Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces

0.26 MB·English

by  Mitsuru Sugimoto

BOUNDEDNESS PROPERTIES OF PSEUDO-DIFFERENTIAL
OPERATORS AND CALDERO´N-ZYGMUND OPERATORS
7
ON MODULATION SPACES
0
0
2
MITSURUSUGIMOTOANDNAOHITOTOMITA
n
a
J Abstract. In this paper, we study the boundedness of pseudo-differential

9 operators with symbols in Sm on the modulation spaces Mp,q. We discuss
ρ,δ
2 theordermfortheboundednessOp(Sm )⊂L(Mp,q(Rn))tobetrue. Wealso
ρ,δ
provetheexistenceofaCaldero´n-Zygmundoperatorwhichisnotboundedon

] themodulationspaceMp,q withq6=2. Thisunboundedness isstilltrueeven
A
if we assume a generalized T(1) condition. These results are induced by the
F unboundedness ofpseudo-differentialoperatorsonMp,q whosesymbolsareof
h. theclassS10,δ with0<δ<1.
t
a
m
[

1. Introduction
1
v The modulation spaces Mp,q introduced by Feichtinger [5, 6] are fundamental
2
function spaces of time-frequency analysis which is originated in signal analysis
3
8 or quantum mechanics. See also [7] or Triebel [22]. Recently these spaces have
1 been also recognizedas a useful tool for the theory of pseudo-differentialoperators
0 (see Gr¨ochenig [10]). The objective of this paper is to discuss the boundedness
7
of pseudo-differential operators and also the unboundedness of Caldero´n-Zygmund
0
operators on modulation spaces.
/
h The boundedness of pseudo-differential operatorson the modulation spaces was
t
a studied by many authors, for example, Gr¨ochenig and Heil [11], Tachizawa [19],
m and Toft [20]. They proved that pseudo-differential operators with symbols in
: some modulation space are Mp,q-bounded, and as a corollary we have the Mp,q-
v
boundedness of pseudo-differential operators with symbols in Ho¨rmander’s class
i
X S0 . A pioneering work of Sjo¨strand [15] should be also mentioned here which
0,0
r proved the L2-boundedness by introducing a symbol class based on the spirit of
a
time-frequency analysis. We note here that L2 =M2,2.
Inthis paper,weconsiderthecaseofgeneralsymbolclassesSm . Firstwerecall
ρ,δ
the result of Caldero´n and Vaillancourt [1], which shows that pseudo-differential
operators with symbols in S0 with 0 < δ < 1 (hence in S0 with 0 δ ρ 1,
δ,δ ρ,δ ≤ ≤ ≤
δ <1)areL2-boundedbyreducingthem tothe caseofS0 . The proofwascarried
0,0
out by “dilation argument” based on the fact
f(λx) L2(R) =λ−n/2 f(x) L2(R), λ>0.
k k k k
2000 Mathematics Subject Classification. 42B20,42B35,47G30.
Keywords andphrases. Caldero´n-Zygmundoperators,modulationspaces,pseudo-differential
operators.
1
2 MITSURUSUGIMOTOANDNAOHITOTOMITA
Wecanconcludethattheoperatornormofσ(X,D)isequaltothatofσ(λX,λ−1D)
fromthisequality. Byfollowingthesameargument,wecanalsoexpectthatpseudo-
differential operators with symbols in S0 with 0 < δ < 1 are Mp,q-bounded.
δ,δ
However,the dilationpropertyofthe modulationspacesMp,q is ratherdifferentas
was investigated in author’s previous paper [17], and due to the property, we can
give the following negative answer to this expectation:
Theorem 1.1. Let 1 < q < , m R, 0 δ ρ 1 and δ < 1. Then we have
the boundedness Op(Sm ) ∞(M2,q∈(Rn)) if≤and≤only≤if m 1/q 1/2δn.
ρ,δ ⊂L ≤−| − |
The “only if” part of Theorem 1.1 is a restricted version of Theorem 3.6 which
says that Op(Sm ) (Mp,q(Rn)) only if m 1/q 1/2δn. This result was
ρ,δ ⊂ L ≤ −| − |
firstprovedin author’spaper [18]. The “if” partis a restrictedversionofTheorem
3.1 in Section 3 which treats the general Mp,q-boundedness with m m(p,q),
≤
where m(p,q) is an index such that m(2,q)= 1/q 1/2δn. We remark that the
boundedness Op(Sm ) (Mp,q(Rn)) with m−<| 1−/q |1/2δn (Proposition 3.2)
ρ,δ ⊂L −| − |
is a straightforwardconsequence ofthe dilation argumentstated inthe above if we
directly use the dilation property (Proposition 2.1) of the modulation spaces. The
main contribution of Theorem 1.1 is the boundedness result with the critical order
for m.
Theorem 1.1 also indicates a difference between Lp spaces and the modulation
spaces Mp,q. Fefferman [4] proved that Op(Sm ) (Lp(Rn)) if m 1/p
ρ,δ ⊂ L ≤ −| −
1/2(1 ρ)n,0 δ ρ 1andδ <1. Moreover,itisknownthatthisorderofmis
| − ≤ ≤ ≤
critical([16,Chapter7,5.12]). We remarkthat,fortheLp-boundednessofpseudo-
differential operators with symbols in Sm , the critical order of m is determined
ρ,δ
by ρ. On the other hand, Theorem 1.1 says that, for the Mp,q-boundedness, the
critical order of m is determined by δ (at least, in the case p=2).
Bydevelopingthe investigationofpseudo-differentialoperators,wecanalsodis-
cuss the boundedness property of Caldero´n-Zygmundoperatorsonthe modulation
spaces. A Caldero´n-Zygmund operator is an L2-bounded linear mapping whose
distributional kernel is a function K(x,y) outside the diagonal x=y satisfying
{ }
K(x,y) C x y −n,
| |≤ | − |
x x′ ǫ
K(x,y) K(x′,y) C | − | for x y >2x x′ ,
| − |≤ x y n+ǫ | − | | − |
| − |
y y′ ǫ
K(x,y) K(x,y′) C | − | for x y >2y y′ ,
| − |≤ x y n+ǫ | − | | − |
| − |
where0<ǫ 1. ItiswellknowthatCaldero´n-ZygmundoperatorsareLp-bounded
≤
for 1 < p < (see, for example, [3, Theorem 5.10]). In this paper, we consider
∞
the problem “Are Caldero´n-Zygmund operators Mp,q-bounded?”. It is easy to
show that the Lp-boundedness of the operators of convolution type induces the
Mp,q-boundedness. Hence,Caldero´n-Zygmundoperatorsofconvolutiontype(Riesz
transforms, for example) are always Mp,q-bounded. We will prove that it is not
generally true if remove the convolution type assumption. The following theorem
is a simplified version of Theorem 4.1 in Section 4.
Theorem 1.2. Let 1<p,q < . If q =2, then there exists a Caldero´n-Zygmund
operator which is not bounded o∞n Mp,q(6Rn).
PSEUDO-DIFFERENTIAL OPERATORS ON MODULATION SPACES 3
Theorem1.2is a directconsequenceofthe unboundedness ofpseudo-differential
operators on the modulation space Mp,q whose symbols are of the class S0 with
1,δ
0 < δ < 1 (see Theorem 3.6). In fact, it is know that such pseudo-differential
operatorsare Caldero´n-Zygmundoperators(see [16, Chapter 7, Sections 1 and2]).
Onthe otherhand,weknowsomeboundednessresultsonthe BesovspacesB˙s,q
p
and Triebel-Lizorkin spaces F˙s,q. By the same reason described above, Caldero´n-
p
Zygmund operators of convolution type are alwaysB˙s,q-bounded. Furthermore, in
p
the case of non-convolutiontype, the following generalizedT(1)condition is useful
to discuss the B˙s,q-boundedness of T:
p
(1.1) T(P)=0 for all polynomials P such that degP ℓ.
≤
This condition, together with an extra condition on the smoothness of the kernel
of T and the weak boundedness property which will be defined in Section 4, induce
the B˙s,q-boundedness of T, where s is determined by the order of the polynomials
p
ℓandthesmoothnessorderofthekernel(seeLemari´e[13],MeyerandCoifman[14,
p.114]). For the F˙s,q-boundedness, we need more conditions on the transpose T∗
p
of T, that is, a smoothness condition of the kernel of T∗ and the condition
(1.2) T∗(P)=0 for all polynomials P such that degP ℓ∗
≤
(see Frazier,TorresandWeiss [8]). We remarkthat the boundedness onthe homo-
geneous spaces (B˙s,q,F˙s,q) induces that on the inhomogeneous spaces (Bs,q,Fs,q)
p p p p
under suitable conditions.
On account of these results, we can expect the Mp,q-boundedness of Caldero´n-
ZygmundoperatorsT whichsatisfy the smoothnessconditions onthe kernels,con-
ditions (1.1)-(1.2), and the weak boundedness property. But even if we assume
thesereasonableconditions,wecanneverprovetheMp,q-boundednessofCaldero´n-
Zygmund operators. See Theorem 4.1 for the detailed statement.
2. Preliminaries
Let (Rn) and ′(Rn) be the Schwartz spaces of all rapidly decreasing smooth
S S
functionsandtempereddistributions,respectively. WedefinetheFouriertransform
f and the inverse Fourier transform −1f of f (Rn) by
F F ∈S
1
f(ξ)=f(ξ)= e−iξ·xf(x)dx and −1f(x)= eix·ξf(ξ)dξ.
F ZRn F (2π)n ZRn
We introdbuce the modulation spaces basedon Gr¨ochenig[9]. Fix a function γ
(Rn) 0 (called the window function). Then the short-time Fourier transform∈
SV f of\f{ } ′(Rn) with respect to γ is defined by
γ
∈S
V f(x,ξ)=(f,M T γ) for x,ξ Rn,
γ ξ x
∈
where M γ(t)=eiξ·tγ(t), T γ(t)=γ(t x) and (, ) denotes the inner product on
ξ x
L2(Rn). We note that, for f ′(Rn),−V f is con·tin·uous on R2n and V f(x,ξ)
γ γ
∈S | |≤
C(1+ x + ξ )N for someconstantsC,N 0([9,Theorem11.2.3]). Let1 p,q
. Th|en| th|e|modulation space Mp,q(Rn)≥consists of all f ′(Rn) such t≤hat ≤
∞ ∈S
q/p 1/q
f = V f = V f(x,ξ)pdx dξ < .
Mp,q γ Lp,q γ
k k k k (ZRn(cid:18)ZRn| | (cid:19) ) ∞
4 MITSURUSUGIMOTOANDNAOHITOTOMITA
We note that M2,2(Rn) = L2(Rn) ([9, Proposition 11.3.1]) and Mp,q(Rn) is a Ba-
nach space ([9, Proposition 11.3.5]). The definition of Mp,q(Rn) is independent of
the choice of the window function γ (Rn) 0 , that is, different window func-
tions yield equivalent norms ([9, Pro∈poSsition \11{.3}.2]). We denote by (Mp,q(Rn))
the space of all bounded linear operators on Mp,q(Rn). L
Inordertostatethedilationpropertyofthemodulationspaces,weintroducethe
indices. For 1 p , p′ is the conjugate exponent of p (that is, 1/p+1/p′ =1).
≤ ≤∞
We define subsets of (1/p,1/q) [0,1] [0,1] in the following way:
∈ ×
I : min(1/q,1/2) 1/p, I∗ : max(1/q,1/2) 1/p,
1 ≥ 1 ≤
I : min(1/p,1/p′) 1/q, I∗ : max(1/p,1/p′) 1/q,
2 ≥ 2 ≤
I : min(1/q,1/2) 1/p′, I∗ : max(1/q,1/2) 1/p′.
3 ≥ 3 ≤
See the following figure:
1/q 1/q
6 6
1 1
@ (cid:0)
@ I∗ (cid:0)
I I 2
1 3 @ (cid:0)
@(cid:0)
1/2 1/2
(cid:0)@
(cid:0) @ I∗ I∗
I 3 1
(cid:0) 2 @
(cid:0) @ - -
0 1/2 1 1/p 0 1/2 1 1/p
µ (p,q) µ (p,q)
1 2
We introduce the indices
µ (p,q)=max 0,1/q min(1/p,1/p′) 1/p,
1
{ − }−
µ (p,q)=min 0,1/q max(1/p,1/p′) 1/p.
2
{ − }−
Then
2/p+1/q if (1/p,1/q) I ,
1
− ∈
µ (p,q)= 1/p if (1/p,1/q) I ,
1  2
− ∈
1/q 1 if (1/p,1/q) I3,
− ∈
−2/p+1/q if (1/p,1/q)∈I1∗,
µ (p,q)= 1/p if (1/p,1/q) I∗,
2 − ∈ 2
1/q 1 if (1/p,1/q) I∗.
− ∈ 3
Wedefine the dilationoperatorΛa byΛaf(x)=f(ax),wherea>0. The following
proposition plays an important role in the proofs of Theorem 3.1 and Proposition
3.2.
Proposition 2.1 ([17, Theorem 1.1]). Let 1 p,q . Then the following are
≤ ≤ ∞
true:
(1) There exists a constant C >0 such that
Λaf Mp,q Canµ1(p,q) f Mp,q for all f Mp,q(Rn) and a 1.
k k ≤ k k ∈ ≥
PSEUDO-DIFFERENTIAL OPERATORS ON MODULATION SPACES 5
(2) There exists a constant C >0 such that
Λaf Mp,q Canµ2(p,q) f Mp,q for all f Mp,q(Rn) and 0<a 1.
k k ≤ k k ∈ ≤
The optimality of the power of a in Proposition 2.1 is also discussed in [17].
3. The boundedness of pseudo-differential operators
Let m R and 0 δ ρ 1. The symbol class Sm consists of all σ
∈ ≤ ≤ ≤ ρ,δ ∈
C∞(Rn Rn) such that
×
∂α∂βσ(x,ξ) C (1+ ξ )m−ρ|α|+δ|β|
| ξ x |≤ α,β | |
for all α,β ∈ Zn+ = {0,1,...}n. We denote by |·|Sρm,δ,N, N = 0,1,..., the semi-
norms on Sm , that is,
ρ,δ
|σ|Sρm,δ,N =|α+mβa|≤xNx,sξu∈pRn(1+|ξ|)−(m−ρ|α|+δ|β|)|∂ξα∂xβσ(x,ξ)|.
For σ Sm , the pseudo-differential operator σ(X,D) is defined by
∈ ρ,δ
1
σ(X,D)f(x)= eix·ξσ(x,ξ)f(ξ)dξ
(2π)n ZRn
forf (Rn). We denote by Op(Sm ) the class ofallpsebudo-differentialoperators
∈S ρ,δ
with symbols in Sm . Given a symbol σ Sm with δ < 1, the symbol σ∗ defined
ρ,δ ∈ ρ,δ
by
1
(3.1) σ∗(x,ξ)=Os- e−iy·ζσ(x+y,ξ+ζ)dydζ
(2π)n ZRnZRn
1
= lim e−iy·ζχ(ǫy,ǫζ)σ(x+y,ξ+ζ)dydζ
ǫ→0(2π)n ZRnZRn
satisfies σ∗ Sm and
∈ ρ,δ
(3.2) (σ(X,D)f,g)=(f,σ∗(X,D)g) for all f,g (Rn),
∈S
where χ (R2n) satisfies χ(0,0) = 1 ([12, Chapter 2, Theorem 2.6]). Note
that oscill∈atoSry integrals are independent of the choice of χ (R2n) satisfying
∈ S
χ(0,0) = 1 ([12, Chapter 1, Theorem 6.4]), and the derivatives of σ∗(x,ξ) can be
written as
1
(3.3) ∂α∂βσ∗(x,ξ)=Os- e−iy·ζ(∂α∂βσ)(x+y,ξ+ζ)dydζ
ξ x (2π)n ZRnZRn ξ x
in virtue of [12, Chapter 1, Theorem 6.6] (see also [12, p.70, (2.23)]).
Our main result on the boundedness of pseudo-differential operators is the fol-
lowing:
Theorem 3.1. Let 1 < p,q < , m R, 0 δ ρ 1 and δ < 1. If
m (µ (p,q) µ (p,q))δn, then∞Op(Sm∈) (M≤p,q(R≤n)).≤
≤− 1 − 2 ρ,δ ⊂L
In order to clarify µ (p,q) µ (p,q), we divide I ,...,I∗ in the following way:
1 − 2 1 3
J : (I I∗) (I I∗), J : (I I∗) (I I∗), J : (I I∗) (I I∗).
1 1∩ 2 ∪ 2∩ 1 2 1∩ 3 ∪ 3∩ 1 3 2∩ 3 ∪ 3∩ 2
See the following figure:
6 MITSURUSUGIMOTOANDNAOHITOTOMITA
1/q
6
1
@ (cid:0)
J J
@ 1 3 (cid:0)
@ (cid:0)
@(cid:0)
1/2 J2 (cid:0)@ J2
(cid:0) @
(cid:0)J J @
3 1
(cid:0) @ -
0 1/2 1 1/p
µ (p,q) µ (p,q)
1 2
−
Then we have
1/p 1/q if (1/p,1/q) J ,
1
| − | ∈
µ (p,q) µ (p,q)= 2/p 1 if (1/p,1/q) J ,
1 2  2
− | − | ∈
1/p+1/q 1 if (1/p,1/q) J3.
| − | ∈
2Letaψnd0,ψψ ∈(ξ)S+(Rn)∞beψsu(2c−hjtξh)a=ts1upfoprψa0ll⊂ξ {ξR:n|ξ.|S≤et2ψ}, s=upψp(2ψ−⊂j ){iξf:j1/21.≤T|ξh|en≤
} 0 j=1 ∈ j · ≥
∞
P
(3.4) ∂αψ C 2−j|α|, suppψ 2j−1 ξ 2j+1 , ψ 1.
j L∞ α j j
k k ≤ ⊂{ ≤| |≤ } ≡
j=0
X
The following weak form of Theorem 3.1 (when δ < 1) is a straightforward conse-
quence of the dilation property of the modulation spaces (Proposition 2.1).
Proposition 3.2. Let 1 < p,q < , m R and 0 δ ρ 1. If m <
(µ (p,q) µ (p,q))δn, then Op(Sm∞) (∈Mp,q(Rn)). ≤ ≤ ≤
− 1 − 2 ρ,δ ⊂L
Proof. Our proof is based on that of [16, Chapter 7, Theorem2]. Let σ Sm . By
∈ ρ,δ
the decomposition (3.4), we have
∞ ∞
σ(x,ξ)= σ(x,ξ)ψ (ξ)= σ (x,ξ).
j j
j=0 j=0
X X
Since σ (X,D)=Λ σ (2−jδX,2jδD)Λ , by Proposition 2.1, we see that
j 2jδ j 2−jδ
∞
σ(X,D)f σ (X,D)f
Mp,q j Mp,q
k k ≤ k k
j=0
X
∞
= Λ σ (2−jδX,2jδD)Λ f
k 2jδ j 2−jδ kMp,q
j=0
X
∞
2jδn(µ1(p,q)−µ2(p,q)) σj(2−jδX,2jδD) L(Mp,q) f Mp,q
≤ k k k k
j=0
X
for all f (Rn). Since Sm Sm and 1+2jδ ξ 2j on suppψ (2jδ ), if we set
∈ S ρ,δ ⊂ δ,δ | |∼ j ·
τ (x,ξ)=2−jmσ (2−jδx,2jδξ), then for any α,β Zn
j j ∈ +
∂α∂βτ (x,ξ) C σ
| ξ x j |≤ α,β| |Sρm,δ,|α+β|
PSEUDO-DIFFERENTIAL OPERATORS ON MODULATION SPACES 7
forallj Z . Hence,byOp(S0 ) (Mp,q(Rn))andm+(µ (p,q) µ (p,q))δn<
∈ + 0,0 ⊂L 1 − 2
0, we have
∞
2jδn(µ1(p,q)−µ2(p,q)) σj(2−jδX,2jδD) L(Mp,q) f Mp,q
k k k k
j=0
X
∞
= 2j(m+(µ1(p,q)−µ2(p,q))δn) τj(X,D) L(Mp,q) f Mp,q
k k k k
j=0
X
∞
C 2j(m+(µ1(p,q)−µ2(p,q))δn) f Mp,q =C f Mp,q
≤  k k k k
j=0
X
for all f (Rn).The proof is complete.  (cid:3)
∈S
InviewofProposition3.2,thenon-trivialpartofTheorem3.1istheboundedness
with the critical order m = (µ (p,q) µ (p,q))δn. The rest of this section is
1 2
devoted to the proof of it. Le−t ϕ (Rn−) be such that
∈S
(3.5) suppϕ [ 1,1]n and ϕ(ξ k) 1.
⊂ − − ≡
k∈Zn
X
By the decompositions (3.4) and (3.5), we have
∞ ∞
1 ψ (ξ)= ψ (ξ) ϕ(2−jδy k) ϕ(2−jδξ ℓ) .
j j
≡ j=0 j=0 k∈Zn − ! ℓ∈Zn − !
X X X X
Hence,
∞
(3.6) ϕ(2−jδy k)ϕ(2−jδξ ℓ)ψ (ξ)=1
j
− −
k∈Znℓ∈Znj=0
X X X
for all (y,ξ) Rn Rn. For σ Sm , we set
∈ × ∈ ρ,δ
(3.7) σk,ℓ(x,ξ)=ϕ(2−jδD k)σ(x,ξ)ϕ(2−jδξ ℓ)ψ (ξ),
j x− − j
where
ϕ(2−jδD k)σ(x,ξ)= −1[ϕ(2−jδ k) σ(,ξ)](x)
x− F1 ·− F1 ·
1
= eix·yϕ(2−jδy k) σ(y,ξ)dy,
(2π)n ZRn − F1
and −1 are the partialFourier transformand inverseFourier transformin the
F1 F1
first variable, respectively.
Lemma 3.3. Let 1 p , m R, 0 δ ρ 1 and σ Sm . Then there
≤ ≤ ∞ ∈ ≤ ≤ ≤ ∈ ρ,δ
exists a constant C >0 such that
σk,ℓ(X,D)f C2jm(1+ k )−n−1 f
k j kLp ≤ | | k kLp
for all j Z , k,ℓ Zn and f (Rn), where σk,ℓ is defined by (3.7).
∈ + ∈ ∈S j
Proof. Using σk,ℓ(X,D)=Λ σk,ℓ(2−jδX,2jδD)Λ , we have
j 2jδ j 2−jδ
σk,ℓ(X,D)f =2−jδn/p σk,ℓ(2−jδX,2jδD)Λ f
k j kLp k j 2−jδ kLp
8 MITSURUSUGIMOTOANDNAOHITOTOMITA
for all f (Rn). Set
∈S
1
Kk,ℓ(x,y)= ei(x−y)·ξσk,ℓ(2−jδx,2jδξ)dξ.
j (2π)n ZRn j
Since
σk,ℓ(2−jδX,2jδD)Λ f(x)= Kk,ℓ(x,y)(Λ f)(y)dy,
j 2−jδ j 2−jδ
ZRn
if
(3.8) Kk,ℓ(x,y) C2jm(1+ k )−n−1(1+ x y )−n−1
| j |≤ | | | − |
for all j Z and k,ℓ Zn, then
+
∈ ∈
σk,ℓ(X,D)f C2−jδn/p2jm(1+ k )−n−1 [(1+ )−n−1] Λ f
k j kLp ≤ | | k |·| ∗| 2−jδ |kLp
C2−jδn/p2jm(1+ k )−n−1 Λ f
≤ | | k 2−jδ kLp
=C2jm(1+ k )−n−1 f
Lp
| | k k
for allj Z , k,ℓ Zn andf (Rn). This is the desiredresult. We prove(3.8).
+
∈ ∈ ∈S
Set τ (x,ξ)=2−jmσ(2−jδx,2jδξ)ψ (2jδξ). Since
j j
σk,ℓ(x,ξ)= eik·(2jδx−z)Φ(2jδx z)σ(2−jδz,ξ)ψ (ξ)dz ϕ(2−jδξ ℓ),
j (cid:18)ZRn − j (cid:19) −
we have
2jm
Kk,ℓ(x,y)= ei(x−y)·ξ eik·(x−z)Φ(x z)τ (z,ξ)dz ϕ(ξ ℓ) dξ,
j (2π)n ZRn (cid:26)(cid:18)ZRn − j (cid:19) − (cid:27)
where Φ= −1ϕ. On the other hand, for any α,β Zn,
F ∈ +
∂α∂βτ (x,ξ) C σ
| ξ x j |≤ α,β| |Sρm,δ,|α+β|
for all j Z . Let α,β Zn be such that α, β n+1. Using
∈ + ∈ + | | | |≤
(x y)αkβKk,ℓ(x,y)
− j
=C 2jm ∂αei(x−y)·ξ ∂βeik·(x−z) Φ(x z)τ (z,ξ)dz ϕ(ξ ℓ)dξ
α,β ZRn(cid:16) ξ (cid:17)(cid:26)ZRn(cid:16) z (cid:17) − j (cid:27) −
=C 2jm Cα1,α2
α,β β1,β2
αβ11++Xαβ22==βα
ei(x−y)·ξ eik·(x−z)∂β1Φ(x z)∂α1∂β2τ (z,ξ)dz ∂α2ϕ(ξ ℓ)dξ,
×ZRn (cid:26)ZRn − ξ z j (cid:27) −
we see that
(x y)αkβKk,ℓ(x,y) C σ sup ∂α˜ϕ sup ∂β˜Φ 2jm.
| − j |≤ α,β| |Sρm,δ,|α+β| α˜≤αk kL1 β˜≤βk kL1!
This implies (3.8). The proof is complete. (cid:3)
For 0 δ <1, we take a sufficiently large integer j such that
0
≤
(3.9) 2j0(1−δ)−3 √n.
≥
We recall that ϕ,ψ (Rn) satisfy ϕ [ 1,1]n and ψ ξ : 2−1 ξ 2 (see
∈ S ⊂ − ⊂ { ≤ | | ≤ }
(3.4) and (3.5)).
PSEUDO-DIFFERENTIAL OPERATORS ON MODULATION SPACES 9
Lemma 3.4. Let 0 δ < 1 and ℓ Zn. If there exists a positive integer j(ℓ)
≤ ∈ ≥
j +1 such that suppϕ(2−j(ℓ)δ ℓ) suppψ(2−j(ℓ) )= , then
0
·− ∩ · 6 ∅
suppϕ(2−jδ ℓ) suppψ(2−j )= for all j j(ℓ) j ,
0
·− ∩ · ∅ | − |≥
where j is defined by (3.9).
0
Proof. Since suppϕ(2−jδ ℓ) ξ 2jδℓ 2jδ√n andsuppψ(2−j ) 2j−1
·− ⊂{| − |≤ } · ⊂ { ≤
ξ 2j+1 , our assumption suppϕ(2−j(ℓ)δ ℓ) suppψ(2−j(ℓ) ) = gives
| |≤ } ·− ∩ · 6 ∅
(3.10) 2j(ℓ)−1 2j(ℓ)δ√n 2j(ℓ)δ ℓ 2j(ℓ)+1+2j(ℓ)δ√n.
− ≤ | |≤
Let j j(ℓ) j . We consider the case j j(ℓ) j . By (3.9), (3.10) and
0 0
| − | ≥ − ≤ −
j(ℓ) j +1, we see that
0
≥
2jδ ℓ 2jδ√n=2jδ−j(ℓ)δ+j(ℓ)δ ℓ 2jδ√n
| |− | |−
2jδ−j(ℓ)δ(2j(ℓ)−1 2j(ℓ)δ√n) 2jδ√n=2jδ+j(ℓ)(1−δ)−1 2jδ+1√n
≥ − − −
2jδ+j(ℓ)(1−δ)−1 2jδ+j0(1−δ)−2 >2jδ+j(ℓ)(1−δ)−1 2jδ+j(ℓ)(1−δ)−2
≥ − −
=2jδ+j(ℓ)(1−δ)−2 =2jδ+j(ℓ)(1−δ)−2−(j+1)+(j+1) =2{(j(ℓ)−j)(1−δ)−3}+j+1
2(j0(1−δ)−3)+j+1 2j+1√n 2j+1.
≥ ≥ ≥
Hence,
suppϕ(2−jδ ℓ) ξ 2jδ ℓ 2jδ√n ξ >2j+1
·− ⊂{| |≥ | |− }⊂{| | }
for all j j(ℓ) j . This implies suppϕ(2−jδ ℓ) suppψ(2−j ) = for all
0
− ≤ − ·− ∩ · ∅
j j(ℓ) j .
0
− ≤−
In the same way, we can prove
suppϕ(2−jδ ℓ) ξ 2jδ ℓ +2jδ√n ξ <2j−1
·− ⊂{| |≤ | | }⊂{| | }
for all j j(ℓ) j . This implies suppϕ(2−jδ ℓ) suppψ(2−j ) = for all
0
j j(ℓ) −j . ≥ ·− ∩ · ∅ (cid:3)
0
− ≥
Let η (Rn) be such that suppη is compact and η(ξ ν) C > 0
for all ξ ∈RSn. It is well known that | ν∈Zn − | ≥
∈ P
1/q
(3.11) f η(D ν)f q ,
k kMp,q ∼ ν∈Znk − kLp!
X
where η(D ν)f = −1[η( ν)f] (see, for example, [22]).
− F ·−
Proposition 3.5. Let 1 p , m R, 0 δ ρ 1, δ < 1 and σ Sm . If
≤ ≤ ∞b ∈ ≤ ≤ ≤ ∈ ρ,δ
m (µ (p, ) µ (p, ))δn, then there exists a constant C >0 such that
1 2
≤− ∞ − ∞
σ(X,D)f C f
Mp,∞ Mp,∞
k k ≤ k k
for all f (Rn).
∈S
aPsro(o3f..5)Inanvdiefw of ((3R.1n1)).,Bwye tehsteimdeactoemsuppoνs∈itZionnkϕ(3(D.6)−, wνe)(hσa(Xve,D)f)kLp,where ϕ is
∈S
∞
σ(x,ξ)= σk,ℓ(x,ξ)
j
k∈Znℓ∈Znj=0
X X X
j0 ∞
= σk,ℓ(x,ξ)+ σk,ℓ(x,ξ),
j j
kX∈ZnℓX∈ZnXj=0 kX∈ZnℓX∈Znj=Xj0+1
10 MITSURUSUGIMOTOANDNAOHITOTOMITA
where σk,ℓ(x,ξ) is defined by (3.7), and j is defined by (3.9). We only consider
j 0
the second sum since the estimate for the first sum can be carried out in a similar
way. By Lemma 3.4, we see that
∞
σk,ℓ(x,ξ)= σk,ℓ(x,ξ).
j j
kX∈ZnℓX∈Znj=Xj0+1 kX∈ZnℓX∈Zn j≥Xj0+1
|j−j(ℓ)|<j0
Since ϕ(2−jδ k) 2jδk+[ 2jδ,2jδ]n and
·− ⊂ −
[σk,ℓ(X,D)f](y)
F j
1
= [eix·ξϕ(2−jδD k)σ(x,ξ)]ϕ(2−jδξ ℓ)ψ(2−jξ)f(ξ)dξ
(2π)n ZRnFx→y x− −
1
= ϕ(2−jδ(y ξ) k) σ(y ξ,ξ)ϕ(2−jδξ ℓ)ψ(2−jξ)fb(ξ)dξ,
(2π)n ZRn − − F1 − −
we see that [σk,ℓ(X,D)f] 2jδ(k+ℓ)+[ 2jδ+1,2jδ+1]n. This gives b
F j ⊂ −
ϕ(D ν) σk,ℓ(X,D)f
−  j 
kX∈ZnℓX∈Zn j≥Xj0+1
 |j−j(ℓ)|<j0 
 
= ϕ(D ν)σk,ℓ(X,D)f.
− j
kX∈Zn ℓ∈ZnX,i=1,...,n j≥Xj0+1
|2jδ(ki+ℓi)−νi|≤2jδ+2 |j−j(ℓ)|<j0
Let η (Rn) be such that suppη is compact, η =1 on suppϕ and η(ξ
∈S | ν∈Zn −
ν) C > 0 for all ξ Rn. Note that σk,ℓ(x,ξ) = σk,ℓ(x,ξ)η(2−jδξ ℓ). By
| ≥ ∈ j j P −
Lemma 3.3, we have
∞
ϕ(D ν) σk,ℓ(X,D)f
(cid:13) −  j (cid:13)
(cid:13)(cid:13) kX∈ZnℓX∈ZnjX0+1 (cid:13)(cid:13)Lp
(cid:13)(cid:13)  ϕ(D(cid:13)(cid:13) ν)σk,ℓ(X,D)f
(cid:13)≤ k (cid:13)− j kLp
kX∈Zn ℓ∈ZnX,i=1,...,n j≥Xj0+1
|2jδ(ki+ℓi)−νi|≤2jδ+2 |j−j(ℓ)|<j0
C σk,ℓ(X,D)[η(2−jδD ℓ)f]
≤ k j − kLp
kX∈Zn ℓ∈ZnX,i=1,...,n j≥Xj0+1
|2jδ(ki+ℓi)−νi|≤2jδ+2 |j−j(ℓ)|<j0
C 2jm(1+ k )−n−1 η(2−jδD ℓ)f .
Lp
≤ | | k − k
kX∈Zn ℓ∈ZnX,i=1,...,n j≥Xj0+1
|2jδ(ki+ℓi)−νi|≤2jδ+2 |j−j(ℓ)|<j0
Sinceη(2−jδD ℓ)f =Λ η(D ℓ)Λ f,µ (p, )= 1/pandm+(µ (p, )
− 2jδ − 2−jδ 1 ∞ − 1 ∞ −
µ (p, ))δn 0, by (3.11) and Proposition 2.1, we see that
2
∞ ≤
2jm(1+ k )−n−1 η(2−jδD ℓ)f
Lp
| | k − k
kX∈Zn ℓ∈ZnX,i=1,...,n j≥Xj0+1
|2jδ(ki+ℓi)−νi|≤2jδ+2 |j−j(ℓ)|<j0
= 2j(m−δn/p)(1+ k )−n−1 η(D ℓ)(Λ f)
| | k − 2−jδ kLp
kX∈Zn ℓ∈ZnX,i=1,...,n j≥Xj0+1
|ℓi−(2−jδνi−ki)|≤4 |j−j(ℓ)|<j0