Fourier-integral-operator approximation of solutions to first-order hyperbolic pseudodifferential equations I: convergence in Sobolev spaces

0.43 MB·English

by  Jerome Le Rousseau

Fourier-integral-operator approximation of
solutions to first-order hyperbolic
pseudodifferential equations I: convergence
5 in Sobolev spaces
0
0
2
n J´erˆome Le Rousseau
a Laboratoire d’analyse, topologie, probabilit´es, CNRS UMR 6632
J
Universit´e Aix-Marseille I, France
7
[email protected]

]
P
A February 1, 2008
.
h
t
a Abstract
m
An approximation Ansatz for the operator solution, U(z′,z), of a hyper-
[
bolicfirst-orderpseudodifferentialequation,∂z+a(z,x,Dx)withRe(a)≥
1 0, is constructed as the composition of global Fourier integral operators
v with complex phases. An estimate of the operator norm in L(H(s),H(s))
1
of these operators is provided which allows to prove a convergence result
0 fortheAnsatztoU(z′,z)in some Sobolev spaceas thenumberofopera-
1
tors in the composition goes to ∞.
1
0
5 AMS 2000 subject classification: 35L05, 35L80, 35S10, 35S30, 86A15.
0
/
h
0 Introduction
t
a
m
: We consider the Cauchy problem
v
i
X (0.1) ∂zu+a(z,x,Dx)u=0, 0<z Z
≤
r (0.2) u =u ,
z=0 0
a |
with Z > 0 and a(z,x,ξ) continuous with respect to (w.r.t.) z with values in
S1(Rn Rn). We denote D = 1∂ . Further assumptions will be made on the
× x i x
symbol a(z,x,ξ). When a(z,x,ξ) is in fact independent of x and z it is natural
to treat such a problem by means of Fourier transformation:
u(z,x′)= exp[i x′ xξ za(ξ)] u (x) d−ξ dx,
0
h − | i−
ZZ
where d−ξ := ( 1 )ndξ. For this to be well defined for all u S(Rn) or some
2π 0 ∈
Sobolev space we shall impose the real part of the principal symbol of a to be
1
non-negative. When the symbol a depends on both x and z we can naively
expect
u(z,x′) u (z,x′):= exp[i x′ xξ za(0,x′,ξ)] u (x) d−ξ dx
1 0
≃ h − | i−
ZZ
for z small and hence approximately solve the Cauchy problem (0.1)-(0.2) for
z [0,z(1)] with z(1) small. If we want to progress in the z direction we have
∈
to solve the Cauchy problem
∂ u+a(z,x,D )u=0, z(1) <z Z
z x
≤
u(z,.) =u (z(1),.),
|z=z(1) 1
which we approximatively solve by
u(z,x′) u (z,x′)
2
≃
:= exp[i x′ xξ (z z(1))a(z(1),x′,ξ)] u (z(1),x) d−ξ dx.
1
h − | i− −
ZZ
Such a procedure then goes on.
If we call the operator with kernel
(z′,z)
G
G (x′,x)= exp[i x′ xξ ]exp[ (z′ z)a(z,x′,ξ)]d−ξ,
(z′,z)
h − | i − −
Z
we then see that the procedure described above involves composing such oper-
ators: with chosen values 0 z(1) z(k) Z we then have
≤ ≤···≤ ≤
u (z,x)= (u )(x),
k+1 G(z,z(k))◦G(z(k),z(k−1))◦···◦G(z(1),0) 0
for z z(k). We then naturally define for P= z(0),z(1),...,z(N) , a subdivi-
≥ { }
sion of [0,Z] with 0=z(0) <z(1) < <z(N) =Z, the operator
P,z
··· W
if 0 z z(1),
(z,0)
G ≤ ≤
k
:=
WP,z  if z(k) z z(k+1).
 G(z,z(k)) G(z(i),z(i−1)) ≤ ≤
i=1
Y
Theproceduredescribedaboveyields (u )asanapproximationAnsatzfor
P,z 0
W
thesolutiontotheCauchyproblem(0.1)-(0.2). Wedenote∆ =sup (z
P i=1,...,N i−
z ). The operator is often referred to as the thin-slab propagator (see
i−1 (z′,z)
G
e.g. [3, 2]).
Note that a similar procedure can be used to show the existence of an evolu-
tion system by approximating it by composition of semigroup solutions of the
Cauchy problem with z ’frozen’ in a(z,x,D ) [19, 11]. Note that the thin-slab
x
propagator is howevernot a semigroupnor an evolutionfamily here (see
(z′,z)
G
Section 3 for simple arguments).
The approximation Ansatz proposed here is a tool to compute approximations
of the exact solution to the Cauchy problem (0.1)-(0.2). Such computations in
2
the case of geophysicalproblems can be found in [3]. In explorationseismology
one is confronted with solving equations of the type
(0.3) (∂ ib(z,x,D ,D )+c(z,x,D ,D ))v =0,
z t x t x
−
(0.4) v(0,.)=v (0,.),
+
where t is time, z is the vertical coordinate and x is the lateral or transverse
coordinate. Theoperatorsbandcareoffirstorder,wihtrealprincipalparts,b
1
andc ,andc (z,x,τ,ξ)isnon-negative. WeseethattheCauchyproblem(0.1)-
1 1
(0.2) studied here is more general. The Cauchy problem (0.3)-(0.4) is obtain
by a (microlocal) decoupling of the up-going and down-going wavefields in the
acoustic wave equation (see Appendix A and [22] for details). The proposed
Ansatz can then be a tool to approximate in practice the exact solution of
the Cauchy problem (0.3)-(0.4) for the purpose of imaging the Earth’s interior
[3, 2]. As explained in Appendix A the operator c works as a damping term
that suppresses singularities in the microlocalregionwhere its symbol does not
vanishes. This effect is actually recoveredin the proposed Ansatz. Geophysists
are not only interested in the convergence of this Ansatz to the exact solution
of the Cauchy problem (0.3)-(0.4) but they expect the wavefront set of the
approximate solution to be close, in some sense, to that of the exact solution
because seismic imaging aims at imaging the singularities in the subsurface
(see for instance [23, 1]). We shall investigate the microlocal properties of the
proposed Ansatz in Part II, written in collaborationwith Gu¨nther Ho¨rmann.
In the presentpaper,we are interestedin the analysisofthe convergenceof the
approximationscheme in Sobolev spaces. Section 1 introduces the Cauchy
P
W
problem we study and the precise assumptions made on the symbol a(z,x,ξ),
especiallyonthe realpart,c , andimaginarypart, b ,ofits principalsymbol.
1 1
−
In Section 2, we shall at first concentrate our study on the operator , yet
(z′,z)
G
to be properly defined. Under some assumptions on a(z,x,ξ), we shall prove
that is a global Fourier integral operator (FIO) with complex phase and
(z′,z)
G
thatitmapsS intoS,S′ intoS′ andH(s) intoH(s) foranys. Anestimation
of will be the first step towards the analysis in Section 3 of
kG(z′,z)k(H(s),H(s))
the convergenceof . Infact weprovethatfor z′ z sufficiently smallthen
P,z
W −
(Theorem 2.22)
1+ z′ z M,
kG(z′,z)k(H(s),H(s)) ≤ | − |
for some constant M. Such an estimate is achieved by the analysis of the
behaviorofthesymbolexp[ ∆c ]asanelementofS0,inparticularas∆=z′ z
− 1 1 −
2
goes to zero.
In Section 3 we study the convergence of the Ansatz (u ) to the solution
P,z 0
W
ofthe Cauchyproblem(0.1)-(0.2)inSobolevspacesas∆ goesto0. Aconver-
P
genceinnormof tothesolutionoperatoroftheCauchyproblem(0.1)-(0.2)
P,z
W
is actually obtained (Theorem 3.10):
lim U(z,0) =0,
∆P→0kWP,z − k(H(s+1),H(s))
with a convergence rate of order 1.
2
At the end Section 3 we relax some regularity property of the symbol a(z,.)
w.r.t. z by the introduction of another, yet natural, Ansatz: following [17], the
3
thin-slab propagator, , is replaced by the operator with kernel
(z′,z) (z′,z)
G G
G (x′,x)= exp[i x′ xξ ]exp[ z′a(s,bx′,ξ)ds]d−ξ.
(z′,z) h − | i − z
Z
R
b
In Part II, we shall focus on the microlocal aspects of the operator and
P,z
W
how it propagates the singularities of the initial condition u . We shall show
0
that the wavefront set of (u )(z,.) converges in some sense to that of the
P,z 0
W
solution u(z,.) of the Cauchy problem (0.1)-(0.2) as ∆ goes to 0.
P
Multi-compositionofFIOs to approximatesolutions ofCauchyproblems where
first proposed in [16] and [15] where the exact solution operator of a first order
hyperbolicsystemisapproximatedwithadifferentAnsatz,up toaregularizing
operator. The technique is basedon the computationand the estimation ofthe
phasefunctionsandamplitudesofthe FIOresultingfromthesemulti-products,
a result know as the Kumano-go-Taniguchi theorem. The technique was then
furtherappliedtoSchr¨odingerequationswithspecificsymbols[12,17]. Inthese
latter works the multi-product in also interpreted as an iterated integral of
Feynman’stypeandconvergenceisstudiedinaweaksense. In[12]aconvergence
result in L2 is proved. This is the type of results sought here for first order
hyperbolic equations. We however do not use the apparatus of multi-phases
andratherfocusonestimatingtheSobolevregularityofeachterminthemulti-
productofFIOsintheproposedAnsatz. WhiletheresultingproductisanFIO,
we do not compute its phase and amplitude.
In this paper, when the constant C is used, its value may change from one line
to the other. If we want to keep track of the value of a constant we shall use
another letter. When we shall write that a function is bounded w.r.t. z and/or
∆ we shall actually mean that z is to be taken in the interval [0,Z] and ∆ in
someinterval[0,∆ ]unlessotherwisestipulated. WeshallgenerallywriteX,
max
X′, X′′, X(1), ..., X(N) for Rn, according to variables, e.g., x, x′, ..., x(N).
In the present paper, symbol spaces are spaces of global symbols; a function
a C∞(Rn Rp) is in Sm (Rn Rp), 0 < ρ 1, 0 δ < 1, if for all
∈ × ρ,δ × ≤ ≤
multi-indices α, β there exists C >0 such that
αβ
∂α∂βa(x,ξ) C (1+ ξ )m−ρ|β|+δ|α|, x Rn, ξ Rp.
| x ξ |≤ αβ | | ∈ ∈
The best possible constants C , i.e.,
αβ
p (a):= sup (1+ ξ )−m+ρ|β|−δ|α| ∂α∂βa(x,ξ),
αβ (x,ξ)∈Rn×Rp | | | x ξ |
define seminorms that turn Sm (Rn Rp) into a Fr´echet space. As usual we
ρ,δ ×
write Sm(Rn Rp) in the case ρ = 1 δ, 1 ρ < 1, and Sm(Rn Rp) in the
ρ × − 2 ≤ ×
case ρ=1, δ =0.
We shalluse, in a standardway,the notation# for the compositionof symbols
of pseudodifferential operators (ψDO). When given an amplitude p(x,y,ξ)
Sm (X X Rn),ρ δ,weshallalsousethenotationσ p (x,ξ)forthesymbo∈l
ofρ,δthe ×pseu×dodiffere≥ntial operator with amplitude p. {Fo}r p Sm (X Rn)
∈ ρ,δ ×
we shall write p∗ for the symbol of the adjoint operator. When composing
ψDOs or computing adjoints of ψDOs we shall make use of the oscillatory
4
integral representation of the resulting symbol instead of asymptotic series for
tworeasons. First,weaimatestimatingoperatornorminL(Hs,Hs)whileusing
asymptotic series representations for symbols yields results up to regularizing
operatorswhichoperatornormscannotbecontrolled. Second,weshallconsider
symbols in Sm, for some m, including the case ρ= 1 for which the asymptotic
ρ 2
formulae of the calculus of ψDOs cease to hold.
For r R we let E(r) be the ψDO with symbol ξ r := (1 + ξ 2)r/2. The
operato∈r E(r) maps H(s)(X) onto H(s−r)(X) unitarhilyifor all s |R|with E(−r)
∈
being the inverse map.
1 The homogeneous first-order hyperbolic equa-
tion
Let s R and Z >0. We consider the Cauchy problem
∈
(1.5) ∂ u+a(z,x,D )u=0, 0<z Z,
z x
≤
(1.6) u =u H(s+1)(Rn),
z=0 0
| ∈
where the symbol a(z,x,ξ) satisfies the following assumption
Assumption 1.1.
a (x,ξ)=a(z,x,ξ)= ib(z,x,ξ)+c(z,x,ξ)
z
−
where b C0([0,Z],S1(Rn Rn)), with real principal symbol b homogeneous of
1
degree 1∈for ξ large enoug×h and c C0([0,Z],S1(Rn Rn)) with non-negative
| | ∈ ×
principal symbol c homogeneous of degree 1 for ξ large enough. Without loss
1
| |
of generality we can assume that b and c are homogeneous of degree 1 for
1 1
ξ 1.
| |≥
In Section 3 we shall further make the following assumption.
Assumption 1.2. The symbol a(z,.) is assumed to be in L([0,Z],S1(Rn
Rn)), i.e. Lipschitz continuous w.r.t. z with values in S1(Rn Rn), in the sens×e
×
that,
a(z′,x,ξ) a(z,x,ξ)=(z′ z)a˜(z′,z,x,ξ), 0 z z′ Z
− − ≤ ≤ ≤
with a˜(z′,z,x,ξ) bounded w.r.t. z′ and z with values in S1(Rn Rn).
×
Weaker assumptions will also be formulated in Section 3, for instance by the
introduction of another approximating Ansatz.
Wedenotebya = ib +c theprincipalsymbolofaandwriteb=b +b with
1 1 1 1 0
b C0([0,Z],S0(R−n Rn)) andc=c +c withc C0([0,Z],S0(Rn Rn)).
0 1 0 0
∈ × ∈ ×
Assumption 1.1 ensures that the hypotheses (i)-(iii) of Theorem 23.1.2 of [8]
are satisfied. Then there exists a unique solution in C0([0,Z],H(s+1)(Rn))
C1([0,Z],H(s)(Rn)) to the Cauchy problem (1.5)-(1.6). ∩
5
Furthermore, we have the following energy estimate [8, Lemma 23.1.1] for any
function in C1([0,Z],H(s)(Rn)) C0([0,Z],H(s+1)(Rn))
∩
(1.7) sup exp[ λz] u(z,.) u(0,.)
− k kH(s) ≤k kH(s)
z∈[0,Z]
Z
+2 exp[ λz] ∂ u+a (x,D )u dz,
− k z z x kH(s)
Z0
with λ large enough (λ solely depending on s).
By Proposition 9.3 in [5, Chapter VI] the family of operators (a ) gen-
z z∈[0,Z]
erates a strongly continuous evolution system. We denote U(z′,z) the corre-
sponding evolution system:
U(z′′,z′) U(z′,z)=U(z′′,z), Z z′′ z′ z 0.
◦ ≥ ≥ ≥ ≥
Then the Cauchy problem (1.5)-(1.6) reads
∂ U(z,z )u +a(z,x,D )U(z,z )u =0, 0 z <z Z,
z 0 0 x 0 0 0
≤ ≤
U(z ,z )u =u H(s+1)(Rn)
0 0 0 0
∈
while U(z,z )u H(s+1)(Rn) for all z [z ,Z].
0 0 0
∈ ∈
2 The thin-slab propagator. Regularity proper-
ties.
We follow the terminology introduced in [9, Sections 25.4-5] for FIOs with
complex phase. Let z′,z [0,Z] with z′ z and let ∆ := z′ z. Define
φ C∞(X′ X Rn)∈as ≥ −
(z′,z)
∈ × ×
(2.8) φ (x′,x,ξ):= x′ xξ +i∆a (z,x′,ξ)
(z′,z) 1
h − | i
= x′ xξ +∆b (z,x′,ξ)+i∆c (z,x′,ξ).
1 1
h − | i
Remark 2.1. The function φ is assumed to be homogeneous of degree 1
(z′,z)
only when ξ 1. This however is not an obstacle to the subsequent analysis,
| |≥
e.g., FIO properties,since to define such operatorsthe phase function need not
be homogeneous of degree 1 for small ξ . In the subsequent results concerning
| |
the phase function and FIOs one will then assume that ξ is large enough, i.e.,
| |
ξ 1.
| |≥
Lemma 2.2. φ is anon-degeneratecomplex phase function of positive type
(z′,z)
(at any point (x′,x ,ξ ) where ∂ φ =0).
0 0 0 ξ (z′,z)
Proof. Note that, by Assumption 1.1, Im(φ ) 0 and φ is homogeneous
(z′,z)
≥
of degree 1; ∂ φ = 0 implies ξ = 0. Thus, φ is a phase function of positive
x
type. Inspecting the partial derivatives of ∂ φ w.r.t. x we conclude that the
ξ
differentials d(∂ φ),...,d(∂ φ) are linearly independent.
ξ1 ξn
6
With a (z,.) S0(X Rn) we have exp[ ∆a (z,.)] S0(X Rn) by Lemma
0 0
∈ × − ∈ ×
18.1.10 in [8]. We define
(2.9) g (x,ξ):=exp[ ∆a (z,x,ξ)].
(z′,z) 0
−
We shall keep this notation (for this symbol and others in the sequel) but it
will be useful however to consider this symbol to depend on the parameters z
and∆insteadofz andz′ inthe followinganalysis. Note thatg is bounded
(z′,z)
w.r.t. z and C∞ w.r.t. ∆ with values in S0(X Rn). Hence, we may define a
distribution kernel G (x′,x) D′(X′ X) ×
(z′,z)
∈ ×
G (x′,x)= exp[i x′ xξ ]exp[ ∆a(z,x′,ξ)]d−ξ
(z′,z)
h − | i −
Z
= exp[iφ (x′,x,ξ)]g (x′,ξ)d−ξ
(z′,z) (z′,z)
Z
as an oscillatory integral. We denote the associated operator by . This
(z′,z)
G
operatorisoftenreferredtoasthethin-slabpropagator(seee.g.[3,2]). Weshow
that is a global FIO in Rn.
(z′,z)
G
Define α:=(x′,x,ξ′,ξ) and
u (α,θ)=∂ φ (x′,x,θ)+ξ = θ +ξ ,
θj xj (z′,z) j − j j
u (α,θ)=∂ φ (x′,x,θ) ξ′ =θ ξ′ +i∆∂ a (z,x′,θ),
ξj x′j (z′,z) − j j − j xj 1
u (α,θ)=∂ φ (x′,x,θ)=x′ x +i∆∂ a (z,x′,θ),
xj θj (z′,z) j − j ξj 1
where j = 1,...,n. We denote by Jˆ the ideal in C∞(R5n) generated by
(z′,z)
the functions u ,u , and u , and we let J be the subset of the functions
θj ξj xj (z′,z)
in Jˆ that are independent of θ.
(z′,z)
Lemma 2.3. There exists ∆ > 0, such that, for all z′,z [0,Z], with z′ > z
1
∈
and ∆=z′ z ∆ , the ideal J is generated by the functions
1 (z′,z)
− ≤
(2.10) v (α)=∂ φ (x′,x,ξ) ξ′,
ξj x′j (z′,z) − j
=ξ ξ′ +i∆∂ a (z,x′,ξ)=u ,
j − j xj 1 ξj|θ=ξ
v (α)=∂ φ (x′,x,ξ)=x′ x +i∆∂ a (z,x′,ξ)=u
xj ξj (z′,z) j − j ξj 1 xj|θ=ξ
j =1,...,n.
SomeofthekeyargumentsoftheproofareclosetothatintheproofofTheorem
25.4.4 in [9].
Proof. The ideal Jˆ is also generated by the functions
(z′,z)
u , u˜ :=u +u =ξ ξ′ +i∆∂ a (z,x′,θ), u ,
θj ξj θj ξj j − j xj 1 xj
j = 1,...,n. We define ν := (x′,ξ′,θ), µ := (x,ξ). We set a point (ν ,µ ) =
0 0
(x′,ξ′,θ ,x ,ξ )wherethese generatorsvanishandweworkina neighborhood
0 0 0 0 0
of this point. (Note that θ =ξ .) Since z a (z,.) S1(X Rn) is bounded
0 0 1
7→ ∈ ×
we have that ∆ >0 such that for 0 ∆ ∆ , and all z [0,Z],
1 1
∃ ≤ ≤ ∈
det∂(u ,...,u ,u˜ ,...,u˜ ,u ,...,u )/∂ν =0.
θ1 θn ξ1 ξn x1 xn 6
7
By Theorem 7.5.7 in [10] we have
x′ x u x˜(µ)
ξ′−ξ = Q(ν,µ) P(ν,µ) u˜x + ξ˜(µ)
 −θ  (cid:18) 0 In (cid:19) uξθ   ξ 
−
     
where P is a C∞ 2n n matrix and Q is a C∞ 2n 2n matrix and the
functions x˜, ξ˜ are also×C∞ in a neighborhood of (ν ,µ×). As the functions
0 0
w (ν,µ):=x′ x x˜(µ),w (ν,µ):=ξ′ ξ ξ˜(µ),w (ν,µ):=θ ξhavelinearly
x ξ θ
− − − − −
independent differentials, Lemma 7.5.8 in [10] proves that they generate Jˆ
(z′,z)
and the proof of that lemma shows that Q is invertible in a neighborhood of
(ν ,µ ). Letting θ =ξ we have
0 0
w (ν,µ) u (x′,x,ξ) v (α)
Q(x′,x,ξ,µ)−1 x = x = x .
w (ν,µ) u˜ (x′,x,ξ) v (α)
ξ ξ ξ
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
We thus obtained that Jˆ is generated by the functions u , v , v , j =
(z′,z) θj xj ξj
1,...,n. We then see that J is generated by v , v , j =1,...,n. In fact,
(z′,z) xj ξj
using Theorem7.5.7in [10]again,anyC∞ functionh(α) canbe locallywritten
in the form
h(α)= (h (α′,µ)v (α′,µ)+h (α′,µ)v (α′,µ))+r(µ),
xj xj ξj ξj
1≤i≤n
X
with α′ =(x′,ξ′) provided that 0 ∆ ∆ . If h J then r J and
1 (z′,z) (z′,z)
Lemma 7.5.10 in [10] implies that≤N ≤N, C >∈0: ∈
N
∀ ∈ ∃
r(µ) C max(Im x˜(µ), Im ξ˜(µ))N,
N
≤ | | | |
locally. Therefore, Theorem 7.5.12 in [10] yields r I(w ,w ) = I(v ,v );
x ξ x ξ
∈
which in turn implies g I(v ,v ) and thereby completes the proof.
x ξ
∈
AsthePoissonbrackets(forthesymplectic2-formσ′ σ onT∗(X′ X),where
− ×
σ′ and σ are the symplectic 2-forms on T∗(X′) and T∗(X) respectively) of
any two of the functions in (2.10) vanish identically we obtain that the ideal
generated by these functions is globally a conic canonical ideal in the sense of
[9, Definition 25.4.1. and Section 25.5]. The phase function φ thus defines
(z′,z)
J(z′,z) intheneighborhoodofanypointofJ(z′,z)R: itthusgloballydefinesJ(z′,z),
which is then of positive type. Therefore the operator is a global FIO
(z′,z)
G
with complex phase (see Definitions 25.4.9. and 25.5.1. in [9]).
Proposition 2.4. There exists ∆ >0 such that if 0 ∆=z′ z ∆ then
1 1
≤ − ≤
the operator is a global Fourier integral operator with complex phase and
(z′,z)
G
G I0(X′ X,(J )′,Ω1/2 ).
(z′,z) ∈ × (z′,z) X′×X
We denote the half density bundle on X′ X by Ω1/2 . Note that (J )′
× X′×X (z′,z)
stands for the twisted canonical ideal, i.e. a Lagrangian ideal (see Section 25.5
in [9]).
Note that, with the following analysis,we could alsoconsider as a global
(z′,z)
FIOwithrealphasewithamplitudeinS0(X′ X Rn)(seee.gG. [20]). However
1 × ×
2
8
such consideration would be rather technical as one usually restricts oneself to
thetypeSmwithρ> 1 forFIOs(seetheremarkattheendofSection25.1in[9];
ρ 2
seeAlso [18, pages391-392]). Viewing the thin-slabpropagator asa FIO
(z′,z)
G
with complex phase is alsoa goodframework to understand the propagationof
singularities in Part II. We shall however make this interpretation for in
(z′,z)
G
Proposition 2.25, below, to apply a result of Kumano-go [13, Theorem 2.5].
Wenowestablishsomeglobalcontinuitypropertiesoftheoperator stated
(z′,z)
G
in a slightly more general form (for similar results with global symbols see for
instance [13], where phase functions are real and other conditions are imposed
on the phase function).
Lemma 2.5. Let A be an FIO with a kernel of the form
K (x,y)= exp[iϕ(x,ξ) i y ξ ]σ (x,ξ)d−ξ D′(Rn Rn),
A A
− h | i ∈ ×
Z
whereσ Sm(Rn Rn)andϕ C∞(Rn Rn)issuchthatIm(ϕ(x,ξ)) 0and
A
ϕishomo∈geneous o×fdegree1in∈ξ, for ξ la×rgeenough, and∂ ϕ S1(Rn≥ Rn).
| | xi ∈ ×
Furthermore, for all i = 1,...,n we assume ∂ ϕ(x,ξ) = x +f (x,ξ) where
ξi i i
f S0(Rn Rn). Then A maps S into S continuously.
i
∈ ×
Proof. Let u S. We then have
∈
Au(x) σ (x,ξ)(1+ ξ )−m (1+ ξ )muˆ(ξ) d−(ξ)
A
| |≤ | | | || | | |
Z
C sup σ (x,ξ)(1+ ξ )−m sup (1+ ξ )m+n+1uˆ(ξ),
A
≤ ξ∈Rn| | | |ξ∈Rn| | | |
whereC = (1+ ξ )−n−1d−ξ. TheoperatorAishencewelldefinedfromS into
C(Rn). If we diffe|re|ntiate we obtain
R
D Au(x)= exp[iϕ(x,ξ)](∂ ϕ(x,ξ)σ (x,ξ) i∂ σ (x,ξ))uˆ(ξ)d−ξ.
xi xi A − xi A
Z
Noting that ∂ ϕ(x,ξ)σ (x,ξ) i∂ σ (x,ξ) Sm+1(Rn Rn) we similarly
xi A − xi A ∈ ×
have
D Au(x) C sup (1+ ξ )m+n+2uˆ(ξ)
| xi |≤ ξ∈Rn| | | |
C′ sup xαDβu(x) for some α,β 0.
≤ x∈Rn| x | ≥
Iterating we find that Au C∞(Rn). Integrating by parts we also have
∈
A(x u)(x)= exp[iϕ(x,ξ)](∂ ϕ(x,ξ)σ (x,ξ) i∂ σ (x,ξ))uˆ(ξ)d−(ξ)
j ξi A − ξi A
Z
=x Au(x)+ exp[iϕ(x,ξ)](f (x,ξ)σ (x,ξ) i∂ σ (x,ξ))uˆ(ξ)d−(ξ).
j i A − ξi A
Z
Since f (x,ξ)σ (x,ξ) i∂ σ (x,ξ) Sm(Rn Rn) we obtain
i A − ξi A ∈ ×
x Au(x) C sup xαDβu(x) +C sup xα′Dβ′u(x),
| j |≤ x∈Rn| x | x∈Rn| x |
9
for some α,α′,β,β′ 0. Similar estimates hold for xαDβAu(x) because of
the hypothesis made≥on f , i=1,...,n. The operator|A thxus map|s S into S
i
continuously.
To show continuity from S′ into S′ we shall need the following lemma.
Lemma 2.6. Let j,k non-negative integers, u S(Rn), f Ck+1(Rn) such
∈ ∈
that
0 Imf(x) C , x Rn, f(r)(x) C , x Rn, 1 r k+1.
0 r
≤ ≤ ∈ | |≤ ∈ ≤ ≤
Then we have
(2.11) ωj+k u(x)(Imf(x))jexp[iωf(x)] dx
(cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) C sup Dαu(x)(f′(x(cid:12))2+Imf(x))|α|/2−k, ω >0,
(cid:12) ≤ x∈Rn| | | (cid:12)|
|αX|≤k
where the constant C is bounded when the function f stays in a domain of
Ck+1(Rn) where C , C ,...,C can be chosen bounded.
0 1 k+1
Proof. TheproofisthesameasthatofTheorem7.7.1in[10]whereu Ck(Rn).
∈ 0
Infactthefurtherassumptionsonf madehereallowtogiveglobalboundsthat
are needed since u S in the present case.
∈
Lemma 2.7. Let A be an FIO with a kernel of the form:
K (x,y)= exp[i x y ξ +iγ(x,ξ)]σ (x,ξ)d−(ξ) D′(Rn Rn),
A A
h − | i ∈ ×
Z
where σ Sm(Rn Rn) and γ S1(Rn Rn) is such that Im(γ(x,ξ)) 0,
A
∈ × ∈ × ≥
and γ is homogeneous of degree 1 in ξ, for ξ large enough. Furthermore, we
| |
assume that there exists d 0 such that
≥
(2.12) Re(∂ γ(x,ξ)) d<1, x Rn, ξ Rn, ξ =1.
x
| |≤ ∈ ∈ | |
Then A maps S′ into S′ continuously.
Observe that the differential of φ(x,ξ):= x y ξ +γ(x,ξ) does not vanish in
R2n Rn 0. The function φis thus acomphle−x ph|aisefunction. The differentials
× \
d(∂ φ),...,d(∂ φ) are linearly independent. Hence φ is a non degenerate
ξ1 ξn
complex phase function of positive type. Note that by (2.12) the function x
h −
y ξ +γ(x,ξ) is an operator phase function in the sense of [6, Definition 1.4.4.].
| i
Proof. Without loss of generality we may assume that γ is homogeneous of
degree 1 for ξ 1. Let At be the transpose of A and let u S, then.
| |≥ ∈
Atu(x)= exp[ i xξ ] exp[i y ξ +iγ(y,ξ)]σ (y,ξ)u(y)dyd−ξ
A
− h | i h | i
Z Z
Define
v(ξ,η)= exp[i y ξ +iγ(y,ξ)]σ (y,η)u(y)dy,
A
h | i
Z
10