On the growth of Sobolev norms for a class of linear Schrödinger equations on the torus with superlinear dispersion
aa r X i v : . [ m a t h . A P ] J un On the growth of Sobolev norms for a class of linearSchr¨odinger equations on the torus with superlineardispersion
Riccardo Montalto ∗ Abstract:
In this paper we consider time dependent Schr¨odinger equations on the one-dimensional torus T := R / (2 π Z ) of the form ∂ t u = i V ( t )[ u ] where V ( t ) is a time dependent, self-adjoint pseudo-differentialoperator of the form V ( t ) = V ( t, x ) | D | M + W ( t ), M > | D | := √− ∂ xx , V is a smooth function uniformlybounded from below and W is a time-dependent pseudo-differential operator of order strictly smaller than M . We prove that the solutions of the Schr¨odinger equation ∂ t u = i V ( t )[ u ] grow at most as t ε , t → + ∞ for any ε >
0. The proof is based on a reduction to constant coefficients up to smoothing remainders of thevector field i V ( t ) which uses Egorov type theorems and pseudo-differential calculus. Keywords:
Growth of Sobolev norms, Linear Schr¨odinger equations, Pseudo-differential operators.
MSC 2010:
Contents V ( t ) In this paper we consider linear Schr¨odinger equations of the form ∂ t u + i V ( t )[ u ] = 0 , x ∈ T (1.1)where T := R / (2 π Z ) is the 1-dimensional torus, V ( t ) is a L self-adjoint, time dependent, pseudo-differentialSchr¨odinger operator of the form V ( t ) := V ( t, x ) | D | M + W ( t ) , | D | := p − ∂ xx , M > . (1.2) ∗ Supported in part by the Swiss National Science Foundation
1e assume that V is a real valued C ∞ function defined on R × T with all derivatives bounded satisfyinginf ( t,x ) ∈ R × T V ( t, x ) > W ( t ) is a time-dependent pseudo differential operator of order strictly smallerthan M . Our main goal is to show that given t ∈ R , s ≥ u ∈ H s ( T ), the Cauchy problem ( ∂ t u + i V ( t )[ u ] = 0 u ( t , x ) = u ( x ) (1.3)admits a unique solution u ( t ) satisfying, for any ε >
0, the bound k u ( t ) k H s ≤ C ( s, ε )(1 + | t − t | ) ε k u k H s for some constant C ( s, ε ) >
0. Here, H s ( T ) denotes the standard Sobolev space on the 1-dimensional torus T equipped with the norm k · k H s .There is a wide literature concerning the problem of estimating the high Sobolev norms of linear partialdifferential equations. For the Schr¨odinger operator V ( t ) = − ∆ + V ( t, x ) on the d -dimensional torus T d , thegrowth ∼ t ε of the k · k H s norm of the solutions of ∂ t u = i V ( t )[ u ] has been proved by Bourgain in [9] forsmooth quasi-periodic in time potentials and in [10] for smooth and bounded time dependent potentials. Inthe case where the potential V is analytic and quasi-periodic in time, Bourgain [9] proved also that k u ( t ) k H s grows like a power of log( t ). Moreover, this bound is optimal, in the sense that he constructed an example forwhich k u ( t ) k H s is bounded from below by a power of log( t ). The result obtained in [10] has been extendedby Delort [11] for Schr¨odinger operators on Zoll manifolds. Furthermore, the logarithmic growth of k u ( t ) k H s proved in [9] has been extended by Wang [19] in dimension 1, for any real analytic and bounded potential.The key idea in these series of papers is to use the so-called spectral gap condition for the operator − ∆.Such a condition states that the spectrum of − ∆ can be enclosed in disjont clusters ( σ j ) j ≥ such that thedistance between σ j and σ j +1 tends to + ∞ for j → + ∞ .All the aforementioned results deal with the Schr¨odinger operator with a multiplicative potential. The firstresult in which the growth of k u ( t ) k H s is exploited for Schr¨odinger operators with unbounded perturbationsis due to Maspero-Robert [17]. More precisely, they prove the growth ∼ t ε of k u ( t ) k H s , for Schr¨odingerequations of the form i ∂ t u = L ( t ) u where L ( t ) = H + P ( t ), H is a time-independent operator of order µ + 1 satisfying the spectral gap condition and P ( t ) is an operator of order ν ≤ µ/ ( µ + 1) (see Theorem1.8 in [17]). The purpose of this paper is to provide a generalization of the result obtained in [17], atleast for Schr¨odinger operators on the 1-dimensional torus, when the order of H is the same as the orderof P ( t ). Note that the operator defined in (1.2) can be written in the form H + P ( t ) where H = | D | M , P ( t ) = ( V ( t, x ) − | D | M + W ( t ) and the operator | D | M fullfills the spectral gap condition since M > P ( t )is strictly smaller than the one of H . This result covers also several applications in higher space dimension.We also mention that in the case of quasi-periodic systems i ∂ t u = L ( ωt )[ u ], L ( ωt ) = H + εP ( ωt ) it isoften possible to prove that k u ( t ) k H s is uniformly bounded in time for ε small enough and for a large setof frequencies ω . The general strategy to deal with these quasi-periodic systems is called reducibility . Itconsists in costructing, for most values of the frequencies ω and for ε small enough, a bounded quasi-periodicchange of variable Φ( ωt ) which transforms the equation i ∂ t u = L ( ωt ) u into a time independent systemi ∂ t v = D v whose solution preserves the Sobolev norms k v ( t ) k H s . We mention the results of Eliasson-Kuksin[12] which proved the reducibility of the Schr¨odinger equation on T d with a small, quasi-periodic in timeanalytic potential and Grebert-Paturel [16] which proved the reducibility of the quantum harmonic oscillatoron R d . Concerning KAM-reducibility with unbounded perturbations, we mention Bambusi [3], [4] for thereducibility of the quantum harmonic oscillator with unbounded perturbations (see also [5] in any dimension),[1], [2], [15] for fully non-linear KdV-type equations, [13], [14] for fully-nonlinear Schr¨odinger equations, [7],[8] for the water waves system and [18] for the Kirchhoff equation. Note that in [1], [2], [15], [7], [8], [18] thereducibility of the linearized equations is obtained as a consequence of the KAM theorems proved for thecorresponding nonlinear equations.We now state in a precise way the main results of this paper. First, we introduce some notations. For anyfunction u ∈ L ( T ), we introduce its Fourier coefficients b u ( ξ ) := 12 π Z T u ( x ) e − i xξ dx , ∀ ξ ∈ Z . (1.4)2or any s ≥
0, we introduce the Sobolev space of complex valued functions H s ≡ H s ( T ), as H s := n u ∈ L ( T ) : k u k H s := X ξ ∈ Z h ξ i s | b u ( ξ ) | < + ∞ o , h ξ i := (1 + | ξ | ) . (1.5)Given two Banach spaces ( X, k · k X ), ( Y, k · k Y ), we denote by B ( X, Y ) the space of bounded linear operatorsfrom X to Y equipped with the usual operator norm k · k B ( X,Y ) . If X = Y , we simply write B ( X ) for B ( X, X ).Given a linear operator
R ∈ B ( L ( T )), we denote by R ∗ the adjoint operator of R with respect to thestandard L inner product h u, v i L := Z T u ( x ) v ( x ) dx , ∀ u, v ∈ L ( T ) . (1.6)We say that the operator R is self-adjoint if R = R ∗ .Given a Banach space ( X, k · k X ), for any k ∈ N , for any −∞ ≤ T < T ≤ + ∞ we consider thespace C k ([ T , T ] , X ) of the k -times continuously differentiable functions with values in X . We denote by C kb ([ T , T ] , X ) the space of functions in C k ([ T , T ] , X ) having bounded derivatives, equipped with the norm k u k C kb ([ T ,T ] ,X ) := max j =1 ,...,k sup t ∈ [ T ,T ] k ∂ jt u ( t ) k X . (1.7)For any domain Ω ⊂ R d , we also denote by C ∞ b (Ω) the space of the C ∞ functions on Ω with all the derivativesbounded.Since the equation we deal with is a Hamiltonian PDE, we briefly describe the Hamiltonian formalism. Wedefine the symplectic form Ω : L ( T ) × L ( T ) → R byΩ[ u , u ] := i Z T ( u ¯ u − ¯ u u ) dx , ∀ u , u ∈ L ( T ) . (1.8)Given a family of linear operators R : R → B ( L ) such that R ( t ) = R ( t ) ∗ for any t ∈ R , we define thetime-dependent quadratic Hamiltonian associated to R as H ( t, u ) := hR ( t )[ u ] , u i L x = Z T R ( t )[ u ] u dx , ∀ u ∈ L ( T ) . The Hamiltonian vector field associated to the Hamiltonian H is defined by X H ( t, u ) := i ∇ u H ( t, u ) = i R ( t ) (1.9)where the gradient ∇ u stands for ∇ u := 1 √ ∇ v + i ∇ ψ ) , v = Re( u ) , ψ := Im( u ) . We say that Φ : R → B ( L ( T )) is symplectic if and only ifΩ h Φ( t )[ u ] , Φ( t )[ u ] i = Ω[ u , u ] , ∀ u , u ∈ L ( T ) , ∀ t ∈ R . We recall the classical thing that if X H is a Hamiltonian vector field, then exp( X H ) is symplectic.Let us consider a time dependent vector field X : R → B ( L ( T )) and a differentiable family of invertiblemaps Φ : R → B ( L ( T )). Under the change of variables u = Φ( t )[ v ], the equation ∂ t u = X ( t )[ u ] transformsinto the equation ∂ t v = X + ( t )[ v ] where the push-forward X + ( t ) of the vector field X ( t ) is defined by X + ( t ) := Φ ∗ X ( t ) := Φ( t ) − (cid:16) X ( t )Φ( t ) − ∂ t Φ( t ) (cid:17) , t ∈ R . (1.10)It is well known that if Φ is symplectic and X ( t ) is a Hamiltonian vector field, then the push-forward X + ( t ) = Φ ∗ X ( t ) is still a Hamiltonian vector field.In the next two definitions, we also define time dependent pseudo differential operators on T .3 efinition 1.1 ( The symbol class S m ) . Let m ∈ R . We say that a C ∞ function a : R × T × R → C belongs to the symbol class S m if and only if for any α, β, γ ∈ N there exists a constant C α,β,γ > such that | ∂ αt ∂ βx ∂ γξ a ( t, x, ξ ) | ≤ C α,β,γ h ξ i m − γ , ∀ ( t, x, ξ ) ∈ R × T × R . (1.11) We define the class of smoothing symbols S −∞ := ∩ m ∈ R S m . Definition 1.2. (the class of operators
OP S m ) Let m ∈ R and a ∈ S m . We define the time-dependentlinear operator A ( t ) = Op (cid:0) a ( t, x, ξ ) (cid:1) = a ( t, x, D ) as A ( t )[ u ]( x ) := X ξ ∈ Z a ( t, x, ξ ) b u ( ξ ) e i xξ , ∀ u ∈ C ∞ ( T ) . We say that the operator A is in the class OP S m .We define the class of smoothing operators OP S −∞ := ∩ m ∈ R OP S m . Now, we are ready to state the main results of this paper. We make the following assumptions. (H1)
The operator V ( t ) = V ( t, x ) | D | M + W ( t ) in (1.2) is L self-adjoint for any t ∈ R . (H2) The function V ( t, x ) in (1.2) is in C ∞ b ( R × T , R ), strictly positive and bounded from below, i.e. δ :=inf ( t,x ) ∈ R × T V ( t, x ) > (H3) The operator W ( t ) is a time-dependent pseudo-differential operator W ( t ) = Op( w ( t, x, ξ )), with symbol w ∈ S M − e for some e > Theorem 1.3 ( Growth of Sobolev norms).
Assume the hypotheses (H1) - (H3) . Let s > , u ∈ H s ( T ) , t ∈ R . Then there exists a unique global solution u ∈ C ( R , H s ( T )) of the Cauchy problem ( ∂ t u + i V ( t )[ u ] = 0 u ( t , x ) = u ( x ) (1.12) and for any ε > there exists a constant C ( s, ε ) > such that k u ( t ) k H s ≤ C ( s, ε )(1 + | t − t | ε ) k u k H s , ∀ t ∈ R . (1.13)This theorem will be proved in Section 5 and it will be deduced by the following Theorem 1.4 ( Normal-form theorem).
Assume the hypotheses (H1) - (H3) . For any K > there existsa time-dependent symplectic differentiable invertible map t
7→ T K ( t ) satisfying sup t ∈ R kT K ( t ) ± k B ( H s ) + sup t ∈ R k ∂ t T K ( t ) ± k B ( H s +1 ,H s ) < + ∞ , ∀ s ≥ such that the following holds: the vector field i V ( t ) is transformed, by the map T K , into the vector field i V K ( t ) := ( T K ) ∗ (i V )( t ) = i (cid:16) λ K ( t, D ) + W K ( t ) (cid:17) (1.15) where λ K ( t, D ) := Op( λ K ( t, ξ )) is a space-diagonal operator with symbol λ K which satisfies λ K ∈ S M , λ K ( t, ξ ) = λ K ( t, ξ ) , ∀ ( t, ξ ) ∈ R × R (1.16) and W K ( t ) = Op (cid:16) w K ( t, x, ξ ) (cid:17) , w K ∈ S − K (1.17) is L self-adjoint.
4n the remaining part of the section, we shall explain the main ideas needed to prove Theorems 1.3, 1.4.In order to prove Theorem 1.3, we need to estimate the Sobolev norm k u ( t ) k H s , s >
0, for the solutions u ( t ) of(1.12). Choosing the integer K ≃ s in Theorem 1.4, we transform the PDE ∂ t u = i V ( t ) u into the PDE ∂ t v =iOp (cid:0) λ s ( t, ξ ) (cid:1) v + O ( | D | − s ) v . The Hamiltonian structure guarantees that the symbol λ s ( t, ξ ) is real. Writingthe Duhamel formula for the latter equation, one easily gets that k v ( t ) k H s . s k v ( t ) k H s + | t − t |k v ( t ) k L which implies that the same estimate holds for u ( t ), i.e. k u ( t ) k H s . s k u ( t ) k H s + | t − t |k u ( t ) k L (in thispaper, we use the standard notation A . s B if and only if A ≤ C ( s ) B for some constant C ( s ) > k u ( t ) k L = k u ( t ) k L (since V ( t ) is self-adjoint), applying the classical interpolation Theorem 2.8, oneobtains the growth ∼ | t − t | ε of the Sobolev norm k u ( t ) k H s for any s, ε > normal form procedure, which transforms the vector field i V ( t )into another one which is an arbitrarily regularizing perturbation of a space diagonal vector field. Such aprocedure is developed in Section 3 and it is based on symbolic calculus and Egorov type Theorems (seeTheorems 2.14-2.16). We describe below our method in more detail.1. Reduction of the highest order.
Our first aim is to transform the vector field i V ( t ) = i (cid:0) V ( t, x ) | D | M + W ( t ) (cid:1) into another vector field i V ( t ) whose highest order is x -independent, i.e. V ( t ) = λ ( t ) | D | M + W ( t ) with W ∈ OP S M − e . This is done in Section 3.1. In order to achieve this purpose, we transformthe vector field i V ( t ) by means of the time 1-flow map of the transport equation ∂ τ u = b α ( τ ; t, x ) ∂ x u + ( ∂ x b α )2 u , b α ( τ ; t, x ) := − α ( t, x )1 + τ α x ( t, x ) , τ ∈ [0 , α ( t, x ) is a function in C ∞ b ( R × T , R ) (to be determined) satisfying inf ( t,x ) ∈ R × T (cid:16) ∂ x α )( t, x ) (cid:17) >
0. This condition guarantees that T → T , x x + α ( t, x ) is a diffeomorphism of the torus with inversegiven by T → T , y y + e α ( t, y ) and e α ∈ C ∞ b ( R × T , R ) satisfying inf ( t,y ) ∈ R × T (cid:16) ∂ y e α )( t, y ) (cid:17) > V ( t ), V ( t ) = Op (cid:16) v ( t, x, ξ ) (cid:17) is analyzed by usingTheorems 2.14, 2.15 and its final expansion is provided in Lemma 3.2. It turns out that the principalpart of the operator V ( t ) is given by h V ( t, y ) (cid:0) e α y ( t, y ) (cid:1) M i y = x + α ( t,x ) | D | M . The function e α is choosen in such a way that V ( t, y ) (cid:0) e α y ( t, y ) (cid:1) M = λ ( t ) where λ ∈ C ∞ b ( R , R ) isindependent of x (see (3.16)–(3.19)). The hypothesis (H2) on V ( t, x ), i.e. inf ( t,x ) ∈ R × T V ( t, x ) > t ∈ R λ ( t ) > ( t,y ) ∈ R × T (cid:16) ∂ y e α )( t, y ) (cid:17) > ( t,x ) ∈ R × T (cid:16) ∂ x α )( t, x ) (cid:17) >
0, by Lemma 2.12.2.
Reduction of the lower order terms.
After the first reduction described above, we deal with a vector fieldi V ( t ) where V ( t ) = λ ( t ) | D | M + W ( t ) and W ∈ OP S M − ¯ e for some constant ¯ e >
0. The next step is totransform such a vector field into another one of the form i (cid:16) λ ( t ) | D | + µ N ( t, D )+ W N ( t ) (cid:17) where µ N ( t, D )is a time-dependent Fourier multiplier of order M − ¯ e and W N ∈ OP S M − N ¯ e for any integer N > n -th step of such a procedure, we deal with a vector field i V n ( t ), V n ( t ) = λ ( t ) | D | M + µ n ( t, D ) + W n ( t ), µ n ∈ S M − ¯ e , W n ∈ OP S M − n ¯ e . We transform such a vector field by means of the time-1 flow map ofthe PDE ∂ τ u = i G n ( t )[ u ] where G n ( t ) = G n ( t ) ∗ , G n ∈ OP S − n ¯ e . Using Theorem 2.16, the transformed vector field i V n +1 ( t ), V n +1 ( t ) = Op (cid:16) v n +1 ( t, x, ξ ) (cid:17) has the symbolexpansion v n +1 ( t, x, ξ ) = λ ( t ) | ξ | M + µ n ( t, ξ ) + w n ( t, x, ξ ) − M λ ( t ) | ξ | M − ξ∂ x g n ( t, x, ξ ) + O ( | ξ | M − ( n +1)¯ e )5ne then finds g n ( t, x, ξ ) so that g n = g ∗ n ( g ∗ n is the symbol of the adjoint operator ) and which solvesthe equation w n ( t, x, ξ ) − M λ ( t ) | ξ | M − ξ∂ x g n ( t, x, ξ ) = h w n i x ( t, ξ ) + O ( | ξ | M − ( n +1)¯ e )where h w n i x ( t, ξ ) := π R T w n ( t, x, ξ ) dx (see Lemma 3.5). This implies that the transformed symbolhas the form v n +1 ( t, x, ξ ) = λ ( t ) | ξ | M + µ n +1 ( t, ξ ) + O ( | ξ | M − ( n +1)¯ e ) with µ n +1 = µ n + h w n i x .The paper is organized as follows: in Section 2 we provide some technical tools which are needed for theproof of Theorem 1.4. In Section 3 we develop the regularization procedure of the vector field that we usein Section 4 to deduce Theorem 1.4. Finally, in Section 5 we prove Theorem 1.3. Acknowledgements . The author warmly thanks Giuseppe Genovese, Emanuele Haus, Thomas Kappeler,Felice Iandoli and Alberto Maspero for many useful discussions and comments.
In this section, we recall some well-known definitions and results concerning pseudo differential operatorson the torus T . We always consider time dependent symbols a ( t, x, ξ ) depending in a C ∞ way on the wholevariables, see Definitions 1.1, 1.2. Actually the time t is only a parameter, hence all the classical resultsapply without any modification (we refer for instance to [20], [21]).For the symbol class S m given in the definition 1.1 and the operator class OP S m given in the definition 1.2,the following standard inclusions hold: S m ⊆ S m ′ , OP S m ⊆ OP S m ′ , ∀ m ≤ m ′ . (2.1)We define the class of smoothing symbol and smoothing operators S −∞ := ∩ m ∈ R S m , OP S −∞ := ∩ m ∈ R OP S m Theorem 2.1 (Calderon-Vallancourt) . Let m ∈ R and A = a ( t, x, D ) ∈ OP S m . Then for any s ∈ R , forany α ∈ N the operator ∂ αt A ( t ) ∈ B ( H s + m ( T ) , H s ( T )) with sup t ∈ R k ∂ αt A ( t ) k B ( H s + m ,H s ) < + ∞ . Definition 2.2 ( Asymptotic expansion).
Let ( m k ) k ∈ N be a strictly decreasing sequence of real numbersconverging to −∞ and a k ∈ S m k for any k ∈ N . We say that a ∈ S m has the asymptotic expansion P k ≥ a k , i.e. a ∼ X k ≥ a k if for any N ∈ N a − N X k =0 a k ∈ S m N +1 . Given a symbol a ∈ S m , we denote by b a , the Fourier transform with respect to the variable x , i.e. b a ( t, η, ξ ) := 12 π Z T a ( t, x, ξ ) e − i ηx dx , ( t, η, ξ ) ∈ R × Z × R . (2.2) Theorem 2.3 ( Composition).
Let m, m ′ ∈ R and A = a ( t, x, D ) ∈ OP S m , B = b ( t, x, D ) ∈ OP S m ′ .Then the composition operator AB := A ◦ B = σ AB ( t, x, D ) is a pseudo-differential operator in OP S m + m ′ with symbol σ AB ( t, x, ξ ) = X η ∈ Z a ( t, x, ξ + η ) b b ( t, η, ξ ) e i ηx . (2.3) The symbol σ AB has the following asymptotic expansion σ AB ( t, x, ξ ) ∼ X β ≥ β β ! ∂ βξ a ( t, x, ξ ) ∂ βx b ( t, x, ξ ) , (2.4)6 hat is, ∀ N ≥ , σ AB ( t, x, ξ ) = N − X β =0 β !i β ∂ βξ a ( t, x, ξ ) ∂ βx b ( t, x, ξ ) + r N ( t, x, ξ ) where r N := r N,AB ∈ S m + m ′ − N . (2.5) The remainder r N has the explicit formula r N ( t, x, ξ ) := 1( N − N Z (1 − τ ) N − X η ∈ Z ( ∂ Nξ a )( t, x, ξ + τ η ) d ∂ Nx b ( t, η, ξ ) e i ηx dτ . (2.6) Corollary 2.4.
Let m, m ′ ∈ R and let A = Op( a ) , B = Op( b ) . Then the commutator [ A, B ] = Op( a ⋆ b ) ,with a ⋆ b ∈ S m + m ′ − having the following expansion: a ⋆ b = − i { a, b } + r ( a, b ) , { a, b } := ∂ ξ a∂ x b − ∂ x a∂ ξ b ∈ S m + m ′ − , r ( a, b ) ∈ S m + m ′ − . Theorem 2.5 ( Adjoint of a pseudo-differential operator). If A ( t ) = a ( t, x, D ) ∈ OP S m is a pseudo-differential operator with symbol a ∈ S m , then its L -adjoint is the pseudo-differential operator A ∗ =Op( a ∗ ) ∈ OP S m defined by A ∗ = Op( a ∗ ) with symbol a ∗ ( t, x, ξ ) := X η ∈ Z b a ( t, η, ξ − η ) e i ηx . (2.7) The symbol a ∗ ∈ S m admits the asymptotic expansion a ∗ ( t, x, ξ ) ∼ X α ∈ N ( − i) α α ! ∂ αx ∂ αξ a ( t, x, ξ ) (2.8) meaning that for any integer N ≥ , a ∗ ( t, x, ξ ) = N − X α =0 ( − i) α α ! ∂ αx ∂ αξ a ( t, x, ξ ) + r ∗ N ( t, x, ξ ) where r ∗ N := r ∗ N,A ∈ S m − N . The remainder r ∗ N has the explicit formula r ∗ N ( t, x, ξ ) := ( − N ( N − Z (1 − τ ) N − X η ∈ Z d ∂ Nξ a ( t, η, ξ − tη ) η N e i ηx dτ . (2.9)Note that if a ∈ S m is a symbol independent of x (Fourier multiplier) then a ∗ ( t, ξ ) = a ( t, ξ ) , ∀ ( t, ξ ) ∈ R × R . (2.10)We now prove some useful lemmas which we apply in Section 3. Lemma 2.6.
Let A = Op( a ) ∈ OP S m be self-adjoint, i.e. a ( t, x, ξ ) = a ∗ ( t, x, ξ ) and let ϕ ( t, ξ ) be a realFourier multiplier of order m ′ . We define the symbol b ( t, x, ξ ) := ϕ ( t, ξ ) a ( t, x, ξ ) ∈ S m + m ′ . Then b ∗ ( t, x, ξ ) − b ( t, x, ξ ) ∈ S m + m ′ − . Proof.
One has that Op( b ∗ ) = Op( b ) ∗ = Op( ϕ ) ∗ ◦ Op( a ) ∗ . Since Op( a ) is self-adjoint and since λ is real, one has thatOp( b ∗ ) = Op( ϕ ) ◦ Op( a ) . By applying Theorem 2.3, one gets thatOp( b ∗ ) = Op (cid:16) ϕ ( t, ξ ) a ( t, x, ξ ) + r ( t, x, ξ ) (cid:17) , r ∈ S m + m ′ − and then the lemma is proved. 7e define the operator ∂ − x by setting ∂ − x [1] := 0 , ∂ − x [ e i xk ] := e i xk i k , ∀ k ∈ Z \ { } . (2.11)Furthermore, given a symbol a ∈ S m , we define the averaged symbol h a i x by h a i x ( t, ξ ) := 12 π Z T a ( t, x, ξ ) dx , ∀ ( t, ξ ) ∈ R × R . (2.12)The following elementary property holds: a ∈ S m = ⇒ ∂ − x a , h a i x ∈ S m . (2.13)We now prove the following Lemma 2.7.
Let a ∈ S m . Then the following holds: ( i ) h a ∗ i x = ( h a i x ) ∗ = h a i x , ( ii ) ∂ − x ( a ∗ ) = ( ∂ − x a ) ∗ .Proof. Proof of ( i ) . By Theorem 2.5 and since by the definition (2.12) the symbol h a i x is x -independent,one has that ( h a i x ) ∗ ( t, ξ ) = h a i x ( t, ξ ) . (2.14)Moreover by (2.7), (2.12) one gets h a ∗ i x ( t, ξ ) = 12 π Z T (cid:16)X η ∈ Z b a ( t, η, ξ − η ) e i ηx (cid:17) dx = b a ( t, , ξ ) (2.2) = 12 π Z T a ( t, x, ξ ) dx (2.12) = h a i x ( t, ξ ) (2.14) = ( h a i x ) ∗ ( t, ξ )hence the claimed statement follows. Proof of ( ii ) . By (2.11), one has that [ ∂ − x a ( t, , ξ ) = 0 , [ ∂ − x a ( t, η, ξ ) = b a ( t, η, ξ )i η , η ∈ Z \ { } , hence by formula (2.7)( ∂ − x a ) ∗ ( t, x, ξ ) = X η ∈ Z \{ } b a ( t, η, ξ − η )i η e i ηx = X η ∈ Z b a ( t, η, ξ − η ) ∂ − x ( e i ηx )= ∂ − x (cid:16)X η ∈ Z b a ( t, η, ξ − η ) e i ηx (cid:17) = ∂ − x ( a ∗ )( t, x, ξ ) (2.15)which proves item ( ii ).For any α ∈ R , the operator | D | α , acting on 2 π -periodic functions u ( x ) = P ξ ∈ Z b u ( ξ ) e i xξ is defined by | D | α u ( x ) := X ξ ∈ Z \{ } | ξ | α b u ( ξ ) e i xξ . We shall identify the operator | D | α with the operator associated to a Fourier multiplier | ξ | α χ ( ξ ) in S α where χ ∈ C ∞ ( R , R ) is an even cut-off function satisfying χ ( ξ ) := ( | ξ | ≥
10 if | ξ | ≤ . (2.16)Then, for any α ∈ R , | D | α ≡ Op (cid:0) | ξ | α χ ( ξ ) (cid:1) (2.17)since the action of the two operators on 2 π -periodic functions u ∈ L ( T ) coincides.We conclude this section by stating an interpolation theorem, which is an immediate consequence of theclassical Riesz-Thorin interpolation theorem in Sobolev spaces.8 heorem 2.8. Let ≤ s < s and let A ∈ B ( H s ) ∩ B ( H s ) . Then for any s ≤ s ≤ s the operator A ∈ B ( H s ) and k A k B ( H s ) ≤ k A k λ B ( H s ) k A k − λ B ( H s ) , λ := s − ss − s . In this section we study the properties of the flow of some linear pseudo-PDEs. We start with the followinglemma.
Lemma 2.9.
Let A ( τ ; t ) := Op (cid:16) a ( τ ; t, x, ξ ) (cid:17) , τ ∈ [0 , be a smooth τ -dependent family of pseudo differentialoperators in OP S . Assume that A ( τ ; t ) + A ( τ ; t ) ∗ ∈ OP S . Then the following holds. ( i ) Let s ≥ , u ∈ H s ( T ) , τ ∈ [0 , . Then there exists a unique solution u ∈ C b (cid:16) [0 , , H s ( T ) (cid:17) of theCauchy problem ( ∂ τ u = A ( τ ; t )[ u ] u ( t , x ) = u ( x ) (2.18) satisfying the estimate k u k C b ([0 , ,H s ) . s k u k H s . As a consequence, for any τ ∈ [0 , , the flow map Φ( τ , τ ; t ) , which maps the initial datum u ( τ ) = u intothe solution u ( τ ) of (2.18) at the time τ , is in B ( H s ) with sup τ ,τ ∈ [0 , t ∈ R k Φ( τ , τ ; t ) k B ( H s ) < + ∞ for any s ≥ . Moreover, the operator Φ( τ , τ ; t ) is invertible with inverse Φ( τ , τ ; t ) − = Φ( τ, τ ; t ) . ( ii ) For any τ , τ ∈ [0 , , the flow map t Φ( τ , τ ; t ) is differentiable and sup τ ,τ ∈ [0 , t ∈ R k ∂ kt Φ( τ , τ ; t ) k B ( H s + k ,H s ) < + ∞ , ∀ k ∈ N , s ≥ . (2.19) Proof.
Proof of ( i ) . The proof of item ( i ) is classical. We refer for instance to [21], Section 0.8. Proof of ( ii ) . For any τ ∈ [0 , τ , τ ; t ) solves ( ∂ τ Φ( τ , τ ; t ) = A ( τ ; t )Φ( τ , τ ; t )Φ( τ , τ ; t ) = Id . (2.20)By differentiating (2.20) with respect to t , one gets that ∂ t Φ( τ , τ ; t ) solves ( ∂ τ (cid:16) ∂ t Φ( τ , τ ; t ) (cid:17) = A ( τ ; t ) (cid:16) ∂ t Φ( τ , τ ; t ) (cid:17) + (cid:16) ∂ t A ( τ ; t ) (cid:17) Φ( τ , τ ; t ) ∂ t Φ( τ , τ ; t ) = 0 . By Duhamel principle, we then get ∂ t Φ( τ , τ ; t ) = Z ττ Φ( τ , τ ; t )Φ( ζ, τ ; t ) ∂ t A ( ζ ; t )Φ( τ , ζ ; t ) dζ . By item ( i ) and by Theorem 2.1 (using that ∂ t A ∈
OP S ) one gets that ∂ t Φ( τ , τ ; t ) ∈ B ( H s +1 , H s ) withestimates which are uniform with respect to τ , τ ∈ [0 ,
1] and t ∈ R . Hence (2.19) has been proved for k = 1.Iterating the above argument, one can prove the estimate (2.19) for any positive integer k .In the next lemma we prove the global well-posedness for a class of Schr¨odinger type equations. Let ϕ ( t, ξ ) ∈ OP S m be a real Fourier multiplier, i.e. ϕ ∈ S m , ϕ ( t, ξ ) = ϕ ( t, ξ ) , ∀ ( t, ξ ) ∈ R × R . (2.21)Moreover, let us consider a time dependent linear operator t
7→ R ( t ) satisfying R ∈ C b ( R , B ( H s )) , ∀ s ≥ . (2.22)The following lemma holds: 9 emma 2.10. Let s ≥ , u ∈ H s ( T ) , t ∈ R . Then there exists a unique global solution u ∈ C ( R , H s ) ofthe Cauchy problem ( ∂ t u + i ϕ ( t, D ) u + R ( t )[ u ] = 0 u ( t , x ) = u ( x ) . (2.23) Proof.
The local existence follows by a fixed point argument applied to the map F ( u ) := exp (cid:16) − iΦ( t, D ) (cid:17) [ u ] + Z tt exp (cid:16) − i (cid:16) Φ( t, D ) − Φ( τ, D ) (cid:17)(cid:17) R ( τ )[ u ( τ, · )] dτ where Φ( t, D ) := Op(Φ( t, ξ )) , Φ( t, ξ ) := Z tt ϕ ( ζ, ξ ) dζ . Since ϕ ( t, ξ ) is real, then also Φ( t, ξ ) is real, implying that the propagator exp (cid:16) − iΦ( t, D ) (cid:17) is unitary onSobolev spaces. Choosing R := 2 k u k H s , T := 12 kRk C b ( R , B ( H s )) and defining B R,T ( s ) := n u ∈ C b ([ t − T, t + T ] , H s ) : k u k C b ([ t − T,t + T ] ,H s ) ≤ R o one can prove that F : B R,T ( s ) → B R,T ( s )is a contraction. The global well posedness follows from the fact that the solution is bounded on any boundedinterval and then it can be extended to the whole real line. In this section we collect some abstract egorov type theorems, namely we study how a pseudo differentialoperator transforms under the action of the flow of a first order hyperbolic PDE. Let α : R × T → R be a C ∞ function with all the derivatives bounded, satisfying α ∈ C ∞ b ( R × T , R ) , inf ( t,x ) ∈ R × T (cid:0) α x ( t, x ) (cid:1) > . (2.24)We then consider the non-autonomous transport equation ∂ τ u = A ( τ ; t, x, D ) u , A ( τ ; t ) = A ( τ ; t, x, D ) := b α ( τ ; t, x ) ∂ x + ( ∂ x b α )( τ ; t, x )2 , (2.25) b α ( τ ; t, x ) := − α ( t, x )1 + τ α x ( t, x ) , τ ∈ [0 , . (2.26)Note that the condition (3.1) implies thatinf τ ∈ [0 , t,x ) ∈ R × T (cid:0) τ α x ( t, x ) (cid:1) > , hence the function b ∈ C ∞ b ([0 , × R × T ). Then A ( τ ; · ) ∈ OP S , τ ∈ [0 ,
1] is a smooth family of pseudo-differential operators and it is straightforward to see that A ( τ ; t ) + A ( τ ; t ) ∗ = 0. Therefore, the hypothesesof Lemma 2.9 are verified, implying that, for any τ ∈ [0 , τ ; t ) ≡ Φ(0 , τ ; t ), τ ∈ [0 ,
1] of theequation (3.2), i.e. ( ∂ τ Φ( τ ; t ) = A ( τ ; t )Φ( τ ; t )Φ(0; t ) = Id (2.27)10s a well defined map and satisfies all the properties stated in the items ( i ), ( ii ) of Lemma 2.9. Furthermore, A ( τ ; t ) is a Hamiltonian vector field. Indeed A ( τ ; t ) = i e A ( τ ; t ) , e A ( τ ; t ) := − i (cid:16) b α ( τ ; t, x ) ∂ x + ( ∂ x b α )( τ ; t, x )2 (cid:17) and e A ( τ ; t ) = e A ( τ ; t ) ∗ (2.28)implying that the map Φ( τ ; t ) is symplectic. We then have the following Lemma 2.11.
The flow Φ( τ ; t ) given by (3.3) is a symplectic, invertible map satisfying sup τ ∈ [0 , t ∈ R k ∂ kt Φ( τ ; t ) ± k B ( H s + k ,H s ) < + ∞ , ∀ k ∈ N , s ≥ . In order to state Theorem 2.14 of this section, we need some preliminary results.
Lemma 2.12.
Let α ∈ C ∞ b ( R × T , R ) satisfy the condition (3.1) . Then for any t ∈ R , the map ϕ t : T → T , x x + α ( t, x ) is a diffeomorphism of the torus whose inverse has the form ϕ − t : T → T , y y + e α ( t, y ) , (2.29) with e α : R × T → R satisfying e α ∈ C ∞ b ( R × T , R ) , inf ( t,y ) ∈ R × T (cid:0) e α y ( t, y ) (cid:1) > . (2.30) Furthermore, the following identities hold: α x ( t, x ) = 11 + e α y (cid:0) t, x + α ( t, x ) (cid:1) , e α y ( t, y ) = 11 + α x (cid:0) t, y + e α ( t, y ) (cid:1) (2.31) Proof.
The condition (3.1) and the inverse function theorem imply that for any t ∈ R , the map ϕ t : R → R is a C ∞ diffeomorphism with a C ∞ inverse given by ϕ − t : R → R . Since α is 2 π -periodic in x one verifieseasily that ϕ t ( x + 2 π ) = ϕ t ( x ) + 2 π , implying that ϕ t : T → T is a diffeomorphism of the torus. We nowverify that ϕ − t has the form (2.29). In order to see this, it is enough to show that e α ( t, y ) := ϕ − t ( y ) − y is2 π -periodic in y . Let y = ϕ t ( x ). Applying ϕ − t to both sides of the equality ϕ t ( x + 2 π ) = ϕ t ( x ) + 2 π , onegets that x + 2 π = ϕ − t ( y + 2 π ), i.e. ϕ − t ( y ) + 2 π = ϕ − t ( y + 2 π ). This implies that e α ( t, y + 2 π ) = ϕ − t ( y + 2 π ) − y − π = ϕ − t ( y ) + 2 π − y − π = e α ( t, y )and then e α is 2 π -periodic in y . Since y = x + α ( t, x ) ⇐⇒ x = y + e α ( t, y )one has e α ( t, y ) + α ( t, y + e α ( t, y )) = 0 , ∀ ( t, y ) ∈ R × T . (2.32)It follows by the standard implicit function theorem that e α is C with derivatives ∂ y e α ( t, y ) = − α x ( t, y + e α ( t, y ))1 + α x ( t, y + e α ( t, y )) , ∂ t e α ( t, y ) = − α t ( t, y + e α ( t, y ))1 + α x ( t, y + e α ( t, y )) . (2.33)By induction, it can be proved that e α is C ∞ with all the derivatives bounded, namely e α ∈ C ∞ b ( R × T , R ).The identities (2.31) follow easily by (2.33) and then also (2.30) holds. The proof of the lemma is thenconcluded. 11n the next we study the flow of the ODE ( ˙ x ( τ ) = − b α ( τ ; t, x ( τ ))˙ ξ ( τ ) = ∂ x b α ( τ ; t, x ( τ )) ξ ( τ ) (2.34)where b α ( τ ; t, x ) is defined in (2.26). Given τ , τ ∈ [0 , γ τ ,τ ( t, x, ξ ) = ( γ τ ,τ ( t, x ) , γ τ ,τ ( t, x, ξ ))the flow of the ODE (2.34) with initial time τ and final time τ . We point out that the first equation in(2.34) is independent of ξ , hence the first component of the flow is independent of ξ too. We now prove thefollowing lemma concerning the characteristic equation (2.34). Lemma 2.13.
For any τ , τ ∈ [0 , , γ τ ,τ ∈ C ∞ b ( R × T , R ) and γ τ ,τ ∈ S . Furthermore, for any τ ∈ [0 , one has that γ τ , ( t, x ) = x + τ α ( t, x ) , γ τ , ( t, x, ξ ) = (1 + τ α x ( t, x )) − ξ . Proof.
Given τ ∈ [0 , ( ˙ x ( τ ) = − b α ( τ ; t, x ( τ )) , x ( τ ) = x ˙ ξ ( τ ) = ∂ x b α ( τ ; t, x ( τ )) ξ ( τ ) , ξ ( τ ) = ξ . (2.35)Let x ( τ ) = γ τ ,τ ( t, x ), ξ ( τ ) = γ τ ,τ ( t, x, ξ ) be the unique solution of (2.35). The second equation can beintegrated explicitly, leading to ξ ( τ ) = γ τ ,τ ( t, x, ξ ) = exp (cid:16) Z ττ ∂ x b α (cid:0) ζ ; γ τ ,ζ ( t, x ) (cid:1) dζ (cid:17) ξ , ∀ τ ∈ [0 , . (2.36)Note that, since b α is C ∞ with respect to all its variables and all its derivatives are bounded, by the smoothdependence of the flow on the initial data ( x, ξ ) and on the parameter t , one has that γ τ ,τ is C ∞ w.r. to( t, x ) with all bounded derivatives. Hence by (2.36) one gets that γ τ ,τ ∈ S . By differentiating with respectto the initial datum x the first equation in (2.35) one gets that ∂ τ (cid:0) ∂ x γ τ ,τ ( x ) (cid:1) = − ∂ x b α (cid:0) τ ; γ τ ,τ ( x ) (cid:1) ∂ x γ τ ,τ ( x ) , ∂ x γ τ ,τ ( x ) = 1whose solution is given by ∂ x γ τ ,τ ( x ) = exp (cid:16) − Z ττ ∂ x b α (cid:0) ζ ; γ τ ,ζ ( x ) (cid:1) dζ (cid:17) . (2.37)By formulae (2.36), (2.37), one then obtains that γ τ ,τ ( x, ξ ) = (cid:16) ∂ x γ τ ,τ ( x ) (cid:17) − ξ . (2.38)Note that by the definition of b α given in (3.2) and by the first equation in (2.35), one has that ddτ (cid:16) x ( τ ) + τ α ( x ( τ )) (cid:17) = α ( x ( τ )) + (cid:0) τ α x ( x ( τ )) (cid:1) ˙ x ( τ ) = 0implying that x ( τ ) + τ α ( x ( τ )) = x + τ α ( x ) , ∀ τ, τ ∈ [0 , . In particular, for τ = 0, one gets that γ τ , ( x ) = x (0) = x + τ α ( x )and therefore by (2.38) we obtain γ τ , ( x, ξ ) = (1 + τ α x ( x )) − ξ , which proves the claimed statement. 12ow, we are ready to state the Egorov Theorem. Theorem 2.14.
Let m ∈ R , V ( t ) = Op (cid:0) v ( t, x, ξ ) (cid:1) be in the class S m and Φ( τ ; t ) , τ ∈ [0 , be the flowmap of the PDE (3.3) . Then P ( τ ; t ) := Φ( τ ; t ) V ( t )Φ( τ ; t ) − is a pseudo differential operator in the class OP S m , i.e. P ( τ ; t ) = Φ( τ ; t ) V ( t )Φ( τ ; t ) − = Op (cid:0) p ( τ ; t, x, ξ ) (cid:1) with p ( τ, · , · , · ) ∈ S m , τ ∈ [0 , . Furthermore p ( τ ; t, x, ξ ) admits the expansion p ( τ ; t, x, ξ ) = p ( τ ; t, x, ξ ) + p ≥ ( τ ; t, x, ξ ) , p ( τ, · , · , · ) ∈ S m , p ≥ ( τ ; · , · , · ) ∈ S m − and the principal symbol p has the form p ( τ ; t, x, ξ ) := v (cid:16) t, x + τ α ( t, x ) , (1 + τ α x ( t, x )) − ξ (cid:17) , ∀ ( t, x, ξ ) ∈ R × T × R , ∀ τ ∈ [0 , . Proof.
We closely follow Theorem A.0.9 in [21]. A direct calculation shows that P ( τ ; t ) solves the Heisenbergequation ( ∂ τ P ( τ ; t ) = [ A ( τ ; t ) , P ( τ ; t )] P (0; t ) = V ( t ) . (2.39)We then look for a solution P ( τ ; t ) = Op (cid:16) p ( τ ; t, x, ξ ) (cid:17) ∈ OP S m with p ( τ ; t, x, ξ ) ∼ X n ≥ p n ( τ ; t, x, ξ ) , p n ∈ S m − n , ∀ n ≥ . We show how to compute the asymptotic expansion of the symbol p . The operator A ( τ ; t ) in (3.2) hassymbol a = i a + a ,a ( τ ; t, x, ξ ) := b α ( τ ; t, x ) ξ ∈ S , a ( τ ; t, x, ξ ) := ( ∂ x b α )( τ ; t, x )2 ∈ S . (2.40)The symbol of the commutator [ A ( τ ; t ) , P ( τ ; t )] = Op( a ⋆ p ) has the asymptotic expansion g ⋆ p ∼ X n ≥ a ⋆ p n . (2.41)Note that if p n ∈ S m − n , by (2.40) and Corollary 2.4, one has thati a ⋆ p n = { a , p n } + i r ( a , p n ) , { a , p n } ∈ S m − n , r ( a , p n ) ∈ S m − n − a ⋆ p n ∈ S m − n − . (2.42)This implies that a ⋆ p n = { a , p n } + q n , q n := i r ( a , p n ) + a ⋆ p n ∈ S m − n − . We then solve iteratively ( ∂ τ p = { a , p } p (0; t, x, ξ ) = v ( t, x, ξ ) (2.43)and ( ∂ τ p n ( τ ; t, x, ξ ) = { a , p n } + q n − p n (0; t, x, ξ ) = 0 , ∀ n ≥ . (2.44)Using the characteristic method, the solutions of (2.43), (2.44) are given by p ( τ ; t, x, ξ ) := v ( t, γ τ, ( t, x ) , γ τ, ( t, x, ξ )) , ∀ τ ∈ [0 ,
1] (2.45)and p n ( τ ; t, x, ξ ) = Z τ q n − ( ζ ; t, γ τ,ζ ( t, x ) , γ τ,ζ ( t, x, ξ )) dζ , ∀ n ≥ , (2.46)where for any τ, ζ ∈ [0 , γ τ,ζ , γ τ,ζ ) is the flow of the ODE (2.34). The claimed statement then follows byapplying Lemma 2.13 and by setting p ≥ ∼ P n ≥ p n ∈ OP S m − .13n the following we will also need to analyse the operator Φ( τ ; t ) ∂ t (Φ( τ ; t ) − ). The following lemmaholds: Theorem 2.15.
The operator Ψ( τ ; t ) = Φ( τ ; t ) ∂ t (Φ( τ ; t ) − ) , τ ∈ [0 , is a pseudo differential operator inthe class OP S .Proof. First we compute ∂ τ Ψ( τ ; t ). One has ∂ τ Ψ( τ ; t ) = ∂ τ Φ( τ ; t ) ∂ t (cid:0) Φ( τ ; t ) − (cid:1) + Φ( τ ; t ) ∂ t ∂ τ (cid:0) Φ( τ ; t ) − (cid:1) = ∂ τ Φ( τ ; t ) ∂ t (cid:0) Φ( τ ; t ) − (cid:1) − Φ( τ ; t ) ∂ t (cid:16) Φ( τ ; t ) − ∂ τ Φ( τ ; t )Φ( τ ; t ) − (cid:17) (3.3) = A ( τ ; t )Φ( τ ; t ) ∂ t Φ( τ ; t ) − − Φ( τ ; t ) ∂ t (cid:16) Φ( τ ; t ) − A ( τ ; t ) (cid:17) = [ A ( τ ; t ) , Ψ( τ ; t )] − ∂ t A ( τ ; t ) . (2.47)Therefore Ψ( τ ; t ) solves ( ∂ τ Ψ( τ ; t ) = [ A ( τ ; t ) , Ψ( τ ; t )] − ∂ t A ( τ ; t )Ψ(0; t ) = 0 . (2.48)Arguing as in Theorem 2.14, we find that Ψ( τ ; t ) = Op (cid:16) ψ ( τ ; t, x, ξ ) (cid:17) ∈ OP S , by solving (2.48) in decreasingorders and by determing an asymptotic expansion of the symbol ψ of the form ψ ∼ X n ≥ ψ n , ψ n ∈ S − n , ∀ n ≥ . We also state another semplified version of the Egorov theorem in which we conjugate a symbol by meansof the flow of a vector field which is a pseudo differential operator of order strictly smaller than one. Weconsider a pseudo differential operator G ( t ) = Op( g ( t, x, ξ )), with g ∈ S η , G ( t ) = G ( t ) ∗ , η < τ ∈ [0 , G ( τ ; t ) be the flow of the pseudo-PDE ∂ τ u = i G ( t ) u , (2.49)which is a well-defined invertible map by Lemma 2.9. Then Φ G ( τ ; t ) solves ( ∂ τ Φ G ( τ ; t ) = i G ( t )Φ G ( τ ; t )Φ G (0; t ) = Id . (2.50)The following theorem holds. Theorem 2.16.
Let m ∈ R , V ( t ) = Op (cid:0) v ( t, x, ξ ) (cid:1) ∈ OP S m and G ( t ) = Op( g ( t, x, ξ )) , with g ∈ S η , η < .Then for any τ ∈ [0 , , the operator P ( τ ; t ) := Φ G ( τ ; t ) V ( t )Φ G ( τ ; t ) − is a pseudo differential operator oforder m with symbol p ( τ ; · , · , · ) ∈ S m . The symbol p ( τ ; t, x, ξ ) admits the expansion p ( τ ; t, x, ξ ) = v ( t, x, ξ ) + τ { g, v } ( t, x, ξ ) + p ≥ ( τ ; t, x, ξ ) , p ≥ ( τ ; t, x, ξ ) ∈ S m − − η ) . (2.51) Proof.
We show how to compute the asymptotic expansion of the operator P ( τ ; t ) by taking advantage fromthe fact that the order of G ( t ) is strictly smaller than 1. A direct calculation shows that P ( τ ; t ) solves theHeisenberg equation ( ∂ τ P ( τ ; t ) = i[ G ( t ) , P ( τ ; t )] P (0; t ) = V ( t ) . (2.52)We then look for P ( τ ; t ) = Op (cid:16) p ( τ ; t, x, ξ ) (cid:17) ∈ OP S m with p ( τ ; t, x, ξ ) ∼ X n ≥ p n ( τ ; t, x, ξ ) , p n ( τ ; t, x, ξ ) ∈ S m − n (1 − η ) , ∀ n ≥ . G ( t ) , P ( τ ; t )] = Op( g ⋆ p ) has the asymptotic expansion g ⋆ p ∼ X n ≥ g ⋆ p n (2.53)Note that if p n ∈ S m − n (1 − η ) , by Corollary 2.4, one has that g ⋆p n ∈ S m − ( n +1)(1 − η ) . We then solve iteratively ( ∂ τ p ( τ ; t, x, ξ ) = 0 p (0; t, x, ξ ) = v ( t, x, ξ ) (2.54)and ( ∂ τ p n ( τ ; t, x, ξ ) = i g ⋆ p n − p n (0; t, x, ξ ) = 0 , ∀ n ≥ . (2.55)The solutions of (2.54), (2.55) are then given by p ( τ ; t, x, ξ ) := v ( t, x, ξ ) , ∀ τ ∈ [0 ,
1] (2.56)and p n ( τ ; t, x, ξ ) = i Z τ g ⋆ p n − ( ζ ; t, x, ξ ) dζ , ∀ n ≥ . (2.57)In order to determine the expansion (2.51), we analyze the symbol p . By (2.56), (2.57), one gets p ( τ ; t, x, ξ ) = i τ g ⋆ v ( t, x, ξ ) Corollary . τ { g, v } + i τ r ( g, v )and r ( g, v ) ∈ S m + η − ⊆ S m − − η ) therefore the expansion (2.51) is determined by taking p ≥ ( τ ; t, x, ξ ) ∼ i τ r ( g, v ) + X n ≥ p n ( τ ; t, x, ξ ) . V ( t ) In this section we develop the regularization procedure on the vector field i V ( t ) = i (cid:0) V ( t, x ) | D | M + W ( t ) (cid:1) ,see (1.2), which is needed to prove Theorem 1.4. In Section 3.1 we reduce to constant coefficients the highestorder V ( t, x ) | D | M , see Proposition 3.1. Then, in Section 3.2, we perform the reduction of the lower orderterms up to arbitrarily regularizing remainders, see Proposition 3.3. Our first aim is to eliminate the x -dependence from the highest order of the vector field i V ( t ), namely wewant to eliminate the x -dependence from the term V ( t, x ) | D | M . To this aim, let us consider a C ∞ function α : R × T → R (that will be fixed later) satisfying the following ansatz: α ∈ C ∞ b ( R × T , R ) , inf ( t,x ) ∈ R × T (cid:0) α x ( t, x ) (cid:1) > . (3.1)Then, we consider the non-autonomous transport equation ∂ τ u = A ( τ ; t, x, D )[ u ] , A ( τ ; t ) = A ( τ ; t, x, D ) := b α ( τ ; t, x ) ∂ x + ( ∂ x b α )( τ ; t, x )2 ,b α ( τ ; t, x ) := − α ( t, x )1 + τ α x ( t, x ) , τ ∈ [0 , . (3.2)15y Lemma 2.11, the flow Φ( τ ; t ), τ ∈ [0 ,
1] of the equation (3.2), i.e. ( ∂ τ Φ( τ ; t ) = A ( τ ; t )Φ( τ ; t )Φ(0; t ) = Id (3.3)is a well-defined, symplectic, invertible map H s → H s for any s ∈ R . We define Φ( t ) := Φ(1; t ). In order tostate the Proposition below, we introduce the constant¯ e := M − max { M − , , M − e } . (3.4)Note that, by the above definition and using that M >
1, it follows easily that¯ e > , M − ¯ e ≥ M − , , M − e . (3.5) Proposition 3.1.
The symplectic invertible map Φ( t ) = Φ(1; t ) , given by (3.3) , satisfies sup t ∈ R k Φ( t ) ± k B ( H s ) + sup t ∈ R k ∂ t Φ( t ) ± k B ( H s +1 ,H s ) < + ∞ , ∀ s ≥ . (3.6) There exist a function λ ∈ C ∞ b ( R , R ) satisfying inf t ∈ R λ ( t ) > and an operator W ( t ) = Op (cid:16) w ( t, x, ξ ) (cid:17) , w ∈ S M − ¯ e with W ( t ) = W ( t ) ∗ for any t ∈ R , such that (Φ − ) ∗ (i V )( t ) = i V ( t ) with V ( t ) := λ ( t ) | D | M + W ( t ) . (3.8)All the rest of this section is devoted to the proof of the proposition stated above. The property (3.6)follows by applying Lemma 2.9, using that A ( τ ; t ) ∈ OP S and using that, by a direct calculation, A ( τ ; t ) + A ( τ ; t ) ∗ = 0. The push-forward of the vector field i V ( t ) by means of the map Φ( t ) − is then given by i V ( t )with V ( t ) = Φ( t ) V ( t )Φ( t ) − + iΦ( t ) ∂ t (cid:0) Φ( t ) − (cid:1) . (3.9)By applying Lemmata 2.14, 2.15, one has that V ( t ) = Op( v ( t, x, ξ )) ∈ OP S M with v ( t, x, ξ ) = p ( t, x, ξ ) + p ≥ ( t, x, ξ ) , (3.10)with p ( t, x, ξ ) := v (cid:16) t, x + α ( t, x ) , (1 + α x ( t, x )) − ξ (cid:17) , p ≥ ∈ S max { M − , } (2.1) , (3.5) ⊆ S M − ¯ e . (3.11)In the next lemma we compute the expansion of the symbol v ( t, x, ξ ) of the operator V ( t ) defined in(3.9). Lemma 3.2.
The symbol v ( t, x, ξ ) has the form v ( t, x, ξ ) = h V ( t, y ) (cid:0) e α y ( t, y ) (cid:1) M i y = x + α ( t,x ) | ξ | M χ ( ξ ) + w ( t, x, ξ ) , w ∈ S M − ¯ e (3.12) where we recall the definitions (2.16) , (2.17) and y y + e α ( t, y ) is the inverse diffeomorphism of x x + α ( t, x ) . roof. Using (3.9)-(3.11), one obtains v ( t, x, ξ ) = p ( t, x, ξ ) + p ≥ ( t, x, ξ )= v (cid:16) t, x + α ( t, x ) , (1 + α x ( t, x )) − ξ (cid:17) + p ≥ ( t, x, ξ ) (3.13)By (1.2), the symbol of the operator V ( t ) has the form v ( t, x, ξ ) = V ( t, x ) | ξ | M χ ( ξ ) + w ( t, x, ξ ) , w ∈ S M − e , hence, by (3.11), one has p ( t, x, ξ ) = v (cid:16) t, x + α ( t, x ) , (1 + α x ( t, x )) − ξ (cid:17) = V ( t, x + α ( t, x ))(1 + α x ( t, x )) − M | ξ | M χ (cid:16) (1 + α x ( t, x )) − ξ (cid:17) + w p ( t, x, ξ ) (3.14)where w p ( t, x, ξ ) := w (cid:16) t, x + α ( t, x ) , (1 + α x ( t, x )) − ξ (cid:17) ∈ S M − e (3.5) ⊆ S M − ¯ e . (3.15)By using the mean value theorem, one writes χ (cid:16) (1 + α x ( t, x )) − ξ (cid:17) = χ ( ξ ) + w χ ( t, x, ξ ) ,w χ ( t, x, ξ ) := − α x ( t, x ) ξ α x ( t, x ) Z ∂ ξ χ (cid:16) ζ (1 + α x ( t, x )) − ξ + (1 − ζ ) ξ (cid:17) dζ . Since ∂ ξ χ ( ξ ) = 0 for any | ξ | ≥ w χ ∈ OP S −∞ , hence by (3.13), (3.14)one obtains that v ( t, x, ξ ) = V ( t, x + α ( t, x ))(1 + α x ( t, x )) − M | ξ | M χ ( ξ ) + w ( t, x, ξ )where w ( t, x, ξ ) := p ≥ ( t, x, ξ ) + w p ( t, x, ξ )+ V ( t, x + α ( t, x ))(1 + α x ( t, x )) − M | ξ | M w χ ( t, x, ξ ) . Recalling (3.11), (3.15) and that w χ ∈ S −∞ , one obtains that w ∈ S M − ¯ e . Since α satisfies (3.1), we canapply lemma 2.12, obtaining that the diffeomorphism of the torus x x + α ( t, x ) is invertible with inverse y y + e α ( t, y ) and e α ∈ C ∞ b ( R × T , R ) satisfies (2.30). Using the identity (2.31), we then have V ( t, x + α ( x ))(1 + α x ( x )) − M = h V ( t, y ) (cid:0) e α y ( y ) (cid:1) M i y = x + α ( t,x ) and the lemma is proved.We now determine the function e α ( t, y ) so that V ( t, y ) (cid:0) e α y ( t, y ) (cid:1) M = λ ( t ) , (3.16)for some bounded and real-valued C ∞ function λ , to be determined. The equation (3.16) is equivalent tothe equation e α y ( t, y ) = λ ( t ) M V ( t, y ) M − . (3.17)17otice that, by the assumption (H2) , V ( t, y ) does never vanish. We choose λ ( t ) so that the average of theright hand side of the equation (3.17) is 0, hence we set λ ( t ) := (cid:16) π Z T V ( t, y ) − M dy (cid:17) − M . (3.18)Therefore, we solve (3.17) by defining e α ( t, y ) := ∂ − y h λ ( t ) M V ( t, y ) M − i (3.19)(recall the definition (2.11)). Note that by the hypothesis (H2) on V and by the definitions (3.18), (3.19),one has λ ∈ C ∞ b ( R , R ), e α ∈ C ∞ b ( R × T , R ) andinf t ∈ R λ ( t ) > , inf ( t,y ) ∈ R × T (cid:16) e α y ( t, y ) (cid:17) > α satisfies the ansatz (3.1)since x x + α ( t, x ) is the inverse diffeomorphism of y y + e α ( t, y ).Finally, by lemma 3.2 and since e α and λ solve the equation (3.16), we obtain that V ( t ) is given by V ( t ) = λ ( t ) | D | M + W ( t ) , W ( t ) = Op (cid:0) w ( t, x, ξ ) (cid:1) ∈ OP S M − ¯ e . (3.20)Since Φ( t ) is symplectic, the vector field i V ( t ) is Hamiltonian, i.e. V ( t ) is L self-adjoint. Since λ ( t ) | D | M isselfadjoint, then W ( t ) = V ( t ) − λ ( t ) | D | M is self-adjoint too, hence the proof of Proposition 3.1 is concluded. In this Section we transform the vector field i V ( t ), obtained in Proposition 3.1, into another one which isan arbitrarily regularizing perturbation of a space-diagonal operator. This is done in the following Proposition 3.3.
Let N ∈ N . For any n = 1 , . . . , N there exists a linear Hamiltonian vector field i V n ( t ) ofthe form V n ( t ) := λ ( t ) | D | M + µ n ( t, D ) + W n ( t ) , (3.21) where µ n ( t, D ) := Op (cid:16) µ n ( t, ξ ) (cid:17) , µ n ∈ S M − ¯ e , (3.22) W n ( t ) := Op (cid:16) w n ( t, x, ξ ) (cid:17) , w n ∈ S M − n ¯ e , (3.23) with µ n ( t, ξ ) real and W n ( t ) L self-adjoint, i.e. w n = w ∗ n (see Theorem 2.5).For any n ∈ { , . . . , N − } , there exists a symplectic invertible map Φ n ( t ) satisfying sup t ∈ R k Φ n ( t ) ± k B ( H s ) + sup t ∈ R k ∂ t Φ n ( t ) ± k B ( H s +1 ,H s ) < + ∞ , ∀ s ≥ and i V n +1 ( t ) = (Φ − n ) ∗ (i V n )( t ) , ∀ n ∈ { , . . . , N − } . (3.25)The rest of the section is devoted to the proof of the above Proposition. It is proved arguing by induction.Let us describe the induction step. At the n -th step , we deal with a Hamiltonian vector field of the formi V n ( t ) which satisfies the properties (3.21)-(3.23). We look for an operator G n ( t ) of the form G n ( t ) := Op (cid:0) g n ( t, x, ξ ) (cid:1) ∈ OP S − n ¯ e with G n ( t ) = G n ( t ) ∗ (3.26)and we consider the flow Φ G n ( τ ; t ) of the pseudo PDE ∂ τ u = i G n ( t )[ u ] . (3.27)18he flow map Φ G n ( τ ; t ) solves ( ∂ τ Φ G n ( τ ; t ) = i G n ( t )Φ G n ( τ ; t )Φ G n (0; t ) = Id . (3.28)Note that, since G n ( t ) is self-adjoint, i G n ( t ) is a Hamiltonian vector field, implying that Φ G n ( τ ; t ) is symplecticfor any τ ∈ [0 , t ∈ R . Since G n ( t ) ∈ OP S − n ¯ e (2.1) ⊆ OP S and (i G n ( t )) + (i G n ( t )) ∗ = i (cid:0) G n ( t ) − G n ( t ) ∗ (cid:1) = 0,by Lemma 2.9, the maps Φ G n ( τ ; t ) ± satisfy the property (3.24). Note that, since the vector field G n ( t ) doesnot depend on τ , one has Φ G n ( τ ; t ) − = Φ G n ( − τ ; t ). We set Φ n ( t ) := Φ G n (1; t ). The transformed vector fieldis given by (Φ − n ) ∗ (i V n )( t ) = i V n +1 ( t ), where V n +1 ( t ) := Φ n ( t ) V n ( t )Φ n ( t ) − + iΦ n ( t ) ∂ t (cid:0) Φ n ( t ) − (cid:1) . (3.29)Since G n ( t ) is a pseudo-differential operator of order strictly smaller than 1, we can apply Theorem 2.16,obtaining that P n ( t ) = Op (cid:0) p n ( t, x, ξ ) (cid:1) := Φ n ( t ) V n ( t )Φ n ( t ) − ∈ OP S M with p n = v n + { g n , v n } + p n, ≥ , p n, ≥ ∈ S M − n ¯ e (2.1) ⊆ S M − ( n +1)¯ e . (3.30)Furthermore, defining Ψ n ( τ ; t ) := iΦ G n ( τ ; t ) ∂ t (cid:0) Φ G n ( τ ; t ) − (cid:1) , a direct calculation shows thatΨ n ( τ ; t ) = i Z τ S G n ( ζ ; t ) dζ , S G n ( ζ ; t ) := Φ G n ( ζ ; t ) ∂ t G n ( t )Φ G n ( ζ ; t ) − . Since ∂ t G n ( t ) ∈ OP S − n ¯ e , by Theorem 2.16Ψ n ( t ) ≡ Ψ n (1; t ) = iΦ n ( t ) ∂ t (cid:0) Φ n ( t ) − (cid:1) = Op (cid:16) ψ n ( t, x, ξ ) (cid:17) ∈ OP S − n ¯ e . Using that M − ¯ e ≥ n ( t ) = Op (cid:0) ψ n ( t, x, ξ ) (cid:1) ∈ OP S − n ¯ e (2.1) ⊆ OP S M − ( n +1)¯ e . (3.31)In the next lemma, we provide an expansion of the symbol v n +1 ( t, x, ξ ) of the operator V n +1 ( t ) given in(3.29). Lemma 3.4.
The operator V n +1 ( t ) = Op (cid:0) v n +1 ( t, x, ξ ) (cid:1) ∈ OP S M admits the expansion v n +1 ( t, x, ξ ) = λ ( t ) | ξ | M χ ( ξ ) + µ n ( t, ξ ) + w n ( t, x, ξ ) − M λ ( t ) | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ ) + r v n ( t, x, ξ ) (3.32) where r v n ∈ S M − ( n +1)¯ e .Proof. By (3.29)-(3.31), one has v n +1 = v n + { g n , v n } + p n, ≥ + ψ n . (3.33)Since, by the induction hypothesis, v n ( t, x, ξ ) = λ ( t ) | ξ | M χ ( ξ ) + µ n ( t, ξ ) + w n ( t, x, ξ ) , µ n ∈ S M − ¯ e , w n ∈ S M − n ¯ e one has that { g n , v n } = { g n , λ ( t ) | ξ | M χ ( ξ ) } + { g n , µ n } + { g n , w n } = − λ ( t ) ∂ ξ (cid:16) | ξ | M χ ( ξ ) (cid:17) ( ∂ x g n ) + { g n , µ n } + { g n , w n } = − M λ ( t ) | ξ | M − ξχ ( ξ )( ∂ x g n ) − λ ( t ) | ξ | M ( ∂ ξ χ ( ξ ))( ∂ x g n )+ { g n , µ n } + { g n , w n } . (3.34)19sing that ∂ ξ χ ( ξ ) = 0 for | ξ | ≥
1, since g n ∈ S − n ¯ e , µ n ∈ S M − ¯ e , w n ∈ S M − n ¯ e , by Corollary 2.4 one gets λ ( t ) | ξ | M ( ∂ ξ χ ( ξ ))( ∂ x g n ) ∈ S −∞ , { g n , µ n } ∈ S M − ( n +1)¯ e , (3.35) { g n , w n } ∈ S M − n ¯ e (2.1) ⊆ S M − ( n +1)¯ e . (3.36)Thus, (3.33), (3.34) imply the claimed expansion with r v n := − λ ( t ) | ξ | M ( ∂ ξ χ ( ξ ))( ∂ x g n ) + { g n , µ n } + { g n , w n } + p n, ≥ + ψ n . Finally, (3.30), (3.31), (3.35), (3.36) imply that r v n ∈ S M − ( n +1)¯ e . Choice of the symbol g n . In the next lemma, we show that the symbol g n can be chosen in order toeliminate the x -dependence from the term of order M − n ¯ e in the expansion (3.32). Lemma 3.5.
There exists a symbol g n ∈ S − n ¯ e , g n = g ∗ n , such that − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ ) + w n ( t, x, ξ ) − h w n i x ( t, ξ ) ∈ S M − ( n +1)¯ e (3.37) (recall the definition (2.12) ).Proof. Let χ ∈ C ∞ ( R , R ) be a cut-off function satisfying χ ( ξ ) = 1 , ∀| ξ | ≥ ,χ ( ξ ) = 0 , ∀| ξ | ≤ . (3.38)Writing 1 = χ + 1 − χ , one gets that − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ ) + w n ( t, x, ξ ) − h w n i x ( t, ξ )= − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ ) + χ ( ξ ) (cid:0) w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:1) + (cid:0) − χ ( ξ ) (cid:1)(cid:0) w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:1) . (3.39)By the definition of χ given in (3.38), one easily gets that (cid:0) − χ ( ξ ) (cid:1)(cid:0) w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:1) ∈ S −∞ , (3.40)therefore we look for a solution g n of the equation − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ ) + χ ( ξ ) (cid:0) w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:1) ∈ S M − ( n +1)¯ e . (3.41)Since we require that G n = Op( g n ) is self-adjoint, we look for a symbol of the form g n ( t, x, ξ ) = σ n ( t, x, ξ ) + σ ∗ n ( t, x, ξ ) ∈ S − n ¯ e (3.42)with the property that σ ∗ n ( t, x, ξ ) = σ n ( t, x, ξ ) + r n ( t, x, ξ ) , r n ∈ S − n ¯ e . (3.43)Plugging the ansatz (3.42) into the equation (3.41), using (3.43) and since − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x r n )( t, x, ξ ) ∈ S M − − n ¯ e ⊆ S M − ( n +1)¯ e , (3.44)we are led to solve the equation − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x σ n )( t, x, ξ ) + χ ( ξ ) (cid:0) w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:1) = 0 (3.45)whose solution is given by σ n ( t, x, ξ ) := χ ( ξ ) ∂ − x h w n ( t, x, ξ ) − h w n i x ( t, ξ ) i λ ( t ) M | ξ | M − ξ . (3.46)20ince w n , h w n i x ∈ S M − n ¯ e , using that M > χ in(3.38), one gets that σ n ∈ S − n ¯ e and hence also g n = σ n + σ ∗ n ∈ S − n ¯ e . We now use Lemma 2.6 with ϕ ( t, ξ ) = χ ( ξ )2 λ ( t ) M | ξ | M − ξ , a ( t, x, ξ ) = ∂ − x h w n ( t, x, ξ ) − h w n i x ( t, ξ ) i . Recalling that w n = w ∗ n , by Lemma 2.7we have that a = a ∗ , hence we can apply Lemma 2.6, obtaining that the ansatz (3.43) is satisfied. By (3.39),(3.42), (3.43), (3.45) one then gets − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ ) + w n ( t, x, ξ ) − h w n i x ( t, ξ )= (cid:0) − χ ( ξ ) (cid:1)(cid:0) w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:1) − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x r n )( t, x, ξ )and recalling (3.40), (3.44) one then gets (3.37).By Lemmata 3.4, 3.5, the operator V n +1 ( t ) has the form V n +1 ( t ) = λ ( t ) | D | M + µ n +1 ( t, D ) + W n +1 ( t ) , (3.47)where µ n +1 ( t, ξ ) := µ n ( t, ξ ) + h w n i x ( t, ξ ) ∈ S M − ¯ e , (3.48) W n +1 ( t ) := Op (cid:16) r v n ( t, x, ξ ) − λ ( t ) M | ξ | M − ξχ ( ξ )( ∂ x g n )( t, x, ξ )+ w n ( t, x, ξ ) − h w n i x ( t, ξ ) (cid:17) ∈ OP S M − ( n +1)¯ e . (3.49)Since by the induction hypothesis w n = w ∗ n and µ n ( t, ξ ) is real, by (2.10) and Lemma 2.7-( i ), one hasthat h w n i x ( t, ξ ) is real and therefore µ n +1 ( t, ξ ) is real. Furthermore, since Φ n is symplectic and i V n is aHamiltonian vector field, one has that i V n +1 is still a Hamiltonian vector field, meaning that V n +1 is self-adjoint. Using that µ n +1 ( t, ξ ) is a real Fourier multiplier, one has that λ ( t ) | D | M + µ n +1 ( t, D ) is a self-adjointoperator, implying that W n +1 ( t ) = V n +1 ( t ) − λ ( t ) | D | M − µ n +1 ( t, D )is self-adjoint too. Then, the proof of Proposition 3.3 is concluded. Let K ∈ N and let us fix a positive integer N K ∈ N as N K := h M + K ¯ e i + 1 (4.1)so that M − N K ¯ e < − K (for any x ∈ R , we denote by [ x ] its integer part). Then we define T K ( t ) := Φ( t ) − ◦ Φ ( t ) − ◦ . . . ◦ Φ N K − ( t ) − , W K ( t ) := W N K ( t ) , λ K ( t, D ) := λ ( t ) | D | M + µ N K ( t, D ) (4.2)where Φ( t ) = Φ(1; t ) is given by (3.3), λ ( t ) is defined in (3.18) and for any n ∈ { , . . . , N K − } , Φ n ( t ), W n ( t ), µ n ( t, D ) are given in Theorem 3.3. By (3.6), (3.24), using the product rule, one gets that T K satisfiesthe property (1.14). Furthermore, by (3.8), (3.21), (3.25) one obtains (1.15), with λ K ( t, D ), W K ( t ) definedin (4.2), hence the proof of Theorem 1.4 is concluded. Let s > t ∈ R , u ∈ H s ( T ). We fix the constant K ∈ N , appearing in Theorem 1.4, as K = K s := [ s ] + 1 (5.1)21o that K > s . By applying Theorem 1.4, one has that u ( t ) is a solution of the Cauchy problem ( ∂ t u + i V ( t )[ u ] = 0 u ( t ) = u (5.2)if and only if v ( t ) := T − K s ( t ) u ( t ) is a solution of the Cauchy problem ( ∂ t v + i λ K s ( t, D ) v + i W K s ( t )[ v ] = 0 v ( t ) = v , v := T − K s ( t )[ u ] (5.3)with λ K s ( t, D ) = Op( λ K s ( t, ξ )) ∈ OP S M with λ K s ( t, ξ ) = λ K s ( t, ξ ). Since the symbol λ K s ( t, ξ ) is real, wehave λ K s ( t, D ) = λ K s ( t, D ) ∗ . (5.4)Moreover, since K s > s >
0, by (2.1), one has W K s ( t ) ∈ OP S − K s ⊂ OP S − s ⊂ OP S , W K s ( t ) = W K s ( t ) ∗ . (5.5)By applying Lemma 2.23 one gets that there exists a unique global solution v ∈ C ( R , H s ) of the Cauchyproblem (5.3), therefore u ∈ C ( R , H s ) is the unique solution of the Cauchy problem (5.2). In order toconclude the proof, it remains only to prove the bound (1.13). Estimate of v ( t ) . By a standard energy estimate, using (5.4), (5.5), one gets easily that k v ( t ) k L = k v k L , ∀ t ∈ R . (5.6)Writing the Duhamel formula for the Cauchy problem (5.3), one obtains v ( t ) = e iΛ Ks ( t,D ) v + Z tt e i (cid:0) Λ Ks ( t,D ) − Λ Ks ( τ,D ) (cid:1) W K s ( τ )[ v ( τ )] dτ (5.7)where Λ K s ( t, D ) := Op (cid:16) Λ K s ( t, ξ ) (cid:17) , Λ K s ( t, ξ ) := Z tt λ K s ( τ, ξ ) dτ . Since λ K s ( t, ξ ) is real, Λ K s ( t, ξ ) is real too, and therefore the propagator e iΛ Ks ( t,D ) is unitary on H s ( T ).Hence, one has k v ( t ) k H s ≤ k v k H s + (cid:12)(cid:12)(cid:12) Z tt kW K s ( τ )[ v ( τ )] k H s dτ (cid:12)(cid:12)(cid:12) (5.5) , T heorem . , . s k v k H s + (cid:12)(cid:12)(cid:12) Z tt k v ( τ ) k L dτ (cid:12)(cid:12)(cid:12) (5.6) . s k v k H s + | t − t |k v k L . (5.8) Estimate of u ( t ) . Since u ( t ) = T K s ( t )[ v ( t )] and v = T K s ( t ) − [ u ] and T K s ( t ) ± satisfy (1.14), one getsthat k u ( t ) k H s ≃ s k v ( t ) k H s , k u k H s ≃ s k v k H s , k v k L ≃ k u k L , therefore, by (5.8) one deduce that k u ( t ) k H s . s k u k H s + | t − t |k u k L . (5.9) Proof of (1.13) . The estimate (5.9) proves that the propagator U ( t , t ) of the PDE ∂ t u = i V ( t )[ u ], i.e. ( ∂ t U ( t , t ) = i V ( t ) U ( t , t ) U ( t , t ) = Id22atisfies kU ( t , t ) k B ( H s ) . s | t − t | , ∀ s > , ∀ t, t ∈ R . (5.10)Furthermore, since V ( t ) is self-adjoint, the L of the solutions is constant, namely kU ( t , t ) k B ( L ) = 1 , ∀ t, t ∈ R . (5.11)Hence, for any 0 < s < S , by applying Theorem 2.8, one gets that kU ( t , t ) k B ( H s ) ≤ kU ( t , t ) k S − sS B ( L ) kU ( t , t ) k sS B ( H S ) (5.10) , (5.11) . S (1 + | t − t | ) sS . (5.12)Then, for any ε >
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E-mail: [email protected]@math.uzh.ch