cid:37) = ( z , ζ, τ ) ∈ V × R N × R + , with ( ζ, τ ) (cid:44) (0 , q ( (cid:37) ) = q ( (cid:37) ) =
0, whichreads ( − k i ( σ + i ˆ τ σ ) + a ( x , ξ + i ˆ τ ξ ) =
0, for both k = k =
2, meaning that we have( σ + i ˆ τ σ ) = , a ( x , ξ + i ˆ τ ξ ) = . In particular this implies σ = τ σ = τ∂ s ϕ =
0. As here ∂ s ϕ (cid:44) σ = τ =
0. With τ =
0, we have ˆ τ ξ =
0, and we thus obtain a ( x , ξ ) =
0, implying ξ = a ( x , ξ ). (cid:4) Lemma 3.4.
Let L and L be di ff erential operators in V. Let ϕ ∈ C ∞ ( Z ) and set L k ,ϕ = e τϕ L k e − τϕ ,k = , . Assume that char( L ,ϕ ) ∩ char( L ,ϕ ) = ∅ . Then the couple ( L L , ϕ ) satisfies the sub-ellipticity condition of Definition 2.4 in V if and only if both ( L k , ϕ ) , k = , , satisfy this property. Proof . We denote by (cid:96) k , the principal symbols of L k ,ϕ , k = ,
2, and (cid:96) = (cid:96) (cid:96) the principal symbolof e τϕ L L e − τϕ . We observe that { (cid:96), (cid:96) } = | (cid:96) | { (cid:96) , (cid:96) } + | (cid:96) | { (cid:96) , (cid:96) } + f | (cid:96) | | (cid:96) | , for some function f . If ( (cid:96), ϕ ) satisfies the sub-ellipticity condition and if (cid:96) ( (cid:37) ) =
0, with (cid:37) = ( z , ζ, τ ) ∈ V × R N × R + , then (cid:96) ( (cid:37) ) (cid:44) < { (cid:96), (cid:96) } ( (cid:37) ) / i = | (cid:96) | { (cid:96) , (cid:96) } / i , thus yielding thesub-ellipticity condition at (cid:37) for (cid:96) . The same argument applies for (cid:96) .Let us now assume that (cid:96) and (cid:96) both satisfy the sub-ellipticity condition. If (cid:96) ( (cid:37) ) = (cid:96) ( (cid:37) ) = (cid:96) ( (cid:37) ) =
0. Let us assume that (cid:96) ( (cid:37) ) =
0. Then (cid:96) ( (cid:37) ) (cid:44) { (cid:96) , (cid:96) } ( (cid:37) ) / i > { (cid:96), (cid:96) } ( (cid:37) ) / i = | (cid:96) ( (cid:37) ) | { (cid:96) , (cid:96) } ( (cid:37) ) / i > (cid:4) By Lemma 2.5, the couples ( P k , ϕ ) satisfy the sub-ellipticity condition in V . From Lemmata 3.3and 3.4 we deduce the following result. Corollary 3.5.
The couple ( P , ϕ ) satisfies the sub-ellipticity condition of Definition 2.4 in V. Proof of the estimate at { s = } . The proof of Theorem 3.1 uses Lemma 4.3 in [BL15].
Proof of Theorem 3.1.
We denote by a ( (cid:37) ) the principal symbol of ( P ϕ + P ∗ ϕ ) / b ( (cid:37) ) that of( P ϕ − P ∗ ϕ ) / (2 i ). We have a ∈ S , τ and b ∈ S , τ . We set A = Op( a ) and B = Op( b ) and Q a , b ( w ) = Aw , Bw ) + . The sub-ellipticity of ( P , ϕ ) given by Corollary 3.5 reads a ( (cid:37) ) = b ( (cid:37) ) = ⇒ { a , b } > , (cid:37) ∈ V × R N × R + . With Lemma 4.3 in [BL15], we obtain, for some C > C (cid:48) >
0, for τ ≥ ffi cientlylarge, C (cid:107) v (cid:107) ,τ ≤ C (cid:48) (cid:0) (cid:107) Av (cid:107) + + (cid:107) Bv (cid:107) + + | tr( v ) | , / ,τ (cid:1) + τ (cid:0) Q a , b ( v ) − Re B a , b ( v ) (cid:1) , where | B a , b ( v ) | (cid:46) | tr( v ) | , / ,τ , for v = w | Z , with w ∈ C ∞ c ( R d × R ) and supp( w ) ⊂ W . We thusobtain τ − (cid:107) v (cid:107) ,τ (cid:46) (cid:107) ( A + iB ) v (cid:107) + + | tr( v ) | , / ,τ . As we have P ϕ = A + iB mod Ψ , τ , by taking τ su ffi ciently large, with the usual semi-classicalargument (3.1) we obtain(3.2) τ − / (cid:107) v (cid:107) ,τ (cid:46) (cid:107) P ϕ v (cid:107) + + | tr( v ) | , / ,τ . The conclusion of the proof is then classical. (cid:4)
Proof of Corollary 3.2.
Let W (cid:48) be an open set of R N such that W (cid:98) W (cid:48) (cid:98) V and let χ, ˜ χ ∈ C ∞ c ( W (cid:48) )be such that χ ≡ W and ˜ χ ≡ χ ).We may apply estimate (3.2), an equivalent form of the estimate of Theorem 3.1, to the function τ / χ ( z ) Λ − / T ,τ v , for v = w | Z , with w ∈ C ∞ c ( R d × R ) and supp( w ) ⊂ W . Observe that we have χ ( z ) Λ − / T ,τ v = Λ − / T ,τ v + R , − M v , P ϕ χ ( z ) Λ − / T ,τ v = ˜ χ ( z ) P ϕ Λ − / T ,τ v + R , − M v , because of the support of v , with R , − M ∈ Ψ , − M τ , and R , − M ∈ Ψ , − M τ , for any M ∈ N .Setting ˜ v = τ / Λ − / T ,τ v ∈ S ( R N + ), we thus obtain, with (3.2),(3.3) τ − / (cid:107) ˜ v (cid:107) ,τ (cid:46) (cid:107) ˜ χ P ϕ ˜ v (cid:107) + + | tr(˜ v ) | , / ,τ + (cid:107) v (cid:107) , − M ,τ . We then observe that we have τ − / (cid:107) ˜ v (cid:107) ,τ = (cid:107) Λ − / T ,τ v (cid:107) ,τ = (cid:107) v (cid:107) , − / ,τ . We also have | tr(˜ v ) | , / ,τ = τ / | tr( v ) | , ,τ , as [ D s , Λ r T ,τ ] = r ∈ R . Next, as [ ˜ χ P ϕ , Λ − / T ,τ ] ∈ Ψ , − / τ , we have (cid:107) ˜ χ P ϕ ˜ v (cid:107) + (cid:46) τ / (cid:107) Λ − / T ,τ ˜ χ P ϕ v (cid:107) + + τ / (cid:107) v (cid:107) , − / ,τ (cid:46) (cid:107) P ϕ v (cid:107) + + τ / (cid:107) v (cid:107) , − / ,τ . From (3.3), we thus obtain (cid:107) v (cid:107) , − / ,τ (cid:46) (cid:107) P ϕ v (cid:107) + + τ / | tr(˜ v ) | , ,τ + τ / (cid:107) v (cid:107) , − / ,τ . With the usual semi-classical argument (3.1) we conclude the proof, as (cid:107) v (cid:107) , − / ,τ (cid:38) τ / (cid:107) v (cid:107) ,τ . (cid:4)
4. E stimate at the boundary (0 , S ) × ∂ Ω A semi-classical calculus with three parameters.
We set W = R N × R N , N = d +
1, oftenreferred to as phase-space. A typical element of W will be X = ( s , x , σ, ξ ), with s ∈ R , x ∈ R d , σ ∈ R , and ξ ∈ R d . We also write x = ( x (cid:48) , x d ), x (cid:48) ∈ R d − , x d ∈ R , and accordingly ξ = ( ξ (cid:48) , ξ d ).With s and x playing very similar rˆole in the definition of the calculus, we set z = ( s , x ) ∈ R N , z (cid:48) = ( s , x (cid:48) ) ∈ R N − , and z N = x d . We also set ζ = ( σ, ξ ) ∈ R N , ζ (cid:48) = ( σ, ξ (cid:48) ) ∈ R N − , and ζ N = ξ d .In this section, we shall consider a weight function of the form(4.1) ϕ γ,ε ( z ) = e γψ ε ( z ) , ψ ε ( z ) = ψ ( ε z (cid:48) , z N ) , with γ and ε as parameters, satisfying γ ≥ ε ∈ [0 , ψ ∈ C ∞ ( R N ). To define a properpseudo-di ff erential calculus, we assume the following properties of ψ :(4.2) ψ ≥ C > , (cid:107) ψ ( k ) (cid:107) L ∞ < ∞ , k ∈ N . In particular, there exists k > R N ψ ≤ ( k +
1) inf R N ψ. A class of semi-classical symbols.
We introduce the following class of tangential symbolsdepending on the variables z ∈ R N , ζ (cid:48) ∈ R N − and ˆ t ∈ R N . We set ˆ λ T = | ζ (cid:48) | + | ˆ t | . Definition 4.1.
Let m ∈ R . We say that a ( z , ζ (cid:48) , ˆ t ) ∈ C ∞ ( R N + × R N − × R N ) belong to the class S m T , ˆ t if, for all multi-indices α ∈ N N , β ∈ N N − , δ ∈ N N , there exists C α,β,δ > | ∂ α z ∂ βζ (cid:48) ∂ δ ˆ t a ( z , ζ (cid:48) , ˆ t ) | ≤ C α,β,δ ˆ λ m −| β |−| δ | T , ( z , ζ (cid:48) , ˆ t ) ∈ R N + × R N − × R N , | ˆ t | ≥ . If Γ is a conic open set of R N + × R N − × R N , we say that a ∈ S m T , ˆ t in Γ if the above property holdsfor ( z , ζ (cid:48) , ˆ t ) ∈ Γ . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 21
Note that, as opposed to usual semi-classical symbols, we ask for some regularity with respectto the semi-classical parameter that is a vector of R N here.This class of symbols will not be used as such to define a class of pseudo-di ff erential operatorsbut rather to generate other classes of symbols and associated operators in a more refined semi-classical calculus that we present now.4.1.2. Metrics.
For τ ∗ ≥
2, we set M = R N × R N × [ τ ∗ , + ∞ ) × [1 , + ∞ ) × [0 , , M T = R N + × R N − × [ τ ∗ , + ∞ ) × [1 , + ∞ ) × [0 , . We denote by (cid:37) = ( z , ζ, τ, γ, ε ) a point in M and by (cid:37) (cid:48) = ( z , ζ (cid:48) , τ, γ, ε ) a point in M T .We set ˜ τ = τγϕ γ,ε ( z ) ∈ R + . For simplicity, even though ˜ τ is independent of ζ (cid:48) , we shall write˜ τ = ˜ τ ( (cid:37) (cid:48) ), when we wish to keep in mind that ˜ τ is not a simple parameter but rather a function. As ψ > τ ≥ τ ∗ , and γ ≥ τ ≥ τ ∗ . We then set λ τ = λ τ ( (cid:37) ) = | ζ | + ˜ τ ( (cid:37) (cid:48) ) , λ T , ˜ τ = λ T , ˜ τ ( (cid:37) (cid:48) ) = | ζ (cid:48) | + ˜ τ ( (cid:37) (cid:48) ) . The explicit dependences of λ ˜ τ and λ T , ˜ τ upon (cid:37) and (cid:37) (cid:48) are now dropped to ease notation in thissection. Similarly, we shall write ϕ ( z ), or simply ϕ , in place of ϕ γ,ε ( z ).We consider the following metric on phase-space W = R N × R N g = (1 + γε ) | dz (cid:48) | + γ | dz N | + λ − τ | d ζ | , (4.4)for τ ≥ τ ∗ , γ ≥
1, and ε ∈ [0 , g on Ω .)On the phase-space W (cid:48) = R N × R N − adapted to a tangential calculus, we consider the followingmetric: g T = (1 + γε ) | dz (cid:48) | + γ | dz N | + λ − T , ˜ τ | d ζ (cid:48) | , for τ ≥ τ ∗ , γ ≥
1, and ε ∈ [0 , g on W defines a Weyl-H¨ormander pseudo-di ff erential calculus, and that both ϕ and λ ˜ τ have the properties to be used as proper order functions.For a presentation of the Weyl-H¨ormander calculus we refer to [Ler10], [H¨or85b, Sections 18.4–6]and [H¨or79]. Proposition 4.2.
The metric g and the order functions ϕ γ,ε , λ ˜ τ are admissible, in the sense that,the following properties hold (uniformly with respect to the parameters τ , γ , and ε ): (1) g satisfies the uncertainty principle, that is h − g = γ − λ ˜ τ ≥ . (2) ϕ γ,ε , λ ˜ τ and g are slowly varying; (3) ϕ γ,ε , λ ˜ τ and g are temperate. We refer to Appendix A.2.1 for a proof. Similarly, we have the following proposition.
Proposition 4.3.
The metric g T and the order functions ϕ γ,ε , λ T , ˜ τ are admissible. For the tangentialcalculus we have h − g T = (1 + εγ ) − λ T , ˜ τ ≥ . Note that the proof of the uncertainty principle uses that τ ∗ ≥
2. The condition τ ∗ ≥ ffi ce if we chose ψ ≥ ln(2). We preferred not to add this technical condition on the weightfunction ψ .Consequently, ˜ τ ( (cid:37) (cid:48) ) is also an admissible order function for both calculi. Symbols.
Let a ( (cid:37) ) ∈ C ∞ ( R N × R N ), with τ , γ , and ε acting as parameters, and m , r ∈ R , besuch that for all multi-indices α, β ∈ N N , with α = ( α (cid:48) , α N ), we have(4.5) | ∂ α z ∂ βζ a ( (cid:37) ) | ≤ C α,β γ | α N | (1 + εγ ) | α (cid:48) | ˜ τ r λ m −| β | ˜ τ , (cid:37) ∈ M . With the notation of [H¨or85b, Sections 18.4-18.6] we then have a ( (cid:37) ) ∈ S (˜ τ r λ m ˜ τ , g ).Similarly, let a ( (cid:37) (cid:48) ) ∈ C ∞ ( R N + × R N − ), with τ , γ , and ε acting as parameters, and m ∈ R . If forall multi-indices α = ( α (cid:48) , α N ) ∈ N N , β (cid:48) ∈ N N − , we have(4.6) | ∂ α z ∂ β (cid:48) ζ (cid:48) a ( (cid:37) (cid:48) ) | ≤ C α,β (cid:48) γ | α N | (1 + εγ ) | α (cid:48) | ˜ τ r λ m −| β (cid:48) | T , ˜ τ , (cid:37) (cid:48) ∈ M T , we then write a ( (cid:37) (cid:48) ) ∈ S (˜ τ r λ m T , ˜ τ , g T ). Observe that S (˜ τ r λ m T , ˜ τ , g T ) ⊂ S ( λ r + m T , ˜ τ , g T ).The principal symbol associated with a ( (cid:37) (cid:48) ) ∈ S (˜ τ r λ m T , ˜ τ , g T ) is given by its equivalence class in S (˜ τ r λ m T , ˜ τ , g T ) / S ((1 + εγ )˜ τ r λ m − T , ˜ τ , g T ). We denote this principal part by σ ( a ). Often, an homogeneousrepresentative can be selected and the principal part is then identified with this particular repre-sentative of the equivalence class. (Conic sets and homogeneous symbols are precisely defined inSection 4.1.5 below.)We define the following class of symbols, that are polynomial with respect to ξ N , S m , m (cid:48) ˜ τ = m (cid:80) j = S ( λ m + m (cid:48) − j T , ˜ τ , g T ) ζ jN . For a ( (cid:37) ) ∈ S m , m (cid:48) ˜ τ , with a ( (cid:37) ) = (cid:80) mj = a j ( (cid:37) (cid:48) ) ζ jN , with a j ( (cid:37) (cid:48) ) ∈ S ( λ m + m (cid:48) − j T , ˜ τ , g T ), we denote its principalpart by σ ( a )( (cid:37) ) = (cid:80) mj = σ ( a j )( (cid:37) (cid:48) ) ζ jN .For this calculus with parameters to make sense, it is important to check that λ ˜ τ ∈ S ( λ ˜ τ , g ) and λ T , ˜ τ ∈ S ( λ T , ˜ τ , g T ) and ˜ τ ∈ S (˜ τ, g ) ∩ S (˜ τ, g T ). In fact, the latter property implies the first two. Lemma 4.4.
We have ˜ τ = τγϕ γ,ε ∈ S (˜ τ, g ) ∩ S (˜ τ, g T ) . We refer to Section A.2.2 for a proof.4.1.4.
A semi-classical cotangent vector.
We set ˆ τ = τ d z ϕ γ,ε ( z ) = τγϕ γ,ε ( z ) d z ψ ε ( z ) = ˜ τ ( (cid:37) (cid:48) ) d z ψ ε ( z ) ∈ R N . As for ˜ τ , we shall write ˆ τ = ˆ τ ( (cid:37) (cid:48) ), when we wish to keep in mind that ˆ τ is not a constant cotan-gent vector. Note that ˆ τ = (ˆ τ (cid:48) , ˆ τ N ) withˆ τ (cid:48) ( (cid:37) (cid:48) ) = ˜ τ ( y (cid:48) ) d z (cid:48) ψ ε ( z ) = ε ˜ τ ( y (cid:48) ) d z (cid:48) ψ ( ε z (cid:48) , z N ) , ˆ τ N ( (cid:37) (cid:48) ) = ˜ τ ( y (cid:48) ) ∂ z N ψ ( ε z (cid:48) , z N ) . As d z (cid:48) ψ ε ∈ S ( ε, g T ) and ∂ z N ψ ε ∈ S (1 , g T ), we have the following result. Lemma 4.5.
We have ˆ τ (cid:48) ∈ S ( ε ˜ τ, g ) N − ∩ S ( ε ˜ τ, g T ) N − and ˆ τ N ∈ S (˜ τ, g ) ∩ S (˜ τ, g T ) . For later use, we also introduce the following notation:ˆ τ σ = ˆ τ σ ( (cid:37) (cid:48) ) = τ∂ s ϕ γ,ε ( z ) ∈ R , ˆ τ ξ = ˆ τ ξ ( (cid:37) (cid:48) ) = τ d x ϕ γ,ε ( z ) ∈ R N − = R d , (4.7) ˆ τ ξ d = ˆ τ ξ d ( (cid:37) (cid:48) ) = τ∂ x d ϕ γ,ε ( z ) ∈ R , ˆ τ ξ (cid:48) = ˆ τ ξ (cid:48) ( (cid:37) (cid:48) ) = τ d x (cid:48) ϕ γ,ε ( z ) ∈ R N − = R d − . We then have ˆ τ = (ˆ τ σ , ˆ τ ξ ) = (ˆ τ σ , ˆ τ ξ (cid:48) , ˆ τ ξ d ) , ˆ τ (cid:48) = (ˆ τ σ , ˆ τ ξ (cid:48) ) , ˆ τ N = ˆ τ ξ d . (4.8)Even thought the following lemma is very elementary, we state it for futur reference. Lemma 4.6.
Let V be an open set of R N such that ∂ x d ψ ( z ) ≥ C > for z ∈ V. Then, we have | ˆ τ | (cid:16) ˆ τ ξ d (cid:16) ˜ τ, z ∈ V . (4.9) Proof . As (cid:107) ψ (cid:48) (cid:107) ∞ ≤ C , if ∂ x d ψ ≥ C > z ∈ V ⊂ R N , then we have | ˆ τ | (cid:46) ˜ τ (cid:46) ˆ τ ξ d and thus theresult. (cid:4) PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 23
Conic sets and homogeneity.
We recall that a set Γ ⊂ R N + × R N − × R N is said to be conicif ( z , ζ (cid:48) , ˆ t ) ∈ Γ implies that ( z , νζ (cid:48) , ν ˆ t ) ∈ Γ for all ν > κ : M T → R N + × R N − × R N ,(cid:37) (cid:48) = ( z , ζ (cid:48) , τ, γ, ε ) (cid:55)→ ( z , ζ (cid:48) , ˆ τ ( (cid:37) (cid:48) )) . Throughout Section 4 and Appendix B, we shall use the following terminology.
Definition 4.7.
An open subset U of M T is said to be conic if Γ = κ ( U ) is conic in R N + × R N − × R N .A function f : U → E , E a vector space, is said to be homogeneous of degree m if f takes theform f = g ◦ κ with g : R N + × R N − × R N → E such that g ( z , νζ, ν ˆ t ) = ν m g ( z , ζ, ˆ t ), for ν > z , ζ, ˆ τ ) instead of the variables ( z , ζ, τ, γ, ε ), where, as above, ˆ τ = τ d z ϕ γ,ε ( z ) = τγϕ γ,ε ( z ) d z ψ ε ( z ).If U is a conic open subset of M T we shall say that a ∈ S (˜ τ r λ m T , ˜ τ , g T ) in U if property (4.6)holds in U , with a similar terminology for symbols that satisfy the defining property of S m , m (cid:48) ˜ τ in U .In what follows, the following lemma will be used for instance, to generate cuto ff functions. Itwill also be used to obtain symbols with the adapted homogeneity with respect to ζ (cid:48) and ˆ τ . Werefer to Section A.2.3 for a proof. Lemma 4.8.
Let U be a conic open subset of M T and set Γ = κ ( U ) . Assume also that | ˆ τ | (cid:16) ˜ τ in U . Let m ∈ R and ˆ a ( z , ζ (cid:48) , ˆ t ) ∈ S m T , ˆ t in Γ (as given in Definition 4.1). We then have a ( (cid:37) (cid:48) ) = ˆ a ◦ κ ( (cid:37) (cid:48) ) ∈ S ( λ m T , ˜ τ , g T ) in U . In fact, if ˆ a is polynomial in ( ζ (cid:48) , ˆ t ) the assumption | ˆ τ | (cid:16) ˜ τ in U is notneeded. The following lemma is elementary.
Lemma 4.9.
Let U be a conic open subset of M T and let a ∈ S (˜ τ r λ m T , ˜ τ , g T ) in U . Let χ ∈ S (1 , g T ) in M T , with supp( χ ) ⊂ U . Then, χ a ∈ S (˜ τ r λ m T , ˜ τ , g T ) in M T . Operators and Sobolev bounds.
For a ∈ S (˜ τ r λ m ˜ τ , g ) we define the following pseudo-di ff erentialoperator in R N : Op( a ) u ( z ) = (2 π ) − N ∫ R N e iz · ζ a ( z , ζ, τ, γ, ε ) ˆ u ( ζ ) d ζ, u ∈ S ( R N ) , (4.10)where ˆ u is the Fourier transform of u . In the sense of oscillatory integrals, we haveOp( a ) u ( z ) = (2 π ) − N ∫∫ R N e i ( z − y ) · ζ a ( z , ζ, τ, γ, ε ) u ( y ) d ζ dy . The associated class of pseudo-di ff erential operators is denoted by Ψ (˜ τ r λ m ˜ τ , g ). If a is polynomialin the variables ζ and ˆ τ ( (cid:37) (cid:48) ) = ˜ τ d z ψ ε ( z ), we then write Op( a ) ∈ D (˜ τ r λ m ˜ τ , g ).Tangential operators are defined similarly. For a ∈ S (˜ τ r λ m T , ˜ τ , g T ) we setOp T ( a ) u ( z ) = (2 π ) − ( N − ∫∫ R N − e i ( z (cid:48) − y (cid:48) ) · ζ (cid:48) a ( z , ζ (cid:48) , τ, γ, ε ) u ( y (cid:48) , z N ) d ζ (cid:48) dy (cid:48) , (4.11)for u ∈ S ( R N + ), where z ∈ R N + . We write A = Op T ( a ) ∈ Ψ (˜ τ r λ m T , ˜ τ , g T ). We set Λ m T , ˜ τ = Op T ( λ m T , ˜ τ ). We also introduce the following class of operators that act as di ff erential operators in the z N variable and as tangential pseudo-di ff erential operators in the z (cid:48) variables: Ψ m , r ˜ τ = m (cid:80) j = Ψ ( λ m + r − j T , ˜ τ , g T ) D jz N , m ∈ N , r ∈ R , (4.12)that is, Op( a ) ∈ Ψ m , r ˜ τ if a ∈ S m , r ˜ τ . Operators of this class can be applied to functions that are onlydefined on the half-space { z N ≥ } .At places, it will be handy to use the Weyl quantization for tangential operators, namely with a ∈ S (˜ τ r λ m T , ˜ τ , g T ) we defineOp T w ( a ) u ( z ) = (2 π ) − ( N − ∫∫ R N − e i ( z (cid:48) − y (cid:48) ) · ζ (cid:48) a (cid:0) ( z (cid:48) + y (cid:48) ) / , z N , ζ (cid:48) , τ, γ, ε ) u ( y (cid:48) , z N ) d ζ (cid:48) dy (cid:48) . (4.13)This quantification is often advantageous as Op T w ( a ) ∗ = Op T w ( a ), and thus, for the symbol a real,the operator Op T w ( a ) is (formally) selfadjoint. Note that Op T ( a ) − Op T w ( a ) ∈ (1 + εγ ) Ψ (˜ τ r λ m − T , ˜ τ , g T ).We now present some Sobolev-bound type result that we shall use in what follows. We use thefollowing notation (cid:107) . (cid:107) + = (cid:107) . (cid:107) L ( R N + ) , ( ., . ) + = ( ., . ) L ( R N + ) , for the L -norm on the half space R N + and the associated scalar product.We have the following lemma whose proof is similar to that of Lemma 2.7 in [Le 15]. Lemma 4.10.
Let r , m ∈ R and a ∈ S (˜ τ r λ m T , ˜ τ , g T ) . There exists C > such that, for τ su ffi cientlylarge, | (Op T ( a ) u , v ) + | ≤ C (cid:107) Op T (˜ τ r (cid:48) λ m (cid:48) T , ˜ τ ) u (cid:107) + (cid:107) Op T (˜ τ r (cid:48)(cid:48) λ m (cid:48)(cid:48) T , ˜ τ ) v (cid:107) + , u , v ∈ S ( R N + ) . for r = r (cid:48) + r (cid:48)(cid:48) , m = m (cid:48) + m (cid:48)(cid:48) , with r (cid:48) , r (cid:48)(cid:48) ∈ R , m (cid:48) , m (cid:48)(cid:48) ∈ R . This contains the estimate (cid:107) Op T (˜ τ r (cid:48) λ m (cid:48) T , ˜ τ ) Op T ( a ) u (cid:107) + ≤ C (cid:107) Op T (˜ τ r + r (cid:48) λ m + m (cid:48) T , ˜ τ ) u (cid:107) + , u ∈ S ( R N + ) , (4.14)for r , m (cid:48) ∈ R . The proof of Lemma 4.10 relies in the fact that, for r , m ∈ R ,Op T (˜ τ r λ m T , ˜ τ ) Op T (˜ τ − r λ − m T , ˜ τ ) = Id + R , with R ∈ (1 + εγ ) Ψ ( λ − T , ˜ τ , g T ) and (cid:107) R (cid:107) L → L (cid:28) τ large.Note also that we have the following result (see Section A.2.4 for a proof). Lemma 4.11.
We have (cid:107) Op T (˜ τ r λ m T , ˜ τ ) u (cid:107) + (cid:16) (cid:107) Op T ( λ m T , ˜ τ )˜ τ r u (cid:107) + , u ∈ S ( R N + ) , (4.15) and | Op T (˜ τ r λ m T , ˜ τ ) u | z N = + | L ( R N − ) (cid:16) | Op T ( λ m T , ˜ τ )˜ τ r u | z N = + | L ( R N − ) , u ∈ S ( R N − ) , (4.16) for τ chosen su ffi ciently large. We define the following semi-classical Sobolev norms | u | m , ˜ τ = | Λ m T , ˜ τ u | z N = + | L ( R N − ) , m ∈ R , u ∈ S ( R N − ) , (cid:107) u (cid:107) m , ˜ τ (cid:16) m (cid:80) j = (cid:107) Λ m − j T , ˜ τ D jz N u (cid:107) + , m ∈ N , u ∈ S ( R N + ) . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 25
We also set, for m ∈ N and m (cid:48) ∈ R , (cid:107) u (cid:107) m , m (cid:48) , ˜ τ (cid:16) m (cid:80) j = (cid:107) Λ m − j + m (cid:48) T , ˜ τ D jz N u (cid:107) + , u ∈ S ( R N + ) . At the boundary { z N = } we define the following norms, for m ∈ N and m (cid:48) ∈ R , | tr( u ) | m , m (cid:48) , ˜ τ (cid:16) m (cid:80) j = | Λ m − j + m (cid:48) T , ˜ τ D jz N u | z N = + ) | L ( R N − ) , u ∈ S ( R N + ) . The following argument will be used on many occasions in what follows, for r , r (cid:48) , m ∈ R , and (cid:96) > γ r (cid:107) ˜ τ r (cid:48) w (cid:107) m , ˜ τ (cid:28) (cid:107) ˜ τ r (cid:48) + (cid:96) w (cid:107) m , ˜ τ (cid:46) (cid:107) ˜ τ r (cid:48) w (cid:107) m + (cid:96), ˜ τ , (4.17)for τ chosen su ffi ciently large, as γ r (cid:46) ϕ γ,ε = exp( γψ ε ) since ψ ε ≥ C >
0. We have similar suchinequalities for the other norms introduced above.With the above results we deduce the following two propositions.
Proposition 4.12.
Let r , m ∈ R , and a ∈ S (˜ τ r λ m T , ˜ τ , g T ) . Then, for r (cid:48) , m (cid:48) ∈ R , there exists C > such that | ˜ τ r (cid:48) Op T ( a ) u | z N = + | m (cid:48) , ˜ τ ≤ C | ˜ τ r + r (cid:48) u | z N = + | m + m (cid:48) , ˜ τ , u ∈ S ( R N + ) , for τ su ffi ciently large. Proposition 4.13.
Let r , m (cid:48) ∈ R , m ∈ N , and a ∈ ˜ τ r S m , m (cid:48) ˜ τ . Then, for r (cid:48) , m (cid:48)(cid:48)(cid:48) ∈ R and m (cid:48)(cid:48) ∈ N , thereexists C > such that (cid:107) ˜ τ r (cid:48) Op( a ) u (cid:107) m (cid:48)(cid:48) , m (cid:48)(cid:48)(cid:48) , ˜ τ ≤ C (cid:107) ˜ τ r + r (cid:48) u (cid:107) m + m (cid:48)(cid:48) , m (cid:48) + m (cid:48)(cid:48)(cid:48) , ˜ τ , u ∈ S ( R N + ) , for τ su ffi ciently large. Similarly to Lemma 4.11, we have the following equivalences for norms.
Lemma 4.14.
Let m ∈ N and r , m (cid:48) ∈ R . We have, for τ chosen su ffi ciently large, (cid:107) ˜ τ r w (cid:107) m , m (cid:48) , ˜ τ (cid:16) m (cid:80) j = (cid:107) D jz N (˜ τ r w ) (cid:107) , m + m (cid:48) − j , ˜ τ (cid:16) m (cid:80) j = (cid:107) ˜ τ r (cid:48) j Λ m (cid:48)(cid:48) j T , ˜ τ D jz N (˜ τ r (cid:48)(cid:48) j Λ m (cid:48)(cid:48)(cid:48) j T , ˜ τ w ) (cid:107) + , where r = r (cid:48) j + r (cid:48)(cid:48) j , and m + m (cid:48) − j = m (cid:48)(cid:48) j + m (cid:48)(cid:48)(cid:48) j , with r (cid:48) j , r (cid:48)(cid:48) j ∈ R and m (cid:48)(cid:48) j , m (cid:48)(cid:48)(cid:48) j ∈ R , j = , . . . , m.Similarly, we have | tr(˜ τ r w ) | m , m (cid:48) , ˜ τ (cid:16) m (cid:80) j = | D jz N (˜ τ r w ) | z N = + | m + m (cid:48) − j , ˜ τ (cid:16) m (cid:80) j = | ˜ τ r (cid:48) j Λ m (cid:48)(cid:48) j T , ˜ τ (cid:0) D jz N (˜ τ r (cid:48)(cid:48) j Λ m (cid:48)(cid:48)(cid:48) j T , ˜ τ w ) (cid:1) | z N = + | L ( R n − ) . See Section A.2.5 for a proof.
Proposition 4.15 (local tangential Gårding inequality) . Let W , W be two open sets of R N , withW (cid:98) W . Let a ( (cid:37) (cid:48) ) ∈ S (˜ τ r λ m T , ˜ τ , g T ) , with principal part a r , m . If there exist C > and R > suchthat Re a r , m ( (cid:37) (cid:48) ) ≥ C ˜ τ r λ m T , ˜ τ , z ∈ W , ζ (cid:48) ∈ R N − , τ ≥ τ ∗ , λ T , ˜ τ ≥ R , then for any < C (cid:48) < C there exists τ ≥ τ ∗ such that Re(Op T ( a ) u , u ) + ≥ C (cid:48) (cid:107) ˜ τ r / u (cid:107) , m / , ˜ τ , τ ≥ τ . for u = w | Z , with w ∈ C ∞ c ((0 , S ) × R d ) and supp( w ) ⊂ W . In many occurrences, we shall use the following microlocal version of the Gårding inequality.
Proposition 4.16 (microlocal tangential Gårding inequality) . Let U ⊂ M T be a conic open set.Let also χ ( (cid:37) (cid:48) ) ∈ S (1 , g T ) be homogeneous of degree zero and such that supp( χ ) ⊂ U . Let r , m ∈ R and a ( (cid:37) (cid:48) ) ∈ S (˜ τ r λ m T , ˜ τ , g T ) , with principal part a r , m . If there exist C > and R > such that Re a r , m ( (cid:37) (cid:48) ) ≥ C ˜ τ r λ m T , ˜ τ , (cid:37) (cid:48) ∈ U , τ ≥ τ ∗ , λ T , ˜ τ ≥ R , then for any < C (cid:48) < C, M ∈ N , there exist C M and τ ≥ τ ∗ such that Re(Op T ( a ) Op T ( χ ) u , Op T ( χ ) u ) + ≥ C (cid:48) (cid:107) ˜ τ r / Op T ( χ ) u (cid:107) , m / , ˜ τ − C M (cid:107) u (cid:107) , − M , ˜ τ , for u ∈ S ( R N + ) and τ ≥ τ . Local setting and statement of the Carleman estimate.
To explain the construction of thephase function, it is useful to use a particular set of coordinates. We set Z = (0 , S ) × Ω and ∂ Z = (0 , S ) × ∂ Ω .Let z = ( s , x ) ∈ ∂ Z . In a neighborhood V of z in R N , using normal geodesic coordinates forthe x variable, we can express the principal part of the Laplace operator A in the following form A = D x d + R ( x , D x (cid:48) ) , (4.18)where R ( x , D x (cid:48) ) is a tangential di ff erential operator of order 2 with principal symbol r ( x , ξ (cid:48) ), r ( x , ξ (cid:48) ) ≥ C | ξ (cid:48) | , (4.19)where C >
0. We denote by ˜ r ( x , ξ (cid:48) , η (cid:48) ) the associated real symmetric bilinear form. The boundary(0 , S ) × ∂ Ω is locally given by { z N = } = { x d = } .Without any loss of generality we shall assume that V is a bounded open set.We then let ψ ( z ) be defined in R N and fulfilling the properties listed in (4.2) with moreover, ∂ x d ψ ( z ) = ∂ z N ψ ( z ) ≥ C > , z ∈ V , (4.20)and we set ϕ γ,ε ( z ) = exp( γψ ε ( z )) with ψ ε ( z ) = ψ ( ε s , ε x (cid:48) , x d ), for γ ≥ ε ∈ [0 , ϕ in place if ϕ γ,ε for the sake of concision.The main result of this section is the following Carleman estimate. Theorem 4.17.
Let P = D s + A . Let z = ( s , x ) ∈ (0 , S ) × ∂ Ω . Let ϕ ( z ) = ϕ γ,ε ( z ) be defined asabove. There exists an open neighborhood W of z in (0 , S ) × R d , W ⊂ V, and there exist τ ≥ τ ∗ , γ ≥ , ε ∈ (0 , , and C > such that (4.21) γ (cid:80) | α |≤ (cid:107) ˜ τ −| α | e τϕ D α s , x u (cid:107) + + (cid:80) ≤ j ≤ | e τϕ D rx d u | ∂ Z | / − j , ˜ τ ≤ C (cid:16) (cid:107) e τϕ Pu (cid:107) + + (cid:80) j = , | e τϕ D jx d u | ∂ Z | / − j , ˜ τ (cid:17) , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , ε ] , and for u = w | Z , with w ∈ C ∞ c ((0 , S ) × R d ) and supp( w ) ⊂ W. As written in Case (iii) of Section 1.3, the proof we provide of this theorem is based on adecomposition of phase-space in three microlocal regions and the derivation of an adapted estimatein each one of these regions. The definition of these three regions is based on the properties of theroots of the principal symbol of P viewed as a polynomial function of degree four in the variable ξ d . We start with the analysis of those properties in the next section and define the microlocalregions in Section 4.4 below. In section 4.5 we provide a proof scheme for a microlocal Carlemanestimate in each of the three regions. Then, in Sections 4.6–4.8 we precisely state and prove themicrolocal estimate associated with each region. Finally, in Section 4.9 the various microlocalestimates are patched together, to yield the estimate of Theorem 4.17. PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 27
Root properties.
Here, z will be assumed to be an element of V so that all the symbols arewell defined. We write, as in Sections 2 and 3, P = P P , with P k = ( − k iD s + A . Setting P ϕ = e τϕ Pe − τϕ we have P ϕ = Q Q , with Q k = e τϕ P k e − τϕ = ( − k i ( D s + i τ∂ s ϕ ( z )) + A ϕ , (4.22)with, in the selected normal geodesic coordinates, A ϕ = e τϕ Ae − τϕ = ( D x d + i τ∂ x d ϕ ( z )) + R ( x , D x (cid:48) + i τ d x (cid:48) ϕ ( z )) , z = ( s , x ) . In fact, we shall write Q k in the following form Q k = ( D x d + i τ∂ x d ϕ ( z )) + M k , M k = ( − k i ( D s + i τ∂ s ϕ ( z )) + R ( x , D x (cid:48) + i τ d x (cid:48) ϕ ( z )) . (4.23)This form will allow us, when a smooth square root H k of M k is available in the tangential calculusassociated with g T , to write, up to lower order terms, Q k = ( D x d + i τ∂ x d ϕ + iH k )( D x d + i τ∂ x d ϕ − iH k ) , and, then, we shall base our derivation of a Carleman estimate for P on estimates for first-orderfactors. This approach was introduced in the seminal work of A.-P. Calder´on [Cal58]. It hasbeen used recently to address boundary and interface problems in the derivation of Carlemanestimates [LL13, CR14]. Of course, the two smooth square roots, H and H , may not alwaysbe available. Still, on the occurrence of such a case, we shall find that the operators Q and Q will be characterized by perfectly elliptic estimates at the boundary, that is, one can estimate thesemi-classical Sobolev norm of the solution in Ω as well as the counterpart norms for the tracesof normal derivatives of the solution on ∂ Ω (with the natural 1 / Q and Q and the properties of their roots.We denote the principal parts of Q k and M k by q k and m k , which gives, with (cid:37) = ( z , ζ, τ, γ, ε )and ζ = ( σ, ξ ), q k ( (cid:37) ) = (cid:0) ξ d + i τ∂ x d ϕ ( z ) (cid:1) + m k ( (cid:37) (cid:48) ) = (cid:0) ξ d + i ˆ τ ξ d ( (cid:37) (cid:48) ) (cid:1) + m k ( (cid:37) (cid:48) ) , (4.24)with m k ( (cid:37) (cid:48) ) = ( − k i (cid:0) σ + i ˆ τ σ ( (cid:37) (cid:48) ) (cid:1) + r (cid:0) x , ξ (cid:48) + i ˆ τ ξ (cid:48) ( (cid:37) (cid:48) ) (cid:1) , (4.25)recalling the definition of ˆ τ ( (cid:37) (cid:48) ) introduced in Section 4.1.4 and using the notation (4.7)–(4.8).For ˆ t = (ˆ t σ , ˆ t ξ ) ∈ R × R d , with ˆ t ξ = (ˆ t ξ (cid:48) , ˆ t ξ d ) ∈ R d − × R , we setˆ q k ( z , ζ, ˆ t ) = (cid:0) ξ d + i ˆ t ξ d (cid:1) + ˆ m k ( z , ζ (cid:48) , ˆ t ) , ˆ m k ( z , ζ (cid:48) , ˆ t ) : = ( − k i ( σ + i ˆ t σ ) + r ( x , ξ (cid:48) + i ˆ t ξ (cid:48) ) . (4.26)We have q k ( (cid:37) ) = ˆ q k ( z , ζ, ˆ τ ) and m k ( (cid:37) (cid:48) ) = ˆ m k ( z , ζ (cid:48) , ˆ τ ).We now study the roots of ˆ q k ( z , ζ (cid:48) , ξ d , ˆ t ), with ζ (cid:48) = ( σ, ξ (cid:48) ), when viewed as a polynomial inthe variable ξ d , with the other variables, z , ζ (cid:48) , and ˆ t acting as parameters. To that purpose, weintroduce the following quantityˆ µ k ( z , ζ (cid:48) , ˆ t ) : = t ξ d Re ˆ m k ( z , ζ (cid:48) , ˆ t ) − t ξ d + (cid:0) Im ˆ m k ( z , ζ (cid:48) , ˆ t ) (cid:1) . (4.27)We choose ˆ h k ( z , ζ (cid:48) , ˆ t ) ∈ C such thatRe ˆ h k ( z , ζ (cid:48) , ˆ t ) ≥ h k ( z , ζ (cid:48) , ˆ t ) = ˆ m k ( z , ζ (cid:48) , ˆ t ) . (4.28)We may then write(4.29) ˆ q k ( z , ζ, ˆ t ) = ( ξ d + i ˆ t ξ d ) + ˆ h k ( z , ζ (cid:48) , ˆ t ) = (cid:0) ξ d − ˆ ρ k , + ( z , ζ (cid:48) , ˆ t ) (cid:1)(cid:0) ξ d − ˆ ρ k , − ( z , ζ (cid:48) , ˆ t ) (cid:1) , with(4.30) ˆ ρ k , ± ( z , ζ (cid:48) , ˆ t ) = − i ˆ t ξ d ± i ˆ h k ( z , ζ (cid:48) , ˆ t ) . The choice of ˆ h k is unique if ˆ m k (cid:60) R − . The results of this section are yet valid in the case ˆ m k ∈ R − ;however, in the following sections, those results based on the factorization (4.29) will only be usedin settings where ˆ m k ∈ R − does not occur.We give some properties of the roots ˆ ρ k , ± ( z , ζ (cid:48) , ˆ t ). Lemma 4.18.
We assume that ˆ t ξ d ≥ . Let k = or . The roots ˆ ρ k , + ( z , ζ (cid:48) , ˆ t ) and ˆ ρ k , − ( z , ζ (cid:48) , ˆ t ) areboth homogeneous of degree one in ( ζ (cid:48) , ˆ t ) , and such that (4.31) Im ˆ ρ k , − ≤ − ˆ t ξ d ≤ Im ˆ ρ k , + . We also have (4.32) ˆ ρ k , − = ˆ ρ k , + ⇔ ˆ ρ k , − = ˆ ρ k , + = − i ˆ t ξ d ⇔ ˆ m k = . Moreover, if ˆ t ξ d > , we have (4.33) Im ˆ ρ k , + (cid:83) ⇔ ˆ µ k (cid:83) . In particular, if ˆ t ξ d >
0, observe that the root ˆ ρ k , − remains in the lower half complex plane,independently of the values of z , ζ (cid:48) , and ˆ t , while the root ˆ ρ k , + may cross the real line. Proof . The roots can be chosen continuous with respect to ζ and ˆ t and homogeneity comes nat-urally. Observe that Im ˆ ρ k , ± = − ˆ t ξ d ± Re ˆ h k . As Re ˆ h k ≥ ρ k , ± above yields the equivalences in (4.32).Finally, as Im ˆ ρ k , + (cid:83) h k (cid:83) ˆ t ξ d , Lemma 4.19 below implies (4.33), sinceRe ˆ h k ≥ t ξ d > (cid:4) Lemma 4.19.
Let t ∈ C and m = t . We then have, for x ∈ R such that x (cid:44) , | Re t | (cid:83) | x | ⇔ x Re m − x + (Im m ) (cid:83) . Proof . Let t = x + iy . We have Re m = x − y and Im m = xy and we observe that4 x Re m − x + (Im m ) = x + y )( x − x ) , which gives the result. (cid:4) Corollary 4.20.
We assume that ˆ t ξ d > . Let k = or . If C > , there exists C (cid:48) > such that ˆ µ k ( z , ζ (cid:48) , ˆ t ) ≥ C ( | ˆ t | + | ζ (cid:48) | ) ⇒ Im ˆ ρ k , + ( z , ζ (cid:48) , ˆ t ) ≥ C (cid:48) ˆ λ T , ˆ λ T = ( | ˆ t | + | ζ (cid:48) | ) / , for ( z , ζ (cid:48) , ˆ t ) ∈ V ∩ R N + × R N − × R N . Proof . We consider the compact set (recall that V is bounded) C = { ( z , ζ (cid:48) , ˆ t ) ∈ V ∩ R N + × R N − × R N ; ˆ λ T = } . The inequality ˆ µ k ≥ C yields a compact set K of C . By (4.33) in Lemma 4.18, we have Im ˆ ρ k , + ≥ C (cid:48) > K , and we conclude by homogeneity. (cid:4) Proposition 4.21.
We assume that ˆ t ξ d ≥ . Let k = or , we have the following properties: (1) There exist θ ∈ (0 , and C > such that ifz ∈ V and | ˆ t | ≤ θ ˆ λ T , then the roots ˆ ρ k , ± are simple and non real, and moreover Im ˆ ρ k , + ≥ C ˆ λ T , Im ˆ ρ k , − ≤ − C ˆ λ T ( z , ζ (cid:48) , ˆ t ) ∈ V × R N − × R N , (4.34) with ˆ λ T = ( | ˆ t | + | ζ (cid:48) | ) / . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 29 (2)
There exists C > such that ≤ ˆ t ξ d ≤ C (cid:0) | ˆ t (cid:48) | + | ζ (cid:48) | (cid:1) , and | ζ (cid:48) | ≤ C | ˆ t | , if ˆ ρ k , + ∈ R , where ˆ t (cid:48) = (ˆ t σ , ˆ t ξ (cid:48) ) . In such case, the value of the imaginary part of the secondroot is prescribed and nonpositive: Im ˆ ρ k , − = − t ξ d . (3) There exists C > such that | ˆ t (cid:48) | / C ≤ | ζ (cid:48) | ≤ C | ˆ t (cid:48) | , if ˆ q k has a double root.Finally, if ˆ t ξ d > and if | ˆ t (cid:48) | / ˆ t ξ d is su ffi ciently small, and if the polynomial ˆ q k , k = or , has adouble root, then both roots of the second symbol, ˆ q k (cid:48) with k (cid:48) (cid:44) k, are in the lower half complexplane. More precisely, there exist C , C > such that if | ˆ t (cid:48) | ≤ C ˆ t ξ d then (4.35) ˆ ρ k , + = ˆ ρ k , − ⇒ Im ˆ ρ k (cid:48) , ± ≤ − C ˆ t ξ d . Proof . Proof of point (1).
Because of homogeneity it is su ffi cient to assume that ( ζ (cid:48) , ˆ t ) is on thesphere S = (cid:8) ˆ λ T = (cid:9) . If ˆ t =
0, then we have ˆ m k = ˆ h k = r ( x , ξ (cid:48) ) + ( − k i σ . Observe that ˆ m k (cid:44) σ = ξ (cid:48) =
0, which cannot hold as | ζ (cid:48) | =
1. Moreover Re ˆ m k ≥
0. Hence, wehave Re ˆ h k >
0. Then we write ˆ q k = ξ d + ˆ h k = ( ξ d + i ˆ h k )( ξ d − i ˆ h k ) , yielding ˆ ρ k , − = − i ˆ h k and ˆ ρ k , + = i ˆ h k , which gives Im ˆ ρ k , − < ρ k , + >
0. As S ∩ { ˆ t = } iscompact we find that Im ˆ ρ k , − ≤ − C < ρ k , + ≥ C >
0, for some C >
0, for | ζ (cid:48) | = t =
0. Then, using a compactness argument once more, using the continuity of the roots, thereexist θ ∈ (0 ,
1) such thatIm ˆ ρ k , − ( z , ζ (cid:48) , ˆ t ) ≤ − C (cid:48) < , Im ˆ ρ k , + ( z , ζ (cid:48) , ˆ t ) ≥ C (cid:48) > , if z ∈ V and | ˆ t | ≤ θ , recalling that V is bounded. We then obtain (4.34) in V by homogeneity. Inparticular this excludes having double roots and real roots. Proof of point (2).
Observe that the inequality | ζ (cid:48) | ≤ C | ˆ t | , in the case of a real root is simply anotherformulation of part of point (1). Next, we observe that | ˆ m k | (cid:46) | ˆ t (cid:48) | + | ζ (cid:48) | implies | Re ˆ h k | (cid:46) | ˆ t (cid:48) | + | ζ (cid:48) | .Since having ˆ ρ k , + ∈ R is equivalent to Re ˆ h k = ˆ t ξ d by (4.30), we thus obtain ˆ t ξ d (cid:46) | ˆ t (cid:48) | + | ζ (cid:48) | . AsIm ˆ ρ k , − = − ˆ t ξ d − Re ˆ h k , we then have Im ˆ ρ k , − = − t ξ d . Proof of point (3)
The equation ˆ m k =
0, which is equivalent to having a double root, reads r ( x , ξ (cid:48) ) − r ( x , ˆ t ξ (cid:48) ) − ( − k σ ˆ t σ = , σ − ˆ t σ + ( − k r ( x , ξ (cid:48) , ˆ t ξ (cid:48) ) = , (4.36)with ˜ r ( x , ξ (cid:48) , η (cid:48) ) defined below (4.19). From (4.36), using that r ( x , . ) is uniformly positive definite,we obtain | ξ (cid:48) | (cid:46) | ˆ t ξ (cid:48) | + | σ || ˆ t σ | , | σ | (cid:46) | ˆ t σ | + | ξ (cid:48) || ˆ t ξ (cid:48) | . The sum of the two estimates gives | ζ (cid:48) | (cid:46) | ˆ t (cid:48) | + | σ || ˆ t σ | + | ξ (cid:48) || ˆ t ξ (cid:48) | , and with the Young inequality weobtain | ζ (cid:48) | (cid:46) | ˆ t (cid:48) | . Similarly, from (4.36) we obtain | ˆ t ξ (cid:48) | (cid:46) | ξ (cid:48) | + | σ || ˆ t σ | , | ˆ t σ | (cid:46) | σ | + | ξ (cid:48) || ˆ t ξ (cid:48) | , and the sum of the two estimates gives | ˆ t (cid:48) | (cid:46) | ζ (cid:48) | + | σ || ˆ t σ | + | ξ (cid:48) || ˆ t ξ (cid:48) | , and with the Young inequalitywe obtain | ˆ t (cid:48) | (cid:46) | ζ (cid:48) | .Note that we could deduce that | ζ (cid:48) | (cid:46) | ˆ t | from point (1). Here, we have obtained a sharperestimate. Proof of (4.35).
If ˆ q k has a double root, then | ˆ t (cid:48) | (cid:16) | ζ (cid:48) | by point (3). Let δ ∈ (0 , C = − δ . To have Im ˆ ρ k (cid:48) , ± ≤ − C ˆ t ξ d it su ffi ces to have Im ˆ ρ k (cid:48) , + ≤ − C ˆ t ξ d by Lemma 4.18. With the notation of the proof of that lemma, this reads − ˆ t ξ d + Re ˆ h k (cid:48) ≤ − C ˆ t ξ d , that is 0 ≤ Re ˆ h k (cid:48) ≤ δ ˆ t ξ d .Now as we have | Re ˆ h k (cid:48) | ≤ | ˆ h k (cid:48) | ≤ | ˆ m k (cid:48) | / (cid:46) | ˆ t (cid:48) | + | ζ (cid:48) | , we find that 0 ≤ Re ˆ h k (cid:48) (cid:46) | ˆ t (cid:48) | here. The resultthus follows if we assume that | ˆ t (cid:48) | / ˆ t ξ d is chosen su ffi ciently small. (cid:4) Lemma 4.22.
Assume that | ˆ t (cid:48) | ≤ C ˆ t ξ d for some C > . There exists δ > such that if δ ∈ (0 , δ ) and ˆ µ k ( z , ζ (cid:48) , ˆ t ) ≥ − δ ˆ λ T , with ˆ λ T = | ˆ t | + | ζ (cid:48) | , then the roots of ˆ q k are simple. Proof . Because of homogeneity it is su ffi cient to work on the sphere S = (cid:8) ˆ λ T = (cid:9) . Writingˆ m k = ˆ h k with Re ˆ h k ≥ µ k ≥ − δ reads4 (cid:0) ˆ t ξ d + (Im ˆ h k ) (cid:1)(cid:0) (Re ˆ h k ) − ˆ t ξ d (cid:1) ≥ − δ, using the computation of the proof of Lemma 4.19 with x = ˆ t ξ d . Assume that we have a doubleroot. In such case ˆ m k = | ˆ t (cid:48) | (cid:16) | ζ (cid:48) | by point (3) of Proposition 4.21. We thenhave ˆ h k =
0, yielding 4ˆ t ξ d ≤ δ = δ ˆ λ T (cid:46) δ ˆ t ξ d , using that | ˆ t (cid:48) | ≤ C ˆ t ξ d . Thus, for δ chosen su ffi cientlysmall we reach a contradiction. (cid:4) Lemma 4.23.
Let k = or . If both δ > and | ˆ t (cid:48) | / ˆ t ξ d are su ffi ciently small, there exists C > such that for ( z , ζ (cid:48) , ˆ t ) ∈ V ∩ R N + × R N − × R N ˆ µ k ( z , ζ (cid:48) , ˆ t ) ≥ − δ ˆ λ T ⇒ | ˆ t | ≤ C | ζ (cid:48) | , with ˆ λ T = | ˆ t | + | ζ (cid:48) | . Proof . Because of homogeneity it is su ffi cient to work on the sphere S = (cid:8) ˆ λ T = (cid:9) . Let us now assume that the implication does not hold. Then there exists ( z ( n ) , ζ (cid:48) ( n ) , ˆ t ( n ) ) ∈ V ∩ R N + × S , such that ˆ µ k ( z ( n ) , ζ (cid:48) ( n ) , ˆ t ( n ) ) ≥ − δ and | ˆ t ( n ) | > n | ζ (cid:48) ( n ) | . As ( z ( n ) , ζ (cid:48) ( n ) , ˆ t ( n ) ) lays in a compactset (recall that V is bounded), it converges, up to a subsequence, to ( z ( ∞ ) , ζ (cid:48) ( ∞ ) , ˆ t ( ∞ ) ) ∈ V ∩ R N + × S .We find that ζ (cid:48) ( ∞ ) = m k ( z ( ∞ ) , , ˆ t ( ∞ ) ) = ( − k − i (ˆ t ( ∞ ) σ ) − r ( x , ˆ t ( ∞ ) ξ (cid:48) ) , yieldingˆ µ k ( z ( ∞ ) , , ˆ t ( ∞ ) ) = − t ( ∞ ) ξ d ) r ( x , ˆ t ( ∞ ) ξ (cid:48) ) − t ( ∞ ) ξ d ) + (ˆ t ( ∞ ) σ ) ≤ − , for | ˆ t ( ∞ ) (cid:48) | / ˆ t ( ∞ ) ξ d su ffi ciently small, as we have | ˆ t ( ∞ ) | =
1. For δ su ffi ciently small, we hence reach acontradiction. (cid:4) Microlocal regions.
With the functions ˆ µ k , k = ,
2, introduced in (4.27) we shall defineseveral microlocal regions. Observe first that ˆ µ k is an homogeneous polynomial function of degreefour in ( ζ (cid:48) , ˆ t ). We thus have ˆ µ k ∈ S T , ˆ t in the sense given by Definition 4.1. From Lemma 4.8, wefind that we have ˆ µ k ( z , ζ (cid:48) , ˆ τ ( (cid:37) (cid:48) )) ∈ S ( λ T , ˜ τ , g T ). We thus define µ k ( (cid:37) (cid:48) ) : = λ − T , ˜ τ ( (cid:37) (cid:48) ) ˆ µ k ( z , ζ (cid:48) , ˆ τ ( (cid:37) (cid:48) )) ∈ S (1 , g T ) , k = , , (cid:37) (cid:48) = ( z , ζ (cid:48) , τ, γ, ε ) . (4.37)We recall that ˆ τ = τ d z ϕ γ,ε ( z ) = ˜ τ ( (cid:37) (cid:48) ) d z ψ ε ( z ) and ψ ε ( z ) = ψ ( ε z (cid:48) , z N ) with 0 < ε <
1. Observe thatwe have | ˆ τ ( (cid:37) (cid:48) ) | = ˜ τ ( (cid:37) (cid:48) ) (cid:107) d z ψ ε (cid:107) L ∞ ≤ ˜ τ ( (cid:37) (cid:48) ) (cid:107) d z ψ (cid:107) L ∞ . Thus, having 0 ≤ ˜ τ ≤ δθ λ T , ˜ τ ( (cid:37) (cid:48) ) / (cid:107) d z ψ (cid:107) L ∞ , for δ ∈ (0 , | ˆ τ ( (cid:37) (cid:48) ) | ≤ θ λ T , ˜ τ ( (cid:37) (cid:48) ). The value θ is as introduced in Proposition 4.21. We set θ = θ / (cid:107) d z ψ (cid:107) L ∞ .Let δ ∈ (0 ,
1] and let V be the bounded open neighborhood in R N of z ∈ ∂ Z , introduced inSection 4.2. We set M T , V = V × R N − × [ τ ∗ , + ∞ ) × [1 , + ∞ ) × [0 , k = , F ( V , δ ) = { (cid:37) (cid:48) ∈ M T , V ; z ∈ V , ˜ τ ( (cid:37) (cid:48) ) ≤ δθ λ T , ˜ τ ( (cid:37) (cid:48) ) } , E ( k ) − ( V , δ ) = { (cid:37) (cid:48) ∈ M T , V ; z ∈ V , µ k ( (cid:37) (cid:48) ) ≤ − δ } , E ( k )0 ( V , δ ) = { (cid:37) (cid:48) ∈ M T , V ; z ∈ V , µ k ( (cid:37) (cid:48) ) ≥ − δ } . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 31 δ/ δ µ µ F igure
2. Microlocal region E − . In dark color is the region where χ (1) δ, − ≡
1. In light color is the support of χ (1) δ, − . The boundaries of the associatedregions for χ (2) δ, − are marked dashed. Evidently, we have M T , V = E ( k ) − ( V , δ ) ∪ E ( k )0 ( V , δ ). We now set E − ( V , δ ) = E (1) − ( V , δ ) ∪ E (2) − ( V , δ ) , E ( V , δ ) = E (1)0 ( V , δ ) ∩ E (2)0 ( V , δ ) , and we have M T , V = E − ( V , δ ) ∪ E ( V , δ ). Below, in the text, when no precision is needed, we shalluse the “vague” terminology F , E − , or E , to refer to microlocal regions that take the forms of F ( V , δ ), E − ( V , δ ), E ( V , δ ).We let χ − , χ ∈ C ∞ ( R ), with values in [0 , χ − ≡ −∞ , − , and supp( χ − ) ⊂ ( −∞ , − / ,χ ≡ − , + ∞ ) , and supp( χ ) ⊂ [ − , ∞ ) . Let V (cid:98) V be an open neighborhood of z in R N and let χ V ∈ C ∞ ( R N ) be such that supp( χ V ) ⊂ V and χ V ≡ V . With η ∈ C ∞ c ( − θ , θ ), with values in [0 ,
1] such that η ≡ − θ / , θ / χ δ, F ( (cid:37) (cid:48) ) = η (cid:0) ˜ τ ( (cid:37) (cid:48) ) / ( δλ T , ˜ τ ( (cid:37) (cid:48) )) (cid:1) ∈ S (1 , g T ) . and χ F ( (cid:37) (cid:48) ) = χ V ( z ) χ , F ( (cid:37) (cid:48) ) ∈ S (1 , g T ) . We set χ ( k ) δ, − ( (cid:37) (cid:48) ) = χ V ( z ) (1 − χ / , F ( (cid:37) (cid:48) )) χ − ( µ k ( (cid:37) (cid:48) ) /δ ) ∈ S (1 , g T ) . Observe that we have χ ( k ) δ, − ≡ E ( k ) − ( V , δ ) \ F ( V , / , supp( χ ( k ) δ, − ) ⊂ E ( k ) − ( V , δ/ \ F ( V , / , and thus χ (1) δ, − + χ (2) δ, − ≥ E − ( V , δ ) \ F ( V , / , supp( χ (1) δ, − + χ (2) δ, − ) ⊂ E − ( V , δ/ \ F ( V , / . We finally set χ δ, ( (cid:37) (cid:48) ) = χ V ( z ) (1 − χ / , F ( (cid:37) (cid:48) )) χ ( µ ( (cid:37) (cid:48) ) /δ ) χ ( µ ( (cid:37) (cid:48) ) /δ ) ∈ S (1 , g T ) . µ µ − δ − δ F igure
3. Microlocal region E . In dark color is the region where χ δ, ≡ χ δ, . Observe that we have χ δ, ≡ E ( V , δ ) \ F ( V , /
4) supp( χ δ, ) ⊂ E ( V , δ ) \ F ( V , / , and χ F + χ (1) δ, − + χ (2) δ, − + χ δ, ≥ M T , V . (4.38)With the microlocal cuto ff functions we have just introduced we associate tangential pseudo-di ff erential operators, all in Ψ (1 , g T ), Ξ F = Op T (cid:0) χ F (cid:1) , Ξ ( k ) δ, − = Op T (cid:0) χ ( k ) δ, − (cid:1) , k = , , and Ξ δ, = Op T (cid:0) χ δ, (cid:1) . (4.39)4.5. Proof strategies in the three microlocal regions.
Derivations in all three microlocal regionsrequire first the proof of estimates for various factors and second the concatenation of those es-timates. For this second part, to avoid redundancies, we describe in Appendix B.4, along withproofs, how various type of estimates can be concatenated.The estimate associated with region E − is proven in Section 4.6. In region E − , we have P ϕ = Q Q where at least one of the factors is characterized by a principal symbol with two roots inthe lower half complex plane. This yields for this factor, say Q , a perfectly elliptic estimate atthe boundary { x d = } , as given by Lemma B.1 (see Appendix B.1). For the second operator Q , one can derive an estimate whose form is classical and exhibits a loss of a half derivative, asgiven in Proposition B.10. A proof is provided in Appendix B.5, in particular since the estimateneeds to hold uniformly with respect to all parameters introduced. Finally, the two estimates areconcatenated to obtain an estimate for P ϕ in E − .The estimate associated with region E is proven in Section 4.7. The treatment this regionrequires the most delicate argument and justifies the development of the Weyl-H¨ormander calculusof Section 4.1. Microlocally, in this region we write P ϕ = Q Q and we manage to write each Q k , k = ,
2, in the form Q k = Q k , − Q k , + + (1 + γε ) R , where Q k , − , Q k , + and R are all first-order operators. The operator Q k , − is characterized bya principal symbol with a root in the lower half complex plane. Setting Q − = Q , − Q , − and Q + = Q , + Q , + , with delicate commutator arguments we obtain P ϕ = Q − Q + + (1 + γε ) R where PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 33 R ∈ Ψ , τ . We thus manage gather togethers factors with similar root locations without generat-ing a remainder in γ Ψ , τ . Observe that this latter class for the remainder is obtained if operatorcommutations within the Weyl-H¨ormander calculus are carried invoking usual arguments. Here,to obtain the sharper class (1 + γε ) Ψ , τ , we use the precise forms of the involved operators andsymbols.For Q − we have a perfectly elliptic estimate at the boundary { x d = } , as given by Lemma B.1(see Appendix B.1). For each operator Q k , + a sub-elliptic estimate can be obtained with a traceterm used as an observation as given by Lemma B.6 in Appendix B.3.3. Concatenated together,two such estimates yield an estimate for Q + with a loss of a full derivative and observation termsthat involve both the Dirichlet trace and the Neumann trace of the solution. Concatenating nowthe estimates for Q − and Q + one obtains microlocally an estimate of the form γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ + | tr( v ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + v (cid:107) + + | tr( v ) | , / , ˜ τ . With the form of the remainder term (1 + γε ) R that appeared above in the decomposition of P ϕ one then sees that a similar estimate can be obtained for P ϕ in place of Q − Q + by choosing γ > ffi ciently large and ε > ffi ciently small. Observe that if the remainder term had been in γ Ψ , τ we would not have been able to transform the estimate obtained for Q − Q + into an estimatefor P ϕ .Some technical aspects of the proof in the region E described above require to have ˜ τ ( (cid:37) (cid:48) ) ofthe same order as λ T , ˜ τ ( (cid:37) (cid:48) ). This is however not true in that region. One thus rather considers aregion of the form E \ F , since region F is characterized by ˜ τ ( (cid:37) (cid:48) ) ≤ C λ T , ˜ τ ( (cid:37) (cid:48) ) for a well chosenconstant (see above). A last microlocal region, namely F , thus needs to be considered.The treatment of region F is given in Section 4.8 and has some similarities with what is donein the region E \ F . Yet, the treatment of remainder terms needs not be as refined. The operator P ϕ is written in the form P ϕ = Q − Q + + γ R with R ∈ Ψ , τ and again Q − = Q , − Q , − and Q + = Q , + Q , + . Here also, for Q − we have a perfectly elliptic estimate at the boundary { x d = } .For Q + , the estimate we obtain is very di ff erent from what is done in E \ F . The region F is designed so that the roots associated with the factors Q , + and Q , + are both located in theupper half complex plane. For each of these operators one can thus obtain a microlocal ellipticestimate at the boundary { x d = } with one trace used as an observation term yet without anyloss of derivative as given in Lemma B.4 in Appendix B.3.1. Put together, with a concatenationargument, an estimate for Q + is obtained with observation terms that involve both the Dirichlettrace and the Neumann trace of the solution. This estimate for Q + does not exhibit any loss ofderivative: it is an elliptic estimate. Concatenated together the estimates for Q + and Q − yield alsoan elliptic estimate for Q − Q + with the above two traces as observation terms. The elliptic strengthof this estimate then allows one to handle the remainder term in γ Ψ , τ yielding a similar result for P ϕ in the microlocal region F .As a final step of the proof of Theorem 4.17, we patch together the estimates obtained in theabove three microlocal regions. This is done in Section 4.9.4.6. Microlocal estimate in the region E − . We prove the following estimate.
Proposition 4.24.
Let M ∈ N . Let k = or . For δ ∈ (0 , , there exist τ ≥ τ ∗ , γ ≥ , andC > such that γ (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ + | tr( Ξ ( k ) δ, − v ) | , / , ˜ τ ≤ C (cid:16) (cid:107) P ϕ Ξ ( k ) δ, − v (cid:107) + + | tr( Ξ ( k ) δ, − v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ (cid:17) , (4.40) for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for v ∈ S ( R N + ) . The term (cid:107) v (cid:107) , − M , ˜ τ in the r.h.s. stands as a remainder that will be ’absorbed’ once the estimationsin the di ff erent microlocal regions are patched together. In fact, observe that this term is much weaker than that in the l.h.s. in the Carleman estimate (4.21) of Theorem 4.17. The meaningfulobservation term in the r.h.s. of (4.40) is | tr( v ) | , / , ˜ τ , which is of the strength as the terms in thel.h.s. of (4.21), and can be found in the r.h.s. of that latter estimate. Proof . We have P ϕ = Q Q . We consider the case k =
1. The same proof can be written in thecase k =
2. To ease notation we write χ in place of χ δ, − and Ξ in place of Ξ δ, − .In a conic neighborhood of supp( χ ) ⊂ M T , V , with V introduced in Section 4.2, we have µ ≤− C δ . As (4.20) holds in V we have ˆ τ ξ d ≥ C ˜ τ and thus | ˆ τ ξ | (cid:16) ˜ τ . By Lemma 4.18, both roots of thesymbol q of the operator Q are in the lower half complex plane. Thus,(4.41) the operator Q fulfills the requirements of Lemma B.1.Also, for the operator Q , without any assumption on the position of the roots in the complexplane, we have the following estimate, characterized by the loss of a half derivative and a boundaryobservation term, by Proposition B.10, for (cid:96) ∈ R , γ / (cid:107) ˜ τ − / Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) Q Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ , (4.42)for v ∈ S ( R N + ), for τ ≥ τ ∗ and γ ≥ ffi ciently large, and ε ∈ [0 ,
1] (recall thatsupp( χ ) ⊂ M T , V which gives supp( Ξ v ) ⊂ V (cid:48) (cid:98) V , for some open set V (cid:48) , thus permitting theapplication of Proposition B.10).With (4.41), (4.42), and Proposition B.8, applied with Q − = Q and Q + = Q here, and with α = α = δ = δ =
0, we obtain the result of the proposition, by choosing τ ≥ τ ∗ and γ ≥ ffi ciently large. (cid:4) Microlocal estimate in the region E \ F . We prove the following estimate.
Proposition 4.25.
Let M ∈ N . For δ ∈ (0 , chosen su ffi ciently small and δ ∈ (0 , δ ] , there exist τ ≥ τ ∗ , γ ≥ , ε ∈ (0 , , and C > such that γ (cid:107) ˜ τ − Ξ δ, v (cid:107) , , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ ≤ C (cid:16) (cid:107) P ϕ Ξ δ, v (cid:107) + + | tr( Ξ δ, v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ (cid:17) , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , ε ] , and for v ∈ S ( R N + ) . Before giving the proof of this microlocal estimate we need to provide some additional proper-ties of the symbols m k introduced in Section 4.3 and its square root, h k . Note that the region F isintroduced to isolate the case where ˜ τ ≤ C | ζ (cid:48) | and this permits to exploit the relation | ζ (cid:48) | ≤ C ˜ τ inthe region E \ F . This is used to obtain some symbol properties of h k .We recall the form of the tangential di ff erential operator M k , as introduced in (4.23), M k : = ( − k i ( D s + i τ∂ s ϕ ( z )) + R ( x , D x (cid:48) + i τ d x (cid:48) ϕ ( z )) , whose principal symbol is given by m k ( (cid:37) (cid:48) ) : = ( − k i ( σ + i τ∂ s ϕ ( z )) + r ( x , ξ (cid:48) + i τ d x (cid:48) ϕ ( z )) ∈ S ( λ T , ˜ τ , g T ). Observe that we have the following symbol estimation. Lemma 4.26.
We have ∂ x d m k ∈ S ((1 + εγ ) λ T , ˜ τ , g T ) . Proof . We write m k ( (cid:37) (cid:48) ) = ( − k i (cid:0) σ + i ˆ τ σ ( (cid:37) (cid:48) ) (cid:1) + r (cid:0) x , ξ (cid:48) + i ˆ τ ξ (cid:48) ( (cid:37) (cid:48) ) (cid:1) , with the notation of (4.7). Wethen have ∂ x d m k = − − k ( ∂ x d ˆ τ σ )( σ + i ˆ τ σ ) + i ˜ r ( x , ξ (cid:48) + i ˆ τ ξ (cid:48) , ∂ x d ˆ τ ξ (cid:48) ) + ∂ x d r ( x , ξ (cid:48) + i ˆ τ ξ (cid:48) ) , with ˜ r ( x , ξ (cid:48) , η (cid:48) ) defined below (4.19). By Lemma 4.5, we have ˆ τ (cid:48) = (ˆ τ σ , ˆ τ ξ (cid:48) ) ∈ S ( ε ˜ τ, g T ) N − yielding ∂ x d ˆ τ (cid:48) ∈ S ( εγ ˜ τ, g T ) N − , and as ∂ x d r ( x , ξ (cid:48) + i ˆ τ ξ (cid:48) ) ∈ S ( λ T , ˜ τ , g T ), the result follows. (cid:4) PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 35
Let k = ,
2. If µ k ( (cid:37) (cid:48) ) ≥ − C δ , and for δ > ffi ciently small then m k (cid:44) q k ( (cid:37) ) are simple, by Lemma 4.22 since | ˆ τ (cid:48) | (cid:46) ˆ τ ξ d for z ∈ V ; recall the definition of theoperator Q k , Q k = ( D x d + i τ∂ x d ϕ ( x )) + M k , and its principal symbol q k in (4.22)–(4.24). Lemma 4.27.
Let C , C (cid:48) > and let U δ be a conic open set of M T , V such that µ k ( (cid:37) (cid:48) ) ≥ − C δ and λ T , ˜ τ ≤ C (cid:48) | ˆ τ ( x ) | in U δ . For δ ∈ (0 , and ε > chosen su ffi ciently small, if < δ ≤ δ and ≤ ε ≤ ε , the symbol m k is elliptic and there exists h k ∈ S ( λ T , ˜ τ , g T ) in U δ that is elliptic and thatsatisfies h k = m k and Re h k ≥ . Moreover, we have ∂ x d h k ∈ S ((1 + εγ ) λ T , ˜ τ , g T ) in U δ . The second part of Lemma 4.27 improves, for h k , upon the natural behavior of an arbitraryelement of f ∈ S ( λ T , ˜ τ , g T ) for which we have ∂ x d f ∈ S ( γλ T , ˜ τ , g T ). This is a key aspect of ourproof strategy of the Carleman estimate. In fact, if one chooses ε =
0, that is, a weight function ψ = ψ ( x d ), then one finds directly that ∂ x d m k ∈ S ( λ T , ˜ τ , g T ) and ∂ x d h k ∈ S ( λ T , ˜ τ , g T ), as confirmedby Lemmata 4.26 and 4.27. However, such a weight function is not convex with respect to theboundary { x d = } , which turns out to be an obstruction for the applications of the Carlemanestimate we consider here. If we simply let ψ be of the form ψ ( s , x (cid:48) , x d ) we then obtain ∂ x d h k ∈ S ( γλ T , ˜ τ , g T ) and the proof scheme for the Carleman estimate collapses: the parameter γ needs tobe set large, which yields uncontrolled terms in the derivation. The introduction of the parameter ε , writing ψ ε ( z ) = ψ ( ε s , ε x (cid:48) , x d ) is thus designed to control this behavior and to bring the analysisas “close” as we wish to the case ε = { x d = } . Proof . In V , we have ∂ x d ψ ≥ C > | ˆ τ | (cid:16) ˆ τ ξ d (cid:16) ˜ τ by Lemma 4.6. Next, | ˆ τ (cid:48) | / ˆ τ ξ d canbe made as small as needed by choosing ε > δ ∈ (0 , δ ] and ε > ffi ciently small, by Lemma 4.23 we have | ˆ τ ( x ) | (cid:46) | ζ (cid:48) | and with the additional assumption madehere we obtain(4.43) | ζ (cid:48) | (cid:16) ˜ τ (cid:16) ˜ τ ξ d in U δ . If m k ( (cid:37) (cid:48) ) remains away from a neighborhood of the negative real axis in the complex plane for (cid:37) (cid:48) ∈ U δ , we can then define h k ( (cid:37) (cid:48) ) as the principal square root of m k ( (cid:37) (cid:48) ). Then, it is straightforwardto obtain h k ∈ S ( λ T , ˜ τ , g T ) in U δ . In fact, if we assume | Im m k ( (cid:37) (cid:48) ) | ≤ αλ T , ˜ τ , as we have, recallingthe definition of µ k in (4.37), µ k ( (cid:37) (cid:48) ) λ T , ˜ τ ( (cid:37) (cid:48) ) = τ ξ d Re m k ( (cid:37) (cid:48) ) − τ ξ d + (cid:0) Im m k ( (cid:37) (cid:48) ) (cid:1) it yields, using (4.43), Re m k ( (cid:37) (cid:48) ) ≥ ˆ τ ξ d (1 + O ( δ + α )). By choosing α and δ su ffi ciently small, weobtain Re m k ( (cid:37) (cid:48) ) (cid:38) ˆ τ ξ d in U δ .As m k ( (cid:37) (cid:48) ) is homogeneous of degree two, we find that h k is homogeneous of degree one in U δ .Recalling that z = ( x , s ) remains in a compact domain here, we thus find(4.44) | h k ( (cid:37) (cid:48) ) | (cid:38) λ T , ˜ τ in U δ . Next, as h k = m k (cid:44) U δ we may write, with Lemma 4.26,2 h k ∂ x d h k = ∂ x d m k ∈ S ((1 + εγ ) λ T , ˜ τ , g T ) . which yields the result using the ellipticity estimate (4.44). (cid:4) W ∂ Q x d = z N ∈ R z (cid:48) = ( s , x (cid:48) ) ∈ R × R d − = R N − V V z V F igure
4. Open neighborhoods of z ∈ ∂ Z introduced in the course of theproof of Theorem 4.17. We let χ δ , χ δ, ∈ S (1 , g T ) be supported in M T , V , homogeneous of degree zero, and be such that µ k ≥ − C δ for both k = , χ δ, ≡ χ δ, )and χ δ ≡ χ δ, ). Recalling the notation of Section 4.4 andthe microlocalization symbols constructed there, this can be done as follows, for instance for theconstruction of χ δ, . Let ˆ χ ∈ C ∞ ( R ) be such thatsupp( ˆ χ ) ⊂ [ − , + ∞ ) , ˆ χ ≡ − , + ∞ ) . We also introduce V ⊂ V an open neighborhood of supp( χ V ) in R N + , in particular V (cid:98) V (thelocal geometry is illustrated in Figure 4) and we choose χ V ∈ C ∞ ( R N + ) such that χ V ≡ V , supp( χ V ) ⊂ V . We set χ δ, ( (cid:37) (cid:48) ) = χ V ( z ) (1 − χ / , F ( (cid:37) (cid:48) )) ˆ χ ( µ ( (cid:37) (cid:48) ) /δ ) ˆ χ ( µ ( (cid:37) (cid:48) ) /δ ) ∈ S (1 , g T ) . we have χ δ, ≡ χ δ, ).We choose δ > ffi ciently small so that the results of Lemmata 4.22 and 4.23 apply, that is,on supp( χ δ ) the roots of q k are simple and | ˆ τ ( (cid:37) (cid:48) ) | (cid:46) | ζ (cid:48) | , and also the result of Lemma 4.27 holdsfor U δ a conic neighborhood of supp( χ δ ), for δ ∈ (0 , δ ) and for ε > ffi ciently small.With the value of δ fixed now, to ease notation we now write χ, χ , χ in place of χ δ , χ δ, , χ δ, and Ξ , Ξ in place of Op T ( χ δ, ) , Op T ( χ δ, ). Lemma 4.28.
Let χ = χ or χ and, accordingly, Ξ = Ξ or Ξ . We haveQ k Ξ = Q k , + Q k , − Ξ + (1 + γε ) R Ξ + R − M = Q k , − Q k , + Ξ + (1 + γε ) R (cid:48) Ξ + R (cid:48)− M , where Q k , a = (cid:0) D x d + i ˆ τ ξ d − i a Op T w ( h k χ ) (cid:1) , a ∈ { + , −} , and R , R (cid:48) ∈ Ψ ( λ T , ˜ τ , g T ) and R − M , R (cid:48)− M ∈ Ψ ( λ − M T , ˜ τ , g T ) , for arbitrary large M ∈ N . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 37
Proof . In the proof we shall denote by R j a generic operator in Ψ ( λ j T , ˜ τ , g T ), j ∈ R , whose expres-sion may change from one line to the other.Observe that we have, for any M ∈ N , M k Ξ = M k Op T ( χ ) Ξ + R − M = Op T ( m k χ ) Ξ + (1 + γε ) R Ξ + R − M . With Lemma 4.27 applied with U δ , a conic neighborhood of supp( χ ), we haveOp T ( m k χ ) = Op T ( h k χ ) mod Ψ ((1 + γε ) λ T , ˜ τ , g T ) , using the properties of the tangential calculus (see Proposition 4.3). This yields M k Ξ = Op T ( h k χ ) Ξ + (1 + γε ) R Ξ + R − M = Op T w ( h k χ ) Ξ + (1 + γε ) R Ξ + R − M . We then find Q k Ξ = ( D x d + i ˆ τ ξ d ) Ξ + M k Ξ= (cid:0) D x d + i ˆ τ ξ d + i Op T w ( h k χ ) (cid:1)(cid:0) D x d + i ˆ τ ξ d − i Op T w ( h k χ ) (cid:1) Ξ+ i [ D x d + i ˆ τ ξ d , Op T w ( h k χ )] Ξ + (1 + γε ) R Ξ + R − M , In fact, the order of the operators can be changed and we find Q k Ξ = (cid:0) D x d + i ˆ τ ξ d − i Op T w ( h k χ ) (cid:1)(cid:0) D x d + i ˆ τ ξ d + i Op T w ( h k χ ) (cid:1) Ξ − i [ D x d + i ˆ τ ξ d , Op T w ( h k χ )] Ξ + (1 + γε ) R Ξ + R − M . The following lemma then yields the result. (cid:4)
Lemma 4.29.
Let χ = χ or χ and, accordingly, Ξ = Ξ or Ξ . We have, for a ∈ { + , −} , [ D x d + i ˆ τ ξ d , Q k , a ] Ξ = − i a [ D x d + i ˆ τ ξ d , Op T w ( h k χ )] Ξ = (1 + γε ) R Ξ + R − M with R ∈ Ψ ( λ T , ˜ τ , g T ) andR − M ∈ Ψ ( λ − M T , ˜ τ , g T ) . Proof . We have [ˆ τ ξ d , Op T w ( h k χ )] ∈ Ψ ((1 + γε ) λ T , ˜ τ , g T ) as a consequence of the tangential calculuswe have introduced. We have [ D x d , Op T w ( h k χ )] = Op T w (cid:0) D x d ( h k χ ) (cid:1) . We then write D x d ( h k χ ) = D x d ( h k ) χ + h k D x d ( χ ) . Because of the definition of χ we have D x d χ ( (cid:37) (cid:48) ) ≡ χ ( (cid:37) (cid:48) )). Thus Op T w (cid:0) ( D x d χ ) h k (cid:1) Ξ ∈ Ψ ( λ − M T , ˜ τ , g T ), for any M ∈ N . Next, by Lemma 4.27 we have χ D x d h k ∈ S ((1 + εγ ) λ T , ˜ τ , g T ), whichconcludes the proof. (cid:4) Lemma 4.30.
Let χ = χ or χ and, accordingly, Ξ = Ξ or Ξ . Let k , (cid:96) ∈ { , } and a , b ∈ { + , −} .We have, for any M ∈ N , [ Q k , a , Q (cid:96), b ] Ξ = (1 + γε ) R Ξ + R − M , [ D x d + i ˆ τ ξ d , Q k , a Q (cid:96), b ] Ξ = (1 + γε ) R , Ξ + R , − M , with R ∈ Ψ ( λ T , ˜ τ , g T ) , R , ∈ Ψ , τ , R − M ∈ Ψ , − M ˜ τ , and R , − M ∈ Ψ , − M ˜ τ . Proof . Since [Op T w ( h k χ ) , Op T w ( h l χ )] ∈ Ψ ((1 + γε ) λ T , ˜ τ , g T ), using the properties of the tangentialcalculus (see Proposition 4.3), the result follows from Lemma 4.29. (cid:4) We may now provide a proof of the microlocal estimate for the region E . Proof of Proposition 4.25.
In the proof, we shall denote by R j , k a generic operator in Ψ j , k ˜ τ , j ∈ N , k ∈ R , whose expression may change from one line to the other. We denote by M an arbitrarilylarge integer whose value may change from one line to the other. With the previous lemmata we write, using that χ ≡ χ ), P ϕ Ξ = Q Q Ξ = Q Ξ Q Ξ + R , − M (4.45) = Q , − Q , + Ξ Q , − Q , + Ξ + (1 + γε ) R , Ξ + R , − M = Q , − Q , + Q , − Q , + Ξ + (1 + γε ) R , Ξ + R , − M = Q , − Q , + Q , − Ξ Q , + Ξ + (1 + γε ) R , Ξ + R , − M = Q , − Q , − Q , + Ξ Q , + Ξ + (1 + γε ) R , Ξ + R , − M = Q − Q + Ξ + (1 + γε ) R , Ξ + R , − M , with Q − = Q , − Q , − and Q + = Q , + Q , + .The principal symbol of Q − is q − = q − q , − ∈ S , τ in a conic neighborhood of supp( χ ), whereall the roots of q − have negative imaginary parts. Thus,(4.46) the operator Q − fulfills the requirements of Lemma B.1.For both Q , + and Q , + we have the following estimate, characterized by the loss of a halfderivative and a trace observation, as given by Lemma B.6, for (cid:96), m ∈ R , γ / (cid:107) ˜ τ m − / Ξ v (cid:107) ,(cid:96), ˜ τ (cid:46) (cid:107) ˜ τ m Q k , + Ξ v (cid:107) ,(cid:96), ˜ τ + | tr(˜ τ m Ξ v ) | ,(cid:96) + / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , k = , , for v ∈ S ( R N + ), and for τ ≥ τ ∗ and γ ≥ ffi ciently large, and ε ∈ [0 , α = α =
1, we have the following estimate for the operator Q + ,for M > (cid:96) ∈ R , γ (cid:107) ˜ τ − Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) Q + Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , (4.47)for v ∈ S ( R N + ), and for τ and γ chosen su ffi ciently large.With (4.46) and (4.47), applying now Proposition B.8, and using that, for any M ∈ N , [ D x d + i ˆ τ ξ d , Q + ] Ξ = (1 + γε ) R , Ξ + R , − M by Lemma 4.30, we obtain γ (cid:107) ˜ τ − Ξ v (cid:107) , , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + Ξ v (cid:107) + + | tr( Ξ v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ), and for τ ≥ τ ∗ and γ ≥ ffi ciently large, for ε ∈ [0 , ε ] with ε > ffi ciently small. Finally, with (4.45), we conclude the proof of Proposition 4.25 bychoosing γ large and ε ∈ [0 , ε ] with ε > ffi ciently small. (cid:4) Remark 4.31.
Note that the end of the proof of Proposition 4.25 is a point where the introductionof the second large parameter γ is crucial. Even in the case ε =
0, that is for a weight function thatonly depend on the variable z N , taking γ large is needed to conclude.4.8. Microlocal estimate in the region F . In the region F we have ˜ τ (cid:46) λ T , ˜ τ and the symbolsof the operators Q k are characterized by two simple roots that are separated (see the first item ofProposition 4.21). We prove the following estimate. Proposition 4.32.
Let M ∈ N . There exist τ ≥ τ ∗ , γ ≥ , and C > such that (cid:107) Ξ F v (cid:107) , , ˜ τ + | tr( Ξ F v ) | , / , ˜ τ ≤ C (cid:16) (cid:107) P ϕ Ξ F v (cid:107) + + | tr( Ξ F v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ (cid:17) , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for v ∈ S ( R N + ) . Proof . We write χ = χ F and Ξ = Ξ F , to ease the reading of the proof.We also let χ , χ ∈ S (1 , g T ) be supported in M T , V , homogeneous of degree zero, and be suchthat | ˆ τ ( (cid:37) (cid:48) ) | ≤ θ λ T , ˜ τ ( (cid:37) (cid:48) ) in their support (using the notation of Section 4.4) and such that χ ≡ χ ) and χ ≡ χ ). This can be PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 39 done as follows, for instance for the construction of χ . We introduce V ⊂ V an open set of R N that is a neighborhood of supp( χ V ) in R N + , in particular V (cid:98) V (the local geometry is illustratedin Figure 4) and we choose χ V ∈ C ∞ ( R N + ) such that χ V ≡ V , supp( χ V ) ⊂ V . We set χ ( (cid:37) (cid:48) ) = χ V ( z ) χ , F ∈ S (1 , g T ) , with the function χ δ, F as introduced in Section 4.4. We have | ˆ τ ( (cid:37) (cid:48) ) | ≤ θ λ T , ˜ τ ( (cid:37) (cid:48) ), which leaves“enough room” for a similar construction for χ . We set Ξ = Op T ( χ ).With Proposition 4.21, in a conic neighborhood of supp( χ ) the roots of q k , k = ,
2, are simple,and we may write q k ( (cid:37) ) = q k , + ( (cid:37) ) q k , − ( (cid:37) ) , q k , ± ( (cid:37) ) = ξ d − ρ k , ± ( (cid:37) (cid:48) ) , where ρ k , ± ∈ S ( λ T , ˜ τ , g T ) in a conic neighborhood of supp( χ ) and there we haveIm ρ k , + ≥ C λ T , ˜ τ , Im ρ k , − ≤ − C λ T , ˜ τ . We set Q k , ± = D x d − Op T w ( χρ k , ± ).In the proof we shall denote by R j , k as a generic operator in Ψ j , k ˜ τ , j ∈ N , k ∈ R , whose expressionmay change from one line to the other. Lemma 4.33.
Let
Ξ = Ξ or Ξ . We have, for arbitrary large M ∈ N ,Q k Ξ = Q k , + Q k , − Ξ + γ R , Ξ + R , − M = Q k , − Q k , + Ξ + γ R , Ξ + R , − M . Proof . We have Q k , + Q k , − = D x d − (cid:0) Op T w ( χρ k , + ) + Op T w ( χρ k , − ) (cid:1) D x d + Op T w ( χρ k , + ) Op T w ( χρ k , − ) + γ R , . We thus find, for any M ∈ N , Q k , + Q k , − Ξ = (cid:16) Op T w ( χ ) D x d − (cid:0) Op T w ( χρ k , + ) + Op T w ( χρ k , − ) (cid:1) D x d + Op T w ( χρ k , + ) Op T w ( χρ k , − ) (cid:17) Ξ+ γ R , Ξ + R , − M = Op w ( χ q k ) Ξ + γ R , Ξ + R , − M = Q k Ξ + γ R , Ξ + R , − M . (cid:4) This result yields, for any M ∈ N , P ϕ Ξ = Q Q Ξ = Q Ξ Q Ξ + R , − M (4.48) = Q , − Q , + Ξ Q , − Q , + Ξ + γ R , Ξ + R , − M = Q , − Q , + Q , − Q , + Ξ + γ R , Ξ + R , − M = Q − Q + + γ R , Ξ + R , − M , where Q − = Q , − Q , − and Q + = Q , + Q , + .Both roots of the symbol q − of the operator Q − are in the lower half complex plane in a conicneighborhood of supp( χ ). Then, with Lemma B.1 we have the following perfect elliptic estimate,for any M > (cid:107) Ξ v (cid:107) , , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:46) (cid:107) Q − Ξ v (cid:107) + + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ), for τ ≥ τ ∗ and γ ≥ ffi ciently large, and ε ∈ [0 , The roots of the first-order factor Q k , + , k = M > (cid:96) ∈ R , (cid:107)| Ξ v (cid:107) ,(cid:96), ˜ τ ≤ C (cid:16) (cid:107) Q k , + Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ (cid:17) , for v ∈ S ( R N + ), for τ ≥ τ ∗ and γ ≥ ffi ciently large, and ε ∈ [0 , α = α = δ = δ =
0, we have the following estimates forthe operator Q + , for M > (cid:96) ∈ R , (cid:107) Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) Q + Ξ v (cid:107) ,(cid:96), ˜ τ + | tr( Ξ v ) | ,(cid:96) + / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , (4.50)for v ∈ S ( R N + ), and for τ ≥ τ ∗ and γ ≥ ffi ciently large.Applying now Proposition B.8, with (4.49) and (4.50), we obtain (cid:107) Ξ v (cid:107) , , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + Ξ v (cid:107) + + | tr( Ξ v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ), and for τ ≥ τ ∗ and γ ≥ ffi ciently large, for ε ∈ [0 , τ and γ large. (cid:4) Proof of the Carleman estimate of Theorem 4.17.
We choose W an open neighborhoodof z in R N such that W (cid:98) V (see Figure 4). Let u = w | Z , with w ∈ C ∞ c ((0 , S ) × R d ) andsupp( w ) ⊂ W . We set v = e τϕ u .We collect the di ff erent estimations that we have obtained in Propositions 4.24, 4.25, and 4.32.For some δ = δ ∈ (0 ,
1) to be kept fixed, for τ ≥ τ ∗ , γ ≥
1, and ε ∈ (0 ,
1] we have γ (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ + | tr( Ξ ( k ) δ, − v ) | , / , ˜ τ (cid:46) (cid:107) P ϕ Ξ ( k ) δ, − v (cid:107) + + | tr( Ξ ( k ) δ, − v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , (4.51)for k = ,
2, and γ (cid:107) ˜ τ − Ξ δ, v (cid:107) , , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ (cid:46) (cid:107) P ϕ Ξ δ, v (cid:107) + + | tr( Ξ δ, v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , (4.52)and (cid:107) Ξ F v (cid:107) , , ˜ τ + | tr( Ξ F v ) | , / , ˜ τ (cid:46) (cid:107) P ϕ Ξ F v (cid:107) + + | tr( Ξ F v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , (4.53)for τ ≥ τ , γ ≥ γ , ε ∈ [0 , ε ].We then pick α > α (cid:0) (4.51) + (4.53) (cid:1) + (4.52). We will choose τ su ffi ciently large so that ατ / ≥ Lemma 4.34.
There exists C > such that αγ (cid:80) k = , (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ + γ (cid:107) ˜ τ − Ξ δ, v (cid:107) , , ˜ τ + α (cid:107) Ξ F v (cid:107) , , ˜ τ ≥ C γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ , for τ chosen su ffi ciently large. With a similar, yet simpler, proof, we have the following lemma.
Lemma 4.35.
We have α (cid:80) k = , | tr( Ξ ( k ) δ, − v ) | , / , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ + α | tr( Ξ F v ) | , / , ˜ τ (cid:38) α | tr( v ) | , / , ˜ τ , for τ chosen su ffi ciently large. PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 41
With these two lemmata we obtain(4.54) γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ + α (cid:16) (cid:107) Ξ F v (cid:107) , , ˜ τ + γ (cid:80) k = , (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ (cid:17) + α | tr( v ) | , / , ˜ τ (cid:46) α (cid:16) r.h.s.(4.51) + r.h.s.(4.53) (cid:17) + r.h.s.(4.52) . The next lemma is crucial in the computation of the commutator [ P ϕ , Ξ δ, ]. A proof is givenbelow. Lemma 4.36.
We have [ P ϕ , Ξ δ, ] = Op( g ) + Op( h ) + γ R , − , where g , h ∈ γ Ψ , τ and R , − ∈ Ψ , − τ ,with • g ( (cid:37) ) = for z in a neighborhood of V • h ( (cid:37) ) = (cid:80) j = h j ( (cid:37) (cid:48) ) ξ jd , h j ∈ γ Ψ ( λ − j T , ˜ τ , g T ) , homogeneous of degree − j, and χ (1) δ, − + χ (2) δ, − + χ F ≥ in a conic neighborhood of supp( h j ) in the variables ( ζ, τ, γ, ε ) , for z ∈ V ,j = , . . . , . We have [ P ϕ , Ξ ] ∈ γ R , , for Ξ = Ξ (1) δ, − , Ξ (2) δ, − or Ξ F . Lemma 4.36 gives, for any M ∈ N , (cid:107) Op( g ) v (cid:107) + (cid:46) (cid:107) v (cid:107) , − M , ˜ τ , and we obtain α (cid:80) k = , (cid:107) P ϕ Ξ ( k ) δ, − v (cid:107) + + (cid:107) P ϕ Ξ δ, v (cid:107) + + α (cid:107) P ϕ Ξ F v (cid:107) + (cid:46) (cid:107) P ϕ v (cid:107) + + (cid:107) Op( h ) v (cid:107) + + αγ (cid:107) v (cid:107) , , ˜ τ + γ (cid:107) v (cid:107) , − , ˜ τ . From (4.54) and (4.51)–(4.53) we thus obtain, for α chosen su ffi ciently small (and kept fixed forthe remainder of the proof) and τ chosen su ffi ciently large(4.55) γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ + (cid:107) Ξ F v (cid:107) , , ˜ τ + γ (cid:80) k = , (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ + | tr( v ) | , / , ˜ τ (cid:46) (cid:107) P ϕ v (cid:107) + + | tr( v ) | , / , ˜ τ + (cid:107) Op( h ) v (cid:107) + . We set χ = χ (1) δ, − + χ (2) δ, − + χ F . We have the following lemma whose proof is given below. Lemma 4.37.
Let W be an open set of R N with W (cid:98) V . There exist C > and τ ≥ τ ∗ such that (cid:107) Op T ( h j ) w (cid:107) + ≤ C γ (cid:0) (cid:107) Op T ( χ ) w (cid:107) , − j , ˜ τ + γ (1 + εγ ) (cid:107) w (cid:107) , − j , ˜ τ (cid:1) , for w ∈ S ( R N + ) , supp( w ) ⊂ W and τ ≥ τ . Thus, we obtain (cid:107)
Op( h ) v (cid:107) + ≤ (cid:80) j = (cid:107) Op( h j ) D jx d v (cid:107) + (cid:46) (cid:80) j = γ (cid:107) Op T ( χ ) D jx d v (cid:107) , − j , ˜ τ + γ (1 + εγ ) (cid:107) v (cid:107) , − , ˜ τ . As [Op T ( χ ) , D jx d ] ∈ γ Ψ j − , τ we obtain (cid:107) Op( h ) v (cid:107) + (cid:46) γ (cid:107) Op T ( χ ) v (cid:107) , , ˜ τ + γ (cid:107) v (cid:107) , − , ˜ τ (cid:46) γ (cid:16) (cid:80) k = , (cid:107) Ξ ( k ) δ, − v (cid:107) , , ˜ τ + (cid:107) Ξ F v (cid:107) , , ˜ τ (cid:17) + γ (cid:107) v (cid:107) , − , ˜ τ . Using this estimate in (4.55), for τ chosen su ffi ciently large, we thus obtain γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ + (cid:107) Ξ F v (cid:107) , , ˜ τ + γ (cid:80) k = , (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ + | tr( v ) | , / , ˜ τ (cid:46) (cid:107) P ϕ v (cid:107) + + | tr( v ) | , / , ˜ τ . The end of the proof of Theorem 4.17 is then classical. (cid:4)
Proof of Lemma 4.34.
With Lemma 4.14 we may write X = αγ (cid:80) k = , (cid:107) ˜ τ − / Ξ ( k ) δ, − v (cid:107) , , ˜ τ + γ (cid:107) ˜ τ − Ξ δ, v (cid:107) , , ˜ τ + α (cid:107) Ξ F v (cid:107) , , ˜ τ (cid:38) (cid:80) j = (cid:16) αγ (cid:80) k = , (cid:107) ˜ τ − / Λ − j T , ˜ τ D jx d Ξ ( k ) δ, − v (cid:107) + + γ (cid:107) ˜ τ − Λ − j T , ˜ τ D jx d Ξ δ, v (cid:107) + + α (cid:107) Λ − j T , ˜ τ D jx d Ξ F v (cid:107) + (cid:17) yielding X (cid:38) γ (cid:80) j = (cid:16) (cid:80) k = , (cid:107) ˜ τ − Λ − j T , ˜ τ D jx d Ξ ( k ) δ, − v (cid:107) + + (cid:107) ˜ τ − Λ − j T , ˜ τ D jx d Ξ δ, v (cid:107) + + (cid:107) ˜ τ − Λ − j T , ˜ τ D jx d Ξ F v (cid:107) + (cid:17) , as α ≥ αγ ˜ τ − / ≥ γ ˜ τ − using, on the one hand, that ( τϕ ) − / = γ ˜ τ − / ≤ τ ≥ τ ∗ ≥ ϕ ≥
1, and, on the other hand, that ατ / ≥ α ˜ τ / = α ( τγϕ ) / ≥ γ / since ϕ ≥ h = χ (1) δ, − + χ (2) δ, − + χ δ, + χ F ∈ S (1 , g T ), X (cid:38) γ (cid:80) j = (cid:107) ˜ τ − Λ − j T , ˜ τ D jx d Op T ( h ) v (cid:107) + . As[ D jx d , Op T ( h )] ∈ γ Ψ j − , τ , we obtain X + γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ (cid:38) γ (cid:80) j = (cid:107) ˜ τ − Λ − j T , ˜ τ Op T ( h ) D jx d v (cid:107) + . By the (local) Gårding inequality of Proposition 4.15, as h ( (cid:37) (cid:48) ) ≥ V ∩ R N + that contains supp( v ), we obtain X + γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ (cid:38) γ (cid:80) j = (cid:107) ˜ τ − D jx d v (cid:107) , − j , ˜ τ (cid:16) γ (cid:107) ˜ τ − v (cid:107) , , ˜ τ , We conclude by taking τ su ffi ciently large with the usual semi-classical inequality (4.17). (cid:4) Proof of Lemma 4.36.
Up to γ S , − τ , the principal symbol of [ P ϕ , Ξ δ, ] is given by − i { p ϕ , χ δ, } ,and thus involves derivatives of χ δ, . We recall the form of χ δ, , as introduced in Section 4.4, χ δ, ( (cid:37) (cid:48) ) = χ V ( z ) (1 − χ / , F ( (cid:37) (cid:48) )) χ ( µ ( (cid:37) (cid:48) ) /δ ) χ ( µ ( (cid:37) (cid:48) ) /δ ) . Computing − i { p ϕ , χ δ, } , we obtain the following list of terms. Terms involving derivatives of χ V ( z ): Those terms contribute to the symbol g that van-ishes in a neighborhood of V . Terms involving derivatives of χ / , F ( (cid:37) (cid:48) ): Those terms are supported in { θ λ T , ˜ τ / ≤ ˜ τ ≤ θ λ T , ˜ τ / } , using the notation of Section 4.4. As χ , F = τ ≤ θ λ T , ˜ τ /
2, we see that χ F ( (cid:37) (cid:48) ) = χ V ( z ) χ , F ( (cid:37) (cid:48) ) = z ∈ V .Those terms contribute to the symbol h . Terms involving derivatives of χ ( µ k ( (cid:37) (cid:48) ) /δ ), k = , From the definition of χ we seethat those terms are supported in {− ≤ µ k ( (cid:37) (cid:48) ) /δ ≤ − } . We have χ − ( µ k ( (cid:37) (cid:48) ) /δ ) = χ , F ( (cid:37) (cid:48) ) + (1 − χ / , F ( (cid:37) (cid:48) )) ≥ χ F ( (cid:37) (cid:48) ) + χ ( k ) δ, − ( (cid:37) (cid:48) ) ≥ z ∈ V . Those terms contribute to thesymbol h . (cid:4) Proof of Lemma 4.37.
Let χ W ( z ) ∈ C ∞ c ( V ) be such that χ W ≡ W . Themicrolocal version of the Gårding inequality of Proposition 4.16 gives, by Lemma 4.36,Re(Op T ( χ ) Op T ( χ W h j ) w , Op T ( χ W h j ) w ) + + (cid:107) w (cid:107) , − M , ˜ τ (cid:38) (cid:107) Op T ( χ W h j ) w (cid:107) + . Then, with the Young inequality, we obtain (cid:107) Op T ( χ ) Op T ( χ W h j ) w (cid:107) + + (cid:107) w (cid:107) , − M , ˜ τ , (cid:38) (cid:107) Op T ( χ W h j ) w (cid:107) + . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 43
Since Op T ( χ W h j ) w = Op T ( h j ) w + R , − M w , with R , − M ∈ Ψ , − M ˜ τ , for any M ∈ N , we obtain (cid:107) Op T ( χ ) Op T ( h j ) w (cid:107) + + (cid:107) w (cid:107) , − M , ˜ τ , (cid:38) (cid:107) Op T ( h j ) w (cid:107) + . As [Op T ( χ ) , Op T ( h j )] ∈ γ (1 + εγ ) Ψ ( λ − j T , ˜ τ , g T ), we obtain the sought estimate. (cid:4)
5. S pectral inequality and application
We start this section by stating and proving an interpolation type inequality. Next, we prove thespectral inequality of Theorem 1.3. Finally, as an application, we state a null-controllability resultthat follows from it.5.1.
An interpolation inequality.
Let S > α ∈ (0 , S / Z = (0 , S ) × Ω and we introduce Y = ( α, S − α ) × Ω for some α >
0. As is done in other sections,we denote by z = ( s , x ) ∈ Z , with s ∈ (0 , S ) and x ∈ Ω . We recall that P denotes the augmentedelliptic operator P : = D s + B , where B = ∆ x . Theorem 5.1 (Interpolation inequality) . Let O be a nonempty open subset of Ω . There exist C > and δ ∈ (0 , such that for u ∈ H ( Z ) that satisfiesu ( s , x ) | x ∈ ∂ Ω = , ∂ ν u ( s , x ) | x ∈ ∂ Ω = , s ∈ (0 , S ) , we have (cid:107) u (cid:107) H ( Y ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:80) ≤ j ≤ (cid:107) ∂ js u | s = (cid:107) H − j ( O ) (cid:17) δ . (5.1)First, we provide a local interpolation estimate in a neighborhood of a point of { } × O . Lemma 5.2 (local interpolation near s = . Let x ∈ O , there exist V a neighborhood of (0 , x ) in R × R d , C > , and δ ∈ (0 , such that for u ∈ H ( Z ) we have (cid:107) u (cid:107) H ( V ∩ Z ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:80) ≤ j ≤ (cid:107) ∂ js u | s = (cid:107) H − j ( O ) (cid:17) δ . (5.2)Second, we provide an interpolation estimate with an interior observation, that is, we have anestimate away from the boundary 0 × Ω . Proposition 5.3 (Interpolation with an interior observation) . Let Z be a nonempty open set in Z.There exist C > and δ ∈ (0 , such that for u ∈ H ( Z ) that satisfiesu ( s , x ) | x ∈ ∂ Ω = , ∂ ν u ( s , x ) | x ∈ ∂ Ω = , s ∈ (0 , S ) , we have (cid:107) u (cid:107) H ( Y ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( Z ) (cid:17) δ . (5.3)With these two local interpolation results, whose proofs are given below, we can then write aproof of Theorem 5.1. Proof of Theorem 5.1.
Introducing V as given in Lemma 5.2, we let Z be an open subset of V ∩ Z .With Lemma 5.2 we then have (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( Z ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:32) (cid:107) Pu (cid:107) L ( Z ) + (cid:80) ≤ j ≤ (cid:107) ∂ js u | s = (cid:107) H − j ( O ) (cid:33) δ , (5.4)as we can assume that (cid:107) Pu (cid:107) L ( Z ) ≤ (cid:107) u (cid:107) H ( Z ) otherwise estimate (5.1) is trivial. Applying Proposi-tion 5.3 we have, for some δ (cid:48) ∈ (0 , (cid:107) u (cid:107) H ( Y ) ≤ C (cid:107) u (cid:107) − δ (cid:48) H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( Z ) (cid:17) δ (cid:48) . This, with (5.4), gives (5.1) with δ (cid:48) δ in place of δ . (cid:4) For the proofs of Lemma 5.2 and Proposition 5.3. We shall need the following lemma whoseproof can be found in [Rob95].
Lemma 5.4.
Let A ≥ , B ≥ , and C ≥ . We assume that A ≤ B and that there exist τ > , µ > and ν > such that A ≤ e − ντ B + e µτ C , for τ ≥ τ . (5.5) Then A ≤ KB − δ C δ , where K = max(2 , e µτ ) and δ = ν/ ( ν + µ ) ∈ (0 , . Proof of Lemma 5.2.
Let r > z = ( − r , x ), where r is chosen su ffi ciently small to have B ∩ { s = } ⊂ O with B = B ( z , r ). Let ψ = −| z − z | , with z = ( s , x ). We have ∂ s ψ ( z ) ≤ − C < B . We set ϕ ( z ) = e γψ ( z ) . Let χ ∈ C ∞ ( R d + ) be such that χ ( z ) = | z − z | ≤ r / χ ( z ) = | z − z | ≥ r /
4. We apply the local Carleman estimate of Corollary 3.2 to v = χ u , and we obtain,for γ ≥ ffi ciently large (to be kept fixed in what follows), (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ D α z v (cid:107) L ( B ∩ Z ) (cid:46) (cid:107) e τϕ Pv (cid:107) L ( Z ) + τ / (cid:80) j = | tr( e τϕ D js v | s = + ) | , − j ,τ . (5.6)Note that if γ is fixed we have τ (cid:16) ˜ τ . In { } × O , we have ϕ ≤ e − γ r then τ / (cid:80) j = | tr( e τϕ D js v | s = + ) | , − j ,τ (cid:46) e C τ (cid:80) j = | D js u | s = + | H − j ( O ) , C = (1 + a ) e − γ r , (5.7)for any a >
0. We have Pv = χ Pu + [ P , χ ] u . The term [ P , χ ] is a di ff erential operator of order 3and it is supported in { z ∈ R d + ; 7 r / ≤ | z − z | ≤ r / } . On this set, we have ϕ ≤ e − γ (7 r / . Wethus find (cid:107) e τϕ [ P , χ ] u (cid:107) L ( Z ) (cid:46) e C τ (cid:107) u (cid:107) H ( Z ) , C = e − γ (7 r / . (5.8)In Z , we have ϕ ≤ e − γ r < C ; this implies (cid:107) e τϕ χ Pu (cid:107) L ( Z ) (cid:46) e C τ (cid:107) Pu (cid:107) L ( Z ) . (5.9)In { z ∈ R d + ; | z − z | ≤ r } , χ ≡ u = v , and on this set ϕ ≥ e − γ (3 r ) then we have e C τ (cid:107) u (cid:107) H ( B ( z , r ) ∩ Z ) (cid:46) (cid:80) | α |≤ τ / −| α | (cid:107) e τϕ D | α | z v (cid:107) L ( B ∩ Z ) , C = e − γ (3 r ) . (5.10)Remark that C < C < C , for a > ffi ciently small. Following (5.6)–(5.10) we obtain (cid:107) u (cid:107) H ( B ( z , r ) ∩ Z ) (cid:46) e ( C − C ) τ (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:80) j = | D js u | s = + | H − j ( O ) (cid:17) + e − ( C − C ) τ (cid:107) u (cid:107) H ( Z ) . Applying Lemma 5.4, we obtain the result with V = B ( z , r ). (cid:4) We prove Proposition 5.3 by means of two lemmata. For α (cid:48) ∈ (0 , α ) and a ∈ (0 , Y α (cid:48) , a = ( α (cid:48) , S − α (cid:48) ) × Ω a , where Ω a = { x ∈ Ω , dist( x , ∂ Ω ) > a > } . Lemma 5.5.
Let Z be a nonempty open set in Z. Let α (cid:48) ∈ (0 , α ) and a ∈ (0 , . There exist C > and δ ∈ (0 , such that for u ∈ H ( Z ) , (cid:107) u (cid:107) H ( Y α (cid:48) , a ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( Z ) (cid:17) δ . (5.12) PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 45
Lemma 5.6.
Let ( s , x ) ∈ (0 , S ) × ∂ Ω . There exist δ ∈ (0 , , C > , V a neighborhood of ( s , x ) , α (cid:48) ∈ (0 , α ) , and a ∈ (0 , such that we have (cid:107) u (cid:107) H ( V ∩ Z ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( Y α (cid:48) , a ) (cid:17) δ , (5.13) for u ∈ H ( Z ) satisfyingu ( s , x ) | x ∈ ∂ Ω = , ∂ ν u ( s , x ) | x ∈ ∂ Ω = , s ∈ (0 , S ) . Proof of Proposition 5.3.
We can assume that (cid:107) Pu (cid:107) L ( Z ) ≤ (cid:107) u (cid:107) H ( Z ) , otherwise inequality (5.3) isobvious. In particular, if (5.3) holds for a value δ = δ > δ ∈ [0 , δ ]possibly with a larger constant C = C δ . The same observation can be made for the estimations(5.12) and (5.13).With a compactness argument we can find a finite number of open sets V j , j ∈ J , where esti-mate (5.13) holds for some values δ = δ j ∈ (0 , α (cid:48) j ∈ (0 , α ), and a j ∈ (0 , α, S − α ) × ∂ Ω ⊂ ∪ j ∈ J V j . For a ∈ (0 ,
1) and α (cid:48) ∈ (0 , α ), set ˜ Y α (cid:48) , a = ( α (cid:48) , S − α (cid:48) ) × ˜ Ω a , where ˜ Ω a = { x ∈ Ω , dist( x , ∂ Ω ) < a } .There exists a ∈ (0 ,
1) and α ∈ (0 , α ) such that ˜ Y α , a ⊂ Z ∩ ( ∪ j ∈ J V j ). Applying the localinterpolation estimate (5.13) for each V j , using now δ = min j ∈ J δ j ∈ (0 , , α = min j ∈ J α (cid:48) j ∈ (0 , α ) , and a = min j ∈ J a j ∈ (0 , Y α (cid:48) , a increases as α (cid:48) and a decrease) we obtain(5.14) (cid:107) u (cid:107) H ( ˜ Y α , a ) (cid:46) (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( Y α , a ) (cid:17) δ . Let Z be a nonempty open set in Z . By Lemma 5.5 we obtain, for some δ ∈ (0 , (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( Y α , a ) (cid:46) (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( Z ) (cid:17) δ , as the estimate of (cid:107) Pu (cid:107) L ( Z ) is clear here. Then, estimates (5.14) and (5.15) give (cid:107) u (cid:107) H ( ˜ Y α , a ) (cid:46) (cid:107) u (cid:107) − δ δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( Z ) (cid:17) δ δ . (5.16)Taking a ∈ (0 , a ) and α (cid:48) ∈ (0 , α ), we have Y ⊂ Y α (cid:48) , a ∪ ˜ Y α , a , and, by (5.12) in Lemma 5.5 and(5.16), we obtain (5.3). (cid:4) Proof of Lemma 5.5.
By a compactness argument, it su ffi ces to prove (5.12) with B ( z , R ) in placeof Y α (cid:48) , a where z ∈ Y α (cid:48) , a and 0 < R ≤ min( α (cid:48) , a ) /
2, implying B ( z , R ) ⊂ Z . Let z (0) be in Z and r > B ( z (0) , r ) (cid:98) Z . As Y α (cid:48) , a is connected, there exists a path Γ ⊂ Y α (cid:48) , a from z (0) = Γ (0) to z = Γ (1). Set r = dist( Γ , ∂ Z ). We have r > r = inf( R , r , r / z ( j ) ) j , for j ≥
0, by z ( j ) = Γ ( t j ) where t = t j = inf A j if A j (cid:44) ∅ , A j = ∅ , A j = { σ ∈ ( t j − , Γ ( σ ) (cid:60) B ( z j − , r ) } . The sequence ( z ( j ) ) j is finite by a compactness argument. The construction of the sequence isillustrated in Figure 5. r z (1) z ( j − r z ( j ) z ( j + z ( N − z ( N − Z ∂ Z Γ r = dist( Γ , ∂ Z ) ≥ r z (2) z ( N ) = y z (0) r F igure
5. Construction of the sequence ( z ( j ) ) j , j ∈ J , along the path Γ . Let ( z (0) , · · · , z ( N ) ) be such a sequence with z ( N ) = z . Note that we have B ( z ( j + , r ) ⊂ B ( z ( j ) , r ) ⊂ Z , for j = , · · · , N −
1, because of the choice we made for r above. Now we claim that there exists C > δ ∈ (0 ,
1) such that (cid:107) u (cid:107) H ( B ( z ( j + , r )) ≤ (cid:107) u (cid:107) H ( B ( z ( j ) , r )) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( B ( z ( j ) , r )) (cid:17) δ , (5.17)for j = , . . . , N −
1. This claim is proven below.We assume that (cid:107) Pu (cid:107) L ( Z ) ≤ (cid:107) u (cid:107) H ( Z ) , since otherwise the estimate we wish to prove is obvious.We then have (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( B ( z ( j + , r )) (cid:46) (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( B ( z ( j ) , r )) (cid:17) δ . By induction on j , we find (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( B ( z , r )) (cid:46) (cid:107) u (cid:107) − µ H ( Z ) (cid:16) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( B ( z (0) , r )) (cid:17) µ , (5.18)where µ = δ N .As P is elliptic, and B ( z (0) , r ) (cid:98) Z we have (cid:107) u (cid:107) H ( B ( z (0) , r )) (cid:46) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( Z ) . This estimateand (5.18) give (5.12).To prove estimation (5.17) we apply the local Carleman estimate of Proposition 2.7. We set ψ ( z ) = −| z − z ( j ) | and ϕ ( z ) = e γψ ( z ) and χ ∈ C ∞ c ( B ( z ( j ) , r )) to be such that χ ( z ) = r / < | z − z ( j ) | < r / , | z − z ( j ) | < r / r / < | z − z ( j ) | . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 47
The function v = χ u is supported in the open set B ( z ( j ) , r ) \ B ( z ( j ) , r / ⊂ Z where d ψ does notvanish. For γ ≥ ffi ciently large, by Proposition 2.7, we have (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α z v (cid:107) L ( Z ) (cid:46) (cid:107) e τϕ Pv (cid:107) L ( Z ) . (5.19)We have Pv = χ Pu + [ P , χ ] u and [ P , χ ] is a di ff erential operator of order 3 supported in A ∪ A with A = { z ; 5 r / ≤ | z − z ( j ) | ≤ r / } , A = { z ; 7 r / ≤ | z − z ( j ) | ≤ r / } . We write (cid:107) e τϕ Pv (cid:107) L ( Z ) ≤ (cid:107) e τϕ Pu (cid:107) L ( B ( z ( j ) , r )) + (cid:107) e τϕ [ P , χ ] u (cid:107) L ( A ∪ A ) . Since ϕ decreases as | z − z ( j ) | increases, we find (cid:107) e τϕ Pv (cid:107) L ( Z ) (cid:46) e τ C (cid:107) Pu (cid:107) L ( Z ) + e τ C (cid:107) u (cid:107) H ( B ( z ( j ) , r )) + e τ C (cid:107) u (cid:107) H ( Z ) , (5.20)where C = e − γ (7 r / and C = e − γ (5 r / .As we have χ ≡ B ( z ( j ) , r ) \ B ( z ( j ) , r ) we have e τ C (cid:107) u (cid:107) H ( B ( z ( j ) , r ) \ B ( z ( j ) , r )) ≤ (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α z v (cid:107) L ( Z ) , (5.21)where C = e − γ (3 r ) . Remark that C < C < C .Inequalities (5.19), (5.20), and (5.21) give (cid:107) u (cid:107) H ( B ( z ( j ) , r )) (cid:46) e τ ( C − C ) ( (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( B ( z ( j ) , r )) ) + e − τ ( C − C ) (cid:107) u (cid:107) H ( Z ) . as the estimate on B ( z ( j ) , r ) is clear with such a r.h.s. if τ ≥ τ ∗ ≥
1. We can optimize this lastestimate applying Lemma 5.4, which yields (5.17), and concludes the proof of Lemma 5.5. (cid:4)
Proof of Lemma 5.6.
The proof follows the same ideas as that of estimate (5.17) applying theboundary-type local Carleman estimate of Theorem 4.17. We use local coordinates in a boundedneighborhood V in R N of the point z = ( s , x ) of (0 , S ) × ∂ Ω as introduced in Section 4.2, suchthat this part of the boundary is locally given by { z N = x d = } and Z is locally given by { z N > } ;coordinates can be chosen to have moreover z = ( z (cid:48) , z (cid:48) =
0. We set z (1) = (0 , r ) where r > ψ ∈ C ∞ ( R N ) be such that ψ ( z ) = r − | z − z (1) | if | z − z (1) | ≤ r , r if 4 r ≤ | z − z (1) | . We have ψ ( z ) ≥ r > (cid:107) ψ ( k ) (cid:107) L ∞ < ∞ , k ∈ N , and ∂ ν ψ ( z ) = − ∂ z N ψ ( z ) = z N − r ) ≤ − C < , for | z − z (1) | ≤ r and z N =
0. Upon reducing the open neighborhood V , the weight function ψ fulfills the requirements listed in (4.2) and (4.20).We set ϕ ( z ) = e γψ ε ( z ) , where ψ ε ( z ) = ψ ( ε z (cid:48) , z N ). According to Theorem 4.17, there exist a neigh-borhood W (cid:98) V in R N of z , τ ≥ τ ∗ , γ ≥
1, and ε ∈ (0 ,
1] so that the Carleman estimate (4.21)holds for τ ≥ τ , γ ≥ γ , ε ∈ (0 , ε ] and smooth functions supported in W . We set γ = γ and ε = ε . The geometry of the level sets of the weight function is illustrated in Figure 6.In connection with the weight function ψ ε , we introduce the following anisotropic norm in R N ,that depends on the (now fixed) parameter ε , | z − y | ε = (cid:0) ε | z (cid:48) − y (cid:48) | + ( z N − y N ) (cid:17) / . Note that with γ and ε fixed we have τ (cid:16) ˜ τ . z (1) z N = x d V W ( , S ) × ∂ Ω ψ ε = Cst z (cid:48) = ( s , x (cid:48) ) ∈ R N − rz F igure
6. Geometry near the boundary for the application of the local Car-leman estimate of Theorem 4.17.
We denote by B ε ( z , r ) the ball of radius r centered at z associated with this norm. We have ψ ε ( z ) = r − | z − z (1) | ε if | z − z (1) | ε ≤ r , r if 4 r ≤ | z − z (1) | ε . Let χ ∈ C ∞ c ( R ) be such that χ ( z N ) = | z N | < r , r < | z N | , where r < r /
4. Let also χ ∈ C ∞ c ( B ε ( z (1) , r )) be such that χ ( z ) = | z − z (1) | ε < r , r (cid:48) < | z − z (1) | ε , where r , r (cid:48) are such that 2 r < r < r (cid:48) < r . Observe that if we choose the values of r (cid:48) − r > r > ffi ciently small, then the open set { z ∈ Z ; z N ∈ (0 , r ) } ∩ { z ∈ Z ; | z − z (1) | ε < r (cid:48) } iscontained in W . We now set χ ( z ) = χ ( z ) χ ( z N ). Figure 7 shows, near z , the region where χ ≡ χ (cid:48) ) ∩ Z ⊂ A ∪ A with A = { z ∈ Z ; z N ∈ ( r , r ) and | z − z (1) | ε < r (cid:48) } , A = { z ∈ Z ; z N ∈ (0 , r ) and r < | z − z (1) | ε < r (cid:48) } . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 49 z N = x d W r / r / ( s , x (cid:48) ) ∈ R N − r r z | z − z (1) | ε = r | z − z (1) | ε = r | z − z (1) | ε = r (cid:48) A A ( , S ) × ∂ Ω z (1) F igure
7. Geometry near the boundary for the derivation of the local in-terpolation inequality. The light color region shows where χ ≡
1; the darkcolor region shows where χ varies. Note that the relative scale of the twoaxes has been modified, if compared to Figure 6, for a better display of theregions A and A near z . The Carleman estimate (4.17) applies to v = χ u , by a density argument. As u | z N = + = ∂ ν u | z N = + = γ and ε were fixed above) (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α s , x v (cid:107) L ( W ∩ Z ) (cid:46) (cid:107) e τϕ Pv (cid:107) L ( W ∩ Z ) , τ ≥ τ . (5.22)We have Pv = χ Pu + [ P , χ ] u , where [ P , χ ] is a di ff erential operator of order 3 that is supportedin A ∪ A . On A , we have ϕ ≤ e γ (12 r − (2 r − r ) ) . On A we have ϕ ≤ e γ (12 r − r ) . We thus obtain (cid:107) e τϕ Pv (cid:107) L ( W ∩ Z ) (cid:46) e τ C (cid:0) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( Y α (cid:48) , a ) (cid:1) + e τ C (cid:107) u (cid:107) H ( Z ) , (5.23)where C = e γ (12 r − r ) , C = e γ (12 r − (2 r − r ) ) and 0 < a < r and some α (cid:48) ∈ (0 , α ) (recalling thedefinition of the set Y α (cid:48) , a in (5.11)).We restrict the l.h.s. of (5.22) to V = { z ∈ Z ; z N ∈ (0 , r ) } ∩ { z ∈ Z ; | z − z (1) | < r } , with r = r + r /
2, whose closure is a neighborhood of z in Z . Note that 2 r < r < r . As on this setwe have ϕ ≥ e γ (12 r − r ) and u ≡ v , we obtain e τ C (cid:107) u (cid:107) H ( V ) ≤ (cid:80) | α |≤ τ −| α | (cid:107) e τϕ D α s , x v (cid:107) L ( W ∩ Z ) , (5.24) where C = e γ (12 r − ( r + r / ) . Then (5.22), (5.23) and (5.24) give (cid:107) u (cid:107) H ( V ) (cid:46) e τ ( C − C ) (cid:0) (cid:107) Pu (cid:107) L ( Z ) + (cid:107) u (cid:107) H ( Y α (cid:48) , a ) (cid:1) + e − τ ( C − C ) (cid:107) u (cid:107) H ( Z ) . (5.25)Observe that we have C < C < C . By Lemma 5.4, we obtain the sought local interpolationinequality at the boundary. (cid:4) Spectral inequality.
Let φ j and µ j be eigenfunctions and associated eigenvalues of the bi-Laplace operator B with the clamped boundary conditions, that form a Hilbert basis for L ( Ω ),viz., B φ j = µ j φ j , φ j | ∂ Ω = ∂ ν φ j | ∂ Ω = , ( φ j , φ k ) L ( Ω ) = δ jk , with 0 < µ ≤ µ ≤ · · · ≤ µ j ≤ · · · . We now prove the spectral inequality of Theorem 1.3, namely,for some C > (cid:107) u (cid:107) L ( Ω ) ≤ Ce C µ / (cid:107) u (cid:107) L ( O ) , µ > , u ∈ Span { φ j ; µ j ≤ µ } . (5.26) Proof . We let µ > α , . . . , α n ∈ C with n ∈ N such that µ n ≤ µ < µ n + . We set u ( x ) = (cid:80) µ j ≤ µ α j φ j ( x ) , w ( s , x ) = (cid:80) µ j ≤ µ α j µ − / j f ( µ / j s ) φ j ( x ) , where f ( s ) = γ sin( γ s ) cosh( γ s ) − γ cos( γ s ) sinh( γ s ) where here γ = √ /
2. As D s f = − f , wehave Pv =
0, with P = D s + B . We also have f (0) = f (cid:48) (0) = f (2) (0) = , f (3) (0) = , and f ( s ) = g ( γ s ) , g ( s ) =
12 ( e − s cos( s − π/ − e s cos( s + π/ . Since w ( s , x ) | x ∈ ∂ Ω = ∂ ν w ( s , x ) | x ∈ ∂ Ω =
0, the interpolation inequality of Theorem 5.1 yields (cid:107) w (cid:107) H ( Y ) ≤ C (cid:107) w (cid:107) − δ H ( Z ) (cid:107) ∂ s w | s = (cid:107) δ L ( O ) . Observe that we have ∂ s w | s = = u and (cid:107) w (cid:107) H ( Y ) (cid:38) (cid:107) w (cid:107) L ( Y ) with (cid:107) w (cid:107) L ( Y ) = (cid:80) µ j ≤ µ µ − / j | α j | S − α ∫ α f ( µ / j s ) ds = (cid:80) µ j ≤ µ | α j | γ − µ − / j ( S − α ) γµ / j ∫ αγµ / j g ( s ) ds (cid:38) µ − / (cid:80) µ j ≤ µ | α j | = µ − / (cid:107) u (cid:107) L ( Ω ) , using the following lemma, whose proof is given below. Lemma 5.7.
Let < a < b and t > . There exists C such that ∫ btat g ( s ) ds ≥ C for t ≥ t . We thus obtain (cid:107) u (cid:107) L ( Ω ) (cid:46) µ / (cid:107) w (cid:107) − δ H ( Z ) (cid:107) u (cid:107) δ L ( O ) . (5.27)Next, we estimate (cid:107) w (cid:107) H ( Z ) , with the following lemma, which, from (5.27), allows one to concludethe proof of Theorem 1.3. (cid:4) Lemma 5.8.
There exists C > such that (cid:107) w (cid:107) H ( Z ) ≤ Ce C µ / (cid:107) u (cid:107) L ( Ω ) . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 51
Proof . We have (cid:107) w (cid:107) H ( Z ) (cid:16) (cid:80) k = S ∫ (cid:107) ∂ ks w ( s , . ) (cid:107) H − k ( Ω ) ds (cid:46) (cid:80) k = S ∫ (cid:107) ∂ ks w ( s , . ) (cid:107) H ( Ω ) ds , where H s ( Ω ) denotes the classical Sobolev spaces in Ω . Recalling from (1.15) that, if v | ∂ Ω = ∂ ν v | ∂ Ω =
0, we have (cid:107) v (cid:107) H ( Ω ) (cid:46) (cid:107) ∆ v (cid:107) L ( Ω ) , we find (cid:107) ∂ ks w ( s , . ) (cid:107) H ( Ω ) (cid:46) (cid:107) ∆ (cid:80) µ j ≤ µ α j µ ( k − / j f ( k ) ( µ / j s ) φ j (cid:107) L ( Ω ) = (cid:107) (cid:80) µ j ≤ µ α j µ ( k + / j f ( k ) ( µ / j s ) φ j (cid:107) L ( Ω ) = (cid:80) µ j ≤ µ | α j | µ ( k + / j ( f ( k ) ( µ / j s )) (cid:46) µ e S µ / (cid:80) µ j ≤ µ | α j | . Integrating this estimate over (0 , S ) and summing over k yields the result. (cid:4) Proof of Lemma 5.7.
For s ∈ [ − π/ + k π, k π ], k ∈ N ∗ , we have cos( s + π/ ≥ √ /
2. For t chosen su ffi ciently large, if t ≥ t , there exists k ∈ N such that [ − π/ + k π, k π ] ⊂ [ at , bt ] and | g ( s ) | = | e − s cos( s − π/ − e s cos( s + π/ | ≥
1. Then, ∫ btat g ( s ) ds ≥ π/
2. Finally, there exists C > ∫ btat g ( s ) ds ≥ C for t ∈ [ t , t ], since the function g ( s ) is almost everywherepositive. (cid:4) A null-controllability result for a higher-order parabolic equation.
Let T >
0. We con-sider here the controlled evolution equation on (0 , T ) × Ω with the clamped boundary conditions( ν denotes the outer unit normal to ∂ Ω ): ∂ t y + ∆ y = χ O f , y | (0 , T ) × ∂ Ω = , ∂ ν y | (0 , T ) × ∂ Ω = , y | t = = y ∈ L ( Ω ) , (5.28)where O is an open subset of Ω and χ O ∈ L ∞ ( Ω ) is such that χ O > O . The function f ∈ L ((0 , T ) × Ω ) is the control function here. Well-posedness for this parabolic system isrecalled in Corollary 1.10. One may wonder if one can choose f to drive the solution from itsinitial condition y to zero at final time T . Thanks to the spectral inequality of Theorem 1.3 onecan answer positively to this null-controllability question. Theorem 5.9 (Null-controllability) . There exists C > such that for any y ∈ L ( Ω ) , there existsf ∈ L ((0 , T ) × Ω ) such that the solution to (5.28) vanishes at T = and moreover (cid:107) f (cid:107) L ((0 , T ) × Ω ) ≤ C (cid:107) y (cid:107) L ( Ω ) . The proof can be adapted in a straight forward manner from the proof scheme of [LR95] de-veloped for the heat equation and that is presented in a fairly synthetic way in the survey article[LL12]. 6. R esolvent estimate and application
Using one of the interpolation inequalities proven in Section 5 (Proposition 5.3), we prove theresolvent estimate of Theorem 1.4. Finally, as an application, we state a stabilization result thatfollows from it for the plate equation.6.1.
Resolvent estimate.
Let U ∈ D ( B ) = ( H ( Ω ) ∩ H ( Ω )) × H ( Ω ) and F ∈ H = H ( Ω ) × L ( Ω ), be such that ( i σ Id H −B ) U = F , U = t (cid:16) u , u (cid:17) , F = t (cid:16) f , f (cid:17) , (6.1)for σ (cid:44)
0. Our goal is to find an estimate of the form (cid:107) U (cid:107) H ≤ Ke K | σ | / (cid:107) F (cid:107) H . We have i σ u + u = f , ( − σ − i σα + B ) u = f , with f = ( i σ − α ) f − f . (6.2) Multiplying the second equation by u and an integration over Ω give (cid:104) ( − σ + B ) u , u (cid:105) L ( Ω ) − i σ (cid:107) α / u (cid:107) L ( Ω ) = (cid:104) f , u (cid:105) L ( Ω ) , The first term is real and the second term is purely imaginary. We thus have σ (cid:107) α / u (cid:107) L ( Ω ) = − Im (cid:104) f , u (cid:105) L ( Ω ) . Using that α ≥ δ > O yields δσ (cid:107) u (cid:107) L ( O ) ≤ σ (cid:107) α / u (cid:107) L ( Ω ) ≤ (cid:107) f (cid:107) L ( Ω ) (cid:107) u (cid:107) L ( Ω ) , (6.3)for σ ≥ σ .A key estimate is given by the following lemma. We provide a proof below. Lemma 6.1.
There exists C > such that (cid:107) u (cid:107) H ( Ω ) ≤ Ce C | σ | / (cid:0) (cid:107) f (cid:107) L ( Ω ) + (cid:107) u (cid:107) L ( O ) (cid:1) . Then estimate (6.3) yields (cid:107) u (cid:107) H ( Ω ) (cid:46) e C | σ | / (cid:0) (cid:107) f (cid:107) L ( Ω ) + (cid:107) u (cid:107) L ( Ω ) (cid:107) f (cid:107) L ( Ω ) (cid:1) , σ ≥ σ , and with the Young inequality we obtain (cid:107) u (cid:107) H ( Ω ) (cid:46) e C | σ | / (cid:107) f (cid:107) L ( Ω ) . Using the form of f given in (6.2) we then obtain (cid:107) u (cid:107) H ( Ω ) (cid:46) e C | σ | / (cid:0) (cid:107) f (cid:107) L ( Ω ) + (cid:107) f (cid:107) L ( Ω ) (cid:1) , Finally as u = f − i σ u we obtain (cid:107) u (cid:107) H ( Ω ) + (cid:107) u (cid:107) L ( Ω ) (cid:46) e C | σ | / (cid:0) (cid:107) f (cid:107) L ( Ω ) + (cid:107) f (cid:107) L ( Ω ) (cid:1) , (6.4)yielding the resolvent estimate of Theorem 1.4.6.2. Proof of Lemma 6.1.
Let ρ = exp( i sgn( σ ) π/ ρ = sgn( σ ) i and ρ = −
1. Weset u = exp( s ρ | σ | / ) u and have Qu = e s ρ | σ | / f , with Q = D s + B + α D s , recalling (6.2). Let S > β ∈ (0 , S / Z = (0 , S ) × Ω and Y = ( β, S − β ) × Ω . We then apply theinterpolation inequality of Proposition 5.3: with 0 < β < β < S we have C > δ > (cid:107) u (cid:107) H ( Y ) ≤ C (cid:107) u (cid:107) − δ H ( Z ) (cid:16) (cid:107) Qu (cid:107) L ( Z ) + (cid:107) u (cid:107) L (( β ,β ) × O ) (cid:17) δ . (6.5)Next, we note that we have (cid:107) u (cid:107) H ( Y ) ≥ (cid:107) u (cid:107) L (( β, S − β ) , H ( Ω )) ≥ e − C | σ | / (cid:107) u (cid:107) H ( Ω ) , (cid:107) u (cid:107) H ( Z ) (cid:46) e C | σ | / (cid:107) u (cid:107) H ( Ω ) , (cid:107) u (cid:107) L (( β ,β ) × O ) ≤ e C | σ | / (cid:107) u (cid:107) L ( O ) , yielding with (6.5) (cid:107) u (cid:107) H ( Ω ) ≤ Ce C | σ | / (cid:0) (cid:107) f (cid:107) L ( Z ) + (cid:107) u (cid:107) L ( O ) (cid:1) . This concludes the proof of the estimate of Lemma 6.1.
PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 53
A stabilization result for the plate equation.
Let now ( y , y ) ∈ D ( B k ), k ≥
1, and y be thesolution of the damped plate equation(6.6) ∂ t y + ∆ y + α∂ t y = , y | t = = y , ∂ t y | t = = y , y | [0 , + ∞ ) × ∂ Ω = ∂ ν y | [0 , + ∞ ) × ∂ Ω = , with α a nonnegative function such that y ≥ δ > O , an open subset of Ω . If we set Y = ( y , ∂ t y )we have ( ∂ t + B ) Y =
0. From the resolvent estimate of Theorem 4.17 we obtain the followingenergy decay for the damped plate equation, using the results set in an abstract framework in[BD08].
Theorem 6.2.
With the energy function introduced in (1.16) the solution to the damped plateequation (6.6) satisfies, for some C > ,E ( y )( t ) ≤ C (cid:0) log(2 + t ) (cid:1) k (cid:107)B k Y (cid:107) H , t > , Y = ( y , y ) ∈ D ( B k ) . Among the existing results available in the literature for plate type equations, many of them con-cern the “hinged” boundary conditions. We first mention these result. An important result obtainedin [Jaf90] on the controllability of the plate equation on a rectangle domain with an arbitrary smallcontrol domain. The method relies on the generalization of Ingham type inequalities in [Kah62].An exponential stabilization result, in the same geometry, can be found in [RTT06], using similartechniques. In [RTT06] the localized damping term involves the time derivative ∂ t y as in (6.6).Interior nonlinear feedbacks can be used for exponential stabilization [Teb09]. There, feedbacksare localized in a neighborhood of part of the boundary that fulfills multiplier-type conditions. Ageneral analysis of nonlinear damping that includes the plate equation is provided in [ABA11] un-der multiplier-type conditions. For “hinged” boundary conditions also, with a boundary dampingterm, we cite [ATT07] where, on a square domain, a necessary and su ffi cient condition is pro-vided for exponential stabilization. In [Nou09], a polynomial stabilization rate is obtained if thecondition of [ATT07] is relaxed.For “clamped” boundary conditions, few results are available. We cite [AB06], where a generalanalysis of nonlinearly damped systems that includes the plate equation under multiplier-typeconditions is provided. In [ABPT17], the analysis of discretized general nonlinearly dampedsystem is also carried out, with the plate equation as an application. In [Teb12], a nonlineardamping involving the p- Laplacian is used also under multiplier-type conditions. In [DS15], anexponential decay is obtained in the case of “clamped” boundary conditions, yet with a dampingterm of the Kelvin-Voigt type, that is of the form ∂ t ∆ y , that acts over the whole domain.Theorem 6.2 provides a log-type stabilization result. Optimality is a natural question and onecould be interested in seeking geometries that improve this decay rate, yielding polynomial orexponential rate, in the case of “clamped” boundary conditions, in the spirit of some of the existingresults we cite above. A ppendices A. P roofs of some technical results
A.1.
Proof of the estimate optimality in the case of symbol flatness.
Here, we provide a proofof Proposition 1.6.We have Q ( z , D z ) = q ( z , D z ) + r m − ( z , D z ) + r m − ( z , D z ) with r m − ( z , D z ) homogeneous of degree m − r m − ( z , D z ) of order m − z , ζ , τ ) as in the statement of the proposition, then by homogeneity, as τ (cid:44) ζ ∈ R N , such that q ( z , θ ) = , d z ,ζ q ( z , θ ) = , θ α (cid:44) , with θ = ζ + id ϕ ( z ) . (A.1) Without any loss of generality we may assume that z =
0. Because of the form of (1.14), observealso that there is no loss of generality if we assume that ϕ (0) = w ( z ) = (cid:104) z , θ (cid:105) . We note that ϕ ( z ) − Im w ( z ) = G ( z ) + | z | O (1) , G ( z ) = d z ϕ (0)( z , z ) . We then pick f ∈ C ∞ c ( R N ), f (cid:46)
0, and set u τ ( z ) = e i τ w ( z ) f ( τ / z ). We have (cid:107) e τϕ u τ (cid:107) L ( R N ) = ∫ R N e τ (cid:0) G ( z ) + | z | O (1) (cid:1) | f ( τ / z ) | dz = τ − N / ∫ R N e G ( y ) + τ − / | y | O (1) | f ( y ) | dy (A.2) ∼ τ →∞ τ − N / ∫ R N e G ( y ) | f ( y ) | dy , with the change of variables y = τ / z and the dominated convergence theorem.As we note that e − i τ w ( z ) D α z u τ = ( D z + τθ ) α f (cid:0) τ / z (cid:1) = τ | α | θ α f (cid:0) τ / z (cid:1) + τ | α |− / O (1) , similarly, we find (cid:107) e τϕ D α z u τ (cid:107) L ( R N ) ∼ τ →∞ τ | α |− N / | θ α | ∫ R N e G ( y ) | f ( y ) | dy , (A.3)as we have θ α (cid:44) e − i τ w ( z ) Qe i τ w ( z ) = q ( z , D z + τθ ) + r m − ( z , D z + τθ ) + r m − ( z , D z + τθ ) . With the Taylor formula and homogeneity we observe that q ( z , D z + τθ ) = τ m q ( z , θ ) + τ m − d ζ q ( z , θ )( D z ) + τ m − d ζ q ( z , θ )( D z , D z ) + ∫ (1 − t ) d ζ q ( z , tD z + τθ )( D z , D z , D z ) dt . Next, we write q ( z , θ ) = q (0 , θ ) (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) = + d z q (0 , θ )( z ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = + d z q (0 , θ )( z , z ) + ∫ (1 − t ) d z q ( tz , θ )( z , z , z ) dt , d ζ q ( z , θ )( D z ) = d ζ q (0 , θ )( D z ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = + d ζ d z q (0 , θ )( D z , z ) + ∫ (1 − t ) d ζ d z q ( tz , θ )( D z , z , z ) dt , and d ζ q ( z , θ )( D z , D z ) = d ζ q (0 , θ )( D z , D z ) + ∫ d ζ d z q ( tz , θ )( D z , D z , z ) dt , PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 55 which gives e − i τ w ( z ) Qe i τ w ( z ) = τ m − (cid:18) d z q (0 , θ )( τ / z , τ / z ) + τ − / ∫ (1 − t ) d z q ( tz , θ )( τ / z , τ / z , τ / z ) dt + τ − / d ζ d z q (0 , θ )( D z , τ / z ) + τ − ∫ (1 − t ) d ζ d z q ( tz , θ )( D z , τ / z , τ / z ) dt + τ − d ζ q (0 , θ )( D z , D z ) + τ − / ∫ d ζ d z q ( tz , θ )( D z , D z , τ / z ) dt + τ − m (cid:0) r m − ( z , D z + τθ ) + r m − ( z , D z + τθ ) (cid:1)(cid:19) . We then find e − i τ w ( z ) Qu τ = τ m − (cid:16) d z q (0 , θ )( τ / z , τ / z ) f ( τ / z ) + d ζ d z q (0 , θ ) (cid:0) D z f ( τ / z ) , τ / z (cid:1) + (cid:16) d ζ q (0 , θ )( D z , D z ) f (cid:17) ( τ / z ) + r m − ( z , θ ) f ( τ / z ) + τ − / O (1) (cid:17) . Arguing as for (A.2), we obtain, as τ → ∞ , (cid:107) e τϕ Qu τ (cid:107) L ( R N ) = τ m − − N / ∫ R N e G ( y ) (cid:12)(cid:12)(cid:12) d z q (0 , θ )( y , y ) f ( y ) + d ζ d z q (0 , θ ) (cid:0) D z f ( y ) , y (cid:1) (A.4) + (cid:0) d ζ q (0 , θ )( D z , D z ) f (cid:1) ( y ) + r m − (0 , θ ) f ( y ) (cid:12)(cid:12)(cid:12) dy + O ( τ m − − N / − / ) . The assumed estimate (1.14) along with (A.2)–(A.4) thus implies that δ = (cid:4) Remark A.1.
Observe that if in addition we assume that m ≥ d ϕ ( z ) does not vanish in Ω . In fact, if d ϕ ( z ) = ζ = θ = m ≥ m =
1, it is known that a Carleman estimate with the loss of a half derivative can holdeven if the gradient of the weight function vanishes (see Lemma 8.1.1 in [H¨or63]). For instance,for ϕ ( z ) = z / D z , we have τ / (cid:107) e τϕ u (cid:107) L ( R N ) (cid:46) (cid:107) e τϕ D z u (cid:107) L ( R N ) , for τ > u ∈ C ∞ c ( R N ). Then, for the operator D z , we have τ (cid:107) e τϕ u (cid:107) L ( R N ) (cid:46) (cid:107) e τϕ D z u (cid:107) L ( R N ) , for τ > u ∈ C ∞ c ( R N ). We then have the case of an operator of order m = R N suchthat an estimate with a loss a full derivative holds and yet d ϕ may vanish. Remark A.2.
The reader should observe that the statement of Proposition 1.6 assumes that thesymbol q ( z , ζ + i τ d ϕ ( z )) vanishes at second order at a complex root, that is, for τ >
0. Flatness at areal root may not yield δ =
0. In fact, in R N , N ≥
2, consider the operator Q = ( D z + D z ) m with m ≥ ϕ ( z ) = z . Then q ( ζ + i τ d ϕ ) = ( ζ + ζ + i τ ) m which vanishes (at order m ) for τ = ζ + ζ =
0. Yet, we have the following estimate τ m (cid:107) e τϕ u (cid:107) L ( R N ) ≤ (cid:107) e τϕ Pu (cid:107) L ( R N ) , (A.5)for v ∈ C ∞ c ( R ). This means δ = The proof of (A.5) is as follows. We write e τϕ ( D z + D z ) u = ( D z + i τ + D z ) v with v = e τϕ u .We then have (cid:107) e τϕ ( D z + D z ) u (cid:107) L ( R N ) = (cid:107) ( D z + D z ) v (cid:107) L ( R N ) + (cid:107) τ v (cid:107) L ( R N ) − i τ Re ∫ v ( D z + D z ) v dz = (cid:107) ( D z + D z ) v (cid:107) L ( R N ) + (cid:107) τ v (cid:107) L ( R N ) − τ ∫ ( ∂ z + ∂ z ) | v | dz (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = v ) compact ≥ τ (cid:107) v (cid:107) L ( R N ) = τ (cid:107) e τϕ u (cid:107) L ( R N ) . Multiple applications of this estimate yield (A.5).Note however that we do not claim to have (cid:107) e τϕ Du (cid:107) L ( R N ) (cid:46) (cid:107) e τϕ ( D z + D z ) u (cid:107) L ( R N ) , as D z + D z is not elliptic.A.2. Proofs associated with the semi-classical calculus.
A.2.1.
Proof of Proposition 4.2.
The dual quadratic form of g on W is given by g σ = λ τ | dz | + | d ζ (cid:48) | (1 + γε ) + | d ζ N | γ . We then have, for X = ( z X , ζ X ), as γ ≥ (cid:0) h g ) − ( X ) = inf T ∈W T (cid:44) (cid:16) g σ X ( T ) / g X ( T ) (cid:17) / = min (cid:0) γ − , (1 + γε ) − (cid:1) λ ˜ τ ( X ) ≥ (2 γ ) − λ ˜ τ ( X ) ≥ τϕ ( z X ) / ≥ , as τ ≥ τ ∗ ≥
2. The uncertainty principle is thus fulfilled.For X = ( z X , ζ X ) ∈ W , we write z X = ( z (cid:48) X , ( z X ) N ), with z (cid:48) X ∈ R N − . Similarly, we also write ζ X = ( ζ (cid:48) X , ( ζ X ) N ), with ζ (cid:48) X ∈ R N − .We now prove the slow variations of g and ϕ , λ ˜ τ , namely, there exist K > r >
0, such that ∀ X , Y , T ∈ W , g X ( Y − X ) ≤ r ⇒ g Y ( T ) ≤ Kg X ( T ) , K − ≤ ϕ ( z X ) ϕ ( z Y ) ≤ K , K − ≤ λ ˜ τ ( X ) λ ˜ τ ( Y ) ≤ K , where X = ( z X , ζ X ) and Y = ( z Y , ζ Y ). We thus assume that g X ( Y − X ) ≤ r , with 0 < r < + γε )( | z (cid:48) X − z (cid:48) Y | ) + γ | ( z X ) N − ( z Y ) N | + λ ˜ τ ( X ) − ( | ζ X − ζ Y | ) ≤ Cr . (A.6)We observe that we have ϕ ( z X ) = e γψ ε ( z X ) = ϕ ( z Y ) e γ (cid:0) ψ ε ( z X ) − ψ ε ( z Y ) (cid:1) , where ψ ε ( z X ) = ψ ( ε z (cid:48) X , ( z X ) N ). Note that | ψ ε ( z X ) − ψ ε ( z Y ) | ≤ (cid:0) ε | z (cid:48) X − z (cid:48) Y | + | ( z X ) N − ( z Y ) N | (cid:1) (cid:107) ψ (cid:48) (cid:107) L ∞ . With (A.6), we thus obtain ϕ ( z X ) ≤ ϕ ( z Y ) e Cr (cid:107) ψ (cid:48) (cid:107) L ∞ (cid:46) ϕ ( z Y ) . (A.7)Similarly, we have ϕ ( z Y ) (cid:46) ϕ ( z X ) . (A.8)We also have | ζ Y | ≤ | ζ Y − ζ X | + | ζ X | ≤ Cr λ ˜ τ ( X ) + | ζ X | (cid:46) λ ˜ τ ( X ) . (A.9) PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 57
Next, we write | ζ X | ≤ | ζ Y − ζ X | + | ζ Y | ≤ Cr λ ˜ τ ( X ) + | ζ Y | ≤ Cr (cid:0) τγϕ ( z X ) + | ζ X | (cid:1) + | ζ Y | . Hence, for r su ffi ciently small, with (A.7), we have | ζ X | (cid:46) τγϕ ( z X ) + | ζ Y | (cid:46) λ ˜ τ ( Y ) . (A.10)With (A.7) and (A.10), resp. (A.8) and (A.9), we find λ ˜ τ ( X ) (cid:46) λ ˜ τ ( Y ) , resp. λ ˜ τ ( Y ) (cid:46) λ ˜ τ ( X ) . Then, if T = ( z T , ζ T ) ∈ W we find | ζ T | λ ˜ τ ( Y ) (cid:46) | ζ T | λ ˜ τ ( X ) (cid:46) | ζ T | λ ˜ τ ( Y ) , and this gives g Y ( T ) (cid:46) g X ( T ) (cid:46) g Y ( T ), concluding the proof of the slow variations of λ ˜ τ and g .We now prove the temperance of g , ϕ and λ ˜ τ , namely, there exist K > N >
0, such that ∀ X , Y , T ∈ W , g X ( T ) g Y ( T ) ≤ C (cid:0) + g σ X ( X − Y ) (cid:1) N , ∀ X , Y ∈ W , ϕ ( z X ) ϕ ( z Y ) ≤ C (cid:0) + g σ X ( X − Y ) (cid:1) N , λ ˜ τ ( X ) λ ˜ τ ( Y ) ≤ C (cid:0) + g σ X ( X − Y ) (cid:1) N , where X = ( z X , ζ X ) and Y = ( z Y , ζ Y ). We have g σ X ( X − Y ) = λ ˜ τ ( X ) | z X − z Y | + | ζ (cid:48) X − ζ (cid:48) Y | (1 + γε ) + | ( ζ X ) N − ( ζ Y ) N | γ . We note that | ζ X | ≤ | ζ Y | + | ζ X − ζ Y | ≤ | ζ Y | + | ζ X − ζ Y | γ τγϕ ( z Y )(A.11) ≤ | ζ Y | + (cid:18) | ζ (cid:48) X − ζ (cid:48) Y | + γε + | ( ζ X ) N − ( ζ Y ) N | γ (cid:19) τγϕ ( z Y ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) λ ˜ τ ( Y ) . First, if (1 + εγ ) | z (cid:48) X − z (cid:48) Y | + γ | ( z X ) N − ( z Y ) N | ≤
1, then, arguing as in (A.7), we find ϕ ( z X ) (cid:46) ϕ ( z Y ) , τγϕ ( z X ) (cid:46) λ ˜ τ ( Y ) . Second, if (1 + εγ ) | z (cid:48) X − z (cid:48) Y | + γ | ( z X ) N − ( z Y ) N | ≥
1, we then have 2 | z X − z Y | ≥ /γ . We write, as τ ≥ τ ∗ ≥ ϕ ( z X ) = ˜ τ ( z X ) γτ ≤ λ ˜ τ ( X ) γ (cid:46) | z X − z Y | λ ˜ τ ( X ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) (cid:46) (cid:0) + g σ X ( X − Y ) / ϕ ( z Y ) , using that ϕ ≥
1. We also write τγϕ ( z X ) (cid:46) λ ˜ τ ( X ) ≤ λ ˜ τ ( X ) λ ˜ τ ( Y ) γ (cid:46) | z X − z Y | λ ˜ τ ( X ) λ ˜ τ ( Y ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) λ ˜ τ ( Y ) . In any case, we have ϕ ( z X ) ≤ (cid:0) + g σ X ( X − Y ) / (cid:1) ϕ ( z Y ) (cid:46) (cid:0) + g σ X ( X − Y ) (cid:1) ϕ ( z Y ) , that is, the temperance of ϕ and we have τγϕ ( z X ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) λ ˜ τ ( Y ), which, along with(A.11), yields the temperance of λ ˜ τ : λ ˜ τ ( X ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) λ ˜ τ ( Y ) (cid:46) (cid:0) + g σ X ( X − Y ) (cid:1) λ ˜ τ ( Y ) . For the temperance of g we need to prove(1 + εγ ) | z (cid:48) T | + γ | ( z T ) N | + | ζ T | λ ˜ τ ( X ) (cid:46) (cid:0) + g σ X ( X − Y ) (cid:1) N (cid:16) (1 + εγ ) | z (cid:48) T | + γ | ( z T ) N | + | ζ T | λ ˜ τ ( Y ) (cid:17) , for T = ( z T , ζ T ) ∈ W . To conclude it su ffi ces to prove λ ˜ τ ( Y ) (cid:46) (cid:0) + g σ X ( X − Y ) (cid:1) N λ ˜ τ ( X ) . We have | ζ Y | ≤ | ζ X | + | ζ X − ζ Y | ≤ | ζ X | + | ζ X − ζ Y | γ τγϕ ( z X )(A.12) ≤ | ζ X | + (cid:18) | ζ (cid:48) X − ζ (cid:48) Y | + γε + | ( ζ X ) N − ( ζ Y ) N | γ (cid:19) τγϕ ( z X ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) λ ˜ τ ( X ) . It thus remains to prove τγϕ ( z Y ) (cid:46) (cid:0) + g σ X ( X − Y ) (cid:1) N λ ˜ τ ( X ) . (A.13)First, if (1 + γε ) | z (cid:48) X − z (cid:48) Y | + γ | ( z X ) N − ( z Y ) N | ≤
1, then ϕ ( z Y ) (cid:46) ϕ ( z X ), arguing as in (A.8). Es-timate (A.13) is then clear. Second, if (1 + γε ) | z (cid:48) X − z (cid:48) Y | + γ | ( z X ) N − ( z Y ) N | ≥
1, which implies2 | z X − z Y | ≥ /γ , with (4.3) we write τγϕ ( z Y ) ≤ τγϕ ( z X ) k + (cid:46) λ ˜ τ ( X ) k + ( τγ ) k (cid:46) (cid:16) λ ˜ τ ( X ) τγ (cid:17) k λ ˜ τ ( X ) (cid:46) (cid:16) | z X − z Y | λ ˜ τ ( X ) τ (cid:17) k λ ˜ τ ( X ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) k λ ˜ τ ( X ) , since τ ≥ τ ∗ ≥
1. In any case, we thus have τγϕ ( z Y ) (cid:46) (cid:0) + g σ X ( X − Y ) / (cid:1) k λ ˜ τ ( X ) , which concludes the proof. (cid:4) A.2.2.
Proof of Lemma 4.4.
We have ˜ τ (cid:46) λ ˜ τ (resp. ˜ τ (cid:46) λ T , ˜ τ ) and d ζ ˜ τ =
0. Only di ff erentiationsof ˜ τ with respect to z thus need to be considered. Recalling that ˜ τ = τγϕ γ,ε we find that, for α = ( α (cid:48) , α N ) ∈ N N , we can write ∂ α z ˜ τ ( (cid:37) (cid:48) ) as a linear combination of terms of the form τγ + k ϕ γ,ε ( z ) k (cid:81) j = ∂ α ( j ) z ψ ε ( z ) = τγ + k ε | α (cid:48) | ϕ γ,ε ( z ) k (cid:81) j = ∂ α ( j ) z ψ ( ε z (cid:48) , z N ) , with α (1) + · · · + α ( k ) = α , | α ( j ) | ≥ j = , . . . , k , and k ≤ | α | , implying, as γ ≥ | ∂ α z ˜ τ ( (cid:37) (cid:48) ) | (cid:46) ˜ τ ( (cid:37) (cid:48) ) γ | α | ε | α (cid:48) | (cid:46) ˜ τ ( (cid:37) (cid:48) ) γ α N ( εγ ) | α (cid:48) | , as (cid:107) ψ ( (cid:96) ) (cid:107) L ∞ ≤ C for any (cid:96) ∈ N , which yields the results. (cid:4) A.2.3.
Proof of Lemma 4.8.
For α = ( α (cid:48) , α N ) ∈ N N and β (cid:48) ∈ N N − , we may write ∂ α z ∂ β (cid:48) ζ (cid:48) a ( (cid:37) (cid:48) ) as alinear combination of terms of the form, b ( (cid:37) (cid:48) ) = (cid:0) k (cid:81) j = ∂ α ( j ) z ˆ τ p j ( (cid:37) (cid:48) ) (cid:1) ∂ α ( a ) z ∂ β (cid:48) ζ (cid:48) ∂ α ( b ) ˆ t ˆ a (cid:0) κ ( (cid:37) (cid:48) ) (cid:1) , PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 59 for some α ( b ) ∈ N N , with k = | α ( b ) | , with α = α ( a ) + α (1) + · · · + α ( k ) , | α ( j ) | ≥
1, and where p j ∈ { , . . . , N } , j = , . . . , k . Using Lemma 4.5 and Definition 4.1, and we obtain | b ( (cid:37) (cid:48) ) | (cid:46) k (cid:81) j = (cid:0) λ T , ˜ τ (1 + εγ ) | α ( j ) (cid:48) | γ | α ( j ) N | (cid:1) ( | ˆ τ ( (cid:37) (cid:48) ) | + | ζ (cid:48) | ) m −| β (cid:48) |−| α ( b ) | (cid:46) (1 + εγ ) | α (1) (cid:48) | + ··· + | α ( k ) (cid:48) | γ α (1) N + ··· + α ( k ) N λ k T , ˜ τ ( | ˆ τ ( (cid:37) (cid:48) ) | + | ζ (cid:48) | ) m −| β (cid:48) |−| α ( b ) | (cid:46) (1 + εγ ) | α (cid:48) | γ α N λ k T , ˜ τ ( | ˆ τ ( (cid:37) (cid:48) ) | + | ζ (cid:48) | ) m − k −| β (cid:48) | , as γ ≥
1. If ˆ a is polynomial then the term b ( (cid:37) (cid:48) ) vanishes if m − | β (cid:48) | − | α ( b ) | <
0. Thus if m − | β (cid:48) | −| α ( b ) | ≥ | ˆ τ | (cid:46) ˜ τ in U , we obtain | b ( (cid:37) (cid:48) ) | ≤ (1 + εγ ) | α (cid:48) | γ α N λ m −| β (cid:48) | T , ˜ τ , which yields the result. If ˆ a is not polynomial and if we have ˜ τ (cid:16) | ˆ τ | , we obtain the same estimation,even if m − | β (cid:48) | − | α ( b ) | < (cid:4) A.2.4.
Proof of Lemma 4.11.
By applying (4.14), we have (cid:107) Λ m T , ˜ τ ˜ τ r u (cid:107) + (cid:46) (cid:107) Op T (˜ τ r λ m T , ˜ τ ) u (cid:107) + . Next, we write Op T (˜ τ r λ m T , ˜ τ ) = Op( λ m T , ˜ τ )˜ τ r + γ R , with R ∈ Ψ (˜ τ r λ m − T , ˜ τ , g T ) by the tangential calculuswe have introduced. This yields, as ˜ τ r ∈ S ( λ r T , ˜ τ , g T ), (cid:107) Op T (˜ τ r λ m T , ˜ τ ) u (cid:107) + (cid:46) (cid:107) Op( λ m T , ˜ τ )˜ τ r u (cid:107) + + γ (cid:107) Op T (˜ τ r λ m − T , ˜ τ ) u (cid:107) + , which yields (4.15) by choosing τ su ffi ciently large. Estimation (4.16) follows the same. (cid:4) A.2.5.
Proof of Lemma 4.14.
By definition of the Sobolev norms introduced in Section 4.1.6 wehave (cid:107) ˜ τ r w (cid:107) m , m (cid:48) , ˜ τ (cid:16) m (cid:80) j = (cid:107) D jx d (˜ τ r w ) (cid:107) , m + m (cid:48) − j , ˜ τ = m (cid:80) j = (cid:107) Λ m + m (cid:48) − j T , ˜ τ D jx d (˜ τ r w ) (cid:107) + . Let m (cid:48)(cid:48) j ∈ R . We have [ Λ m (cid:48)(cid:48) j T , ˜ τ , D jx d ] ∈ (cid:80) ji = γ i Ψ ( λ m (cid:48)(cid:48) j T , ˜ τ , g T ) D j − ix d , from the tangential calculus we haveintroduced. With Lemma 4.4 we have [˜ τ r , Λ m (cid:48)(cid:48) j T , ˜ τ ] ∈ (1 + εγ ) Ψ (˜ τ r Λ m (cid:48)(cid:48) j − T , ˜ τ , g T ). With the same lemma,for r (cid:48) j ∈ R we also have [˜ τ r (cid:48) j , D jx d ] ∈ (cid:80) ji = γ i Ψ (˜ τ r (cid:48) j , g T ) D j − ix d . For r = r (cid:48) j + r (cid:48)(cid:48) j , and m + m (cid:48) − j = m (cid:48)(cid:48) j + m (cid:48)(cid:48)(cid:48) j ,with r (cid:48) j , r (cid:48)(cid:48) j ∈ R and m (cid:48)(cid:48) j , m (cid:48)(cid:48)(cid:48) j ∈ R , we thus obtain, by Proposition 4.13, (cid:107) ˜ τ r w (cid:107) m , m (cid:48) , ˜ τ ≥ m (cid:80) j = (cid:107) ˜ τ r (cid:48) j Λ m (cid:48)(cid:48) j T , ˜ τ D jx d (˜ τ r (cid:48)(cid:48) j Λ m (cid:48)(cid:48)(cid:48) j T , ˜ τ w ) (cid:107) + − C (cid:48) m (cid:80) j = j (cid:80) i = γ i (cid:107) ˜ τ r D j − ix d w (cid:107) , m + m (cid:48) − j , ˜ τ − C (cid:48)(cid:48) m (cid:80) j = γ (cid:107) ˜ τ r D jx d w (cid:107) , m + m (cid:48) − j − , ˜ τ ≥ m (cid:80) j = (cid:107) ˜ τ r (cid:48) j Λ m (cid:48)(cid:48) j T , ˜ τ D jx d (˜ τ r (cid:48)(cid:48) j Λ m (cid:48)(cid:48)(cid:48) j T , ˜ τ w ) (cid:107) + − C (cid:48) m − (cid:80) j = m (cid:80) i = γ i (cid:107) ˜ τ r D jx d w (cid:107) , m + m (cid:48) − j − i , ˜ τ − C (cid:48)(cid:48) m (cid:80) j = γ (cid:107) ˜ τ r D jx d w (cid:107) , m + m (cid:48) − j − , ˜ τ . With the argument given in (4.17), we have m − (cid:80) j = m (cid:80) i = γ i (cid:107) ˜ τ r D jx d w (cid:107) , m + m (cid:48) − j − i , ˜ τ + m (cid:80) j = γ (cid:107) ˜ τ r D jx d w (cid:107) , m + m (cid:48) − j − , ˜ τ (cid:28) (cid:107) ˜ τ r w (cid:107) m , m (cid:48) , ˜ τ , for τ chosen su ffi ciently large, and we thus find (cid:107) ˜ τ r w (cid:107) m , m (cid:48) , ˜ τ (cid:38) m (cid:80) j = (cid:107) ˜ τ r (cid:48) j Λ m (cid:48)(cid:48) j T , ˜ τ D jx d (˜ τ r (cid:48)(cid:48) j Λ m (cid:48)(cid:48)(cid:48) j T , ˜ τ w ) (cid:107) + , for τ chosen su ffi ciently large. Similarly, we find that (cid:107) ˜ τ r w (cid:107) m , m (cid:48) , ˜ τ (cid:46) m (cid:80) j = (cid:107) ˜ τ r (cid:48) j Λ m (cid:48)(cid:48) j T , ˜ τ D jx d (˜ τ r (cid:48)(cid:48) j Λ m (cid:48)(cid:48)(cid:48) j T , ˜ τ w ) (cid:107) + , for τ chosen su ffi ciently large. The result for the trace norms is obtained arguing the same. (cid:4) B. E lliptic and sub - elliptic estimates at the boundary (0 , S ) × ∂ Ω B.1.
Roots with negative imaginary part: a perfect elliptic estimate.
For z ∈ ∂ Z , V denotesthe neighborhood introduced in Section 4.2. We recall that M T , V = V × R N − × [ τ ∗ , + ∞ ) × [1 , + ∞ ) × [0 , (cid:96) ( (cid:37) ) ∈ S m , τ , with (cid:37) = ( z , ζ, τ, γ, ε ) and m ≥
1, be polynomial in ζ N with homogeneouscoe ffi cients in ( ζ (cid:48) , ˆ τ ) and L = (cid:96) ( z , D z , τ, γ, ε ). Lemma B.1.
Let U be a conic open subset of M T , V . We assume that, for (cid:96) ( (cid:37) (cid:48) , ζ N ) viewed as apolynomial in ζ N , for (cid:37) (cid:48) ∈ U , • the leading coe ffi cient is ; • all roots of (cid:96) ( (cid:37) (cid:48) , ζ N ) = have negative imaginary part.Let χ ( (cid:37) (cid:48) ) ∈ S (1 , g T ) , be homogeneous of degree zero and such that supp( χ ) ⊂ U . Then, for anyM ∈ N , there exist C > , τ ≥ τ ∗ , γ ≥ such that (cid:107) Op T ( χ ) w (cid:107) m , , ˜ τ + | tr(Op T ( χ ) w ) | m − , / , ˜ τ ≤ C (cid:16) (cid:107) L Op T ( χ ) w (cid:107) + + (cid:107) w (cid:107) m , − M , ˜ τ (cid:17) , for w ∈ S ( R N + ) and τ ≥ τ , γ ≥ γ , ε ∈ [0 , . This lemma can be proven by adapting the proof of [BL15, Lemma 6.5] to the semi-classicalcalculus we use here. For the notion of homogeneity for symbols and conic sets in the presentcalculus, we refer to Section 4.1.5.B.2.
Sub-ellipticity quantification.
For z ∈ ∂ Z , V denotes the neighborhood introduced in Sec-tion 4.2. We let the function ψ be as introduced in Section 4, satisfying (4.2) and (4.20), and werecall that ψ ε ( z ) = ψ ( ε z (cid:48) , z N ) and ϕ ( z ) = exp( γψ ε ( z )). We also recall that λ τ = ˜ τ + | ζ | with˜ τ ( (cid:37) (cid:48) ) = τγϕ ( z ). Proposition B.2.
Let (cid:96) ( z , ζ ) be polynomial of degree m in ζ , with smooth coe ffi cient in z. Weassume that for any M ∈ R N \ { } , the symbol (cid:96) satisfies the simple-characteristic property indirection M in a neighborhood of V (see Definition 2.1). There exist C > and γ ≥ such that, | (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) | + τϕ ( z ) | ψ (cid:48) ε ( z ) | (cid:8) Re (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) , Im (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) (cid:9) ≥ C λ m ˜ τ , for z ∈ V, ζ ∈ R N , τ ≥ τ ∗ , γ ≥ γ and ε ∈ [0 , . Proof . We have 0 < C ≤ | ψ (cid:48) ε ( z ) | ≤ C for z ∈ V and we set K = { M ∈ R N ; C ≤ | M | ≤ C } . As V is assumed bounded (see section 4.2), we consider the compact set C = (cid:8) ( z , ζ, θ, M ); θ + | ζ | = , z ∈ V , ζ ∈ R N , θ ∈ R + , M ∈ K (cid:9) . We define f ( z , ζ, θ, M ) = | (cid:96) ( z , ζ + i θ M ) | + | θ M | (cid:12)(cid:12)(cid:12) (cid:104) (cid:96) (cid:48) ζ ( z , ζ + i θ M ) , M (cid:105) (cid:12)(cid:12)(cid:12) . (B.1) PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 61
As the simple-characteristic property holds in direction M for all M ∈ K and z ∈ V , we have f ( z , ζ, θ, M ) ≥ C > , ( z , ζ, θ, M ) ∈ C . By homogeneity, we obtain f ( z , ζ, θ, M ) ≥ C ( θ + | ζ | ) m , z ∈ V , ζ ∈ R N , θ ∈ R + , M ∈ K . (B.2)We compute the following Poisson bracket, with ˆ τ ( (cid:37) (cid:48) ) = τ d ϕ ( z ), { Re (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) , Im (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) } = i { (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) , (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) } = Θ (cid:96),ϕ ( z , ζ, τ ) , with Θ (cid:96),φ ( z , ζ, t ) : = t (cid:80) j , k ∂ z j z k φ ( z ) ∂ ζ j (cid:96) ( z , ζ + itd φ ( z )) ∂ ζ k (cid:96) ( z , ζ + itd φ ( z )) + Im (cid:80) j ∂ z j (cid:96) ( z , ζ + itd φ ( z )) ∂ ζ j (cid:96) ( z , ζ + itd φ ( z )) . Note that Θ (cid:96),φ ( z , ζ, t ) is homogeneous of degree 2 m − ζ, t ). With ϕ ( z ) = exp( γψ ε ( z )) we obtain Θ (cid:96),ϕ ( z , ζ, τ ) = Θ (cid:96),ψ ε ( z , ζ, ˜ τ ( (cid:37) (cid:48) )) + γ ˜ τ ( (cid:37) (cid:48) ) |(cid:104) (cid:96) (cid:48) ζ ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) , ψ (cid:48) ε ( z ) (cid:105)| . We thus find, with f defined in (B.1), | (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) | + τϕ ( z ) | ψ (cid:48) ε ( z ) | (cid:8) Re (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) , Im (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) (cid:9) (B.3) = | (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) | + τϕ ( z ) | ψ (cid:48) ε ( z ) | Θ (cid:96),ϕ ( z , ζ, τ ) = | (cid:96) ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) | + | ˜ τ ( (cid:37) (cid:48) ) ψ (cid:48) ε ( z ) | |(cid:104) (cid:96) (cid:48) ζ ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) , ψ (cid:48) ε ( z ) (cid:105)| + τϕ ( z ) | ψ (cid:48) ε ( z ) | Θ (cid:96),ψ ε ( z , ζ, ˜ τ ( (cid:37) (cid:48) )) = f ( z , ζ, ˜ τ ( (cid:37) (cid:48) ) , ψ (cid:48) ε ( z )) + τϕ ( z ) | ψ (cid:48) ε ( z ) | Θ (cid:96),ψ ε ( z , ζ, ˜ τ ( (cid:37) (cid:48) )) . Now, as ψ (cid:48) ε ( z ) remains in the compact set K , we find, by (B.2), f ( z , ζ, ˜ τ ( (cid:37) (cid:48) ) , ψ (cid:48) ε ( z )) (cid:38) (ˆ τ ( (cid:37) (cid:48) ) + | ζ | ) m (cid:38) λ m ˜ τ , (B.4)since | ˆ τ ( (cid:37) (cid:48) ) | = | ψ (cid:48) ε | ˜ τ ( (cid:37) (cid:48) ) ≥ C ˜ τ ( (cid:37) (cid:48) ). The homogeneity of Θ (cid:96),ψ ε ( z , ζ, ˜ τ ( (cid:37) (cid:48) )) gives | τϕ ( z ) | ψ (cid:48) ε ( z ) | Θ (cid:96),ψ ε ( z , ζ, ˜ τ ( (cid:37) (cid:48) )) | (cid:46) γ − ˜ τ ( (cid:37) (cid:48) ) λ m − τ (cid:46) γ − λ m ˜ τ . With (B.3) and (B.4), we obtain the result for γ chosen su ffi ciently large. (cid:4) We recall the definition of q k ( (cid:37) ) given in (4.24), we have q k ( y ) = p k ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) with p k ( z , ζ ) = ( − k i σ + ξ d + r ( x , ξ (cid:48) ). From Proposition 2.3 and Proposition B.2, we have the following result,in any dimension N ≥
2, that is, d ≥ Corollary B.3.
Let k = or . There exist C > and γ ≥ such that | q k ( (cid:37) ) | + τϕ ( z ) | ψ (cid:48) ε ( z ) | { Re q k ( (cid:37) (cid:48) ) , Im q k ( (cid:37) ) } ≥ C λ τ , (cid:37) = ( z , ζ, τ, γ, ε ) , for z ∈ V, ζ ∈ R N , τ ≥ τ ∗ , γ ≥ γ , and ε ∈ [0 , , and where ˆ τ ( (cid:37) (cid:48) ) = τγϕ ( z ) d ψ ε ( z ) . B.3.
Estimates for first-order factors.
In this section, we shall assume that U ⊂ M T is a conicopen set where the symbol q k ( (cid:37) ) = p k ( z , ζ + i ˆ τ ( (cid:37) (cid:48) )) can be factorized into two smooth first-orderterms, q k ( (cid:37) ) = q k , − ( (cid:37) ) q k , + ( (cid:37) ) , q k , ± ( (cid:37) ) = ξ d − ρ k , ± ( (cid:37) (cid:48) ) . By Lemma 4.18 we see that q k , − is elliptic, and q k , + may vanish. B.3.1.
A root with a positive imaginary part: an elliptic estimate with a trace term.
Here, wefurther assume that there exists a second conic open set U ⊂ U such that Im ρ k , + ( (cid:37) (cid:48) ) (cid:38) λ T , ˜ τ , for (cid:37) (cid:48) ∈ U . We let χ, χ ∈ S (1 , g T ) be homogeneous of degree zero and such that χ ≡ χ ) , supp( χ ) ⊂ U . With Q k , + = D x d − Op T w ( χ ρ k , + ) we have the following estimation. Lemma B.4.
Let (cid:96) ∈ R and M ∈ N . There exist τ ≥ τ ∗ , γ ≥ , and C > , such that (cid:107) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ ≤ C (cid:16) (cid:107) Q k , + Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | tr(Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) , (B.5) for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for w ∈ S ( R N + ) . Proof . We write Q = A − iB with A = D x d − Op T w ( χ Re ρ k , + ) , B = Op T w ( χ Im ρ k , + ) , both formally selfadjoint.We use a pseudo-di ff erential multiplier technique, following for instance [LL13] and compute,with s = (cid:96) + Q Op T ( χ ) w , − i Λ s T , ˜ τ Op T ( χ ) w ) + = − A Op T ( χ ) w , i Λ s T , ˜ τ Op T ( χ ) w ) + + B Op T ( χ ) w , Λ s T , ˜ τ Op T ( χ ) w ) + = − ( i [ A , Λ s T , ˜ τ ] Op T ( χ ) w , Op T ( χ ) w ) + + B Op T ( χ ) w , Λ s T , ˜ τ Op T ( χ ) w ) + − ( Λ s T , ˜ τ Op T ( χ ) w | x d = + , Op T ( χ ) w | x d = + ) L ( R N − ) ≥ B Op T ( χ ) w , Λ s T , ˜ τ Op T ( χ ) w ) + − C γ (cid:107) Op T ( χ ) w (cid:107) ,(cid:96) + / , ˜ τ − | Op T ( χ ) w | x d = + | ,(cid:96) + / , ˜ τ , which by the (microlocal) Gårding inequality of Proposition 4.16 yields, for any M ∈ N ,Re( Λ (cid:96) T , ˜ τ Q Op T ( χ ) w , − i Λ (cid:96) + T , ˜ τ Op T ( χ ) w ) + + | Op T ( χ ) w | x d = + | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:38) (cid:107) Op T ( χ ) w (cid:107) ,(cid:96) + , ˜ τ , for τ and γ chosen su ffi ciently large. Then, with the Young inequality, we obtain (cid:107) Q Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | Op T ( χ ) w | x d = + | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:38) (cid:107) Op T ( χ ) w (cid:107) ,(cid:96) + , ˜ τ . Finally, observing that we have (cid:107) D x d Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ (cid:46) (cid:107) Q Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + (cid:107) Op T ( χ ) w (cid:107) ,(cid:96) + , ˜ τ allows one to conclude the proof. (cid:4) B.3.2.
Transmitted sub-ellipticity. In U where q k ( (cid:37) ) is smoothly factorized, q k ( (cid:37) ) = q k , − ( (cid:37) ) q k , + ( (cid:37) ),we now describe how the sub-ellipticity property of Corollary B.3 is “transmitted” to the nonel-liptic factor q k , + . Proposition B.5.
Let k = or . There exist γ ≥ , α > , and C > such that αγ ˜ τ − | Im ρ k , + | + (cid:8) ξ d − Re ρ k , + , − Im ρ k , + (cid:9) ≥ C γ ˜ τ − λ T , ˜ τ , (cid:37) (cid:48) ∈ U , (B.6) for γ ≥ γ and α ≥ α . Proof . We write2 i { Re q k , Im q k } = { q k , q k } = | q k , − | { q k , + , q k , + } + | q k , + | { q k , − , q k , − } + i Im (cid:0) { q k , − , q k , + } q k , + q k , − (cid:1) , yielding { Re q k , Im q k } = | q k , − | { Re q k , + , Im q k , + } + | q k , + | { Re q k , − , Im q k , − } + Im (cid:0) { q k , − , q k , + } q k , + q k , − (cid:1) . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 63
We write, for M > | q k , + | (cid:12)(cid:12)(cid:12) { Re q k , − , Im q k , − } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Im (cid:0) { q k , − , q k , + } q k , + q k , − (cid:1)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) γλ ˜ τ | q k , + | + γλ τ | q k , + | (cid:1) ≤ C (cid:48) (1 + M ) γλ ˜ τ | q k , + | + C (cid:48) M − γλ τ . For M > γ ≥ ffi ciently large we obtain, with Corollary B.3, | q k , − ( (cid:37) ) | (cid:16) | q k , + ( (cid:37) ) | + τϕ ( z ) | ψ (cid:48) ε ( z ) | { Re q k , + , Im q k , + } ( (cid:37) ) (cid:17) ≥ C λ τ − C (cid:48) (1 + M )˜ τλ ˜ τ | q k , + | , In U we have | q k , − ( (cid:37) ) | (cid:16) λ ˜ τ , as q k , − is elliptic which gives α | q k , + ( (cid:37) ) | + τϕ ( z ) | ψ (cid:48) ε ( z ) | { Re q k , + , Im q k , + } ( (cid:37) ) ≥ C λ τ , (cid:37) (cid:48) ∈ U , ξ d ∈ R , for α > ffi ciently large. If we now choose ξ d = Re ρ k , + ( (cid:37) (cid:48) ) we then obtain the result. (cid:4) B.3.3.
A root with a vanishing imaginary part: a sub-elliptic estimate with a trace term.
Here,we consider as above a conic open set U ⊂ M T , such that the symbol q k ( (cid:37) ) = p k ( z , ζ + i ˆ τ ( (cid:37) (cid:48) ))can be factorized into two smooth first-order terms, q k ( (cid:37) ) = q k , − ( (cid:37) ) q k , + ( (cid:37) ). We let χ, χ ∈ S (1 , g T )be as above and we recall that Q k , + : = D x d − Op T w ( χ ρ k , + ). We have the following lemma. Lemma B.6.
Let (cid:96), m ∈ R and M ∈ N . There exist τ ≥ τ ∗ , γ ≥ , and C > , such that γ / (cid:107) ˜ τ m − / Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ ≤ C (cid:16) (cid:107) ˜ τ m Q k , + Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | tr(˜ τ m Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) , (B.7) for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for w ∈ S ( R N + ) . Proof . For concision, we write Q in place of Q k , + . We decompose Q according to Q = A + iB with A = D x d − Op T w ( χ Re ρ k , + ) ∈ Ψ , τ , B = − Op T w ( χ Im ρ k , + ) ∈ Ψ , τ = Ψ ( λ T , ˜ τ , g T ) . (B.8)Observe that both A and B are formally selfadjoint.We set w (cid:96), m = ˜ τ m Λ (cid:96) T , ˜ τ Op T ( χ ) w and compute (cid:107) Qw (cid:96), m (cid:107) + = (cid:107) ( A + iB ) w (cid:96), m (cid:107) + = (cid:107) Aw (cid:96), m (cid:107) + + (cid:107) Bw (cid:96), m (cid:107) + + Aw (cid:96), m , iBw (cid:96), m ) + (B.9)From the form of A and B given in (B.8) we find2 Re( Aw (cid:96), m , iBw (cid:96), m ) + = i ([ A , B ] w (cid:96), m , w (cid:96), m ) + − (Op T w ( χ Im ρ k , + ) w (cid:96), m | x d = + , w (cid:96), m | x d = + ) L ( R N − ) . yielding, with (B.9), (cid:107) Qw (cid:96), m (cid:107) + + | tr(˜ τ m Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ (cid:38) (cid:107) Aw (cid:96), m (cid:107) + + (cid:107) Bw (cid:96), m (cid:107) + + i ([ A , B ] w (cid:96), m , w (cid:96), m ) + (cid:38) (cid:107) Aw (cid:96), m (cid:107) + + ( (cid:0) αγ ˜ τ − B + i [ A , B ] (cid:1) w (cid:96), m , w (cid:96), m ) + (cid:38) (cid:107) Aw (cid:96), m (cid:107) + + ( Λ (cid:96) T , ˜ τ ˜ τ m (cid:0) αγ ˜ τ − B + i [ A , B ] (cid:1) ˜ τ m Λ (cid:96) T , ˜ τ Op T ( χ ) w , Op T ( χ ) w ) + , for α = α with α given by Proposition B.5, and for τ such that αγ ˜ τ − ≤
1. As the principalsymbol of Λ (cid:96) T , ˜ τ ˜ τ m (cid:0) αγ ˜ τ − B + i [ A , B ] (cid:1) ˜ τ m Λ (cid:96) T , ˜ τ is given, in a conic neighborhood of supp( χ ), where χ ≡
1, by ˜ τ m λ (cid:96) T , ˜ τ (cid:0) αγ ˜ τ − (Im ρ k , + ) + { ξ d − Re ρ k , + , − Im ρ k , + } (cid:1) ∈ S ( γ ˜ τ m − λ + (cid:96) T , ˜ τ , g T ) , then Proposition B.5 and the (microlocal) Gårding inequality of Proposition 4.16 yield, for any M ∈ N , by choosing τ and γ su ffi ciently large, (cid:107) Qw (cid:96), m (cid:107) + + | tr(˜ τ m Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M (cid:38) (cid:107) Aw (cid:96), m (cid:107) + + γ / (cid:107) ˜ τ m − / Op T ( χ ) w (cid:107) , + (cid:96), ˜ τ . From the form of A in (B.8) we have γ / (cid:107) ˜ τ − / D x d w (cid:96), m (cid:107) + (cid:46) (cid:107) Aw (cid:96), m (cid:107) + + γ / (cid:107) ˜ τ − / w (cid:96), m (cid:107) , , ˜ τ (cid:46) (cid:107) Aw (cid:96), m (cid:107) + + γ / (cid:107) ˜ τ m − / Op T ( χ ) w (cid:107) , + (cid:96), ˜ τ . We thus obtain (cid:107) Qw (cid:96), m (cid:107) + + | tr(˜ τ m Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M (cid:38) γ / (cid:0) (cid:107) ˜ τ m − / Op T ( χ ) w (cid:107) , + (cid:96), ˜ τ + (cid:107) ˜ τ − / D x d w (cid:96), m (cid:107) + (cid:1) (cid:38) γ / (cid:107) ˜ τ m − / Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ , by choosing τ su ffi ciently large and using Lemma 4.14. This concludes the proof. (cid:4) B.4.
Estimate concatenations.
Let U be on conic open set of M T . Let χ ( (cid:37) (cid:48) ) ∈ S (1 , g T ) behomogeneous of degree zero such that supp( χ ) ⊂ U . Let ρ ( k ) ( (cid:37) (cid:48) ) ∈ S ( λ T , ˜ τ , g T ), k = ,
2, behomogeneous of degree one in U and define Q ( k ) = D x d − Op T w ( χ ρ ( k ) ). The operators Q k , ± , k = ,
2, defined in what precedes and in Section 4 are of this form. Above, for such operators, weproved some microlocal estimates of the form(B.10) γ α k / (cid:107) ˜ τ α k ( m − / Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + δ k | tr(˜ τ m α k Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ ≤ C (cid:16) (cid:107) ˜ τ m α k Q ( k ) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + (1 − δ k ) | tr(˜ τ m α k Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) , with δ k = (1 − α k )(1 − β k ) and α k , β k ∈ { , } , (cid:96), m ∈ R , and where χ ∈ S (1 , g T ), homogeneous ofdegree zero and such that χ ≡ χ ).If α k = β k = (cid:107) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | tr(Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ ≤ C (cid:16) (cid:107) Q ( k ) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) . This is a perfect elliptic estimate that holds if ρ ( k ) is in the lower half complex plane –see Lemma B.1.If α k = β k = (cid:107) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ ≤ C (cid:16) (cid:107) Q ( k ) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | tr(Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) . This is an elliptic estimate, yet with a trace observation term in the r.h.s., that holds if ρ ( k ) is in theupper half complex plane –see Lemma B.4.Finally, if α k =
1, independently of the value of β k we have γ / (cid:107) ˜ τ m − / Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ ≤ C (cid:16) (cid:107) ˜ τ m Q ( k ) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | tr(˜ τ m Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) . This estimate is characterized by the loss of a half derivative and a boundary observation term inthe r.h.s.; such an estimate is proven in Lemma B.6 when the root ρ ( k ) may cross the real axis.We shall now describe how such estimates can be concatenated, as this is often done in thecourse of the proof of Theorem 4.17. Proposition B.7.
Let (cid:96) ∈ R and M ∈ N . Let Q ( k ) be defined as above, for k = , . Let τ ≥ τ ∗ , γ ≥ and C > such that estimate (B.10) holds, with (cid:96), m ∈ R , with α k , β k ∈ { , } , for bothk = and , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for w ∈ S ( R N + ) . We assume that α ≤ α and − δ ≤ − δ . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 65
Let χ ∈ S (1 , g T ) , be homogeneous of degree zero and such that χ ≡ on supp( χ ) . There exist τ ≥ τ ∗ , γ ≥ and C > such that the following estimate for the second-order operator Q (1) Q (2) holds, γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + | tr(Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ ≤ C (cid:16) (cid:107) Q (1) Q (2) Op T ( χ ) w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Op T ( χ ) w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for w ∈ S ( R N + ) . Note that the assumptions made on α k and 1 − δ k , k = ,
2, imply that Q (1) yields an estimate ofbetter quality than that associated with Q (2) . Proof . We introduce χ ∈ S (1 , g T ) that is such that χ ≡ χ ) and χ ≡ χ ). Forconcision, we write Ξ = Op T ( χ ) and Ξ = Op T ( χ ). Here, M will denote an arbitrary large integerwhose value may change from one line to the other.Using Q (2) Ξ w as the unknown function in the estimate (B.10) for Q (1) , with m = γ α / (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + δ | tr( Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ (B.11) (cid:46) γ α / (cid:107) ˜ τ − α / Ξ Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + δ | tr( Ξ Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ + | tr( w ) | , − M , ˜ τ (cid:46) (cid:107) Q (1) Ξ Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ + | tr( w ) | , − M , ˜ τ (cid:46) (cid:107) Q (1) Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ . Observe now that we can write, using that D x d − Q (2) ∈ Ψ ( λ T , ˜ τ , g T ),(1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) δ γ α / | tr(˜ τ − α / Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) δ | tr( Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ . With this estimate and (B.11), we thus obtain γ α / (cid:16) (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:17) (B.12) (cid:46) (cid:107) Q (1) Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ . Up to creating error terms, we shall now modify this inequality to be able to apply the estimate(B.10) associated with Q (2) . We write (cid:107) ˜ τ − α / Q (2) Ξ D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ D x d Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) ˜ τ − α / Q (2) D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / D x d Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ + | tr( w ) | , − M , ˜ τ (cid:46) (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ + | tr( w ) | , − M , ˜ τ + γ (cid:107) ˜ τ − α / Ξ w (cid:107) ,(cid:96), ˜ τ , using that [ D x d , Q (2) ] ∈ γ Ψ , τ and using Lemma 4.14. Hence with (B.12) we have γ α / (cid:16) (cid:107) ˜ τ − α / Q (2) Ξ D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ D x d Ξ w ) | ,(cid:96) + / , ˜ τ (B.13) + (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96) + , ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:17) (cid:46) γ α / (cid:16) (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + γ (cid:107) ˜ τ − α / Ξ w (cid:107) ,(cid:96), ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ + | tr( w ) | , − M , ˜ τ (cid:46) (cid:107) Q (1) Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + γ + α / (cid:107) ˜ τ − α / Ξ w (cid:107) ,(cid:96), ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ We write, with Lemma 4.14, for τ chosen su ffi ciently large, (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96), ˜ τ (cid:16) (cid:107) ˜ τ − ( α + α ) / D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96) + , ˜ τ (B.14) (cid:46) (cid:107) ˜ τ − ( α + α ) / Ξ D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96) + , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ , and | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:16) | tr(˜ τ − α / D x d Ξ w ) | ,(cid:96) + / , ˜ τ + | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (B.15) (cid:46) | tr(˜ τ − α / Ξ D x d Ξ w ) | ,(cid:96) + / , ˜ τ + | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + | tr( w ) | , − M , ˜ τ . Applying now estimate (B.10) associated with Q (2) to D x d Ξ w and w , with m = − α /
2, using that α = α α , we obtain γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ D x d Ξ w (cid:107) ,(cid:96), ˜ τ + δ γ α / | tr(˜ τ − α / Ξ D x d Ξ w ) | ,(cid:96) + / , ˜ τ (B.16) (cid:46) γ α / (cid:16) (cid:107) ˜ τ − α / Q (2) Ξ D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ D x d Ξ w ) | ,(cid:96) + / , ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ and γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96) + , ˜ τ + δ γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (B.17) (cid:46) γ α / (cid:16) (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96) + , ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ . With (B.14)–(B.17), we achieve γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96), ˜ τ + δ γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) γ α / (cid:16) (cid:107) ˜ τ − α / Q (2) Ξ D x d Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ D x d Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) ˜ τ − α / Q (2) Ξ w (cid:107) ,(cid:96) + , ˜ τ + (1 − δ ) | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ . Combining this latter estimate with (B.13) we obtain γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96), ˜ τ + δ γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) Q (1) Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + γ + α / (cid:107) ˜ τ − α / Ξ w (cid:107) ,(cid:96), ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ , which, with the usual semi-classical inequality (4.17) γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96), ˜ τ + δ γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) Q (1) Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ . Let us now consider two cases:
PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 67
Case α = Then δ = α =
1. We thus have the term | tr( Ξ w ) | ,(cid:96) + / , ˜ τ in the r.h.s.of the estimation and the sought result then holds. Case α = Then we write | tr( Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) | tr( Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + | tr( Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) | tr( Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + δ | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ . which leads to δ | tr( Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) δ | tr( Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ + δ | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ . Recalling that the term δ | tr( Q (2) Ξ w ) | ,(cid:96) + / , ˜ τ can be found in the l.h.s. of (B.11), We thusobtain γ α / (cid:107) ˜ τ − α / Ξ w (cid:107) ,(cid:96), ˜ τ + ( δ + δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ (cid:46) (cid:107) Q (1) Q (2) Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ . If δ + δ > α =
0. If δ + δ = | tr( Ξ w ) | ,(cid:96) + / , ˜ τ can be found in the r.h.s. of the estimation and can thus be“artificially” added in the l.h.s..This concludes the proof of Proposition B.7. (cid:4) We now show how to obtain microlocal estimates for some products of two factors of ordertwo.
Proposition B.8.
Let assume that Q − ( z , D z , τ, γ, ε ) ∈ Ψ , τ fulfills the requirement of Lemma B.1in some conic open subset U . Let Q + ( z , D z , τ, γ, ε ) ∈ Ψ , τ be such that, there exist τ ≥ τ ∗ , γ ≥ and C > such that, for (cid:96) ∈ { , , } and all χ ∈ S (1 , g T ) , homogeneous of degree zero, with supp( χ ) ⊂ U , for Ξ = Op T ( χ ) , γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) ,(cid:96), ˜ τ + | tr( Ξ w ) | ,(cid:96) + / , ˜ τ (B.18) ≤ C (cid:16) (cid:107) Q + Ξ w (cid:107) ,(cid:96), ˜ τ + (1 − δ ) | tr( Ξ w ) | ,(cid:96) + / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | ,(cid:96) + / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for w ∈ S ( R N + ) , where α , α ∈ { , } and δ , δ ∈ { , } with α ≤ α , − δ ≤ − δ and moreover δ k = if α k = , k = , . We also assume thatQ + Ξ = D x d Ξ + T , Ξ with T , ∈ Ψ , τ .Let M ∈ N and let χ ∈ S (1 , g T ) be as above. In the case α + α = , we furthermore assumethat, for any M ∈ N , [ D x d + i ˆ τ ξ d , Q + ] Op T ( χ ) = (1 + εγ ) R , Op T ( χ ) + R , − M , with R , ∈ Ψ , τ and R , − M ∈ Ψ , − M ˜ τ , if χ ∈ S (1 , g T ) is homogeneous of degree 0 and such that χ ≡ in a conicneighborhood of supp( χ ) and supp( χ ) ⊂ U .There exist τ ≥ τ ∗ , γ ≥ , ε ∈ (0 , , and C > such that γ ( α + α ) / (cid:107) ˜ τ − ( α + α ) / Ξ w (cid:107) , , ˜ τ + | tr( Ξ w ) | , / , ˜ τ (B.19) ≤ C (cid:16) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | , / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:17) , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , ε ] , and for w ∈ S ( R N + ) . In the case α + α ≤ , we can take ε = . In Section 4, for example, this proposition will be applied to Q + = Q , + Q , + for which anestimation of the form of (B.18) will hold by Proposition B.7. Note that this proposition, in thecase α + α =
2, is one instance where it is important to take ε > ffi ciently small. Proof . We introduce χ ∈ S (1 , g T ) that is such that χ ≡ χ ) and supp( χ ) ⊂ U . Forconcision, we write Ξ = Op T ( χ ) and Ξ = Op T ( χ ). Here, M will denote an arbitrary large integerwhose value may change from one line to the other.Using Q + Ξ w as the unknown function in the estimate of Lemma B.1 for the operator Q − : (cid:107) Q + Ξ w (cid:107) , , ˜ τ + | tr( Q + Ξ w ) | , / , ˜ τ (B.20) (cid:46) (cid:107) Ξ Q + Ξ w (cid:107) , , ˜ τ + | tr( Ξ Q + Ξ w ) | , / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ (cid:46) (cid:107) Q − Ξ Q + Ξ w (cid:107) + + (cid:107) w (cid:107) , − M , ˜ τ (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (cid:107) w (cid:107) , − M , ˜ τ . Combining (B.18), for (cid:96) =
2, with (B.20) we find (cid:107) Q + Ξ w (cid:107) , , ˜ τ + | tr( Q + Ξ w ) | , / , ˜ τ + | tr( Ξ w ) | , / , ˜ τ (B.21) (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | , / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ . We now make the following claim whose proof is given below.
Lemma B.9.
There exists C > such that | tr( Ξ v ) | , / , ˜ τ ≤ C (cid:0) | tr( Q + Ξ w ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:1) . This gives(B.22) (cid:107) Q + Ξ w (cid:107) , , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ α / | tr(˜ τ − α / Ξ w ) | , / , ˜ τ + (cid:107) w (cid:107) , − M , ˜ τ . First, we treat the case α + α ≤
1. As α ≤ α then α =
0. We write (cid:80) j = (cid:16) (cid:107) Q + Ξ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ D jx d Ξ v ) | , / − j , ˜ τ (cid:17) (cid:46) (cid:80) j = (cid:16) (cid:107) Q + D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( D jx d Ξ v ) | , / − j , ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ (cid:46) (cid:80) j = (cid:16) (cid:107) D jx d Q + Ξ w (cid:107) , − j , ˜ τ + | tr( D jx d Ξ v ) | , / − j , ˜ τ (cid:17) + γ (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ (cid:46) (cid:107) Q + Ξ w (cid:107) , , ˜ τ + | tr( Ξ v ) | , / , ˜ τ + γ (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . With (B.22) we then find (cid:80) j = (cid:16) (cid:107) Q + Ξ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ D jx d Ξ v ) | , / − j , ˜ τ (cid:17) (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + γ (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 69
Now, applying (B.18) with (cid:96) = − j , we obtain (cid:80) j = (cid:16) γ α / (cid:107) ˜ τ − α / Ξ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) (B.23) (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + γ (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . With Lemma 4.14, we write, for τ chosen su ffi ciently large, γ α / (cid:107) ˜ τ − α / Ξ w (cid:107) , , ˜ τ + | tr( Ξ w ) | , / , ˜ τ (cid:16) (cid:80) j = (cid:16) γ α / (cid:107) ˜ τ − α / D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( D jx d Ξ w ) | , / − j , ˜ τ (cid:17) (cid:46) (cid:80) j = (cid:16) γ α / (cid:107) ˜ τ − α / Ξ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ . Finally, using (B.23) we obtain γ α / (cid:107) ˜ τ − α / Ξ w (cid:107) , , ˜ τ + | tr( Ξ w ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + γ (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ , and taking τ su ffi ciently large, as 0 ≤ α ≤
1, we achieve the sought estimate.Second, we treat the case α + α =
2, that is, α = α =
1. We set ˜ D x d = D x d + i ˆ τ ξ d ∈ Ψ , τ .We use the further assumption made in this case, namely, for any M ∈ N , [ ˜ D x d , Q + ] Ξ = (1 + εγ ) R , Ξ + R , − M with R , ∈ Ψ , τ and R , − M ∈ Ψ , − M ˜ τ . We write (cid:80) j = (cid:16) (cid:107) Q + Ξ ˜ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ ˜ D jx d Ξ v ) | , / − j , ˜ τ (cid:17) (cid:46) (cid:80) j = (cid:16) (cid:107) Q + ˜ D jx d Ξ Ξ w (cid:107) , − j , ˜ τ + | tr( ˜ D jx d Ξ v ) | , / − j , ˜ τ (cid:17) + (cid:107) w (cid:107) , − M , ˜ τ (cid:46) (cid:80) j = (cid:16) (cid:107) ˜ D jx d Q + Ξ Ξ w (cid:107) , − j , ˜ τ + | tr( ˜ D jx d Ξ v ) | , / − j , ˜ τ (cid:17) + (1 + εγ ) (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ (cid:46) (cid:107) Q + Ξ w (cid:107) , , ˜ τ + | tr( Ξ v ) | , / , ˜ τ + (1 + εγ ) (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . With (B.22) we then find (cid:80) j = (cid:16) (cid:107) Q + Ξ ˜ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ ˜ D jx d Ξ v ) | , / − j , ˜ τ (cid:17) (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ / | ˜ τ − / tr( Ξ w ) | , / , ˜ τ + (1 + εγ ) (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . Now, applying (B.18) with (cid:96) = − j , we obtain (cid:80) j = (cid:16) γ (cid:107) ˜ τ − Ξ ˜ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ ˜ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) (B.24) (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ / | ˜ τ − / tr( Ξ w ) | , / , ˜ τ + (1 + εγ ) (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . Now, as [ ˜ D x d , ˜ τ − ] ∈ γ Ψ , − τ , we have (cid:80) j = (cid:16) γ (cid:107) ˜ D jx d ˜ τ − Ξ w (cid:107) , − j , ˜ τ + | tr( ˜ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) (cid:46) (cid:80) j = (cid:16) γ (cid:107) ˜ τ − ˜ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( ˜ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) + γ (cid:107) ˜ τ − Ξ w (cid:107) , (cid:46) (cid:80) j = (cid:16) γ (cid:107) ˜ τ − Ξ ˜ D jx d Ξ w (cid:107) , − j , ˜ τ + | tr( Ξ ˜ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) + γ (cid:107) ˜ τ − Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ , yielding with (B.24), as γ ˜ τ − (cid:46) (cid:80) j = (cid:16) γ (cid:107) ˜ D jx d ˜ τ − Ξ w (cid:107) , − j , ˜ τ + | tr( ˜ D jx d Ξ w ) | , / − j , ˜ τ (cid:17) (B.25) (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ / | ˜ τ − / tr( Ξ w ) | , / , ˜ τ + (1 + εγ ) (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . As D x d − ˜ D x d = T ∈ Ψ , τ , observe that we have (cid:107) D x d ˜ τ − Ξ w (cid:107) , , ˜ τ (cid:46) (cid:107) ˜ D x d ˜ τ − Ξ w (cid:107) , , ˜ τ + (cid:107) ˜ τ − Ξ w (cid:107) , , ˜ τ , meaning that we have (cid:107) ˜ τ − Ξ w (cid:107) , , ˜ τ (cid:46) (cid:107) ˜ D x d ˜ τ − Ξ w (cid:107) , , ˜ τ + (cid:107) ˜ τ − Ξ w (cid:107) , , ˜ τ . Next, we write (cid:107) D x d ˜ τ − Ξ w (cid:107) , , ˜ τ (cid:46) (cid:107) D x d ˜ D x d ˜ τ − Ξ w (cid:107) , , ˜ τ + (cid:107) D x d T ˜ τ − Ξ w (cid:107) , , ˜ τ (cid:46) (cid:107) ˜ D x d ˜ τ − Ξ w (cid:107) , , ˜ τ + (cid:107) ˜ D x d ˜ τ − Ξ w (cid:107) , , ˜ τ + (cid:107) ˜ τ − Ξ w (cid:107) , , ˜ τ , and thus (cid:107) ˜ τ − Ξ w (cid:107) , , ˜ τ (cid:46) (cid:80) j = (cid:107) ˜ D jx d ˜ τ − Ξ w (cid:107) , − j , ˜ τ . Similarly, we find | tr( Ξ w ) | , / , ˜ τ (cid:46) (cid:80) j = | tr( ˜ D jx d Ξ w ) | , / − j , ˜ τ . With (B.25) we thus obtain γ (cid:107) ˜ τ − Ξ w (cid:107) , , ˜ τ + | tr( Ξ w ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + Ξ w (cid:107) + + (1 − δ ) | tr( Ξ w ) | , / , ˜ τ + (1 − δ ) γ / | ˜ τ − / tr( Ξ w ) | , / , ˜ τ + (1 + εγ ) (cid:107) Ξ w (cid:107) , + (cid:107) w (cid:107) , − M , ˜ τ . Then, taking γ su ffi ciently large and ε > ffi ciently small we obtain the sought estimate. (cid:4) Proof of Lemma B.9.
Recalling that Q + Ξ = D x d Ξ + T , Ξ , where T , ∈ Ψ , τ , we have | tr( Ξ v ) | , / , ˜ τ (cid:16) | tr( D x d Ξ v ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ = | tr(( Q + − T , ) Ξ v ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:46) | tr( Q + Ξ v ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ . PECTRAL INEQUALITY AND RESOLVENT ESTIMATE FOR THE BI-LAPLACE OPERATOR 71
We then write | tr( Ξ v ) | , / , ˜ τ (cid:16) | tr( D x d Ξ v ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ = | tr( D x d ( Q + − T , ) Ξ v ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ (cid:46) | tr( Q + Ξ v ) | , / , ˜ τ + | tr( Ξ v ) | , / , ˜ τ . Combining the two estimates yields the result. (cid:4)
B.5.
An Estimate for Q k . We recall that Q k = (cid:0) D x d + i ˆ τ ξ d ( (cid:37) (cid:48) ) (cid:1) + ( − k i (cid:0) D s + i ˆ τ σ ( (cid:37) (cid:48) ) (cid:1) + r (cid:0) x , D x (cid:48) + i ˆ τ ξ (cid:48) ( (cid:37) (cid:48) ) (cid:1) , with k = ,
2. For this operator we have the following estimation.
Proposition B.10.
Let V (cid:48) (cid:98)
V. Let (cid:96) ∈ R . There exist τ ≥ τ ∗ , γ ≥ and C > such that γ / (cid:107) ˜ τ − / v (cid:107) ,(cid:96), ˜ τ + | tr( v ) | ,(cid:96) + / , ˜ τ ≤ C (cid:16) (cid:107) Q k v (cid:107) ,(cid:96), ˜ τ + | tr( v ) | ,(cid:96) + / , ˜ τ (cid:17) , k = , , for τ ≥ τ , γ ≥ γ , ε ∈ [0 , , and for v = w | R N + , with w ∈ C ∞ c ( R N ) and supp( w ) ⊂ V (cid:48) . The open neighborhood V is that introduced in Section 4.2. Proof . Let k be equal to 1 or 2. We write Q in place of Q k for concision. We also write µ in placeof µ k .We need to define microlocalization symbols and operators as in Section 4.4 and use some ofthe symbols introduced therein. Let χ V (cid:48) ∈ C ∞ ( R N ) be such that supp( χ V (cid:48) ) ⊂ V and χ V (cid:48) ≡ V (cid:48) .For δ ∈ (0 , χ δ, − ( (cid:37) (cid:48) ) = χ V (cid:48) ( z ) χ − ( µ ( (cid:37) (cid:48) ) /δ ) ∈ S (1 , g T ) ˜ χ δ, ( (cid:37) (cid:48) ) = χ V (cid:48) ( z ) (cid:0) − χ − ( µ ( (cid:37) (cid:48) ) /δ ) (cid:1) ∈ S (1 , g T ) , for χ − defined in Section 4.4, and observe that χ δ, − + χ δ, = M T , V (cid:48) . We set Ξ δ, − = Op T ( χ δ, − )and Ξ δ, = Op T ( ˜ χ δ, ).In a conic neighborhood of supp( χ δ, − ) ⊂ M T , V we have µ ≤ − C δ . As (4.20) holds in V we haveˆ τ ξ d ≥ C ˜ τ and thus | ˆ τ ξ | (cid:16) ˜ τ . Thus, by Lemma 4.18, both roots of the symbol q of the operator Q are in the lower half complex plane. Then, with Lemma B.1 we have the following perfect ellipticestimate, for any M > (cid:107) Ξ δ, − v (cid:107) , , ˜ τ + | tr( Ξ δ, − v ) | , / , ˜ τ (cid:46) (cid:107) Q Ξ δ, − v (cid:107) + + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ), for τ ≥ τ ∗ , γ ≥ ffi ciently large, and ε ∈ [0 , χ δ , χ δ, ∈ S (1 , g T ) supported in M T , V , homogeneous of degree zero, be such that µ ≥ − C δ on their supports and χ δ, ≡ χ δ, ) and χ δ ≡ χ δ, ).We choose δ > ffi ciently small so that the result of Lemma 4.22 applies, that is, on supp( χ δ )the roots of q are simple. We have q ( (cid:37) ) = q − ( (cid:37) ) q + ( (cid:37) ) , q ± ( (cid:37) ) = ξ d − ρ ± ( (cid:37) (cid:48) ) . We set Q ± : = D x d − Op T w ( χ ρ ± ).We shall denote by R j , k as a generic operator in Ψ j , k ˜ τ , j ∈ N , k ∈ R , whose expression maychange from one line to the other. We denote by M an arbitrary large integer whose value maychange from one line to the other. We have with a proof similar to that of Lemma 4.33, Q Ξ δ, = Q − Q + Ξ δ, + γ R , Ξ δ, + R , − M . (B.27) In a conic neighborhood of supp( ˜ χ δ, ), the root of the symbol of Q − is in the lower half complexplane. Then, with Lemma B.1, we have the following perfect elliptic estimate, for any M > (cid:107) Ξ δ, v (cid:107) , , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ (cid:46) (cid:107) Q − Ξ δ, v (cid:107) + + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ), for τ ≥ τ ∗ , γ ≥ ffi ciently large, and ε ∈ [0 , Q + we have the following estimate, characterized by the loss of a half derivative and a traceobservation, as given by Lemma B.6, γ / (cid:107) ˜ τ m − / Ξ δ, v (cid:107) ,(cid:96), ˜ τ (cid:46) (cid:107) ˜ τ m Q + Ξ δ, v (cid:107) ,(cid:96), ˜ τ + | tr(˜ τ m Ξ δ, v ) | ,(cid:96) + / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ) and (cid:96) ∈ R , and for τ and γ chosen su ffi ciently large, and ε ∈ [0 , α = α = δ =
1, and δ =
0, we have thefollowing estimates for the operator Q − Q + , for M > (cid:96) ∈ R , γ / (cid:107) ˜ τ − / Ξ δ, v (cid:107) , , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ (cid:46) (cid:107) Q − Q + Ξ δ, v (cid:107) + + | tr( Ξ δ, v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , for v ∈ S ( R N + ), and for τ and γ chosen su ffi ciently large. With (B.27) we thus obtain γ / (cid:107) ˜ τ − / Ξ δ, v (cid:107) , , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ (cid:46) (cid:107) Q Ξ δ, v (cid:107) + + | tr( Ξ δ, v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , (B.29)for τ chosen su ffi ciently large with the usual semi-classical inequality (4.17).Using that χ δ, − + χ δ, = M T , V (cid:48) we obtain, with (B.26) and (B.29) γ / (cid:107) ˜ τ − / v (cid:107) , , ˜ τ + | tr( v ) | , / , ˜ τ (cid:46) γ / (cid:107) ˜ τ − / Ξ δ, − v (cid:107) , , ˜ τ + γ / (cid:107) ˜ τ − / Ξ δ, v (cid:107) , , ˜ τ + | tr( Ξ δ, − v ) | , / , ˜ τ + | tr( Ξ δ, v ) | , / , ˜ τ (cid:46) (cid:107) Q Ξ δ, − v (cid:107) + + (cid:107) Q Ξ δ, v (cid:107) + + | tr( Ξ δ, v ) | , / , ˜ τ + (cid:107) v (cid:107) , − M , ˜ τ , for v = w | R N + , with w ∈ C ∞ c ( R N ) and supp( w ) ⊂ V (cid:48) . Observing that [ Q , Ξ δ, − ] and [ Q , Ξ δ, ] areboth in γ Ψ , τ we conclude the proof with the usual semi-classical inequality (4.17) for τ chosensu ffi ciently large. (cid:4) Conflict of interest:
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