Shubin type Fourier integral operators and evolution equations
aa r X i v : . [ m a t h . A P ] M a r SHUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTIONEQUATIONS
MARCO CAPPIELLO, REN´E SCHULZ, AND PATRIK WAHLBERG
Abstract.
We study the Cauchy problem for an evolution equation of Schr¨odinger type. TheHamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudo-differential perturbation of negative order from Shubin’s class. We prove that the propagatoris a Fourier integral operator of Shubin type of order zero. Using results for such operatorsand corresponding Lagrangian distributions, we study the propagator and the solution, andderive phase space estimates for them. Introduction
In this article we study the propagator and solution to the Cauchy problem(CP) (cid:26) ∂ t u ( t, x ) + i ( q w ( x, D ) + p w ( x, D )) u ( t, x ) = 0 , t > , x ∈ R d ,u (0 , · ) = u ∈ S ′ ( R d ) , where q w ( x, D ) is the Weyl quantization of a real homogeneous quadratic form on T ∗ R d and p w ( x, D ) is a pseudodifferential perturbation operator with complex-valued Shubin type symbol p of negative order. Particular examples of interest are perturbations to the free Schr¨odingerequation and the quantum harmonic oscillator.The Shubin class Γ m , m ∈ R , introduced in [27], is defined as the space of all functions a ∈ C ∞ ( R d ) that satisfy estimates of the form | ∂ αx ∂ βξ a ( x, ξ ) | . (1 + | x | + | ξ | ) m −| α + β | , ( x, ξ ) ∈ R d , α, β ∈ N d . Differently from H¨ormander symbols, the elements of Γ m exhibit a symmetric behavior in thedecay with respect to x and ξ . An interesting example is the symbol a ( x, ξ ) = | x | + | ξ | ∈ Γ forthe harmonic oscillator operator. The theory of pseudodifferential operators with symbols in theShubin classes has been developed in [27] and widely applied to the study of several classes ofpartial differential equations, see e.g. [1–5, 14, 15, 18, 21, 23, 25, 26, 28]. Helffer and Robert [14, 15]introduced Fourier integral operators (FIOs) with Shubin type amplitudes and phase functionsthat are generalized quadratic. Similar oscillatory integrals have been considered by Asada andFujiwara [1], see also [2].Concerning the Cauchy problem (CP), the case when q ( x, ξ ) = | x | + | ξ | and p = 0 is sincelong well known, see e.g. [10, 12, 14]. More generally, in the unperturbed case p = 0, the solutionoperator to the equation (CP) is a metaplectic operator, see e.g. [10]. Namely it is the uniqueone-parameter continuous group of metaplectic operators µ t , associated with the Hamiltonianflow χ t = e tF of q and chosen such that µ = I , where F = J Q is the real 2 d × d matrix Mathematics Subject Classification.
Primary: 53S30. Secondary: 35S10, 35A22.
Key words and phrases.
Fourier integral operator, Schr¨odinger equation, semigroup, perturbation.R. Schulz gratefully acknowledges support of the project “Fourier Integral Operators, symplectic geometryand analysis on noncompact manifolds” received by the University of Turin in form of an “I@Unito” fellowshipas well as institutional support by the University of Hannover. determined by the symmetric matrix Q defining q , q ( x, ξ ) = h ( x, ξ ) , Q ( x, ξ ) i , and the symplecticmatrix(1.1) J = (cid:18) I d − I d (cid:19) . We consider now the problem (CP) under the presence of a non-vanishing perturbation p .Recently the problem has been studied in [8] assuming the symbol p belong to a weightedmodulation space of Sj¨ostrand type, whose elements are not necessarily smooth, see also [9, 30].The authors proved that the equation (CP) admits a propagator given by the composition of ametaplectic operator and a Weyl pseudodifferential operator with symbol in the same modulationspace as p .In this paper we prove a similar statement in a different setting, namely the following result. Theorem 1.1. If δ > and p ∈ Γ − δ then the Cauchy problem (CP) has a propagator of theform µ t a wt ( x, D ) where a t ∈ Γ for t > . With respect to [8] we assume more regularity on the symbol of the perturbation and weobtain a stronger conclusion on the regularity of a t . Moreover, the fact that a t is a Shubinsymbol allows us to obtain additional results in terms of propagation of singularities and phaseestimates for the solution. For this we take advantage of some recent results for a class of FIOswith quadratic phase functions and Shubin amplitudes, cf. [6,7]. In these papers we proved phasespace estimates for an FBI type transform of the kernels of the operators, see [28] for similarestimates in a particular case. We also proved that every operator in the class can be writtenas the composition of a metaplectic operator and a pseudodifferential operator with Shubinsymbol and vice versa. As a byproduct of the analysis we derived a new notion of Lagrangiandistributions in the Shubin framework which generalizes the properties of the kernels of FIOs.Under the assumptions of Theorem 1.1 the propagator of (CP) belongs to this class of FIOsfor each t >
0. This opens up the possibility to study the singularities of solutions to (CP) indetail, proving propagation results for Lagrangian type singularities and phase space estimatesfor the solution, see Theorems 4.5 and 4.9 below.The paper is organized as follows. In Section 2 we recall the technical tools for our analysis, inparticular aspects of pseudodifferential quantization, metaplectic and symplectic analysis, Shubintype FIOs, and properties of an FBI type phase space transform which is a fundamental tool.In Section 3 we construct a parametrix to (CP) and prove that the propagator is a Shubin typeFIO. Finally in Section 4 we study the singularities of propagators and solutions to (CP) anddeduce phase space estimates for them.2.
Preliminaries on microlocal analysis in Shubin’s class
Basic notation.
The gradient operator with respect to x ∈ R d is denoted ∇ x . The symbols S ( R d ) and S ′ ( R d ) denote the Schwartz space of rapidly decaying smooth functions and thetempered distributions, respectively. The notation f ( x ) . g ( x ) means f ( x ) Cg ( x ) for some C > x in the domain of f and of g . We write ( f, g ) for the sesquilinear pairing, conjugatelinear in the second argument, between a distribution f and a test function g , as well as the L scalar product if f, g ∈ L ( R d ). The linear pairing of a distribution f and a test function g is written h f, g i . The symbols T x u ( x ) = u ( x − x ) and M ξ u ( x ) = e i h x,ξ i u ( x ), where h· , ·i denotes the inner product on R d , are used for translation by x ∈ R d and modulation by ξ ∈ R d ,respectively, applied to functions or distributions. For x ∈ R d we use h x i := p | x | , andPeetre’s inequality is h x + y i s C s h x i s h y i | s | , x, y ∈ R d , C s > , s ∈ R . HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 3
We write ¯d x = (2 π ) − d d x for the dual Lebesgue measure, denote by M d × d ( R ) the space of d × d matrices with real entries, and by GL( d, R ) ⊆ M d × d ( R ) the group of invertible matrices.The orthogonal projection on a linear subspace Y ⊆ R d is denoted π Y . The symbol L ( H ) standsfor the space of linear continuous operators on a Hilbert space H . An integral transform of FBI type.
The following integral transform has been used exten-sively in [6, 7] and is used also in this article. For more information see [6].
Definition 2.1.
Let u ∈ S ′ ( R d ) and let g ∈ S ( R d ) \ { } . The transform u
7→ T g u is defined by T g u ( x, ξ ) = (2 π ) − d/ ( u, T x M ξ g ) , x, ξ ∈ R d . If u ∈ S ( R d ) then T g u ∈ S ( R d ) by [13, Theorem 11.2.5]. The adjoint T ∗ g is defined by( T ∗ g U, f ) = ( U, T g f ) for U ∈ S ′ ( R d ) and f ∈ S ( R d ). When U is a polynomially boundedmeasurable function we write T ∗ g U ( y ) = (2 π ) − d/ Z R d U ( x, ξ ) T x M ξ g ( y ) d x d ξ, where the integral is defined weakly so that ( T ∗ g U, f ) = ( U, T g f ) L for f ∈ S ( R d ). Proposition 2.2. [13, Theorem 11.2.3]
Let u ∈ S ′ ( R d ) and let g ∈ S ( R d ) \ . Then T g u ∈ C ∞ ( R d ) and there exists N ∈ N such that |T g u ( x, ξ ) | . h ( x, ξ ) i N , ( x, ξ ) ∈ R d . We have u ∈ S ( R d ) if and only if for any N > |T g u ( x, ξ ) | . h ( x, ξ ) i − N , ( x, ξ ) ∈ R d . The transform T g is related to the short-time Fourier transform [13] V g u ( x, ξ ) = (2 π ) − d/ ( u, M ξ T x g ) , x, ξ ∈ R d , viz. T g u ( x, ξ ) = e i h x,ξ i V g u ( x, ξ ). If g, h ∈ S ( R d ) then T ∗ h T g u = ( h, g ) u, u ∈ S ′ ( R d ) , and thus k g k − L T ∗ g T g u = u for u ∈ S ′ ( R d ) and g ∈ S ( R d ) \
0, cf. [13].Finally we recall the definition of the Gabor wave front set which describes global singularitiesof tempered distributions in phase space, cf. [18, 25, 26].
Definition 2.3. If u ∈ S ′ ( R d ) and g ∈ S ( R d ) \ z ∈ T ∗ R d \ z / ∈ WF( u )if there exists an open cone V ⊆ T ∗ R d \ z , such that for any N ∈ N there exists C V,g,N > |T g u ( z ) | C V,g,N h z i − N when z ∈ V .The Gabor wave front set is hence a closed conic subset of T ∗ R d \
0. If u ∈ S ′ ( R d ) then W F ( u ) = ∅ if and only if u ∈ S ( R d ) [18, Proposition 2.4]. Weyl pseudodifferential operators.
We use pseudodifferential operators in the Weyl calculuswith Shubin amplitudes [21, 27]. Recall that a ∈ C ∞ ( R N × R N ) is a Shubin amplitude of order m ∈ R , denoted a ∈ Γ m ( R N × R N ), if it satisfies the estimates(2.1) | ∂ αx ∂ βξ a ( x, ξ ) | . h ( x, ξ ) i m −| α + β | , ( α, β ) ∈ N N × N N , ( x, ξ ) ∈ R N × R N . We write Γ m = Γ m ( R d ) and observe that T m ∈ R Γ m = S ( R d ). The space Γ m is a Fr´echet spacewith respect to the seminorms that are the best constants hidden in (2.1). MARCO CAPPIELLO, REN´E SCHULZ, AND PATRIK WAHLBERG
To a Shubin amplitude a ∈ Γ m one associates its pseudodifferential Weyl quantization, whichis the operator a w ( x, D ) with Schwartz kernel(2.2) K a ( x, y ) = Z R d e i h x − y,ξ i a (( x + y ) / , ξ ) ¯d ξ ∈ S ′ ( R d )interpreted as an oscillatory integral. Then a w ( x, D ) is a continuous operator on S ( R d ) thatextends uniquely to a continuous operator on S ′ ( R d ). If a ∈ S ( R d ) then a w ( x, D ) : S ′ ( R d ) → S ( R d ) is continuous when S ′ ( R d ) is equipped with its strong topology. Conversely, any con-tinuous linear operator from S ′ ( R d ), endowed with the strong topology, to S ( R d ) may berepresented as a w ( x, D ) for some a ∈ S ( R d ) [29].For a ∈ S ′ ( R d ) and f, g ∈ S ( R d ) we have(2.3) ( a w ( x, D ) f, g ) = (2 π ) − d/ ( a, W ( g, f ))where W ( g, f ) is the Wigner distribution [10, 13] W ( g, f )( x, ξ ) = (2 π ) − d/ Z R d g ( x + y/ f ( x − y/ e − i h y,ξ i d y ∈ S ( R d ) . The Weyl product a b : Γ m × Γ m → Γ m + m is the continuous product (cf. [27]) on thesymbol level corresponding to composition of operators:( a b ) w ( x, D ) = a w ( x, D ) b w ( x, D ) . There is a scale of Sobolev spaces Q s ( R d ), s ∈ R , defined by Q s ( R d ) = { u ∈ S ′ ( R d ) : v ws ( x, D ) u ∈ L ( R d ) } , where v s ( x, ξ ) = h ( x, ξ ) i s , which is adapted to the Shubin calculus. We have (cf. [27, Corol-lary 25.2])(2.4) S ( R d ) = \ s ∈ R Q s ( R d ) , S ′ ( R d ) = [ s ∈ R Q s ( R d ) . The Weyl quantization of Γ m yields continuous maps(2.5) a w ( x, D ) : Q s ( R d ) → Q s − m ( R d ) , s ∈ R , and the Q s → Q s − m operator norm of a w ( x, D ) can be estimated by a finite linear combinationof seminorms of a ∈ Γ m .We use the description of Q s in terms of localization operators [21, Proposition 1.7.12]. Let ψ = π − d/ e −| x | / , x ∈ R d . A localization operator A a with symbol a ∈ S ′ ( R d ) is defined by( A a u, f ) = ( a T ψ u, T ψ f ) , u, f ∈ S ( R d ) . In terms of the localization operator A s := A v s , the space Q s ( R d ) is the Hilbert modulationspace of all u ∈ S ′ ( R d ) such that A s u ∈ L ( R d ), equipped with the norm k u k Q s = k A s u k L .It is possible to express localization operators as pseudodifferential operators (cf. [21, Sec-tion 1.7.2]) writing A a = b w ( x, D ) where(2.6) b = π − d e −|·| ∗ a. HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 5
Metaplectic operators.
We view T ∗ R d ∼ = R d × R d as a symplectic vector space equipped withthe canonical symplectic form(2.7) σ (( x, ξ ) , ( x ′ , ξ ′ )) = h x ′ , ξ i − h x, ξ ′ i , ( x, ξ ) , ( x ′ , ξ ′ ) ∈ T ∗ R d . The real symplectic group Sp( d, R ) ⊆ GL(2 d, R ) is the set of matrices that leaves σ invariant.An often occurring symplectic matrix is J ∈
Sp( d, R ) defined in (1.1).The metaplectic group [12, 20] Mp( d ) is a group of unitary operators on L ( R d ), which is a(connected) double covering of the symplectic group Sp( d, R ). In fact the two-to-one projection π : Mp( d ) → Sp( d, R ) has kernel is ± I . Each operator µ ∈ Mp( d ) is a homeomorphism on S and on S ′ . The metaplectic covariance of the Weyl calculus reads(2.8) µ − a w ( x, D ) µ = ( a ◦ χ µ ) w ( x, D ) , a ∈ S ′ ( R d ) , where µ ∈ Mp( d ) and χ µ = π ( µ ) (cf. [12, Theorem 215], [10]). Fourier integral operators with Shubin amplitudes.
In [7] we have introduced a class ofFourier integral operators (FIOs) with quadratic phase functions and Shubin amplitudes. Thespace of Shubin type FIOs of order m ∈ R associated with χ ∈ Sp( d, R ), denoted I m ( χ ), consistsof those operators K whose kernels admit oscillatory integral representations of the form K a,ϕ ( x, y ) = Z R N e iϕ ( x,y,θ ) a ( x, y, θ ) d θ, ( x, y ) ∈ R d , where a ∈ Γ m ( R d × R N ). The phase function ϕ is a real quadratic form on R d + N whichparametrizes the twisted graph Lagrangian(2.9) Λ ′ χ = { ( x, y, ξ, − η ) ∈ T ∗ R d : ( x, ξ ) = χ ( y, η ) } ⊆ T ∗ R d corresponding to χ ∈ Sp( d, R ). (Cf. [7, Definitions 3.5 and 4.1].) We will not use this represen-tation here but merely recall the following result, see also [8, Theorem 1.3] for a related resultwhere certain modulation spaces are used as amplitudes. Theorem 2.4. [7, Theorem 4.15] If χ ∈ Sp( d, R ) and K ∈ I m ( χ ) then there exist b ∈ Γ m such that for any µ ∈ Mp( d ) such that χ = π ( µ ) K = b w ( x, D ) µ = µ ( b ◦ χ ) w ( x, D ) . Conversely, for any b ∈ Γ m we have b w ( x, D ) µ ∈ I m ( χ ) . This means that FIOs in I m ( χ ) admit factorization into a pseudodifferential operator and ametaplectic operator corresponding to χ . The factorization is uniquely determined by the orderof arrangement. In particular I m ( I ), where I ∈ GL(2 d, R ) is the identity matrix, is the spaceof pseudodifferential operators with Shubin amplitudes of order m ∈ R . A kernel of the form K ,ϕ , i.e. trivial amplitude, corresponds to the operator C ϕ µ where χ = π ( µ ) and C ϕ ∈ C \ Theorem 2.5. [7, Proposition 4.10]
Let χ j ∈ Sp( d, R ) and suppose K j ∈ I m j ( χ j ) , for j = 1 , .Then K K ∈ I m + m ( χ χ ) . We state the mapping properties of FIOs with respect to the Shubin–Sobolev spaces Q s andthe Gabor wave front set respectively. Proposition 2.6. [7, Proposition 4.16 and Corollary 5.4]
Suppose χ ∈ Sp( d, R ) and K ∈ I m ( χ ) . Then K : Q s ( R d ) → Q s − m ( R d ) is continuous for all s ∈ R . For all u ∈ S ′ ( R d ) wehave WF( K u ) ⊆ χ WF( u ) . MARCO CAPPIELLO, REN´E SCHULZ, AND PATRIK WAHLBERG Parametrix and propagator
Consider the initial value Cauchy problem associated with a real homogeneous quadratic form q ∈ Γ defined by q ( x, ξ ) = h ( x, ξ ) , Q ( x, ξ ) i where ( x, ξ ) ∈ R d and Q ∈ M d × d ( R ) is symmetric,and a negative order complex-valued perturbation p ∈ Γ − δ where δ > (cid:26) ∂ t u ( t, x ) + i ( q w ( x, D ) + p w ( x, D )) u ( t, x ) = 0 , t > , x ∈ R d ,u (0 , · ) = u ∈ S ′ ( R d ) . The free evolution.
We first discuss the solution operator (propagator) in the unperturbedcase p = 0. First we treat the propagator as a group on L ( R d ), then on Q s ( R d ).Thus we consider q w ( x, D ) as an unbounded operator in L ( R d ). The closure of − iq w ( x, D )equipped with the domain S equals its maximal realization, denoted M q [19, pp. 425–26]. Theclosure generates a strongly continuous group R ∋ t e − itq w ( x,D ) of unitary operators on L . The group gives the unique solution e − itq w ( x,D ) u ∈ C ([0 , ∞ ) , L ) ∩ C ((0 , ∞ ) , L ) for u ∈ D ( M q ) ⊆ L , see [22, Theorem 4.1.3].The propagator is a time-parametrized group of metaplectic operators, given for t ∈ R by(3.1) e − itq w ( x,D ) = µ t ∈ Mp( d ) , t ∈ R (see e.g. [7, 8, 10, 12, 23]). In fact consider the one-parameter group of symplectic matrices χ t = e tF ∈ Sp( d, R ) where F = J Q ∈ M d × d ( R ). By the unique path lifting theorem (cf. [12,Corollary 355]), there is a unique continuous lifting of R ∋ t χ t ∈ Sp( d, R ) into R ∋ t µ t ∈ Mp( d ) such that π ( µ t ) = χ t for t ∈ R and µ = I . By [12, Corollary 355] µ t satisfies (3.1). Remark . Williamson’s symplectic diagonalization theorem (see e.g. [12, Theorem 93]) impliesthat if Q ∈ M d × d ( R ) is (strictly) positive definite then there exists a matrix χ ∈ Sp( d, R ) suchthat χ t Qχ = (cid:18) Λ 00 Λ (cid:19) where Λ = diag( λ , · · · , λ d ) with λ j , j = 1 , . . . , d positive numbers such that {± iλ j } dj =1 areeigenvalues of F . This gives( q ◦ χ )( x, ξ ) = h χ ( x, ξ ) , Qχ ( x, ξ ) i = d X j =1 λ j ( x j + ξ j )and thus(3.2) ( q ◦ χ ) w ( x, D ) = d X j =1 λ j ( x j − ∂ j )which is a weighted sum of one-dimensional harmonic oscillators. Picking µ ∈ Mp( d ) such that χ = π ( µ ), and conjugating the equation ∂ t u ( t, x ) + iq w ( x, D ) u ( t, x ) = 0 in the x variable with µ using (2.8), leads to the modified Hamiltonian (3.2). We will not pursue this direction, however,since our main concern is to express our results using χ t ∈ Sp( d, R ), and the relation between χ and χ t is not transparent.Next we fix s ∈ R and consider q w ( x, D ) as an unbounded operator in Q s ( R d ). In this case µ t is in general no longer unitary but we still have the following result. Proposition 3.2.
For s ∈ R the group R ∋ t µ t is a strongly continuous group of operatorson Q s ( R d ) whose generator is a closed extension of − iq w ( x, D ) , considered as an unboundedoperator in Q s ( R d ) with domain S ( R d ) . HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 7
Proof.
By [12, Prop. 400], µ t is for fixed t ∈ R a homeomorphism on Q s . First we prove auniform bound for k µ t k L ( Q s ) over − T t T where T > A s = a w ( x, D ) where a ( z ) = π − d ( e −|·| ∗ v s )( z ) with z ∈ R d . This implies a ∈ Γ s and a is elliptic by [26, Proposition 2.3]. Since k µ t f k L = k f k L for all t ∈ R and f ∈ L we obtain for u ∈ Q s using (2.8) k µ t u k Q s = k A s µ t u k L = k µ t µ − t a w ( x, D ) µ t u k L = k ( a ◦ χ t ) w ( x, D ) u k L k ( a ◦ χ t ) w ( x, D ) A − s k L ( L ) k A s u k L = k ( a ◦ χ t ) w ( x, D ) A − s k L ( L ) k u k Q s . Indeed, by [21, Proposition 1.7.12] the inverse of A s exists and A − s = b w ( x, D ) where b ∈ Γ − s .We have(3.3) | ∂ α ( a ◦ χ t )( z ) | C α e | t | k F k ( | s | +2 | α | ) h z i s −| α | , z ∈ R d , α ∈ N d . The set of symbols a ◦ χ t ∈ Γ s is thus uniformly bounded over t ∈ [ − T, T ]. Hence ( a ◦ χ t ) b ∈ Γ is uniformly bounded over t ∈ [ − T, T ]. By the Calder´on–Vaillancourt theorem (see e.g. [10,Theorem 2.73]) k ( a ◦ χ t ) w ( x, D ) A − s k L ( L ) < ∞ uniformly over t ∈ [ − T, T ]. We have shown(3.4) sup | t | T k µ t k L ( Q s ) < ∞ , s ∈ R . Next let f ∈ S and write as above A s ( µ t − I ) f = µ t ( a ◦ χ t ) w ( x, D ) f − a w ( x, D ) f = µ t ( a ◦ χ t − a ) w ( x, D ) f + ( µ t − I ) a w ( x, D ) f. Since µ t is unitary on L we obtain for | t | k ( µ t − I ) f k Q s = k A s ( µ t − I ) f k L k ( a ◦ χ t − a ) w ( x, D ) f k L + k ( µ t − I ) a w ( x, D ) f k L . We have a ◦ χ t − a → s + ν as t → ν >
0. To wit, this follows from the proofof [17, Proposition 18.1.2] modified from H¨ormander to Shubin symbols. This giveslim t → k ( a ◦ χ t − a ) w ( x, D ) f k L k f k Q s + ν = 0 . Combining with the known strong continuity of µ t on L we have shownlim t → k ( µ t − I ) f k Q s = 0 , f ∈ S ( R d ) . Finally, combining (3.4) with the fact that S is dense in Q s we can use [11, Proposition I.5.3]to conclude that µ t is a strongly continuous group on Q s .Consider finally the final statement of the proposition: The generator of the group µ t actingon Q s is a closed extension of − iq w ( x, D ), considered as an unbounded operator on Q s ( R d ) withdomain S ( R d ). This claim is a consequence of [22, Theorem 1.2.4 and Corollary 1.2.5]. (cid:3) Both sides of the equality (3.1) may thus be interpreted as a strongly continuous group on Q s .The operators µ t are not necessarily unitary if s = 0. Due to (2.4) we may allow u ∈ S ′ ( R d ). Infact for some s ∈ R we then have u ∈ Q s +2 . The group µ t acting on Q s has a closed generatorthat is an extension of − iq w ( x, D ) considered an unbounded operator in Q s ( R d ) with domain S ( R d ). Abusing notation we denote also the generator by − iq w ( x, D ). It follows from [22, MARCO CAPPIELLO, REN´E SCHULZ, AND PATRIK WAHLBERG
Definition 1.1.1] and (2.5) that Q s +2 ⊆ D ( q w ( x, D )). Again [22, Theorem 4.1.3] implies that µ t u is the unique solution to (CP) in C ([0 , ∞ ) , Q s ) ∩ C ((0 , ∞ ) , Q s ). We summarize: Proposition 3.3.
For s ∈ R the equation (CP) with p = 0 is solved uniquely by the stronglycontinuous group of operators e − itq w ( x,D ) = µ t on Q s ( R d ) , and for each t ∈ R it is an FIO in I ( χ t ) . We have for u ∈ Q s +2 the unique solution e − itq w ( x,D ) u ∈ C ([0 , ∞ ) , Q s ) ∩ C ((0 , ∞ ) , Q s ) . Construction of a parametrix to the perturbed equation.
We will now consider(CP) with a nonzero complex-valued perturbation p ∈ Γ − δ . As a first step we note that theperturbation operator is bounded p w ( x, D ) : Q s ( R d ) → Q s + δ ( R d ) and compact p w ( x, D ) : Q s ( R d ) → Q s ( R d ) [27, Proposition 25.4]. Perturbation theory (see e.g. [8], [11, Theorems III.1.3and III.1.10]) gives the following conclusion.Let s ∈ R . The solution to (CP) for u ∈ Q s +2 ( R d ) is T t u where(3.5) T t = µ t C t , t > . Here C t is a strongly continuous semigroup of operators on Q s ( R d ) with operator norm estimate(3.6) k C t k L ( Q s ) M e t ( ω + M k p w ( x,D ) k L ( Qs ) ) , t > , where M > ω >
0, and C t = id + ∞ X n =1 ( − i ) n Z t Z t · · · Z t n − P t · · · P t n d t n · · · d t with convergence in the L ( Q s ) norm. In this formula P t = p wt ( x, D ) = ( p ◦ χ t ) w ( x, D ). Theintegrals are Bochner integrals of operator-valued functions. The propagator (3.5) is a stronglycontinuous semigroup of operators on Q s .By [13, Corollary 11.2.6 and Lemma 11.3.3] the Q s norms for s > S ( R d ). Thus C t : S → S is continuous.We show that C t is a pseudodifferential operator. First we use results in [8] to prove that C t has a pseudodifferential operator symbol in a space larger than Γ . By [16, Remark 2.18] wehave Γ − δ ⊆ Γ = \ s > M ∞ , ⊗ v s ( R d )where M ∞ , ⊗ v s denotes a Sj¨ostrand modulation space [13] with the weight v s ( z ), z ∈ R d , and whereΓ denotes the space of smooth symbols whose derivatives are in L ∞ . From [8, Theorem 4.1] itfollows that C t = c wt ( x, D ) where(3.7) c t ∈ \ s > M ∞ , ⊗ v s = Γ , t > . By duality c wt ( x, D ) extends uniquely to a continuous operator on S ′ ( R d ).The outcome of this argument is that the propagator (3.5) is of the form(3.8) T t = µ t c wt ( x, D ) , t > . If u ∈ S ′ ( R d ) then u ∈ Q s +2 for some s ∈ R . Again, by [22, Theorem 4.1.3], T t u is theunique solution to (CP) in C ([0 , ∞ ) , Q s ) ∩ C ((0 , ∞ ) , Q s ).Our objective is to improve (3.7) into(3.9) c t ∈ Γ , t > , which implies that the propagator T t is an FIO of order zero for all t >
0. This improvementwill prove Theorem 1.1.
HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 9
The strategy to prove (3.9) is as follows. We first construct an FIO parametrix { K t } t > tothe equation (CP), that is a family of operators { K t } t > , where K t ∈ I ( χ t ) for t >
0, whichsatisfies(3.10) (cid:26) ∂ t K t u + i ( q w ( x, D ) + p w ( x, D )) K t u = g ( t ) , t > , K u = u , u ∈ S ′ ( R d ) , for a function g ∈ C ([0 , ∞ ) , S ( R d )). (The function g will turn out to depend on u .) We thenprove that K t − T t = R t is regularizing, which implies that T t = K t − R t ∈ I ( χ t ) is an FIO.Thus we start by proving the following result. Theorem 3.4.
The Cauchy problem (CP) admits an FIO parametrix K t ∈ I ( χ t ) for t > such that K = I . The proof is carried out in several steps.
Lemma 3.5.
Let
T > and n > . The family of Weyl symbols p t · · · p t n ∈ Γ − δn , t j ∈ [0 , T ] , j n, is uniformly bounded in Γ − δn , and [0 , T ] n ∋ ( t , t , . . . , t n ) p t · · · p t n ∈ Γ − ( δ − ν ) n is contin-uous for any ν > .Proof. We have(3.11) | ∂ αz p t ( z ) | C α e t k F k ( δ +2 | α | ) h z i − δ −| α | , z ∈ R d , α ∈ N d , which proves that p t ∈ Γ − δ uniformly over t ∈ [0 , T ]. The estimates (3.11) combined with thecontinuity of t ∂ αz p t ( z ) also show that(3.12) p t ∈ C ([0 , T ] , Γ − δ + ν )if ν >
0. The result is thus a consequence of the continuity of the Weyl product on the spacesΓ m (see e.g. [21, Theorem 1.2.16]). (cid:3) Fix
T >
0. For t ∈ [0 , T ] we set b t, = 1, and for n > b t,n = ( − i ) n Z t Z t · · · Z t n − p t · · · p t n d t n · · · d t . In particular, b ,n = 0 for n >
1. By Lemma 3.5, b t,n makes sense as an integral and b t,n ∈ Γ − δn .From (2.3) it follows that integration commutes with the Weyl product so that b wt,n ( x, D ) = ( − i ) n Z t Z t · · · Z t n − P t · · · P t n d t n · · · d t , n > . Using (3.11), the recursion(3.14) b t,n = − i Z t p t b t ,n − d t , n > , and induction, one shows(3.15) t b t,n ∈ C ([0 , T ] , Γ − δn ) , n > . Hence(3.16) | ∂ αz b t,n ( z ) | C α h z i − δn −| α | , α ∈ N d , t T. We also have for n > | ∂ αz ∂ t b t,n ( z ) | C α h z i − δn −| α | , α ∈ N d , t T. In fact (3.14) gives for n > t > ∂ t b t,n = − i p t b t,n − . Hence (3.12) and (3.15) show that(3.19) t ∂ t b t,n ∈ C ([0 , T ] , Γ − δn + ν )provided ν >
0. (Note that the continuity extends to include the end points of the interval[0 , T ].) Lemma 3.5 and (3.16) imply that ∂ t b t,n ∈ Γ − δn uniformly over t ∈ [0 , T ]. This proves(3.17). By differentiating (3.18) and using ∂ t ( p t ( z )) = h ( ∇ p )( χ t z ) , F χ t z i ∈ Γ − δ we also obtain(3.20) ∂ t b t,n ∈ Γ − δn uniformly over t ∈ [0 , T ] for all n > b t,n asymptotically. Lemma 3.6.
Let
T > and a t,n ∈ C ([0 , T ] , Γ m n ) where ( m n ) n ∈ N is a decreasing sequencetending to −∞ as n → ∞ . Assume that ∂ t a t,n ∈ Γ m n uniformly over t ∈ [0 , T ] for every n ∈ N . Then there exists a symbol function t a t ∈ C ([0 , T ] , Γ m ) such that for every N ∈ N \ wehave a t − N X n =0 a t,n ∈ C ([0 , T ] , Γ m N +1 ) . We write a t ∼ ∞ X n =0 a t,n . The proof is a variant of the proof of [27, Proposition 3.5] and other similar results for symbolsdepending on a parameter. The essential modification of the standard proof to obtain continuityis to use | ∂ αz ( a t + s,n ( z ) − a t,n ( z )) | | s | sup θ | ∂ αz ∂ t a t + θs,n ( z ) | . We omit further details except for the statement that the symbol a t is constructed as(3.21) a t = ∞ X n =0 ψ n a t,n where ψ ( z ) = 1, and for n > ψ n ( z ) = ψ ( z/r n ) where ψ ∈ C ∞ ( R d ), ψ ( z ) = 0 for | z | ψ ( z ) = 1 for | z | >
2, and ( r n ) n > a sufficiently rapidly increasing sequence of positive reals. Thisshows that we can extend Lemma 3.6 to take into account also higher order derivatives withrespect to t by modifying the constants r n .Applying Lemma 3.6 to the sequences ( b t,n ) n > defined in (3.13) and ( ∂ t b t,n ) n > simul-taneously, and using (3.15), (3.16), (3.17), (3.19) and (3.20) we obtain b t ∈ C ([0 , T ] , Γ ) ∩ C ([0 , T ] , Γ − δ ) such that b t ∼ ∞ X n =0 b t,n , ∂ t b t ∼ ∞ X n =1 ∂ t b t,n . Note that b w ( x, D ) = I . Lemma 3.7.
For
T > we have r t = ∂ t b t + ip t b t ∈ C ([0 , T ] , S ( R d )) . HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 11
Proof.
Let N ∈ N \
0. By (3.18) and Lemma 3.6 we have ∂ t b t + i N X n =1 p t b t,n − = ∂ t b t − N X n =1 ∂ t b t,n ∈ C ([0 , T ] , Γ − δ ( N +1) ) . On the other hand we also observe that ip t b t − i N X n =1 p t b t,n − = ip t b t − N − X n =0 b t,n ! ∈ C ([0 , T ] , Γ − δ ( N +1)+ ν )again using Lemma 3.6 and (3.12). Since N > (cid:3)
Proof of Theorem 3.4.
Weyl quantization of Lemma 3.7 gives ∂ t b wt ( x, D ) = − ip wt ( x, D ) b wt ( x, D ) + r wt ( x, D ) , t > . For u ∈ S ′ ( R d ) we define K u = u and for t > v t = K t u = µ t b wt ( x, D ) u ∈ S ′ ( R d ) . For t > i∂ t v t = i∂ t ( µ t b wt ( x, D ) u )= q w ( x, D ) v t + µ t ( p wt ( x, D ) b wt ( x, D ) + ir wt ( x, D )) u = q w ( x, D ) v t + p w ( x, D ) v t + iµ t r wt ( x, D ) u . If we set g ( t ) = µ t r wt ( x, D ) u it remains to show g ∈ C ([0 , ∞ ) , S ( R d )). Writing g ( t + s ) − g ( t ) = µ t + s (cid:0) r wt + s ( x, D ) − r wt ( x, D ) (cid:1) u + µ t ( µ s − I ) r wt ( x, D ) u the latter claim is a consequence of Proposition 3.2, Lemma 3.7, (2.5), and the fact that by (2.4) u ∈ Q m for some m ∈ R . We have thus shown that K t is a parametrix to (CP). ✷ The propagator to the perturbed equation as an FIO.
In order to show (3.9) it re-mains to connect the parametrix with the solution operator (3.8) (cf. the proof of [14, Proposition3.1.1]). Let u ∈ S ′ ( R d ). Then for some s ∈ R we have u ∈ Q s +2 ( R d ) ⊆ D (cid:0) q w ( x, D )+ p w ( x, D ) (cid:1) where q w ( x, D ) + p w ( x, D ) is considered an unbounded operator in Q s ( R d ). Lemma 3.8. If T > then (3.23) t µ t r wt ( x, D ) u ∈ C ([0 , T ] , Q s ) , (3.24) t v t ∈ C ([0 , T ] , Q s ) , and (3.25) t ∂ t v t ∈ C ((0 , T ) , Q s ) . Proof.
Property (3.23) is a consequence of Proposition 3.2, Lemma 3.7 and (2.5). Lemma 3.6gives b t ∈ C ([0 , T ] , Γ ). Hence property (3.24) follows from Proposition 3.2 and (2.5).Finally we show (3.25). From (3.22) we obtain for t > ∂ t v t = − iq w ( x, D ) µ t b wt ( x, D ) u − ip w ( x, D ) µ t b wt ( x, D ) u + µ t r wt ( x, D ) u . In order to prove (3.25) we must show t p w ( x, D ) µ t b wt ( x, D ) u ∈ C ((0 , T ) , Q s ) , (3.26) t q w ( x, D ) µ t b wt ( x, D ) u ∈ C ((0 , T ) , Q s ) . (3.27)by virtue of (3.23). Claim (3.26) is a consequence of (3.24), p ∈ Γ − δ and (2.5). Finally claim(3.27) is a consequence of q ∈ Γ , (2.5) and Proposition 3.2. (cid:3) From Lemma 3.8 we may conclude that t K t u ∈ C ([0 , T ] , Q s ) ∩ C ((0 , T ) , Q s ) . From Proposition 3.2 combined with b t ∈ C ([0 , T ] , Γ ) and (2.5) we obtain K t u ∈ Q s +2 ⊆ D (cid:0) q w ( x, D ) + p w ( x, D ) (cid:1) for t >
0. Since [0 , T ] ∋ t K t u solves (3.10) with u ∈ Q s +2 ( R d ), itis a classical solution according to [22, Definition 4.2.1].Assembling the knowledge from (3.8) and Theorem 3.4 gives the following conclusion. Themap t ( K t − T t ) u solves the equation (cid:26) ∂ t ( K t − T t ) u + i ( q w ( x, D ) + p w ( x, D ))( K t − T t ) u = µ t r wt ( x, D ) u , t > , ( K − T ) u = 0and t ( K t − T t ) u ∈ C ([0 , T ] , Q s ) ∩ C ((0 , T ) , Q s ) . Combining (3.23) in Lemma 3.8 with [22, Corollary 4.2.2] gives, invoking (3.8), R t u := ( K t − T t ) u = Z t T t − s µ s r ws ( x, D ) u d s = µ t Z t µ − s c wt − s ( x, D ) µ s r ws ( x, D ) u d s. Lemma 3.9.
For t > the kernel of the operator µ − t R t belongs to S ( R d ) .Proof. By [29, Eq. (50.17) p. 525 and Theorem 51.6] the conclusion follows if we can show thatthe operator Z t µ − s c wt − s ( x, D ) µ s r ws ( x, D ) d s is continuous S ′ ( R d ) → S ( R d ) when S ′ ( R d ) is equipped with its strong topology [24, Sec-tion V.7]. By Proposition 3.2 and (3.6), intersected over s >
0, it suffices to show that r wt ( x, D ) : S ′ ( R d ) → S ( R d ) is continuous uniformly over t ∈ [0 , T ], when S ′ ( R d ) is equippedwith the strong topology.Let K t ∈ S ( R d ) denote the kernel of r wt ( x, D ). According to (2.2) K t and r t are related bythe composition of a linear change of variables and partial inverse Fourier transform. These arecontinuous operators on S ( R d ) and thus Lemma 3.7 implies K t ∈ C ([0 , T ] , S ( R d )).We use the seminorms on g ∈ S ( R d ) k g k α,β = sup x ∈ R d (cid:12)(cid:12) x α ∂ β g ( x ) (cid:12)(cid:12) , α, β ∈ N d . Let u ∈ S ′ ( R d ) and α, β ∈ N d . We have(3.28) k r wt ( x, D ) u k α,β = sup x ∈ R d (cid:12)(cid:12) h x α ∂ βx K t ( x, · ) , u i (cid:12)(cid:12) = sup g ∈ B |h g, u i| , where B = { x α ∂ βx K t ( x, · ) ∈ S ( R d ) , x ∈ R d } ⊆ S ( R d ) . Let γ, κ ∈ N d be arbitrary. We havesup g ∈ B k g k γ,κ = sup x,y ∈ R d (cid:12)(cid:12) x α y γ ∂ βx ∂ κy K t ( x, y ) (cid:12)(cid:12) = k K t k ( α,γ ) , ( β,κ ) < ∞ where the latter seminorm bound is uniform over t ∈ [0 , T ]. Thus B ⊆ S ( R d ) is a boundedset which implies that u sup g ∈ B |h g, u i| is a seminorm on S ′ ( R d ) endowed with the strongtopology. HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 13
Since α, β ∈ N d are arbitrary, (3.28) combined with [24, Theorem V.2] thus prove the claimthat r wt ( x, D ) : S ′ ( R d ) → S ( R d ) is continuous uniformly over t ∈ [0 , T ], when S ′ ( R d ) isequipped with the strong topology. (cid:3) Combining Lemma 3.9 with the fact that an operator has kernel in S ( R d ) if and only if itsWeyl symbol belongs to the same space, we obtain R t = µ t a wt ( x, D )where a t ∈ S ( R d ) and t >
0. This gives finally T t = K t − R t = µ t ( b wt ( x, D ) − a wt ( x, D )) , t > , which, in view of (3.8), implies that c t = b t − a t ∈ Γ which is the sought improvement to (3.7).Thus (3.9) has at long last been proved. This means that we have finally obtained our mainresult Theorem 1.1, which can be alternatively phrased: If δ > and p ∈ Γ − δ then the Cauchyproblem (CP) has an FIO propagator T t ∈ I ( χ t ) for t > . Singularities of the solutions
In this section we will discuss propagation of singularities under equation (CP). Theorem 1.1and Proposition 2.6 give the following result.
Proposition 4.1.
Let T t ∈ I ( e tF ) be the propagator to Cauchy problem (CP) . If u ∈ S ′ ( R d ) then for t > T t u ) ⊆ e tF WF( u ) . Remark . This result can also be seen as a consequence of [19, Proposition 2.2], and [25,Theorem 5.1] combined with (3.7).A more refined concept of singularities are (Shubin) Γ-Lagrangian distributions [7] which wenow explain. A Lagrangian linear subspace Λ ⊆ T ∗ R d is a space of dimension d on which therestriction of σ vanishes. If Λ is Lagrangian then so is χ Λ for each χ ∈ Sp( d, R ). An exampleof a Lagrangian is Λ = R d × { } , and all other Lagrangians may be obtained by application ofelements χ ∈ Sp( d, R ) to Λ [7].A Lagrangian Λ ⊆ T ∗ R d may be parametrized in the form(4.1) Λ = { ( X, AX + Z ) ∈ T ∗ R d , X ∈ Y, Z ∈ Y ⊥ } ⊆ T ∗ R d where Y ⊆ R d is a linear subspace and A ∈ M d × d ( R ) is a symmetric matrix that leaves Y invariant, see [23]. It then automatically leaves Y ⊥ invariant so can be written A = A Y + A Y ⊥ where A Y = π Y Aπ Y and A Y ⊥ = π Y ⊥ Aπ Y ⊥ .Note that the subspace Y ⊆ R d is uniquely determined by Λ, but the matrix A is not. In fact A Y is uniquely determined, but A Y ⊥ can be any matrix such that Y ⊆ Ker A Y ⊥ and A Y ⊥ leaves Y ⊥ invariant.The topological space of Γ-Lagrangian distributions of order m ∈ R with respect to a La-grangian Λ ⊆ T ∗ R d is a subspace denoted I m Γ ( R d , Λ) ⊆ S ′ ( R d ) [7, Definition 6.3]. By [7, Corol-lary 6.12] this space can be defined as follows. Definition 4.3.
A distribution u ∈ S ′ ( R d ) satisfies u ∈ I m Γ ( R d , Λ) if there exist χ ∈ Sp( d, R )that maps χ : R d × { } → Λ isomorphically, and a ∈ Γ m ( R d ) such that u = µa where µ ∈ Mp( d )satisfies π ( µ ) = χ .By [7, Proposition 6.7] we have WF( u ) ⊆ Λ when u ∈ I m Γ ( R d , Λ). Kernels of FIOs are Γ-Lagrangian distributions associated with the twisted graph Lagrangian (2.9) of χ in T ∗ R d [7,Theorem 7.2]. We have the following result on the action of FIOs on Γ-Lagrangian distributions. Theorem 4.4. [7, Theorem 6.11]
Suppose χ ∈ Sp( d, R ) , K ∈ I m ′ ( χ ) and let Λ ⊆ T ∗ R d be aLagrangian. Then K : I m Γ ( R d , Λ) → I m + m ′ Γ ( R d , χ Λ) is continuous. As a consequence we obtain the following result which can be seen as a refinement of Propo-sition 4.1.
Theorem 4.5.
Suppose δ > and p ∈ Γ − δ . Let T t ∈ I ( e tF ) be the propagator to Cauchyproblem (CP) , let Λ ⊆ T ∗ R d be a Lagrangian and let u ∈ I m Γ ( R d , Λ) . Then for all t > T t u ∈ I m Γ ( R d , e tF Λ) . Phase space estimates on the solutions.
In this final section we derive phase spaceestimates for the propagator and solutions to (CP). The estimates for the propagator will berelative to the underlying twisted graph Lagrangian (2.9) of a symplectic matrix χ ∈ Sp( d, R ).For this purpose we adapt the integral transform T g (cf. Definition 2.1) to a matrix χ ∈ Sp( d, R )and to a Lagrangian Λ ⊆ T ∗ R d respectively, see [7, Section 5 and Proposition 6.14]. Definition 4.6.
We define the following phase factor adjusted versions of the transform T g where g ∈ S ( R d ) \ χ ∈ Sp( d, R ) and u ∈ S ′ ( R d ) then T χg ⊗ g u ( z, ζ ) = e − i ( h z,ζ i + σ ( χ ( z , − ζ ) , ( z ,ζ ))) T g ⊗ g u ( z, ζ ) , ( z, ζ ) ∈ T ∗ R d . (2) If Λ ⊆ T ∗ R d is a Lagrangian parametrized by Y ⊆ R d and A ∈ M d × d ( R ) as in (4.1) and u ∈ S ′ ( R d ) then T Λ g u ( x, ξ ) = e − i ( h π Y ⊥ x,ξ i + h x,Ax i ) T g u ( x, ξ ) , ( x, ξ ) ∈ T ∗ R d . We have the following characterization of the kernels of FIOs (cf. [7, Theorem 5.2] and [28]).
Proposition 4.7.
Let K ∈ S ′ ( R d ) , χ ∈ Sp( d, R ) and g ∈ S ( R d ) \ . Then K is the kernel ofan FIO in I m ( χ ) if and only if the estimates | L · · · L k T χg ⊗ g K ( z, ζ ) | . (1 + dist(( z, ζ ) , Λ ′− χ )) m − k (1 + dist(( z, ζ ) , Λ ′ χ )) − N , hold for all k, N ∈ N , where ( z, ζ ) ∈ T ∗ R d and L j = h a j , ∇ z,ζ i with a j ∈ Λ ′ χ for j = 1 , , . . . , k . We also have the following phase space characterization of Lagrangian distributions [7, Propo-sition 6.14].
Proposition 4.8.
Let Λ ⊆ T ∗ R d be a Lagrangian parametrized by Y ⊆ R d and A ∈ M d × d ( R ) asin (4.1) , and let V ⊆ T ∗ R d be a subspace transversal to Λ . A distribution u ∈ S ′ ( R d ) satisfies u ∈ I m Γ ( R d , Λ) if and only if for any g ∈ S ( R d ) \ and for any k, N ∈ N we have (cid:12)(cid:12) L · · · L k T Λ g u ( x, ξ ) (cid:12)(cid:12) . (1 + dist(( x, ξ ) , V )) m − k (1 + dist(( x, ξ ) , Λ)) − N , with ( x, ξ ) ∈ T ∗ R d , where L j = h b j , ∇ x,ξ i are first order differential operators with b j ∈ Λ , j = 1 , . . . , k . Applying Propositions 4.7 and 4.8 to Theorems 1.1 and 4.5 respectively, we obtain the followingphase space estimates for the propagator T t and the solution to (CP), see also [8, 28] for relatedresults in different symbol classes. Note that our regularity assumptions allow for a preciseestimate also of the derivatives of the propagator and solutions. Theorem 4.9.
Let δ > , p ∈ Γ − δ , g ∈ S ( R d ) \ , χ t = e tF , and denote by T t ∈ I ( χ t ) , t > , the propagator to the Cauchy problem (CP) . HUBIN TYPE FOURIER INTEGRAL OPERATORS AND EVOLUTION EQUATIONS 15 (1) The kernel K t of the propagator T t satisfies the estimates for t > | L · · · L k T χ t g ⊗ g K t ( z, ζ ) | . (1 + dist(( z, ζ ) , Λ ′− χ t )) − k (1 + dist(( z, ζ ) , Λ ′ χ t )) − N , ( z, ζ ) ∈ T ∗ R d , for all k, N ∈ N , where L j = h a j , ∇ z,ζ i with a j ∈ Λ ′ χ t for j = 1 , , . . . , k .(2) Suppose Λ ⊆ T ∗ R d is a Lagrangian and set Λ t = χ t Λ . If u ∈ I m Γ ( R d , Λ) then thesolution T t u to (CP) satisfies for t > , k, N ∈ N and ( x, ξ ) ∈ T ∗ R d (cid:12)(cid:12) L · · · L k T Λ t g ( T t u )( x, ξ ) (cid:12)(cid:12) . (1 + dist(( x, ξ ) , V t )) m − k (1 + dist(( x, ξ ) , Λ t )) − N , where L j = h b j , ∇ x,ξ i are first order differential operators with b j ∈ Λ t , j = 1 , . . . , k and V t is a subspace transversal to Λ t . References
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E-mail address : marco.cappiello[AT]unito.it Leibniz Universit¨at Hannover, Institut f¨ur Analysis, Welfenplatz 1, D–30167 Hannover, Germany
E-mail address : rschulz[AT]math.uni-hannover.de Department of Mathematics, Linnæus University, SE–351 95 V¨axj¨o, Sweden
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