Fourier integral operators with weighted symbols
aa r X i v : . [ m a t h . A P ] D ec FOURIER INTEGRAL OPERATORS WITH WEIGHTEDSYMBOLS
ELONG OUISSAM AND SENOUSSAOUI ABDERRAHMANE
Abstract.
The paper contains a survey of a class of Fourier integral opera-tors defined by symbols with tempered weight. These operators are bounded(respectively compact) in L if the weight of the amplitude is bounded (re-spectively tends to 0). Introduction
A Fourier integral operator or FIO for short has the following form[ I ( a, φ ) f ]( x ) = Z Z R ny × R Nθ e iφ ( x,y,θ ) a ( x, y, θ ) f ( y ) dy dθ, f ∈ S ( R n ) (1.1)where φ is called the phase function and a is the symbol of the FIO I ( a, φ ).The study of FIO was started by considering the well known class of symbols S mρ,δ introduced by H¨ormander which consists of functions a ( x, θ ) ∈ C ∞ ( R n × R N )that satisfy | ∂ αθ ∂ βx a ( x, θ ) ≤ C α,β (1 + | θ | ) m − ρ | α | + δ | β | , with m ∈ R , ≤ ρ, δ ≤
1. For the phase function one usually assumes that φ ( x, θ ) ∈ C ∞ ( R n × R N \
0) is positively homogeneous of degree 1 with respect to θ and φ does not have critical points for θ = 0.Later on, other classes of symbols and phase functions were studied. In ([ ? ])and [ ? ], D. Robert and B. Helffer treated the symbol class Γ µρ (Ω) that consists ofelements a ∈ C ∞ (Ω) such that for any multi-indices ( α, β, γ ) ∈ N n × N n × N N , there exists C α,β,γ > , | ∂ αx ∂ βy ∂ γθ a ( x, y, θ ) ≤ C α,β,γ λ µ − ρ ( | α | + | β | + | γ | ) ( x, y, θ ) , where Ω is an open set of R n × R n × R N , µ ∈ R and ρ ∈ [0 ,
1] and they consideredphase functions satisfying certain properties. In ([ ? ]), Messirdi and Senoussaouitreated the L boundedness and L compactness of FIO with symbol class justdefined. These operators are continuous (respectively compact) in L if the weightof the symbol is bounded (respectively tends to 0). Noted that in H¨ormander’sclass this result is not true in general. In fact, in ([ ? ]) the author gave an exampleof FIO with symbol belonging to T <ρ< S ρ, that cannot be extended as a boundedoperator on L ( R n ). Mathematics Subject Classification.
Key words and phrases.
Fourier integral operators, pseudodifferential operators,symbol and phase, boundedness and compactness.
The aim of this work is to extend results obtained in ([ ? ]), we save hypothesison the phase function but we consider symbols with weight ( m, ρ ) (see below).So in the second section we define symbol and phase functions used in this paperand we give the sense of the integral (1.1) by using the known oscillatory integralmethod developed by H¨ormander. A special case of phase functions treated here isdiscussed in the preliminaries, in the third section. The last section is devoted totreat the L boundedness and L compactness of FIO.2. Preliminaries
Definition 1.
A continuous function m : R n → [0 , + ∞ [ is called a tempered weighton R n if ∃ C > , ∃ l ∈ R ; m ( x ) ≤ C m ( x ) (1 + | x − x | ) l , ∀ x, x ∈ R n . Functions of the form λ p ( x ) = (1 + | x | ) p , p ∈ R , define tempered weights. Definition 2.
Let Ω be an open set in R n , ρ ∈ [0 , and m a tempered weight. Afunction a ∈ C ∞ (Ω) is called symbol with weight ( m, ρ ) or ( m, ρ ) -weighted symbolon Ω if ∀ α ∈ N n , ∃ C α > | ∂ αx a ( x ) | ≤ C α m ( x ) (1 + | x | ) − ρ | α | , ∀ x ∈ Ω . We note S mρ (Ω) the space of symbols with weight ( m, ρ ). Proposition 2.1.
Let m and l be two tempered weights. ( i ) If a ∈ S mρ then ∂ αx ∂ βθ a ∈ S mλ − ρ | α + β | ρ ; ( ii ) If a ∈ S mρ and b ∈ S lρ then ab ∈ S mlρ ; ( iii ) If ρ ≤ δ , S mδ ⊂ S mρ ; ( iv ) Let a ∈ S mρ . If there exists C > and µ ∈ R such that | a | ≥ C λ µ uni-formly on Ω then a ∈ S mλ − µ ρ .Proof. For the proof we use Leibniz formula. ( ii ) is obtained by Leibniz formulaand by induction we prove ( iv ). (cid:3) Now, we consider the class of Fourier integral operators[ I ( a, φ ) f ]( x ) = Z Z R ny × R Nθ e iφ ( x,y,θ ) a ( x, y, θ ) f ( y ) dy c dθ, f ∈ S ( R n ) (2.1)where c dθ = (2 π ) − n dθ , a ∈ S mρ and φ be a phase function which satisfies thefollowing hypothesis(H1) φ ∈ C ∞ ( R nx × R ny × R Nθ , R ) ( φ is a real function);(H2) For all ( α, β, γ ) ∈ N n × N n × N N , there exists C α,β,γ > | ∂ αx ∂ βy ∂ γθ φ ( x, y, θ ) | ≤ C α,β,γ λ (2 −| α |−| β |−| γ | ) ( x, y, θ );(H3) There exists K , K > K λ ( x, y, θ ) ≤ λ ( ∂ y φ, ∂ θ φ, y ) ≤ K λ ( x, y, θ ) , ∀ ( x, y, θ ) ∈ R nx × R ny × R Nθ ;(H3 ∗ ) There exists K ∗ , K ∗ > K ∗ λ ( x, y, θ ) ≤ λ ( x, ∂ θ φ, ∂ x φ ) ≤ K ∗ λ ( x, y, θ ) , ∀ ( x, y, θ ) ∈ R nx × R ny × R Nθ . OURIER INTEGRAL OPERATORS WITH WEIGHTED SYMBOLS 3
To give a meaning to the right hand side of (2.1) we use the oscillatory integralmethod. So we consider g ∈ S ( R nx × R ny × R Nθ ) such that g (0) = 1. Let a ∈ S m , wedefine a σ ( x, y, θ ) = g ( xσ , yσ , θσ ) a ( x, y, θ ) , σ > . Theorem 2.2.
Let a ∈ S m ( R n ) and φ be a phase function which satisfies ( H − ( H . Then (1) For all f ∈ S ( R n ) , lim σ → + ∞ [ I ( a σ , φ ) f ]( x ) exists for every point x ∈ R n andis independent of the choice of the function g . We define [ I ( a, φ ) f ]( x ) := lim σ → + ∞ [ I ( a σ , φ ) f ]( x );(2) I ( a, φ ) defines a linear continuous operator on S ( R n ) and S ′ ( R n ) respec-tively.Proof. Let χ ∈ C ∞ ( R ), supp χ ⊂ [ − ,
2] such that χ ≡ − , ε >
0, put ω ε ( x, y, θ ) = χ (cid:18) |∇ y φ | + |∇ θ φ | ε λ ( x, y, θ ) (cid:19) . • In supp ω ε , |∇ y φ | + |∇ θ φ | ≤ ε λ ( x, y, θ ). Using ( H
3) we have K λ ( x, y, θ ) ≤ ελ ( x, y, θ ) + | y | So for ε sufficiently small, fixed at value ε , we obtain λ ( x, y, θ ) ≤ C ( ε ) | y | , ∀ ( x, y, θ ) ∈ supp ω ε . (2.2)Consequently | I ( ω ε a σ , φ ) f ( x ) | ≤ Z Z R ny × R Nθ | a σ ( x, y, θ ) | | f ( y ) | dy c dθ ≤ Z Z R ny × R Nθ m ( x, y, θ ) | f ( y ) | dy c dθ Using the definition of the tempered weight, there exists
C > l ∈ R such that | I ( ω ε a σ , φ ) f ( x ) | ≤ C m (0 , , Z Z R ny × R Nθ λ l ( x, y, θ ) | f ( y ) | dy c dθ. and since f ∈ S ( R n ), we deduce that I ( ω ε a σ , φ ) f is absolutely convergent on supp ω ε . By the Lebesgue’s dominated convergence theorem we can see easily that I ( ω ε a, φ ) f = lim σ → + ∞ I ( ω ε a σ , φ ) f. • In supp (1 − ω ε ), we have supp (1 − ω ε ) ⊂ Ω = { ( x, y, θ ) : |∇ y φ | + |∇ θ φ | ≥ ε λ ( x, y, θ ) } Consider the differential operator L = 1 i ( |∇ y φ | + |∇ θ φ | ) n X j =1 ∂φ∂y j ∂∂y j + N X k =1 ∂φ∂θ k ∂∂θ k . A basic calculus shows that
L e iφ = e iφ . O. ELONG AND A. SENOUSSAOUI,
Lemma 2.3.
For any b ∈ C ∞ ( R ny × R Nθ ) and any k ∈ N we have ( t L ) k [(1 − ω ε ) b ] = X | α | + | β |≤ k g ( k ) αβ ∂ αy ∂ βθ ((1 − ω ε ) b ) where t L is the transpose operator of L and g ( k ) α,β ∈ S λ − k (Ω ) .Proof. The transpose operator t L has the following form t L = n X j =1 F j ∂∂y j + N X j =1 G j ∂∂θ j + H, where F j = − i ( |∇ y φ | + |∇ θ φ | ) ∂φ∂y j ,G j = − i ( |∇ y φ | + |∇ θ φ | ) ∂φ∂θ j ,H = − i ( |∇ y φ | + |∇ θ φ | ) n X j =1 ∂ φ∂y j + N X k =1 ∂ φ∂θ k . In Ω , |∇ y φ | + |∇ θ φ | ≥ ε λ ( x, y, θ ) } therefore using hypothesis ( H
2) we find F j , G j ∈ S λ − (Ω ) and H ∈ S λ − (Ω ).The lemma is deduced by induction on k . (cid:3) I ((1 − ω ε ) a σ , φ ) f ( x ) = Z Z R ny × R Nθ e iφ ( x,y,θ ) ( t L ) k [(1 − ω ε ) a σ f ( y )] dy c dθ. (2.3)Consequently, for k large enough, the integral (2.3) converges when σ → Z Z R ny × R Nθ e iφ ( x,y,θ ) ( t L ) k [(1 − ω ε ) af ( y )] dy c dθ. To prove the second part, we use again the lemma (2.3). (cid:3) Preliminaries
In the sequel we study the special phase function φ ( x, y, θ ) = S ( x, θ ) − yθ. (3.1)where S satisfies(G1) S ∈ C ∞ ( R nx × R nθ , R ),(G2) There exists δ > x,θ ∈ R n | det ∂ S∂x∂θ ( x, θ ) | ≥ δ . (G3) For all ( α, β ) ∈ N n × N n , there exist C α,β >
0, such that | ∂ αx ∂ βθ S ( x, θ ) | ≤ C α,β λ ( x, θ ) (2 −| α |−| β | ) . OURIER INTEGRAL OPERATORS WITH WEIGHTED SYMBOLS 5
Lemma 3.1. If S satisfies (G1), (G2) and (G3), then S satisfies (H1), (H2) and(H3). Also there exists C > such that for all x, x ′ , θ ∈ R n , | x − x ′ | ≤ C | ( ∂ θ S )( x, θ ) − ( ∂ θ S )( x ′ , θ ) | . (3.2) Proposition 3.2. If S satisfies (G1) and (G3), then there exists a constant ǫ > such that the phase function φ given in (3.1) belongs to S λ (Ω φ,ǫ ) where Ω φ,ǫ = (cid:8) ( x, θ, y ) ∈ R n ; | ∂ θ S ( x, θ ) − y | < ǫ ( | x | + | y | + | θ | ) (cid:9) . Proof.
We have to show that: There exists ǫ >
0, such that for all α, β, γ ∈ N n ,there exist C α,β,γ > | ∂ αx ∂ βy ∂ γθ φ ( x, y, θ ) | ≤ C α,β,γ λ ( x, y, θ ) (2 −| α |−| β |−| γ | ) , ∀ ( x, y, θ ) ∈ Ω φ,ǫ . (3.3)If | β | = 1, then | ∂ αx ∂ βy ∂ γθ φ ( x, θ, y ) | = | ∂ αx ∂ γθ ( − θ ) | = ( | α | 6 = 0 | ∂ γθ ( − θ ) | if α = 0;If | β | >
1, then | ∂ αx ∂ βy ∂ γθ φ ( x, y, θ ) | = 0.Hence the estimate (3.3) is satisfied.If | β | = 0, then for all α, γ ∈ N n ; | α | + | γ | ≤
2, there exists C α,γ > | ∂ αx ∂ γθ φ ( x, y, θ ) | = | ∂ αx ∂ γθ S ( x, θ ) − ∂ αx ∂ γθ ( yθ ) | ≤ C α,γ λ ( x, y, θ ) (2 −| α |−| γ | ) . If | α | + | γ | >
2, one has ∂ αx ∂ γθ φ ( x, y, θ ) = ∂ αx ∂ γθ S ( x, θ ). In Ω φ,ǫ we have | y | = | ∂ θ S ( x, θ ) − y − ∂ θ S ( x, θ ) | ≤ √ ǫ ( | x | + | y | + | θ | ) / + Cλ ( x, θ ) , with C >
0. For ǫ sufficiently small, we obtain a constant C > | y | ≤ Cλ ( x, θ ) , ∀ ( x, y, θ ) ∈ Ω φ,ǫ . (3.4)This inequality leads to the equivalence λ ( x, θ, y ) ≃ λ ( x, θ ) in Ω φ,ǫ (3.5)thus the assumption ( G
3) and (3.5) give the estimate (3.3). (cid:3)
Using (3.5), we have the following result.
Proposition 3.3. If ( x, θ ) → a ( x, θ ) belongs to S mk ( R nx × R nθ ) , then ( x, y, θ ) → a ( x, θ ) belongs to S ˜ mk ( R nx × R ny × R nθ ) ∩ S ˜ mk (Ω φ,ǫ ) , k ∈ { , } . L -boundedness and L -compactness of F The main result is as follows.
Theorem 4.1.
Let F be the integral operator of distribution kernel K ( x, y ) = Z R n e i ( S ( x,θ ) − yθ ) a ( x, θ ) c dθ (4.1) where c dθ = (2 π ) − n dθ , a ∈ S mk ( R nx,θ ) , k = 0 , and S satisfies ( G , (G2) and (G3).Then F F ∗ and F ∗ F are pseudodifferential operators with symbol in S m k ( R n ) , k = 0 , , given by σ ( F F ∗ )( x, ∂ x S ( x, θ )) ≡ | a ( x, θ ) | | (det ∂ S∂θ∂x ) − ( x, θ ) | σ ( F ∗ F )( ∂ θ S ( x, θ ) , θ ) ≡ | a ( x, θ ) | | (det ∂ S∂θ∂x ) − ( x, θ ) | O. ELONG AND A. SENOUSSAOUI, we denote here a ≡ b for a, b ∈ S p k ( R n ) if ( a − b ) ∈ S p λ − k ( R n ) and σ stands forthe symbol.Proof. If u ∈ S ( R n ), then F u ( x ) is given by F u ( x ) = Z R n K ( x, y ) u ( y ) dy = Z R n Z R n e i ( S ( x,θ ) − yθ ) a ( x, θ ) u ( y ) dy c dθ = Z R n e iS ( x,θ ) a ( x, θ ) (cid:16) Z R n e − iyθ u ( y ) dy (cid:17)c dθ = Z R n e iS ( x,θ ) a ( x, θ ) F u ( θ ) c dθ. (4.2)Here F is a continuous linear mapping from S ( R n ) to S ( R n ) (by Theorem 2.2). Let v ∈ S ( R n ), then h F u, v i L ( R n ) = Z R n (cid:16) Z R n e iS ( x,θ ) a ( x, θ ) F u ( θ ) c dθ (cid:17) v ( x ) dx = Z R n F u ( θ ) (cid:16) Z R n e − iS ( x,θ ) a ( x, θ ) v ( x ) dx (cid:17)c dθ thus h F u ( x ) , v ( x ) i L ( R n ) = (2 π ) − n hF u ( θ ) , F (( F ∗ v ))( θ ) i L ( R n ) where F (( F ∗ v ))( θ ) = Z R n e − iS ( e x,θ ) a ( e x, θ ) v ( e x ) d e x. (4.3)Hence, for all v ∈ S ( R n ),( F F ∗ v )( x ) = Z R n Z R n e i ( S ( x,θ ) − S ( e x,θ )) a ( x, θ ) a ( e x, θ ) d e x c dθ. (4.4)The main idea to show that F F ∗ is a pseudodifferential operator, is to use the factthat ( S ( x, θ ) − S ( e x, θ )) can be expressed by the scalar product h x − e x, ξ ( x, e x, θ ) i after considering the change of variables ( x, e x, θ ) → ( x, e x, ξ = ξ ( x, e x, θ )).The distribution kernel of F F ∗ is K ( x, ˜ x ) = Z R n e i ( S ( x,θ ) − S (˜ x,θ )) a ( x, θ ) a (˜ x, θ ) c dθ. We obtain from (3.2) that if | x − e x | ≥ ǫ λ ( x, e x, θ ) (where ǫ > | ( ∂ θ S )( x, θ ) − ( ∂ θ S )( e x, θ ) | ≥ ǫ C λ ( x, e x, θ ) . (4.5)Choosing ω ∈ C ∞ ( R ) such that ω ( x ) ≥ , ∀ x ∈ R ω ( x ) = 1 if x ∈ [ − ,
12 ]supp ω ⊂ ] − , OURIER INTEGRAL OPERATORS WITH WEIGHTED SYMBOLS 7 and setting b ( x, ˜ x, θ ) := a ( x, θ ) a (˜ x, θ ) = b ,ǫ ( x, ˜ x, θ ) + b ,ǫ ( x, ˜ x, θ ) b ,ǫ ( x, ˜ x, θ ) = ω ( | x − ˜ x | ǫλ ( x, ˜ x, θ ) ) b ( x, ˜ x, θ ) b ,ǫ ( x, ˜ x, θ ) = [1 − ω ( | x − ˜ x | ǫλ ( x, ˜ x, θ ) )] b ( x, ˜ x, θ ) . We have K ( x, e x ) = K ,ǫ ( x, e x ) + K ,ǫ ( x, e x ), where K j,ǫ ( x, ˜ x ) = Z R n e i ( S ( x,θ ) − S (˜ x,θ )) b j,ǫ ( x, ˜ x, θ ) c dθ, j = 1 , . We will study separately the kernels K ,ǫ and K ,ǫ .On the support of b ,ǫ , inequality (4.5) is satisfied and we have K ,ǫ ( x, e x ) ∈ S ( R n × R n ) . Indeed, using the oscillatory integral method, there is a linear partial differentialoperator L of order 1 such that L (cid:0) e i ( S ( x,θ ) − S (˜ x,θ )) (cid:1) = e i ( S ( x,θ ) − S (˜ x,θ )) where L = − i | ( ∂ θ S )( x, θ ) − ( ∂ θ S )( e x, θ ) | − n X l =1 [( ∂ θ l S )( x, θ ) − ( ∂ θ l S )( e x, θ )] ∂ θ l . The transpose operator of L is t L = n X l =1 F l ( x, e x, θ ) ∂ θ l + G ( x, e x, θ )where F l ( x, e x, θ ) ∈ S λ − (Ω ǫ ), G ( x, e x, θ ) ∈ S λ − (Ω ǫ ), F l ( x, e x, θ ) = i | ( ∂ θ S )( x, θ ) − ( ∂ θ S )( e x, θ ) | − (( ∂ θ l S )( x, θ ) − ( ∂ θ l S )( e x, θ )) ,G ( x, e x, θ ) = i n X l =1 ∂ θ l (cid:2) | ( ∂ θ S )( x, θ ) − ( ∂ θ S )( e x, θ ) | − (( ∂ θ l S )( x, θ ) − ( ∂ θ l S )( e x, θ )) (cid:3) , Ω ǫ = (cid:8) ( x, ˜ x, θ ) ∈ R n : | ∂ θ S ( x, θ ) − ∂ θ S (˜ x, θ ) | > ǫ C λ ( x, ˜ x, θ ) (cid:9) . On the other hand we prove by induction on q that( t L ) q b ,ǫ ( x, ˜ x, θ ) = X | γ |≤ q, γ ∈ N n g γ,q ( x, ˜ x, θ ) ∂ γθ b ,ǫ ( x, ˜ x, θ ) , g ( q ) γ ∈ S λ − q (Ω ǫ ) , and so, K ,ǫ ( x, ˜ x ) = Z R n e i ( S ( x,θ ) − S (˜ x,θ )) ( t L ) q b ,ǫ ( x, ˜ x, θ ) c dθ. Using Leibniz’s formula, (G3) and the form ( t L ) q , we can choose q large enoughsuch that for all α, α ′ , β, β ′ ∈ N n , ∃ C α,α ′ ,β,β ′ > x, e x ∈ R n | x α e x α ′ ∂ βx ∂ β ′ e x K ,ǫ ( x, e x ) | ≤ C α,α ′ ,β,β ′ . O. ELONG AND A. SENOUSSAOUI,
Next, we study K ǫ : this is more difficult and depends on the choice of theparameter ǫ . It follows from Taylor’s formula that S ( x, θ ) − S ( e x, θ ) = h x − e x, ξ ( x, e x, θ ) i R n ,ξ ( x, e x, θ ) = Z ( ∂ x S )( e x + t ( x − e x ) , θ ) dt. We define the vectorial function e ξ ǫ ( x, e x, θ ) = ω (cid:0) | x − ˜ x | ǫλ ( x, ˜ x, θ ) (cid:1) ξ ( x, e x, θ ) + (cid:0) − ω ( | x − ˜ x | ǫλ ( x, ˜ x, θ ) ) (cid:1) ( ∂ x S )( e x, θ ) . We have e ξ ǫ ( x, e x, θ ) = ξ ( x, e x, θ ) on supp b ,ǫ . Moreover, for ǫ sufficiently small, λ ( x, θ ) ≃ λ ( e x, θ ) ≃ λ ( x, e x, θ ) on supp b ,ǫ . (4.6)Let us consider the mapping R n ∋ ( x, e x, θ ) → ( x, e x, e ξ ǫ ( x, e x, θ )) (4.7)for which Jacobian matrix is I n I n ∂ x e ξ ǫ ∂ e x e ξ ǫ ∂ θ e ξ ǫ . We have ∂ e ξ ǫ,j ∂θ i ( x, e x, θ )= ∂ S∂θ i ∂x j ( e x, θ ) + ω (cid:0) | x − ˜ x | ǫλ ( x, ˜ x, θ ) (cid:1)(cid:0) ∂ξ j ∂θ i ( x, e x, θ ) − ∂ S∂θ i ∂x j ( e x, θ ) (cid:1) − | x − ˜ x | ǫλ ( x, ˜ x, θ ) ∂λ∂θ i ( x, ˜ x, θ ) λ − ( x, ˜ x, θ ) ω ′ (cid:0) | x − ˜ x | ǫλ ( x, ˜ x, θ ) (cid:1)(cid:0) ξ j ( x, e x, θ ) − ∂S∂x j ( e x, θ ) (cid:1) . Thus, we obtain (cid:12)(cid:12) ∂ e ξ ǫ,j ∂θ i ( x, e x, θ ) − ∂ S∂θ i ∂x j ( e x, θ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ω ( | x − ˜ x | ǫλ ( x, ˜ x, θ ) ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ξ j ∂θ i ( x, e x, θ ) − ∂ S∂θ i ∂x j ( e x, θ ) (cid:12)(cid:12) + λ − ( x, ˜ x, θ ) (cid:12)(cid:12) ω ′ ( | x − ˜ x | ǫλ ( x, ˜ x, θ ) ) (cid:12)(cid:12)(cid:12)(cid:12) ξ j ( x, e x, θ ) − ∂S∂x j ( e x, θ ) (cid:12)(cid:12) . Now it follows from (G3), (4.6) and Taylor’s formula that (cid:12)(cid:12) ∂ξ j ∂θ i ( x, e x, θ ) − ∂ S∂θ i ∂x j ( e x, θ ) (cid:12)(cid:12) ≤ Z (cid:12)(cid:12) ∂ S∂θ i ∂x j ( e x + t ( x − e x ) , θ ) − ∂ S∂θ i ∂x j ( e x, θ ) (cid:12)(cid:12) dt ≤ C | x − e x | λ − ( x, ˜ x, θ ) , C > OURIER INTEGRAL OPERATORS WITH WEIGHTED SYMBOLS 9 (cid:12)(cid:12) ξ j ( x, e x, θ ) − ∂S∂x j ( e x, θ ) (cid:12)(cid:12) ≤ Z (cid:12)(cid:12) ∂S∂x j ( e x + t ( x − e x ) , θ ) − ∂S∂x j ( e x, θ ) (cid:12)(cid:12) dt ≤ C | x − e x | , C > . (4.9)From (4.8) and (4.9), there exists a positive constant C > | ∂ e ξ ǫ,j ∂θ i ( x, e x, θ ) − ∂ S∂θ i ∂x j ( e x, θ ) | ≤ Cǫ, ∀ i, j ∈ { , . . . , n } . (4.10)If ǫ < δ e C , then (4.10) and (G2) yields the estimate δ / ≤ − e Cǫ + δ ≤ − e Cǫ + det ∂ S∂x∂θ ( x, θ ) ≤ det ∂ θ e ξ ǫ ( x, e x, θ ) , (4.11)with e C >
0. If ǫ is such that (4.6) and (4.11) hold, then the mapping given in (4.7)is a global diffeomorphism of R n . Hence there exists a mapping θ : R n × R n × R n ∋ ( x, e x, ξ ) → θ ( x, e x, ξ ) ∈ R n such that e ξ ǫ ( x, e x, θ ( x, e x, ξ )) = ξθ ( x, e x, e ξ ǫ ( x, e x, θ )) = θ∂ α θ ( x, e x, ξ ) = O (1) , ∀ α ∈ N n \{ } (4.12)If we change the variable ξ by θ ( x, e x, ξ ) in K ,ǫ ( x, e x ), we obtain K ,ǫ ( x, e x ) = Z R n e i h x − ˜ x,ξ i b ,ǫ ( x, ˜ x, θ ( x, e x, ξ )) (cid:12)(cid:12) det ∂θ∂ξ ( x, e x, ξ ) (cid:12)(cid:12)c dξ. (4.13)From (4.12) we have, for k = 0 ,
1, that b ,ǫ ( x, ˜ x, θ ( x, e x, ξ )) | det ∂θ∂ξ ( x, e x, ξ ) | belongsto S m k ( R n ) if a ∈ S mk ( R n ).Applying the stationary phase theorem (c.f. [ ? ] ) to 4.13, we obtain the expres-sion of the symbol of the pseudodifferential operator F F ∗ , σ ( F F ∗ ) = b ,ǫ ( x, ˜ x, θ ( x, e x, ξ )) (cid:12)(cid:12) det ∂θ∂ξ ( x, e x, ξ ) (cid:12)(cid:12) | e x = x + R ( x, ξ )where R ( x, ξ ) belongs to S m λ − k ( R n ) if a ∈ S mk ( R n ), k = 0 , x = x , we have b ,ǫ ( x, ˜ x, θ ( x, e x, ξ )) = | a ( x, θ ( x, x, ξ )) | where θ ( x, x, ξ ) is theinverse of the mapping θ → ∂ x S ( x, θ ) = ξ . Thus σ ( F F ∗ )( x, ∂ x S ( x, θ )) ≡ | a ( x, θ ) | (cid:12)(cid:12) det ∂ S∂θ∂x ( x, θ ) (cid:12)(cid:12) − . From (4.2) and (4.3), we obtain the expression of F ∗ F : ∀ v ∈ S ( R n ),( F ( F ∗ F ) F − ) v ( θ ) = Z R n e − iS ( x,θ ) a ( x, θ )( F ( F − v ))( x ) dx = Z R n e − iS ( x,θ ) a ( x, θ ) (cid:16) Z R n e iS ( x, e θ ) a ( x, e θ )( F ( F − v ))( e θ ) c d e θ (cid:17) dx = Z R n Z R n e − i ( S ( x,θ ) − S ( x, ˜ θ )) a ( x, θ ) a ( x, e θ ) v (˜ θ ) c d e θdx . Hence the distribution kernel of the integral operator F ( F ∗ F ) F − is e K ( θ, e θ ) = Z R n e − i ( S ( x,θ ) − S ( x, ˜ θ )) a ( x, θ ) a ( x, ˜ θ ) c dx. We remark that we can deduce e K ( θ, e θ ) from K ( x, e x ) by replacing x by θ . Onthe other hand, all assumptions used here are symmetrical on x and θ ; therefore, F ( F ∗ F ) F − is a nice pseudodifferential operator with symbol σ ( F ( F ∗ F ) F − )( θ, − ∂ θ S ( x, θ )) ≡ | a ( x, θ ) | (cid:12)(cid:12) det ∂ S∂x∂θ ( x, θ ) (cid:12)(cid:12) − . Thus the symbol of F ∗ F is given by (c.f. [ ? ]) σ ( F ∗ F )( ∂ θ S ( x, θ ) , θ ) ≡ | a ( x, θ ) | (cid:12)(cid:12) det ∂ S∂x∂θ ( x, θ ) (cid:12)(cid:12) − . (cid:3) Corollary 4.2.
Let F be the integral operator with the distribution kernel K ( x, y ) = Z R n e i ( S ( x,θ ) − yθ ) a ( x, θ ) c dθ where a ∈ S m ( R nx,θ ) and S satisfies (G1), (G2) and (G3). Then, we have: (1) For any bounded tempered weight m , F can be extended as a bounded linearmapping on L ( R n )(2) For any m such that lim | x | + | θ |→∞ m ( x, θ ) = 0 , F can be extended as a compactoperator on L ( R n ) .Proof. It follows from Theorem 4.1 that F ∗ F is a pseudodifferential operator withsymbol in S m ( R n ).(1) Since m is bounded, we can apply the Cald´eron-Vaillancourt theorem (see [ ? ])for F ∗ F and obtain the existence of a positive constant γ ( n ) and an integer k ( n )such that k ( F ∗ F ) u k L ( R n ) ≤ γ ( n ) Q k ( n ) ( σ ( F F ∗ )) k u k L ( R n ) , ∀ u ∈ S ( R n )where Q k ( n ) ( σ ( F F ∗ )) = X | α | + | β |≤ k ( n ) sup ( x,θ ) ∈ R n (cid:12)(cid:12) ∂ αx ∂ βθ σ ( F F ∗ )( ∂ θ S ( x, θ ) , θ ) (cid:12)(cid:12) Hence, for all u ∈ S ( R n ), k F u k L ( R n ) ≤ k F ∗ F k / L ( L R n )) k u k L ( R n ) ≤ ( γ ( n ) Q k ( n ) ( σ ( F F ∗ ))) / k u k L ( R n ) . Thus F is also a bounded linear operator on L ( R n ).(2) If lim | x | + | θ |→∞ m ( x, θ ) = 0, the compactness theorem (see [ ? ]) shows that theoperator F ∗ F can be extended as a compact operator on L ( R n ). Thus, the Fourierintegral operator F is compact on L ( R n ). Indeed, let ( ϕ j ) j ∈ N be an orthonormalbasis of L ( R n ), then k F ∗ F − n X j =1 h ϕ j , . i F ∗ F ϕ j k → n → + ∞ . OURIER INTEGRAL OPERATORS WITH WEIGHTED SYMBOLS 11
Since F is bounded, for all ψ ∈ L ( R n ), (cid:13)(cid:13) F ψ − n X j =1 h ϕ j , ψ i F ϕ j (cid:13)(cid:13) ≤ (cid:13)(cid:13) F ∗ F ψ − n X j =1 h ϕ j , ψ i F ∗ F ϕ j (cid:13)(cid:13)(cid:13)(cid:13) ψ − n X j =1 h ϕ j , ψ i ϕ j (cid:13)(cid:13) , it follows that k F − n X j =1 h ϕ j , . i F ϕ j k → n → + ∞ (cid:3) Example 4.3.
We consider the function given by S ( x, θ ) = X | α | + | β | =2 , α,β ∈ N n C α,β x α θ β , for ( x, θ ) ∈ R n where C α,β are real constants. This function satisfies (G1), (G2) and (G3). ELONG OuissamLaboratory of Fundamental and Applied Mathematics of Oran LMFAO, Department ofMathematics, University of Oran Ahmed Benbella, Oran, Algeria
E-mail address : elong [email protected] SENOUSSAOUI AbderrahmaneLaboratory of Fundamental and Applied Mathematics of Oran LMFAO, Department ofMathematics, University of Oran Ahmed Benbella, Oran, Algeria
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