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Rend. Sem. Mat. Univ. Pol. Torino Vol. 61, 4 (2003) M. Cappiello PSEUDODIFFERENTIAL PARAMETRICES OF INFINITE ORDER FOR SG-HYPERBOLIC PROBLEMS Abstract. In this paper we consider a class of symbols of infinite order and develop a global calculus for the related pseudodifferential operators in the functional frame of the Gelfand-Shilov spaces of type S. As an application, we construct a parametrix for the Cauchy problem associated to an operator with principal part Dtm and lower order terms given by SGoperators, cf. Introduction. We do not assume here Levi conditions on the lower order terms. Giving initial data in Gelfand-Shilov spaces, we are able to prove the well-posedness for the problem and to give an explicit expression of the solution. 1. Introduction In this work, we study a class of pseudodifferential operators of infinite order, namely with symbol p(x, ξ ) satisfying, for every ε > 0, exponential estimates of the form (1) sup sup α,β∈Nn (x,ξ )∈R2n C −|α|−|β| (α!)−µ (β!)−ν hξ i|α| hxi|β| · h i 1 1 · exp −ε(|x| θ + |ξ | θ ) Dξα Dxβ p(x, ξ ) < +∞ 1 1 where hξ i = (1+|ξ |2) 2 , hxi = (1+|x|2) 2 , for some µ, ν, θ ∈ R such that µ > 1, ν > 1, θ ≥ µ + ν − 1 and C positive constant independent of α, β. Operators of infinite order were studied by L. Boutet de Monvel [2] in the analytic class and by L. Zanghirati [32] in the Gevrey classes G θ (), ⊂ Rn , θ > 1. In our work we develop a global calculus for the symbols defined in (1). The functional frame is given by the GelfandShilov space Sθ (Rn ), θ > 1 (denoted by Sθθ (Rn ) in [10]). This space makes part of a larger class of spaces of functions denoted by Sµν (Rn ), µ > 0, ν > 0, µ + ν ≥ 1. More precisely, Sµν (Rn ) is defined as the space of all functions u ∈ C ∞ (Rn ) satisfying the following condition: there exist positive constants A, B such that sup sup A−|α| B −|β| (α!)−µ (β!)−ν x α u (β) (x) < +∞. α,β∈Nn x∈Rn Such spaces and the corresponding spaces of ultradistributions have been recently studied in different contexts by A. Avantaggiati [1], by S. Pilipovic [24] following the approach applied by H. Komatsu [17], [18] to the theory of ultradistributions and by S. 411 412 M. Cappiello Pilipovic and N. Teofanov [25], [26] in the theory of modulation spaces. The space Sθ (Rn ) which we will consider in the paper corresponds to the case µ = ν = θ and it can be regarded as a global version of the Gevrey classes G θ (Rn ), θ > 1. Sections 2,3 are devoted to the presentation of the calculus. In Section 4, as an application we construct a parametrix for the Cauchy problem ( P(t, x, Dt , Dx )u = f (t, x) (t, x) ∈ [0, T ] × Rn (2) k Dt u(s, x) = gk (x) x ∈ Rn , k = 0, ..., m − 1 T > 0, s ∈ [0, T ], where P(t, x, Dt , Dx ) is a weakly hyperbolic operator with one constant multiple characteristic of the form P(t, x, Dt , Dx ) = Dtm + (3) m X m− j a j (t, x, Dx )Dt . j =1 For every fixed t ∈ [0, T ], we assume a j (t, x, Dx ), j = 1, ..., m are SGpseudodifferential operators of order ( pj, q j ), with p, q ∈ [0, 1[, p + q < 1 i.e. their symbols a j (t, x, ξ ) satisfy estimates of the form (4) sup Dξα Dxβ a j (t, x, ξ ) ≤ C |α|+|β|+1 (α!)µ (β!)ν hξ i pj −|α| hxiq j −|β| t∈[0,T ] for all (x, ξ ) ∈ R2n , with µ, ν, C as in (1). We also assume continuity of a j (t, x, ξ ) with respect to t ∈ [0, T ]. SG-operators were studied by H.O. Cordes [7], C. Parenti [23], E. Schrohe [29] and applied in different contexts to PDEs. Recently, S. Coriasco and L. Rodino [9] treated their application to the solution of a global Cauchy problem for hyperbolic systems or equations with constant multiplicities; under assumptions of Levi type, namely p = 0, q = 0 for (3), (4), they obtained well-posedness in the Schwartz spaces S(Rn ), S 0 (Rn ). In our paper, arguing under the weaker assumption 0 ≤ p + q < 1, we follow a different approach based on the construction of a parametrix of infinite order. This method has been applied by L. Cattabriga and D. Mari [4], L. Cattabriga and L. Zanghirati [6] to the solution of a similar problem in the local context of the Gevrey spaces G θ (), ⊂ Rn . In Section 5 of our work we start from initial data in Sθ (Rn ), and find a global solution in C m ([0, T ], Sθ (Rn )) , with 1 p + q < θ1 ≤ µ+ν−1 . Analogous results are obtained replacing Sθ (Rn ) with its dual. We emphasize that our pseudodifferential approach, beside giving well-posedness, provides an explicit expression for the solution. Moreover, it seems possible to extend the present techniques to global Fourier integral operators, which would allow to treat general SG-hyperbolic equations with constant multiplicities. Let us give an example representative of our results in the Cauchy problem, showing the sharpness of the bound 1 θ > p + q in the frame of the Gelfand-Shilov spaces. E XAMPLE 1. Let p, q ∈ [0, 1[ such that p + q < 1 and consider the problem m qm pm (t, x) ∈ [0, T ] × R Dt u − x D x u = 0 (5) u(0, x) = c0 (x) x ∈R j Dt u(0, x) = 0 j = 1, ..., m − 1 413 Pseudodifferential parametrices where pm, qm are assumed to be positive integers, c0 (x) ∈ C ∞ (R) and it satisfies the estimate sup x α Dxβ c0 (x) ≤ C α+β+1 (α!β!)θ , (6) x∈R 1 > p + q, θ i.e. c0 (x) ∈ Sθ (R). Under these hypotheses, it is easy to verify that the solution of the problem (5) is given by ∞ pm X (x qm Dx ) j c0 (x) j m t u(t, x) = ( j m)! j =0 which is well defined thanks to the condition (6) and belongs to Sθ (R) for every fixed t. We remark that in the critical case θ1 = p + q the solution is defined only for t belonging to a bounded interval depending on the initial datum c 0 ∈ S 1 (R). We ( p+q) also emphasize that from the expression of the solution we have that the solvability of the problem is guaranteed when c0 (x) satisfies the weaker condition pm sup (x qm Dx ) j c0 (x) ≤ C j +1 ( j !)( p+q)m , x∈R which would characterize a function space larger than Sθ (R), θ1 ≥ p + q. In the sequel we shall prefer to keep data in the Gelfand spaces Sθ (Rn ), because well established in literature and particularly suitable to construct a global pseudo-differential calculus. Let us recall some basic results concerning the space Sθ (Rn ). We refer to [10],[11],[20] for proofs and details. Let θ > 1 and A, B be positive integers and denote by Sθ,A,B (Rn ) the space of all functions u in C ∞ (Rn ) such that sup sup A−|α| B −|β| (α!β!)−θ x α u (β) (x) < +∞. α,β∈Nn x∈Rn We may write Sθ (Rn ) = [ Sθ,A,B (Rn ). A,B∈Z+ P ROPOSITION 1. Sθ,A,B (Rn ) is a Banach space endowed with the norm (7) kuk A,B = sup sup A−|α| B −|β| (α!β!)−θ |x α u (β) (x)|. α,β∈Nn x∈Rn By Proposition 1, we can give to Sθ (Rn ) the topology of inductive limit of an increasing sequence of Banach spaces. We remark that this topology is equivalent to the one given in [10] and that all the statements of this section hold in both the frames. Let us give a characterization of the space Sθ (Rn ), providing another equivalent topology to Sθ (Rn ), cf. the proof of Theorem 2 below. 414 M. Cappiello P ROPOSITION 2. Sθ (Rn ) is the space of all functions u ∈ C ∞ (Rn ) such that 1 θ sup sup B −|β| (β!)−θ ea|x| |Dxβ u(x)| < +∞ β∈Nn x∈Rn for some positive a, B. P ROPOSITION 3. The following statements hold: (i) Sθ (Rn ) is closed under the differentiation; (ii) G θ0 (Rn ) ⊂ Sθ (Rn ) ⊂ G θ (Rn ), where G θ (Rn ) is the space of the Gevrey functions of order θ and G θ0 (Rn ) is the space of all functions of G θ (Rn ) with compact support. We shall denote by Sθ0 (Rn ) the dual space, i.e. the space of all linear continuous forms on Sθ (Rn ). From (ii) of Proposition 3, we deduce the following important result. T HEOREM 1. There exists an isomorphism between L(Sθ (Rn ), Sθ0 (Rn )), space of all linear continuous maps from Sθ (Rn ) to Sθ0 (Rn ), and Sθ0 (R2n ), which associates to every T ∈ L(Sθ (Rn ), Sθ0 (Rn )) a distribution K T ∈ Sθ0 (R2n ) such that hT u, vi = hK T , v ⊗ ui for every u, v ∈ Sθ (Rn ). The distribution K T is called the kernel of T. Finally we give a result concerning the action of the Fourier transformation on Sθ (Rn ). P ROPOSITION 4. The Fourier transformation is an automorphism of Sθ (Rn ) and it extends to an automorphism of Sθ0 (Rn ). 2. Symbol classes and operators. Let µ, ν, θ be real numbers such that µ > 1, ν > 1, θ ≥ max{µ, ν}. ∞ (R2n ; C) the Fréchet space of D EFINITION 1. For every C > 0 we denote by 0µνθ all functions p(x, ξ ) ∈ C ∞ (R2n ) satisfying the following condition: for every ε > 0 k pkε,C = sup α,β∈Nn sup (x,ξ )∈R2n C −|α|−|β| (α!)−µ (β!)−ν hξ i|α| hxi|β| · i h 1 1 · exp −ε(|x| θ + |ξ | θ ) Dξα Dxβ p(x, ξ ) < +∞ endowed with the topology defined by the seminorms k · kε,C , for ε > 0. We set ∞ ∞ (R2n ; C) 0µνθ (R2n ) = lim 0µνθ −→ C→+∞ with the topology of inductive limit of an increasing sequence of Fréchet spaces. 415 Pseudodifferential parametrices ∞ (R2n ) is closed under the differentiation and the sum It is easy to verify that 0µνθ and the product of its elements. In the sequel, we will also consider SG-symbols of finite order which are defined as follows, cf. Introduction. Let m 1 , m 2 ∈ R and let µ, ν be positive real numbers such that µ > 1, ν > 1. m 1 ,m 2 D EFINITION 2. For C > 0, we denote by 0µν (R2n ; C) the Banach space of all ∞ 2n functions p ∈ C (R ) such that k pkC = sup sup α,β∈Nn (x,ξ )∈R2n C −|α|−|β| (α!)−µ (β!)−ν hξ i−m 1 +|α| hxi−m 2 +|β| · · Dξα Dxβ p(x, ξ ) < +∞ endowed with the norm k · kC and define m 1 ,m 2 m 1 ,m 2 (R2n ; C). 0µν (R2n ) = lim 0µν −→ C→+∞ We have obviously m 1 ,m 2 ∞ 0µν (R2n ) ⊂ 0µνθ (R2n ) for all θ ≥ max{µ, ν} and for all m 1 , m 2 ∈ R. ∞ (R2n ), we consider the associated pseudodifferential Given a symbol p ∈ 0µνθ operator Z −n (8) Pu(x) = (2π) eihx,ξ i p(x, ξ )û(ξ )dξ, u ∈ Sθ (Rn ). Rn The integral (8) is absolutely convergent in view of Propositions 2 and 4. L EMMA 1. Given t > 0, let m t (η) = ∞ X ηj , ( j !)t j =0 η ≥ 0. Then, for every > 0 there exists a constant C = C(t, ) > 0 such that (9) 1 t C −1 e(t−)η ≤ m t (η) ≤ Ce(t+)η 1 t for every η ≥ 0. See [16] for the proof. In the following we shall denote for t, ζ > 0, x ∈ Rn , m t,ζ (x) = m t (ζ hxi2 ). T HEOREM 2. The map ( p, u) → Pu defined by (8) is a bilinear and separately ∞ (R2n ) × S (Rn ) to S (Rn ) and it extends to a bilinear and continuous map from 0µνθ θ θ ∞ 2n separately continuous map from 0µνθ (R ) × Sθ0 (Rn ) to Sθ0 (Rn ). 416 M. Cappiello ∞ (R2n ) and show that u → Pu is continuous from S (Rn ) to Proof. Let us fix p ∈ 0µνθ θ itself. Basing on Proposition 2, we fix B ∈ Z+ , a > 0 and consider the bounded set F determined by C 1 > 0 1 θ sup ea|x| |u (β) (x)| ≤ C1 B |β| (β!)θ x∈Rn for all u ∈ F, β ∈ Nn . To prove the continuity with respect to u, we need to show that there exist A1 , B1 ∈ N \ {0} and a positive constant C 2 such that |α| |β| sup x α Dxβ Pu(x) ≤ C2 A1 B1 (α!β!)θ x∈Rn for all α, β ∈ Nn and for all u ∈ F. We observe that for every ζ ∈ R+ , 1 m 2θ,ζ (x) ∞ X j =0 ζj (1 − 1ξ ) j eihx,ξ i = eihx,ξ i . ( j !)2θ Thus, fixed α, β ∈ Nn , we have x (2π) α Dxβ Pu(x) −n = (2π) xα m 2θ,ζ (x) −n α x X β1 +β2 =β ∞ X β1 +β2 =β β! β1 !β2 ! β! X ζ j β1 !β2 ! ( j !)2θ j =0 Z Z Rn eihx,ξ i ξ β1 Dxβ2 p(x, ξ )û(ξ )dξ = eihx,ξ i (1−1ξ ) j ξ β1 Dxβ2 p(x, ξ )û(ξ ) dξ. Rn By Proposition 4, there exist a, B, C > 0 independent of u ∈ F and for all ε > 0 there exists Cε > 0 such that, for ζ < C1 ∞ |α| 1 X α β θ x D Pu(x) ≤ Cε |x| (Cζ ) j · eε|x| x m 2θ,ζ (x) j =0 · X β1 +β2 =β β! |β | B 2 (β2 !)ν β1 !β2 ! Z 1 θ Rn |ξ ||β1 | e−(a−ε)|ξ | dξ. Hence, for ε sufficiently small,using Lemma 1 and standard estimates for binomial and factorial coefficients, we conclude that there exist C 2 , A1 , B1 > 0 depending only on ζ, θ, ε such that |α| |β| sup x α Dxβ Pu(x) ≤ C2 A1 B1 (α!β!)θ . x∈Rn This concludes the first part of the proof. To prove the second part we observe that, for u, v ∈ Sθ (Rn ), Z Z Rn Pu(x)v(x)dx = Rn û(ξ ) pv (ξ )dξ 417 Pseudodifferential parametrices where pv (ξ ) = (2π)−n Z Rn eihx,ξ i p(x, ξ )v(x)dx Furthermore, by the same argument of the first part of the proof, it follows that the map v → pv is linear and continuous from Sθ (Rn ) to itself. Then, by Proposition 4 we can define, for u ∈ Sθ0 (Rn ), Pu(v) = û( pv ), v ∈ Sθ (Rn ). This is a linear continuous map from Sθ0 (Rn ) to itself and it extends P. The same argument used before allows to prove the continuity of the map p → Pu for a fixed u in Sθ (Rn ) or in its dual. ∞ (Rn ) the space of all operators of the form (8) defined by a We denote by O P Sµνθ ∞ (R2n ). symbol of 0µνθ As a consequence of Theorems 1 and 2, there exists a unique distribution K in Sθ0 (R2n ) such that Z Z Z hK , v ⊗ ui = (2π)−n eihx−y,ξ i p(x, ξ )u(y)v(x)dydξ dx, u, v ∈ Sθ (Rn ). We may write formally (10) K (x, y) = (2π)−n Z Rn eihx−y,ξ i p(x, ξ )dξ. ∞ (R2n ). For k ∈ (0, 1), define: T HEOREM 3. Let p ∈ 0µνθ k = {(x, y) ∈ R2n : |x − y| > khxi}. Then the kernel K of P defined by (10) is in C ∞ (k ) and there exist positive constants C, a depending on k such that h i β γ D D y K (x, y) ≤ C |β|+|γ |+1 (β!γ !)θ exp −a(|x| θ1 + |y| θ1 ) (11) x for every (x, y) ∈ k and for every β, γ ∈ Nn . L EMMA 2. For any given R > 0, we may find a sequence ψ N (ξ ) ∈ C0∞ (Rn ), ∞ P N = 0, 1, 2, ... such that ψ N = 1 in Rn , N=0 suppψ0 ⊂ {ξ : hξ i ≤ 3R} suppψ N ⊂ {ξ : 2R N µ ≤ hξ i ≤ 3R(N + 1)µ }, N = 1, 2, ... 418 M. Cappiello −|α| α Dξ ψ N (ξ ) ≤ C |α|+1 (α!)µ R sup(N µ , 1) and for every α ∈ Nn and for every ξ ∈ Rn . Proof. Let φ ∈ C0∞ (Rn ) such that φ(ξ ) = 1 if hξ i ≤ 2, φ(ξ ) = 0 if hξ i ≥ 3 and α Dξ φ(ξ ) ≤ C |α|+1 (α!)µ for all α ∈ Nn and for all ξ ∈ Rn . We may then define ξ ψ0 (ξ ) = φ R ξ ξ −φ , ψ N (ξ ) = φ R(N + 1)µ RNµ N ≥ 1. Proof of Theorem 3. Let us consider a sequence {ψ N } N≥0 as in Lemma 2. We observe that, by the condition θ ≥ µ, ∞ Z X n N=0 R ihx,ξ i ψ N (ξ ) p(x, ξ )û(ξ ) dξ < +∞ e for every x ∈ Rn . Then we have, for u, v ∈ Sθ (Rn ), hK , v ⊗ ui = with K N (x, y) = (2π) so we may decompose −n Z ∞ X N=0 Rn K = hK N , v ⊗ ui eihx−y,ξ i p(x, ξ )ψ N (ξ )dξ ∞ X KN . N=0 Let k ∈ (0, 1) and (x, y) ∈ k . Let h ∈ {1, ..., n} such that |x h − yh | ≥ nk hxi. Then, for every α, γ ∈ Nn , Z (−1)|γ | X α α γ eihx−y,ξ i ξ β+γ ψ N (ξ )Dxα−β p(x, ξ )dξ = Dx D y K N (x, y) = (2π)n β Rn β≤α Z (−1)|γ |+N X α −N (x − y ) eihx−y,ξ i DξNh ξ β+γ ψ N (ξ )Dxα−β p(x, ξ ) dξ = h h n (2π) β Rn β≤α 419 Pseudodifferential parametrices ∞ Z (−1)|γ |+N (x h − yh )−N X α X ζ j eihx−y,ξ i λh j Nαβγ (x, ξ )dξ · β (2π)n m 2θ,ζ (x − y) ( j !)2θ Rn j =0 β≤α with λh j Nαβγ (x, ξ ) = (1 − 1ξ ) j DξNh ξ β+γ ψ N (ξ )Dxα−β p(x, ξ ) . (12) Let eh be the h-th vector of the canonical basis of Rn and βh = hβ, eh i, γh = hγ , eh i. Developing in the right-hand side of (12) we obtain that λh j Nαβγ (x, ξ ) = X (−i ) N1 N1 +N2 +N3 =N N1 ≤βh +γh N! (βh + γh )! · · N1 !N2 !N3 ! (βh + γh − N1 )! i N ·(1 − 1ξ ) j ξ β+γ −N1 eh DξNh2 ψ N (ξ )Dξh3 Dxα−β p(x, ξ ) . Hence, for ε > 0, λh j Nαβγ (x, ξ ) ≤ Cε h X N1 +N2 +N3 =N N1 ≤βh +γh j ·(N2 !N3 !)µ [(α − β)!]ν C2 ( j !)2θ (βh + γh )! N! |α−β|+N2 +N3 · C1 · N1 !N2 !N3 ! (βh + γh − N1 )! 1 RNµ N2 i h 1 1 hξ i|β|+|γ |−N1 −N3 exp ε(|x| θ + |ξ | θ ) . We observe that on the support of ψ N , 2R N µ ≤ hξ i ≤ 3R(N + 1)µ . Thus, from standard factorial inequalities,since θ ≥ max{µ, ν}, it follows that N h i 1 µ 1 ε|x| θ λh j Nαβγ (x, ξ ) ≤ Cε C |α|+|γ | (α!γ !)θ C j ( j !)2θ C3 θ (N + 1) θ exp ε(3R) e 1 2 R with C3 independent of R. From these estimates, choosing ζ < α γ D D y K N (x, y) ≤ C 0 C |α|+|γ | (α!γ !)θ x ε 1 C4 R N 1 C2 , we deduce that h i 1 1 1 exp ε|x| θ − cζ θ |x − y| θ with C4 = C4 (k) independent of R. Finally, the condition θ ≥ ν implies that there exists ak > 0 such that i h 1 1 1 1 sup exp ak (|x| θ + |y| θ ) − cζ ν |x − y| ν ≤ 1. (x,y)∈k Then, choosing R sufficiently large, we obtain the estimates (11). D EFINITION 3. A linear continuous operator T from Sθ (Rn ) to itself is said to be θ−regularizing if it extends to a linear continuous map from Sθ0 (Rn ) to Sθ (Rn ). By Theorem 1 it follows that an operator T is θ−regularizing if and only if its kernel belongs to Sθ (R2n ). 420 M. Cappiello 3. Symbolic calculus and composition formula In this section, we develop a symbolic calculus for operators of the form (8) defined by ∞ (R2n ). From now on we will assume the more restrictive condition symbols from 0µνθ µ > 1, ν > 1, θ ≥ µ + ν − 1. (13) which will be crucial for the composition of our operators. We emphasize that the condition (13) appears also in the local theory of pseudodifferential operators in Gevrey classes and it is necessary to avoid a loss of Gevrey regularity occurring in the composition formula, see [3], [4], [13], [15], [32] where µ = 1, ν = θ and in the stationary phase method, see [12]. To simplify the notations, we set, for t ≥ 0 n o Q t = (x, ξ ) ∈ R2n : hxi < t, hξ i < t Q et = R2n \ Q t . 2n D EFINITION 4. Let B, C > 0. We shall denote by FS ∞ µνθ (R ; B, C) the space P ∞ 2n of all formal sums p j (x, ξ ) such that p j (x, ξ ) ∈ C (R ) for all j ≥ 0 and for every ε > 0 (14) j ≥0 sup sup sup j ≥0 α,β∈Nn (x,ξ )∈Q e B j µ+ν−1 C −|α|−|β|−2 j (α!)−µ (β!)−ν ( j !)−µ−ν+1· i h 1 1 ·hξ i|α|+ j hxi|β|+ j exp −ε(|x| θ + |ξ | θ ) Dξα Dxβ p j (x, ξ ) < +∞. ∞ (R2n ; B, C) obtained from FS ∞ (R2n ; B, C) by quotiConsider the space F Sµνθ µνθ enting by the subspace X 2n E= p j (x, ξ ) ∈ FS ∞ (R ; B, C) : supp( p ) ⊂ Q ∀ j ≥ 0 . µ+ν−1 j Bj µνθ j ≥0 ∞ (R2n ; B, C) by formal By abuse of notation, we shall denote the elements of F Sµνθ P sums of the form p j (x, ξ ). The arguments in the following are independent of j ≥0 ∞ (R2n ; B, C) is a Fréchet space the choice of representative. We observe that F Sµνθ endowed with the seminorms given by the left-hand side of (14), for ε > 0. We set ∞ F Sµνθ (R2n ) = lim −→ ∞ F Sµνθ (R2n , B, C). B,C→+∞ ∞ (R2n ) can be identified with an element A symbol p ∈ 0µνθ where p0 = p, p j = 0 ∀ j ≥ 1. P j ≥0 ∞ (R2n ), p j of F Sµνθ 421 Pseudodifferential parametrices P D EFINITION 5. We say that two sums are equivalent we write j ≥0 for all ε > 0 sup sup P sup N∈Z+ α,β∈Nn (x,ξ )∈Q e B N µ+ν−1 pj ∼ P p j (x, ξ ), j ≥0 j ≥0 qj j ≥0 ! P ∞ (R2n ) q j (x, ξ ) from F Sµνθ if there exist constants B, C > 0 such that C −|α|−|β|−2N (α!)−µ (β!)−ν ( j !)−µ−ν+1hξ i|α|+N hxi|β|+N · i X 1 1 α β θ θ ( p j − q j ) < +∞. · exp −ε(|x| + |ξ | ) Dξ Dx j <N h P T HEOREM 4. Given a sum j ≥0 such that p∼ X ∞ (R2n ), there exists p ∈ 0 ∞ (R2n ) p j ∈ F Sµνθ µνθ pj ∞ F Sµνθ (R2n ). in j ≥0 Proof. Let ϕ ∈ C ∞ (R2n ), 0 ≤ ϕ ≤ 1 such that ϕ(x, ξ ) = 0 if (x, ξ ) ∈ Q 1 , ϕ(x, ξ ) = 1 if (x, ξ ) ∈ Q e2 and (15) We define: δ γ Dx Dξ ϕ(x, ξ ) ≤ C |γ |+|δ|+1 (γ !)µ (δ!)ν ϕ0 (x, ξ ) = ϕ and ϕ j (x, ξ ) = ϕ 1 R j µ+ν−1 x, 2 2 x, ξ R R ∀(x, ξ ) ∈ R2n . 1 R j µ+ν−1 ξ , j ≥ 1. We want to prove that if R is sufficiently large, p(x, ξ ) = (16) X ϕ j (x, ξ ) p j (x, ξ ) j ≥0 ∞ (R2n ) and p ∼ is well defined as an element of 0µνθ P j ≥0 m,∞ p j in F Sµνθ (R2n ). First of all we observe that the sum (16) is locally finite so it defines a function p ∈ C ∞ (R2n ). Consider X X α β α−γ γ α β Dξ Dx p(x, ξ ) = Dxβ−δ Dξ p j (x, ξ ) · Dxδ Dξ ϕ j (x, ξ ). γ δ γ ≤α j ≥0 δ≤β 422 M. Cappiello Choosing R ≥ B where B is the constant in Definition 4, we can apply the estimates (14) and obtain iX h 1 1 α β H j αβ (x, ξ ) Dξ Dx p(x, ξ ) ≤ C |α|+|β|+1 α!β!hxi−|β| hξ i−|α| exp ε(|x| θ + |ξ | θ ) j ≥0 where H j αβ (x, ξ ) = X [(α − γ )!]µ−1 [(β − δ)!]ν−1 δ≤β ·C 2 j −|γ |−|δ| ( j !) γ !δ! γ ≤α µ+ν−1 · γ hxi|δ|− j hξ i|γ |− j Dxδ Dξ ϕ j (x, ξ ) . γ Now the condition (15) and the fact that D xδ Dξ ϕ j (x, ξ ) = 0 in Q e2R j µ+ν−1 for (δ, γ ) 6= (0, 0) imply that j C2 |α|+|β|+1 H j αβ (x, ξ ) ≤ C1 (α!)µ−1 (β!)ν−1 R where C2 is independent of R. Enlarging R, we obtain that X |α|+|β|+1 H j αβ (x, ξ ) ≤ C3 (α!)µ−1 (β!)ν−1 ∀(x, ξ ) ∈ R2n j ≥0 ∞ (R2n ). from which we deduce that p ∈ 0µνθ P It remains to prove that p ∼ p j . Let us fix N ∈ N \ {0}. We observe that if j ≥0 (x, ξ ) ∈ Q e2R N µ+ν−1 , then p(x, ξ ) − X j <N p j (x, ξ ) = X ϕ j (x, ξ ) p j (x, ξ ). j ≥N Thus we have X α β Dξ Dx ϕ j (x, ξ ) p j (x, ξ ) ≤ j ≥N h iX 1 1 C |α|+|β|+1 α!β!hxi−|β|−N hξ i−|α|−N exp ε(|x| θ + |ξ | θ ) H j Nαβ (x, ξ ) j ≥N where H j Nαβ (x, ξ ) = X [(α − γ )!]µ−1 [(β − δ)!]ν−1 γ ≤α γ !δ! · δ≤β ·C 2 j −|γ |−|δ| ( j !) µ+ν−1 Arguing as above we can estimate 2N+|α|+|β|+1 H j Nαβ (x, ξ ) ≤ C4 and this concludes the proof. γ hxi|δ|+N− j hξ i|γ |+N− j |Dxδ Dξ ϕ j (x, ξ )|. (N!)µ+ν−1 (α!)µ−1 (β!)ν−1 423 Pseudodifferential parametrices ∞ (R2n ) such that p ∼ 0. Then the operator P is P ROPOSITION 5. Let p ∈ 0µνθ θ−regularizing. To prove this assertion we need a preliminary result. L EMMA 3. Let M, r, %, B be positive numbers, % ≥ 1. We define h(λ) = inf M r N (N!)r 1 0≤N≤Bλ % λ rN % , λ ∈ R+ . Then there exist positive constants C, τ such that 1 % h(λ) ≤ Ce−τ λ , λ ∈ R+ . Proof. See Lemma 3.2.4 in [27] for the proof. Proof of Proposition 5. It is sufficient to prove that if p ∼ 0, then the kernel of P Z eihx−y,ξ i p(x, ξ )dξ K (x, y) = (2π)−n Rn belongs to Sθ (R2n ). By Definition 5, there exist B, C > 0 and for all ε > 0 there exists a positive constant C ε such that, for every (x, ξ ) ∈ R2n h i 1 1 α β Dξ Dx p(x, ξ ) ≤ Cε C |α|+|β| (α!)µ (β!)ν hξ i−|α| hxi−|β| exp ε(|x| θ + |ξ | θ ) · · inf 1 0≤N≤(B −1 hξ ihxi) µ+ν−1 C 2N (N!)µ+ν−1 . hξ i N hxi N Applying Lemma 3 with % = r = µ + ν − 1, λ = hξ ihxi and taking into account the 1 1 1 1 condition θ ≥ µ+ν −1, and the obvious estimate |x| θ +|ξ | θ ≤ chξ i θ hxi θ , we obtain that for all ε > 0 h i 1 1 α β (17) Dξ Dx p(x, ξ ) ≤ Cε0 C |α|+|β| (α!)µ (β!)ν exp −(τ − ε)(|x| θ + |ξ | θ ) for a certain positive τ. For 0 < ε < τ, it follows that p ∈ Sθ (R2n ). By Theorem 3, it is sufficient to show that there exists k ∈ (0, 1) such that i h 1 1 γ sup C −|α|−|γ | (α!γ !)−θ exp a(|x| θ + |y| θ ) Dxα D y K (x, y) < +∞ (x,y)∈R2n \k for some positive constants a, C. From the estimates (17) we obtain, for τ 0 < τ, Z 1 1 α γ X α |α|−|β| 0 θ ν −τ 0 |x| θ D D y K (x, y) ≤ C [(α − β)!] e |ξ ||β|+|γ | e−τ |ξ | dξ. x β Rn β≤α 424 M. Cappiello Now, for every ε > 0 there exists a positive constant C = C(ε) such that |ξ ||β|+|γ | ≤ C |β|+|γ |+1 (β!γ !)θ eε|ξ | 1 θ Furthermore, we observe that there exists C k0 > 0 such that in R2n \ k − 1 τ0 1 τ0 1 1 τ0 1 |x| θ ≤ k θ + k θ |x| θ − Ck0 |y| θ . 2 2 2 So we can conclude that there exist ak > 0 for which h i 1 1 γ sup exp ak (|x| θ + |y| θ ) Dxα D y K (x, y) ≤ C |α|+|γ |+1 (α!γ !)θ R2n \k and this concludes the proof. Let us give now the main results of this section. ∞ (Rn ) and let t P be its transpose P ROPOSITION 6. Let P = p(x, D) ∈ O P Sµνθ defined by ht Pu, vi = hu, Pvi, (18) u ∈ Sθ0 (Rn ), v ∈ Sθ (Rn ). Then, t P = Q + R, where R is a θ−regularizing operator and Q = q(x, D) is in ∞ (Rn ) with O P Sµνθ q(x, ξ ) ∼ X X j ≥0 |α|= j (α!)−1 ∂ξα Dxα p(x, −ξ ) ∞ (R2n ). in F Sµνθ ∞ (Rn ). Then P Q = T HEOREM 5. Let P = p(x, D), Q = q(x, D) ∈ O P Sµνθ ∞ T + R where R is θ−regularizing and T = t (x, ξ ) ∈ O P Sµνθ (Rn ) with t (x, ξ ) ∼ X X (α!)−1 ∂ξα p(x, ξ )Dxα q(x, ξ ) j ≥0 |α|= j ∞ (R2n ). in F Sµνθ To prove these results it is convenient to enlarge the class of our operators by considering more general classes of symbols. Let µ, ν, θ be real numbers satisfying the condition (13). 3n D EFINITION 6. We shall denote by 5∞ µνθ (R ; C) the Fréchet space of all functions a(x, y, ξ ) ∈ C ∞ (R3n ) such that for every ε > 0 sup sup α,β∈Nn (x,y,ξ )∈R3n 1 |β+γ | 2 C −|α|−|β|−|γ | (α!)−µ (β!γ !)−ν hξ i|α| |x|2 + |y|2 · 425 Pseudodifferential parametrices i h 1 1 1 γ ·hx − yi−|β+γ | exp −ε(|x| θ + |y| θ + |ξ | θ ) Dξα Dxβ D y a(x, y, ξ ) < +∞. We set ∞ 3n 3n 5∞ µνθ (R ) = lim 5µνθ (R , C). −→ C→+∞ It is immediate to verify the following relations: 3n i) if a(x, y, ξ ) ∈ 5∞ µνθ (R ), then the function (x, ξ ) → a(x, x, ξ ) belongs to ∞ 2n 0µνθ (R ). ∞ (R2n ), then p((1−τ )x +τ y, ξ ) ∈ 5∞ (R3n ) for every τ ∈ [0, 1]. ii) if p(x, ξ ) ∈ 0µνθ µνθ 3n Given a ∈ 5∞ µνθ (R ), we can associate to a a pseudodifferential operator defined by Z (19) Au(x) = (2π)−n eihx−y,ξ i a(x, y, ξ )u(y)dydξ, u ∈ Sθ (Rn ). R2n We remark that the integral written above is not absolutely convergent in general. Let us give a more precise meaning to (19). L EMMA 4. Let χ ∈ Sθθ (Rn ), χ(0) = 1. Then, for every x ∈ Rn and u ∈ Sθ (Rn ), the function Z (20) Iχ,δ (x) = (2π)−n eihx−y,ξ i a(x, y, ξ )χ(δξ )u(y)dydξ R2n has a limit when δ → 0+ and this limit is independent of χ. Proof. We remark that for every positive ζ, η the following relations hold: 1 (21) m 2θ,ζ (x) 1 (22) m 2θ,η (ξ ) ∞ X p=0 ∞ X q=0 ζp (1 − 1ξ ) p eihx,ξ i = eihx,ξ i ( p!)2θ ηq (1 − 1 y )q eihx−y,ξ i = eihx−y,ξ i . (q!)2θ Substituting (21) in (20) and integrating by parts, we obtain ∞ (2π)−n X ζ p Iχ,δ (x) = · m 2θ,ζ (x) ( p!)2θ p=0 · Z R2n h i eihx,ξ i (1 − 1ξ ) p e−ihy,ξ i a(x, y, ξ )χ(δξ ) u(y)dydξ = Z ∞ (2π)−n X ζ p eihx−y,ξ i λ p,δ (x, y, ξ )dydξ m 2θ,ζ (x) ( p!)2θ R2n p=0 426 M. Cappiello where λ p,δ (x, y, ξ ) = p X p r r=0 (−1)r X |α|=r r! · α1 !...αn ! X 2α 2α−β (−i y)β ∂ξ [a(x, y, ξ )χ(δξ )] u(y) · β β≤2α Applying (22) we obtain that Iχ,δ (x) = · Z R2n eihx−y,ξ i ∞ ∞ (2π)−n X ζ p X ηq · m 2θ,ζ (x) ( p!)2θ (q!)2θ p=0 1 m 2θ,η (ξ ) q=0 (1 − 1 y )q λ p,δ (x, y, ξ )dydξ. The hypotheses on a, u, χ imply that there exist C 1 , C2 , C3 > 0 and for all ε > 0, there exists Cε > 0 such that 1 1 1 (1 − 1 y )q λ p,δ (x, y, ξ ) ≤ Cε C p C q ( p!q!)2θ eε|x| θ e−(C3 −ε)|y| θ eε|ξ | θ . 1 2 Hence, choosing ζ < C11 , η < C12 and ε sufficiently small, we can re-arrange the sums under the integral sign and obtain an estimate independent of δ. By Lebesgue’s dominated convergence theorem, it turns out that lim Iχ,δ (x) = δ→0+ (2π)−n m 2θ,ζ (x) Z R2n eihx−y,ξ i p ∞ X ∞ X X ζ p ηq X p r! · (−1)r 2θ α1 !...αn ! r ( p!q!) p=0 q=0 r=0 |α|=r X 2α 2α−β · (1 − 1 y )q ∂ξ (−i y)β a(x, y, ξ )u(y) dydξ. β β≤2α From Lemma 4 we deduce the following natural definition. 3n n D EFINITION 7. Given a ∈ 5∞ µνθ (R ), we define, for every u ∈ Sθ (R ) Z −n (23) Au(x) = (2π) lim eihx−y,ξ i a(x, y, ξ )χ(δξ )u(y)dydξ δ→0+ R2n with χ ∈ Sθ (Rn ), χ(0) = 1. ∞ We denote by O P S µνθ (Rn ) the space of all operators of the form (19) defined by 3n an amplitude of 5∞ µνθ (R ). Theorems 2 and 3 extend without relevant changes in the proofs to these operators; details are left to the reader. The next theorem states a relation between operators (19) and the elements of ∞ (Rn ). O P Sµνθ 427 Pseudodifferential parametrices 3n T HEOREM 6. Let A be an operator defined by an amplitude a ∈ 5∞ µνθ (R ). Then we may write A = P + R,Pwhere R is a θ−regularizing operator and P = p(x, D) ∈ ∞ (Rn ), with p ∼ p j , where O P Sµνθ j ≥0 p j (x, ξ ) = (24) X |α|= j (α!)−1 ∂ξα D αy a(x, y, ξ )|y=x . Proof. Let χ ∈ C ∞ (R2n ) such that 1 if |x − y| ≤ 41 hxi (25) χ(x, y) = 0 if |x − y| ≥ 12 hxi and β γ D D y χ(x, y) ≤ C |β|+|γ |+1 (β!γ !)ν x Nn for all β, γ ∈ and (x, y) ∈ R2n . We may decompose a as the sum of two elements ∞ 3n of 5µνθ (R ) writing a(x, y, ξ ) = χ(x, y)a(x, y, ξ ) + (1 − χ(x, y))a(x, y, ξ ). Furthermore, it follows from Theorem 3 that (1 − χ(x, y))a(x, y, ξ ) defines a θ−regularizing operator. Hence, eventually perturbing A with a θ−regularizing operator, we can assume that a(x, y, ξ ) is supported on R2n \ 1 × Rn , where 1 is 2 2 defined as in Theorem 3.P ∞ (R2n ). By Theorem 4 It is trivial to verify that p j defined by (24) belongs to F Sµνθ j ≥0 we can find a sequence ϕ j ∈ C ∞ (R2n ) depending on a parameter R such that p(x, ξ ) = X ϕ j (x, ξ ) p j (x, ξ ) j ≥0 ∞ (R2n ) for R large and p ∼ defines an element of 0µνθ P j ≥0 ∞ (R2n ). Let p j in F Sµνθ P = p(x, D). To prove the Theorem it is sufficient to show that the kernel K (x, y) of A − P is in Sθ (R2n ). We can write a(x, y, ξ ) − p(x, ξ ) = (1 − ϕ0 (x, ξ ))a(x, y, ξ ) ∞ X X + (ϕ N − ϕ N+1 )(x, ξ ) a(x, y, ξ ) − p j (x, ξ ) . j ≤N N=0 Consequently, (26) K (x, y) = K (x, y) + ∞ X N=0 K N (x, y) 428 M. Cappiello where K (x, y) = (2π) K N (x, y) = (2π)−n Z Rn −n Z eihx−y,ξ i (1 − ϕ0 (x, ξ ))a(x, y, ξ )dξ, Rn eihx−y,ξ i (ϕ N −ϕ N+1 )(x, ξ ) a(x, y, ξ ) − X j ≤N A power expansion in the second argument gives for N = 1, 2, ... a(x, y, ξ ) = X |α|≤N (α!)−1 (y − x)α ∂ yα a(x, x, ξ ) + with wα (x, y, ξ ) = (N + 1) Z 1 0 X |α|=N+1 p j (x, ξ ) dξ. (α!)−1 (y − x)α wα (x, y, ξ ) ∂ yα a(x, x + t (y − x), ξ )(1 − t) N dt. In view of our definition of the p j (x, ξ ), integrating by parts, we obtain that K N (x, y) = W N (x, y) + (2π)−n · Z 1 · β!(α − β)! α−β α ∂ y a)(x, x, ξ )dξ, eihx−y,ξ i Dξ (ϕ N − ϕ N+1 )(x, ξ )(Dξ W N (x, y) = (2π)−n Z X 1≤|α|≤N 06 =β≤α β Rn where · X X X |α|=N+1 β≤α α−β β Rn 1 · β!(α − β)! eihx−y,ξ i Dξ (ϕ N − ϕ N+1 )(x, ξ )Dξ wα (x, y, ξ )dξ for all N = 1, 2, ... Using an absolute convergence argument, we may re-arrange the sums under the integral sign. We also observe that X N≥|α| β β Dξ (ϕ N − ϕ N+1 )(x, ξ ) = Dξ ϕ|α| (x, ξ ). Then we have K =K+ X α6 =0 Iα + ∞ X WN N=0 where Iα (x, y) = (2π)−n X 06 =β≤α 1 β!(α − β)! Z β Rn α−β α ∂ y a(x, x, ξ )dξ eihx−y,ξ i Dξ ϕ|α| (x, ξ )Dξ 429 Pseudodifferential parametrices and we may write W0 (x, y) for K 0 (x, y). To conclude the proof, we want to show that ∞ P P W N ∈ Sθ (R2n ). First of all, we have to estimate the derivatives of K Iα , K, α6 =0 N=0 for (x, ξ ) ∈ supp(1 − ϕ0 (x, ξ )), i.e. for hxi ≤ R, hξ i ≤ R. We have X γ !δ! k h δ γ −n k h · x y Dx D y K (x, y) = (2π) x y γ !γ 1 2 !δ1 !δ2 !δ3 ! γ1 +γ2 =γ δ +δ +δ =δ 1 · (−1)|γ1 | |x||k| |y||h| Z Rn 2 3 γ eihx−y,ξ i ξ γ1 +δ1 Dxδ2 D y 2 a(x, y, ξ )Dxδ3 (1 − ϕ0 (x, ξ ))dξ ≤ X γ1 +γ2 =γ γ !δ! C |γ2 |+|δ2 |+|δ3 | (γ2 !δ2 !δ3 !)ν hx − yi|γ2 +δ2 | · γ1 !γ2 !δ1 !δ2 !δ3 ! δ1 +δ2 +δ3 =δ h iZ 1 1 · exp ε(|x| θ + |y| θ ) 1 θ hξ i≤R hξ i|γ1 +δ1 | eεhξ i dξ. Now, a(x, y, ξ ) is supported on R2n \ 1 × Rn and in this region |y| ≤ 2 there exist constants C 1 , C2 > 0 depending on R such that γ |γ |+|δ| sup x k y h Dxδ D y K (x, y) ≤ C1 R |k|+|h| C2 (γ !δ!)θ , 3 2 hxi so, (x,y)∈R2n so K ∈ Sθ (R2n ). Consider now γ x k y h Dxδ D y Iα (x, y) = (2π)−n · Z · X 1 β!(α − β)! h X δ1 +δ2 +δ3 =δ X δ1 +δ2 +δ3 =δ α−β eihx−y,ξ i ξ γ +δ1 Dxδ2 Dξ ϕ|α| (x, ξ )Dξ 06 =β≤α Rn 06 =β≤α 1 β!(α − β)! β Rn (2π)−n Z X Dxδ3 ∂ yα a(x, x, ξ )dξ = δ! (−1)|γ | (−i )h x k · δ1 !δ2 !δ3 ! α−β β δ! (−1)|γ | x k y h · δ1 !δ2 !δ3 ! e−ihy,ξ i ∂ξh eihx,ξ i ξ γ +δ1 Dxδ2 Dξ ϕ|α| (x, ξ )Dξ β i Dxδ3 ∂ yα a(x, x, ξ ) dξ. We need the estimates for (x, ξ ) ∈ suppDξ ϕ|α| (x, ξ ) ⊂ Q 2R|α|µ+ν−1 \ Q R|α|µ+ν−1 . Then, there exist C 1 , C2 , C3 > 0 such that k h δ γ |h|+|k|+1 |α| |γ |+|δ| C2 C3 (k!h!γ !δ!)θ (α!)ν hxi−|α| · x y Dx D y Iα (x, y) ≤ C1 · X 06 =β≤α (β!)µ−1 [(α − β)!]µ−1 1 R|α|µ+ν−1 |β| Z hξ i≤2R|α|µ+ν−1 hξ i−|α−β| dξ 430 M. Cappiello with C2 independent of R. Now, if (x, ξ ) ∈ Q 2R|α|µ+ν−1 \ Q R|α|µ+ν−1 , we have that X |α| C2 (α!)ν hxi−|α| · Z 06 =β≤α (β!)µ−1 [(α − β)!]µ−1 hξ i≤2R|α|µ+ν−1 hξ i−|α−β| dξ ≤ C4 R 1 R|α|µ+ν−1 |β| · |α| with C4 independent of R. Finally, we conclude that |α| C4 k h δ γ |γ |+|δ| . (k!h!γ !δ!)θ x y Dx D y Iα (x, y) ≤ C |h|+|k|+1 C2 R 2n (x,y)∈R sup Choosing R > C 4 , we obtain that P α6 =0 Iα ∈ Sθ (R2n ). Arguing as for Iα , we can prove that also N C k h δ γ |h|+|k|+1 |γ |+|δ| C2 (h!k!γ !δ!)θ x y Dx D y W N (x, y) ≤ C1 R (x,y)∈R2n sup with C independent of R, which gives, for R sufficiently large, that ∞ P W N is in N=0 Sθ (R2n ). This concludes the proof. Proof of Proposition 6. By the formula (18), t P is defined by Z t Pu(x) = (2π)−n eihx−y,ξ i p(y, −ξ )u(y)dydξ, u ∈ Sθ (Rn ). R2n ∞ Thus, t P ∈ O P S µνθ (Rn ) with amplitude p(y, −ξ ). By Theorem 6, t P = Q + R ∞ (Rn ), with where R is θ−regularizing and Q = q(x, D) ∈ O P Sµνθ q(x, ξ ) ∼ X X j ≥0 |α|= j (α!)−1 ∂ξα Dxα p(x, −ξ ). Proof of Theorem 5. We can write Q =t (t Q). Then, by Theorem 6 and Proposition 6, Q = Q 1 + R1 , where R1 is θ−regularizing and Z −n (27) Q 1 u(x) = (2π) eihx−y,ξ i q1 (y, ξ )u(y)dydξ R2n ∞ (R2n ), q (y, ξ ) ∼ with q1 (y, ξ ) ∈ 0µνθ 1 that d Q 1 u(ξ ) = Z Rn P α (α!)−1 ∂ξα D αy q(y, −ξ ). From (27) it follows e−ihy,ξ i q1 (y, ξ )u(y)dy, u ∈ Sθ (Rn ) 431 Pseudodifferential parametrices from which we deduce that Z P Qu(x) = (2π)−n R2n eihx−y,ξ i p(x, ξ )q1 (y, ξ )u(y)dydξ + P R1 u(x). 3n We observe that p(x, ξ )q1 (y, ξ ) ∈ 5∞ µνθ (R ), then we may apply Theorem 6 and obtain that P Qu(x) = T u(x) + Ru(x) ∞ (Rn ) with wher R is θ−regularizing and T = t (x, D) ∈ O P Sµνθ X t (x, ξ ) ∼ (α!)−1 ∂ξα p(x, ξ )Dxα q(x, ξ ). α R EMARK 1. Definitions analogous to 4 and 5 can be given for formal sums of m ,m elements of 0µν1 2 (R2n ). Furthermore, under the condition (13), all the results of this section can be extended to the corresponding operators. 4. Construction of a parametrix for the problem (2) Let µ, ν be real numbers such that µ > 1, ν > 1 and consider the operator in (3) where we assume that a j (t, x, Dx ), j = 1, operators of the form ..., m are pseudodifferential pj,q j (8) with symbols a j (t, x, ξ ) ∈ C [0, T ], 0µν (R2n ) , for some nonnegative p, q such that p + q ∈ [ 0, 1 [ . We want to construct a parametrix for the problem (2).We start by considering the homogeneous h equation. Namely, let θ be a real number such that θ ≥ µ + ν − 1 and ∞ (Rn ), t, s in [0, T ] p + q ∈ 0, θ1 [ . We want to find an operator E(t, s) ∈ O P Sµνθ such that (t, s) ∈ [0, T ]2 , x ∈ Rn P(t, x, Dt , Dx )E(t, s) = R(t, s) j (28) Dt E(s, s) = 0 j = 0, ..., m − 2 m−1 Dt E(s, s) = i I where I is the identity operator and R(t, s) has its kernel in C [0, T ], Sθ (R2n ) . In order to construct the parametrix above, we want to apply the results obtained in Sections 2, 3. To be more precise, we need to reformulate these results for operators with symbols depending with a certain regularity on some parameters. The proofs follow the same arguments of the previous sections. Denote by I a compact pluri-interval of Rd . 3n T HEOREM 7. Let a ∈ C(I, 5∞ µνθ (R )). Then, the operator Z −n A(t)u(s, ·)(x) = (2π) eihx−y,ξ i a(t, x, y, ξ )u(s, y)dydξ R2n 432 M. Cappiello defines a linear continuous map from C (I, Sθ (Rn )) to C I 2 , Sθ (Rn ) which extends to a linear continuous map from C I, Sθ0 (Rn ) to C I 2 , Sθ0 (Rn ) . Furthermore, if 3n a ∈ C k (I, 5∞ µνθ (R )), k ∈ N, then Dtk A(t)u(s, ·)(x) = (2π)−n Z R2n eihx−y,ξ i Dtk a(t, x, y, ξ )u(s, y)dydξ for all x ∈ Rn , (t, s) ∈ I 2 . P ∞ (R2n ) , j ≥ 0 such that p j belongs P ROPOSITION 7. i) Let p j ∈ C I, 0µνθ j ≥0 ∞ (R2n ) , set of the bounded functions from I to F S ∞ (R2n ). Then, there to B I, F Sµνθ µνθ P ∞ 2n ∞ 2n p j in F Sµνθ (R ) uniformly with reexists p in C I, 0µνθ (R ) such that p ∼ j ≥0 spect to t ∈ I. ∞ (R2n ) , p(t) ∼ 0 uniformly with respect to t ∈ I in ii) Let p(t) ∈ C I, 0µνθ ∞ (R2n ). Then the operator P(t) has its kernel in C I, S (R2n ) . F Sµνθ θ ∞ (R2n ) . Then there exists P ROPOSITION 8. Let p(t) ∈ C I, 0µνθ ∞ (Rn ), t ∈ I, with symbol q(t, x, ξ ) ∼ Q(t) = q(t, x, Dx ) in O P Sµνθ P P α α −1 ∞ (R2n ) uniformly with respect to t ∈ I, such (α!) ∂ξ Dx p(t, x, −ξ ) in F Sµνθ j ≥0 |α|= j that t P = Q + R, where R has its kernel in C I, Sθ (R2n ) . ∞ (Rn ) T HEOREM 8. Let P(t) = p(t, x, D), Q(t, s) = q(t, s; x, D) ∈ O P Sµνθ P P for t, s ∈ I, such that p(t, x, ξ ) ∼ p j (t; x, ξ ), q(t, s; x, ξ ) ∼ q j (t, s; x, ξ ) j ≥0 j ≥0 ∞ (R2n ) uniformly with respect to t, s ∈ I. Assume that p j ∈ in F Sµνθ ∞ 2n 2 ∞ 2n C I, 0µνθ (R ) , q j ∈ C I , 0µνθ (R ) . Then P Q(t, s) = B(t, s) + R(t, s), where R has its kernel in C I 2 , Sθθ (R2n and B(t, s) = b(t, s; x, D) is in ∞ (R2n ) with O P Sµνθ b(t, s; x, ξ ) ∼ X X (α!)−1 ∂ξα ph (t, x, ξ )Dxα qk (t, s; x, ξ ) j ≥0 h+k+|α|= j ∞ (R2n ) uniformly with respect to (t, s) ∈ I 2 . in F Sµνθ Following a standard argument based on Theorem 8, we can now construct the symbol e(t, s; x, ξ ) of E(t, s) starting from Then we will its asymptotic expansion. k ∞ 2n prove the regularity of e, namely Dt e ∈ C [0, T ], 0µνθ (R ) for all s ∈ [0, T ], k = 0, ..., m with the aid of Proposition 7. For every (x, ξ ) ∈ R2n , let eh (t, s; x, ξ ), h ≥ 0 be the solutions of the following 433 Pseudodifferential parametrices Cauchy problems for ordinary differential equations ! m P m− j Dm + e0 = 0 a j (t, x, ξ )Dt t j =1 (29) j Dt e0 (s, s; x, ξ ) = 0 D m−1 e (s, s; x, ξ ) = i 0 t (t, s) ∈ [0, T ]2 j = 0, ..., m − 2 and for h ≥ 1, ! m D m + P a (t, x, ξ )D m− j e = d (t, s; x, ξ ) h h j t t (30) j =1 j Dt eh (s, s; x, ξ ) = 0 (t, s) ∈ [0, T ]2 j = 0, ..., m − 1 where dh (t, s; x, ξ ) = − m X h X X m− j (α!)−1 ∂ξα a j (t, x, ξ )Dxα Dt eh−l (t, s; x, ξ ). j =1 l=1 |α|=l We want to prove that ∞ (31) Dtk eh ∈ C [0, T ]2 , 0µνθ (R2n ) h ≥ 0, k = 0, ..., m and (32) X h≥0 ∞ Dtk eh ∈ B [0, T ]2 , F Sµνθ (R2n ) k = 0, ..., m. pj,q j L EMMA 5. Let the functions a j belong to C [0, T ], 0µν (R2n ) and let e0 be defined by (29). Then, there exist positive constants C, c such that α β k (33) Dξ Dx Dt e0 (t, s; x, ξ ) ≤ C |α|+|β| (α!)µ (β!)ν hξ i−|α| hxi−|β| · · exp chξ i p hxiq |t − s| (34) |α+β|m X hξ i pi hxiqi i=min(|α+β|,1) |t − s|i+m−1−k (i + m − 1 − k)! k = 0, ..., m−1, α β m Dξ Dx Dt e0 (t, s; x, ξ ) ≤ C |α|+|β| (α!)µ (β!)ν hξ i−|α| hxi−|β| · · exp chξ i p hxiq |t − s| for every (t, s) ∈ [0, T ]2 , (x, ξ ) ∈ R2n . (|α+β|+1)m X i=1 hξ i pi hxiqi |t − s|i−1 (i − 1)! 434 M. Cappiello Proof. Let k = 0, ..., m − 1. For α = β = 0, (33) follows directly from the initial data of (29) and from well known estimates for the solution of the Cauchy problem for ordinary differential equations. See also [4] and [14]. Let us now assume that (33) holds for |α + β| = N and let l ∈ {1, ..., n}. By (29), it follows that Dξl e0 is a solution of the problem ! m m D m + P a (t, x, ξ )D m− j D e = − P D a (t, x, ξ )D m− j e t j j ξl 0 t j =1 j =1 Dt Dξl e0 (s, s; x, ξ ) = 0 ξl j 0 t j = 0, ..., m − 1 so we have that Dξl e0 (t, s; x, ξ ) = − Z s t e0 (t, τ ; x, ξ ) m X Dξl a j (τ, x, ξ )Dτm− j e0 (τ, s; x, ξ )dτ. j =1 This remark allows to estimate the left-hand side of (33) inductively for every α, β ∈ Nn . The estimate (34) easily follows from (33) and (29). pj,q j L EMMA 6. Let the functions a j belong to C [0, T ], 0µν (R2n ) and let eh , h ≥ 1 be the solutions of (30). Then, there exist positive constants C, c such that, for every α, β ∈ Nn , (t, s) ∈ [0, T ]2 , k = 0, ..., m − 1, (x, ξ ) ∈ R2n , we have α β k (35) Dξ Dx Dt eh (t, s; x, ξ ) ≤ C |α|+|β|+2h [(|α| + h)!]µ [(|β| + h)!]ν (h!)−1 · ·hξ i−|α|−h hxi−|β|−h exp chξ i p hxiq |t − s| and (36) (|α+β|+2h)m X i=1 hξ i pi hxiqi |t − s|i+m−1−k (i + m − 1 − k)! α β m Dξ Dx Dt eh (t, s; x, ξ ) ≤ C |α|+|β|+2h+1 [(|α| + h)!]µ [(|β| + h)!]ν (h!)−1 · ·hξ i−|α|−h hxi−|β|−h exp chξ i p hxiq |t − s| for every h ≥ 1, (t, s) ∈ [0, T ]2 , (x, ξ ) ∈ R2n . (|α+β|+2h+1)m X i=1 hξ i pi hxiqi Proof. First of all, we observe that Z t eh (t, s; x, ξ ) = e0 (t, τ ; x, ξ )dh (τ, s; x, ξ )dτ, s |t − s|i−1 (i − 1)! h ≥ 1. From the initial data of (29), it turns out that, for all α, β ∈ Nn , k = 0, ..., m − 1, Z t α β k α β Dξ Dx Dt eh (t, s; x, ξ ) = Dξ Dx Dtk e0 (t, τ ; x, ξ )dh (τ, s; x, ξ )dτ, h ≥ 1 s 435 Pseudodifferential parametrices which we can easily estimate by induction on h ≥ 1, obtaining (35). The estimate (36) immediately follows from (35) and (30). pj,q j L EMMA 7. Let the functions a j (t, x, ξ ) belong to C [0, T ], 0µν (R2n , j = 1, ..., m. Then, the solutions eh of (29), (30) satisfy the conditions (31) and (32). Proof. We observe that for all k = 0, ..., m − 1, h ≥ 0, (|α+β|+2h)m X i=0 hξ i pi hxiqi |t − s|m−1−k |t − s|i+m−1−k ≤ exp hξ i p hxiq |t − s| . (i + m − 1 − k)! (m − 1 − k)! h Then, by the condition p + q ∈ 0, θ1 [ and the obvious estimate hξ i p hxiq ≤ C1 (|x| p+q + |ξ | p+q + C2 ), it follows immediately that there exists C 1 > 0 and for every ε > 0 there exists C ε > 0 such that (37) sup sup α,β∈Nn (x,ξ )∈R2n −|α|−|β|−2h C1 (α!)−µ (β!)−ν (h!)−µ−ν+1 hξ i|α|+h hxi|β|+h · i h 1 1 |t − s|m−1−k · exp −ε(|x| θ + |ξ | θ ) Dξα Dxβ Dtk eh (t, s; x, ξ ) ≤ Cε (m − 1 − k)! for every (t, s) ∈ [0, T ]2 , k = 0, ..., m − 1. Analogously, we obtain that there exists C2 > 0 and for all ε > 0 there exists C ε0 > 0 such that (38) sup sup α,β∈Nn (x,ξ )∈R2n −|α|−|β|−2h C2 (α!)−µ (β!)−ν (h!)−µ−ν+1 hξ i|α|+h hxi|β|+h · i h 1 1 · exp −ε(|x| θ + |ξ | θ ) Dξα Dxβ Dtm eh (t, s; x, ξ ) ≤ Cε0 . ∞ (R2n ) for all k = The estimates (37), (38) imply that Dtk eh ∈ C [0, T ]2 , 0µνθ 0, ..., m − 1. The continuity of Dtm eh follows from the relations (29), (30). Furthermore, (37) and (38) give directly (32). T HEOREM 9. Let P(t, x, Dt , Dx ) be defined by (3), where a j (t, x, ξ ) belong to pj,q j C [0, T ], 0µν (R2n ) , j = 1, ..., m. Then, for every (t, s) ∈ [0, T ]2 , there exists ∞ (Rn ) satisfying (28) with symbol e(t, s; x, ξ ) such that E(t, s) ∈ O P Sµνθ j ∞ Dt e ∈ C [0, T ]2 , 0µνθ (R2n ) j = 0, ..., m. 436 M. Cappiello P eh and applying i) of Proposition 7, we can construct a symbol Proof. Starting from h≥0 ∞ (R2n ) . The same argument can be repeated for the derivatives e ∈ C [0, T ]2 , 0µνθ of e. By construction, the corresponding operator E satisfies (28). As an immediate consequence of Theorem 9, we obtain a parametrix for the inhomogeneous equation. C OROLLARY 1. Let f ∈ C ([0, T ], Sθ (Rn )) and s ∈ [0, T ]. Under the same hypotheses of Theorem 9 the function Z t u(t, x) = E(t, τ ) f (τ, ·)(x)dτ s (Rn )) Cm and is in ([0, T ], Sθ ( Rt P(t, x, Dt , Dx )u = f (t, x) + s R(t, τ ) f (τ, ·)(x)dτ (t, x) ∈ [0, T ] × Rn Dtk u(s, x) = 0 k = 0, ..., m − 1, x ∈ Rn where E(t, τ ), R(t, τ ) are the same of Theorem 9. The same result holds when we replace Sθ (Rn ) with Sθ0 (Rn ). 5. Existence and uniqueness With the help of the parametrix constructed in the previous section, we are able to prove existence and uniqueness of the solution for the problem (2). For sake of simplicity we prove the existence only for regular data, but we remark that the result holds when we replace Sθ (Rn ) with Sθ0 (Rn ). T HEOREM 10. Let f ∈ C ([0, T ], Sθ (Rn )) and gk ∈ Sθ (Rn ), k = 0, ..., m − 1. Under the same hypotheses of Theorem 9, for any given s ∈ [0, T ], there exists a unique function u ∈ C m ([0, T ], Sθ (Rn )) such that ( P(t, x, Dt , Dx )u = f (t, x) (t, x) ∈ [0, T ] × Rn x ∈ Rn , k = 0, ..., m − 1. Dtk u(s, x) = gk (x) Proof: Let us start by considering the case in which gk (x) = 0, k = 0, ..., m − 1. We shall find h ∈ C ([0, T ], Sθ (Rn )) such that, for every given s ∈ [0, T ], the function Z t u(t, x) = E(t, τ ) [ f (τ, ·) + h(τ, ·)] (x)dτ s belonging to Cm (39) ( ([0, T ], Sθ (Rn )) , is a solution of the Cauchy problem P(t, x, Dt , Dx )u = f (t, x) (t, x) ∈ [0, T ] × Rn Dtk u(s, x) = 0 k = 0, ..., m − 1, x ∈ Rn . 437 Pseudodifferential parametrices Hypotheses and notations are the same as in Corollary 1, in particular E(t, τ ) is the parametrix in Theorem 9. To this end, for g ∈ C ([0, T ], Sθ (Rn )) , define Z t Rg(t, x) = R(t, τ )g(τ, ·)(x)dτ (t, x) ∈ [0, T ] × Rn s where R(t, τ ) is the operator with kernel K R in C [0, T ]2 , Sθ (R2n ) appearing in (28). By Corollary 1,we have to find a function h ∈ C ([0, T ], Sθ (Rn )) such that h(t, x) + Rh(t, x) + R f (t, x) = 0 for every (t, x) ∈ [0, T ] × Rn . To conclude, it is then sufficient to show that the ∞ P series (−1)ν Rν f (t, ·) converges in Sθ (Rn ) to a function h(t, ·) in C ([0, T ], Sθ (Rn )) ν=1 uniformly with respect to t ∈ [0, T ]. Now we have that Z R(t, τ ) f (τ, ·)(x) = K R (t, τ, x, y) f (τ, y)dy (t, τ ) ∈ [0, T ]2 , x ∈ Rn . Rn Using the notations of the Introduction, we deduce that there exist positive integers A, B for which Z t Z (40) kR f (t, ·)k A,B,n ≤ sup kK R (t, s, ·, ·)k A,B,2n | f (τ, y)|dy dτ ≤ Rn s [0,T ]2 Z sup kK R (t, s, ·, ·)k A,B,2n [0,T ]2 t s k f (τ, ·)k A,B,n dτ. In particular, from (40) we deduce that kR f (t, ·)k A,B,n ≤ sup kK R (t, s, ·, ·)k A,B,2n · sup k f (t, ·)k A,B,n · |t − s|. [0,T ]2 [0,T ] Arguing by induction, let us suppose that for a fixed ν > 1 !ν kRν f (t, ·)k A,B,n ≤ sup kK R (t, s, ·, ·)k A,B,2n [0,T ]2 sup k f (t, ·)k A,B,n [0,T ] |t − s|ν . ν! Then, we have kR ν+1 f (t, ·)k A,B,n ≤ sup kK R (t, s, ·, ·)k A,B,2n [0,T ]2 sup kK R (t, s, ·, ·)k A,B,2n [0,T ]2 !ν+1 sup kK R (t, s, ·, ·)k A,B,2n [0,T ]2 Z t s kRν f (τ, ·)k A,B,n dτ ≤ sup k f (t, ·)k A,B,n [0,T ] !ν+1 Z t s sup k f (t, ·)k A,B,n [0,T ] |τ − s|ν dτ ≤ ν! |t − s|ν+1 . (ν + 1)! 438 Hence, M. Cappiello ∞ P (−1)ν Rν f (t, ·) converges in Sθ (Rn ) uniformly with respect to t in [0, T ]. ν=1 This gives solution to the problem with zero initial data. It is now standard to obtain a result of existence of the solution for a homogeneous problem with non-zero initial data. In fact, let gk ∈ Sθ (Rn ), k = 0, ..., m − 1 and let v(t, x) = m−1 X k=0 i k (t − s)k gk (x) . k! Then, arguing as before, we may construct a function h ∈ C ([0, T ], Sθ (Rn )) such that Z t E(t, τ ) [Pv(τ, ·) + h(τ, ·)] dτ u(t, x) = v(t, x) − s is a solution of the Cauchy problem ( P(t, x, Dt , Dx )u = 0 (41) Dtk u(s, x) = gk (x) (t, x) ∈ [0, T ] × Rn k = 0, ..., m − 1, x ∈ Rn . The existence of a solution for the problem (2) directly follows from the existence for (39) and (41). To conclude the proof of Theorem 10, we want to show that if u in j C m [0, T ], Sθ0 (Rn ) is such that Dt u(s, ·) = 0 for some s ∈ [0, T ] and Pu(t, ·) = 0 for all t ∈ [0, T ], then u(t, ·) = 0 on [0, T ]. The argument we will follow is the same developed in [4], [6], so we will give only the main lines of the proof and leave the details to the reader. Let us consider the transpose t P of the operator P given by t P =t am − Dt (t am−1 − Dt (... − Dt (t a1 − Dt )...)) where t a j is the transpose of the operator a j , j = 1, ..., m. By Proposition 8 we can pj,q j write t a j (t, x, Dx ) = b j (t, x, Dx ) + r j (t, x, Dx ), where b j ∈ O P Sµν (Rn ) and r j 2n are θ−regularizing operators with kernel in C [0, T ], Sθ (R ) . Given f ∈ C ([0, T ], Sθ (Rn )) and s0 ∈ [0, T ], we want to prove the existence of a function v ∈ C m ([0, T ], Sθ (Rn )) , such that t (t, x) ∈ [0, T ] × Rn P(t, x, Dt , Dx )v = f (t, x) (42) v(s0 , x) = 0 t a j − Dt (... − Dt (t a1 − Dt )...) v(s0 , x) = 0 j = 0, ..., m − 1. If such a function exists, then, given u as before, we can write Z s0 Z s0 Z s0 hu(t, ·), f (t, ·)idt = hu(t, ·),t Pv(t, ·)idt = hPu(t, ·), v(t, ·)idt = 0 s s s from which it follows that u = 0. The existence of a solution of the problem (42) can be obtained from the following lemma. 439 Pseudodifferential parametrices b j (t, x, ξ ), j = 1, ..., m, be as before and assume that b j ∼ P L EMMA 8.∞Let 2n b j,r in F Sµνθ (R ) uniformly with respect to t ∈ [0, T ].Then for every (t, s) in r≥0 ∞ (Rn ),k, j = [0, T ]2 there exists a m × m matrix of operators F k, j (t, s) ∈ O P Sµνθ 0, ..., m − 1 such that: ∞ (R2n ) together with their i) their symbols f kj (t, s; x, ξ ) belong to C [0, T ]2 , 0µνθ first order derivatives; ii) F kj (s, s) = −i δ kj I j, k = 0, ..., m − 1 k,0 k, j −1 k, j k, j iii) b j F − Dt F = F +R j = 1, ..., m −1, k = 0, ..., m −1 and bm F k,0 − Dt F k,m−1 = R k,m where R k, j , R k,m have their kernels in C [0, T ]2 , Sθ (R2n ) . Proof. The lemma can be proved following the arguments in [4],[6] combined with the global results obtained in the previous sections. We omit the details for sake of brevity. 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