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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(3),837-846
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.123086
[‡©Žf3Besov˜mþ
k.5
éé鬬¬kkk
úô“‰ŒÆêÆX,úô7u
ÂvFϵ2023c28F¶¹^Fϵ2023c34F¶uÙFϵ2023c313F
Á‡
3©·‚•ÄÌaáuH¨ormanderaS
m
ρ,1
ž[‡©ŽfT
a
3Besov˜mþk.5.é
u0 ≤ρ≤1,p≥1,-
m
0
=m
0
(ρ,p)=
(
n(ρ−1)/p,1≤p≤2;
n(ρ−1)/2,p≥2.
XJa∈S
m
ρ,1
…s>m−m
0
,·‚y²[ ‡©ŽfT
a
´Besov˜mB
s
p,q
B
s−m+m
0
p,q
k.Ž f.
ù‡(Jí2Stein˜‡(J.
'…c
[‡©Žf§H¨ormandera§Besov˜m
OntheBoundednessof
Pseudo-DifferentialOperators
onBesovSpaces
ShiyingCai
DepartmentofMathematics,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Feb.8
th
,2023;accepted:Mar.4
th
,2023;published:Mar.13
th
,2023
©ÙÚ^:é¬k.[‡©Žf3Besov˜mþk.5[J].A^êÆ?Ð,2023,12(3):837-846.
DOI:10.12677/aam.2023.123086
é¬k
Abstract
In this note, we consider the boundedness of the pseudo-differential operator T
a
whose
symbolabelongstoH¨ormanderclassS
m
ρ,1
onBesovspaces.Let0 ≤ρ≤1,p≥1
m
0
=m
0
(ρ,p)=
(
n(ρ−1)/p,1≤p≤2;
n(ρ−1)/2,p≥2.
Ifa∈S
m
ρ,1
ands>m−m
0
,thenthepseudo-differentialoperatorT
a
isboundedfromB
s
p,q
to
B
s−m+m
0
p,q
.AndourworkistogeneralizearesultofStein.
Keywords
Pseudo-DifferentialOperator,H¨ormanderClass,BesovSpace
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.ïÄµÚyG
[‡©Žf3 ‡©•§¥k2•A^.Kohn-NirenbergÚH¨ormander©O3[1,2]m©XÚ
ïÄùaŽf.˜„ó,˜‡[‡©Žf/ªþXe½Â
T
φ,a
f(x) =
Z
R
n
e
iφ(x,ξ)
a(x,ξ)
ˆ
f(ξ)dξ,
Ù¥
b
fL«fFourierC†,a¡•Ì.φ¡•ƒ ¼ê.φ(x,ξ)=x·ξž,´[‡©Žf.éu
Ìa,•-‡˜a˜m´H¨ormander3[3]Ú?H¨ormanderaS
m
ρ,δ
.·‚`aáuS
m
ρ,δ
(m∈R,0 ≤
ρ,δ≤1)´•é¤kõ-•Iα,β,a÷v
sup
ξ∈R
n
(1+|ξ|)
−m+ρ|α|−δ|β|
|∂
β
x
∂
α
ξ
a(x,ξ)|= A
α,β
<+∞.
3[‡©ŽfnØ¥,§3Lebesgue˜mL
p
ÚHardy˜mH
1
þk.5´•-‡¯Kƒ
˜.y3ù‡¯K®²k¿©ïÄ.{óƒ,XJa∈S
m
ρ,δ
,δ<1,@om≤min{0,n(ρ−
δ)/2}ž,@oT
a
3L
2
þk.,äNŒ±ëwH¨ormander[4],Hounie[5].?˜Ú,éua∈S
m
ρ,1
,
DOI:10.12677/aam.2023.123086838A^êÆ?Ð
é¬k
Rodino3[6]¥y²m<n(ρ−1)/2žT
a
3L
2
þk..Óž¦E˜‡~fa∈S
n(ρ−1)/2
ρ,1
¦
T
a
3L
2
þØ´k..aq‡~a∈S
0
1,1
Œ±ëwChing[7]ÚStein[8]P.272.éuà:œ
/,3™Ñ‡ùÂ¥,Steiny²a∈S
n(ρ−1)/2
ρ,δ
…0≤δ<ρ=1½ö0<δ=ρ<1ž,T
a
´
f(1,1)k.…lH
1
L
1
k..
´
AlvarezÚHounie[9]í2ù˜(J,¦‚y²XJa∈S
m
ρ,δ
,
0 <ρ≤1,0 ≤δ<1…m=
n
2
(ρ−1+min{0,ρ−δ}),@oT
a
´f(1,1)k.…lH
1
L
1
k..L
p
þ
k.5Œ±ÏLFefferman-Stein)ÛŠ•{‰Ñ.?˜Ú,éuÌaáuS
m
ρ,1
½ö•Œ o
÷H¨ormanderaL
∞
S
m
ρ
ž,Guo-Zhu3[10]¥y²ˆ«.œ/k.5½ö‰ÑÃ.‡~.
Xc¤ã,a∈S
n(ρ−1)/2
ρ,1
žT
a
3L
2
þ™7´k.,éuù«œ/,A~, Stein3[8]¥
k˜‡k(J.
2.©Ì‡(J
½nA([8]P.253,Proposition6)XJa∈S
m
1,1
…γ>m,@o[‡©ŽfT
a
´lLipschitz˜
mΛ
γ
Λ
γ−m
k.N.
·‚3©¥̇8´lA‡•¡í2Steinù‡(J.Äk·‚0eLittlewood-
Paley©)ÚBesov˜m.ùpB
r
L«R
n
¥±:•¥%Œ»•r¥.šK¼êη∈C
∞
c
(B
2
)¦
éξ∈B
1
ðkη(ξ) = 1¿½Â
bϕ(ξ) = η(ξ),cϕ
j
(ξ) = η(2
−j
ξ)−η(2
1−j
ξ),j∈Z.
N´wÑcϕ
j
|8á3‚{ξ: 2
j−1
<|ξ|<2
j+1
}S…ØJy±eª
bϕ(ξ)+
∞
X
j=1
cϕ
j
(ξ) = η(ξ)+
∞
X
j=1
[η(2
−j
ξ)−η(2
1−j
ξ)] = 1,∀ξ∈R
n
;
ϕ
j
(x) = 2
jn
ϕ
0
(2
j
x),∀x∈R
n
,j∈Z.
S
0
(R
n
)L«…O2Â¼ê˜ m.éus∈R,0 <p≤∞,0<q<∞,·‚½ÂšàgBesov˜mÚà
gBesov˜m,
B
s
p,q
= {f∈S
0
: kfk
B
s
p,q
= kϕ∗fk
p
+(
∞
X
j=1
(2
js
kϕ
j
∗fk
p
)
q
)
1
q
<∞};
˙
B
s
p,q
= {f∈S
0
: kfk
˙
B
s
p,q
= (
∞
X
j=−∞
(2
js
kϕ
j
∗fk
p
)
q
)
1
q
<∞}.
éus∈R,0 <p≤∞,q= ∞,ü‡˜m½Â?•
B
s
p,∞
= {f∈S
0
: kfk
B
s
p,∞
= kϕ∗fk
p
+sup
1≤j<∞
2
js
kϕ
j
∗fk
p
<∞};
˙
B
s
p,∞
= {f∈S
0
: kfk
˙
B
s
p,∞
= sup
j∈Z
2
js
kϕ
j
∗fk
p
<∞}.
DOI:10.12677/aam.2023.123086839A^êÆ?Ð
é¬k
3©¥,••B,q= ∞ž·‚•æ^q<∞žLˆª,Ø2ƒ±«©.,,••{Bå
„,±·‚P
∆
j
f= ϕ
j
∗f,S
0
f= ϕ∗f.
éuàgBesov˜m,duõ‘ª•",eü‡2¼êA??u˜‡õ‘ª,Kü
‡2¼ê@•´ƒÓ.
¯¤±•,Lipschitz˜mÚL
2
Ñ´šàgBesov˜mA~,äN/`,B
s
∞,∞
= Λ
s
,B
0
2,2
= L
2
.
éu0 ≤ρ≤1,1 ≤ρ≤∞,½Â
m
0
=m
0
(ρ,p)=





n(ρ−1)/p,1≤p≤2;
n(ρ−1)/2,p≥2.
e¡´·‚̇½n
½n1XJa∈S
m
ρ,1
…s>m−m
0
,@oéu[‡©ŽfT
a
·‚k
kT
a
fk
B
s−m+m
0
p,q
≤Ckfk
B
s
p,q
(2.1)
Ù¥~êC=•6un,ρ,p,m,sÚa3S
m
ρ,1
¥,Œ‰ê.
51.XJρ=1Ká=km
0
=0,dž2-p=q=∞,s=γ,|^B
s
∞,∞
=Λ
s
·‚Ò½
nA.
52.ùpXJKs>m−m
0
b(Ø™7¤á.~X-p=q=2,m=m
0
= n(ρ−1)/2,s=
m−m
0
=0,KdžB
0
2,2
=L
2
,·‚®²•ù«œ/eT
a
3L
2
þ™7´k.,Ïd½n¥
(ØØ2¤á.
3©¥,·‚^i1CL«˜‡~ê,§=•6un,ρ,p,m,sÚa3S
m
ρ,1
¥,Œ‰ê,
…3ØÓ/•§L«äNꊌUؘ,·‚˜„Ø2•[`².,,·‚bOŽ¥Ñy
fÑáuSchwartz˜m,Ïd9È©Ñ´ýéŒÈ,2|^È—5*Ð˜„œ/.
3.̇½ny²
Äk·‚‰Ñ˜‡š~Ä:Ún,ù‡(J3ƒ'©z¥ÄÑk,•å„,·‚ùp•
‰Ñy².
Ún1a∈S
m
ρ,1
,0 ≤ρ≤1,@oé?¿m∈R,p∈[1,∞],[‡©ŽfT
a
ÚS
0
EÜ÷v
kT
a
S
0
fk
p
≤Ckfk
p
.(3.1)
y²w,T
a
S
0
fŒL«•
T
a
S
0
f(x) =
Z
R
n
e
ix·ξ
a(x,ξ)η(ξ)
b
f(ξ)dξ=
Z
R
n
e
k(x,x−y)f(y)dy
DOI:10.12677/aam.2023.123086840A^êÆ?Ð
é¬k
Ù¥
e
k(x,z) =
R
R
n
e
iz·ξ
a(x,ξ)η(ξ)dξ.|^FourierC†Ä5ŸN´
(1+|z|)
2n
|
e
k(x,z)|=C
X
|α|≤2n
|z
α
Z
R
n
e
iz·ξ
a(x,ξ)η(ξ)dξ|= C
X
|α|≤2n
|
Z
R
n
e
iz·ξ
∂
α
ξ
[a(x,ξ)η(ξ)]dξ|
≤C
X
|α|≤2n
|
Z
R
n
|∂
α
ξ
[a(x,ξ)η(ξ)]|dξ≤C.
dYoungØªá=Œ
kT
a
S
0
fk
p
≤

Z
R
n
(
Z
R
n
e
k(x,x−y)f(y)dy)
p
dx

1
p
≤C

Z
R
n
(
Z
R
n
(1+|x−y|)
−2n
|f(y)|dy)
p
dx

1
p
≤Ckfk
p
.
ù‡Úny..2
·‚‡?Ue[11]¥·K2.3e¡ù‡Ún.
Ún2a∈S
m
ρ,1
,0 ≤ρ≤1,@oj≥1,1 ≤p≤∞ž,[‡©ŽfT
a
Ú∆
j
EÜ÷v
kT
a
∆
j
fk
p
≤C2
j(m−m
0
)
kfk
p
.(3.2)
y²Pa
j
(x,ξ)=(η(2
−j
ξ)−η(2
1−j
ξ))a(x,ξ),K{ξ: a
j
(x,ξ)6=0}⊂{ξ:2
j−1
<|ξ|<2
j+1
}.
j≥1ž,é?¿õ-•Iαw,k
|∂
α
ξ
a
j
(x,ξ)|≤C2
j(m−ρ|α|)
.
Šâ½ÂT
a
∆
j
ŒL«•
T
a
∆
j
f(x)=
Z
R
n
e
ix·ξ
a(x,ξ)
d
∆
j
f(ξ)dξ
=
Z
R
n
e
ix·ξ
(η(2
−j
ξ)−η(2
1−j
ξ))a(x,ξ)
b
f(ξ)dξ
=
Z
R
n
k
j
(x,x−y)f(y)dy
Ù¥k
j
(x,z) =
R
R
n
e
iz·ξ
a
j
(x,ξ)dξ.
σ
j
(z) = 2
jnρ
(1+2
jρ
|z|)
−3n
,Kw,kkσ
j
k
1
= C.
1≤p≤2ž,pÝêp
0
=
p
p−1
≥2.džé?¿x∈R
n
,|^FourierC†Ä5Ÿ
ÚHausdorff-YoungØª,·‚Œ±
DOI:10.12677/aam.2023.123086841A^êÆ?Ð
é¬k


Z
R
n
|
k
j
(x,x−y)
σ
1
p
j
(x−y)
|
p
0
dy


1
p
0
=2
−
jnρ
p

Z
R
n
|k
j
(x,z)(1+2
jρ
|z|)
3n
p
|
p
0
dz

1
p
0
≤2
−
jnρ
p

Z
R
n
|k
j
(x,z)(1+2
jρ
|z|)
3n
|
p
0
dz

1
p
0
≤C2
−
jnρ
p
X
|α|≤3n

Z
R
n
|2
jρ|α|
z
α
k
j
(x,z)|
p
0
dz

1
p
0
=C2
−
jnρ
p
X
|α|≤3n
Z
R
n




2
jρ|α|
z
α
Z
R
n
e
iz·ξ
a
j
(x,ξ)dξ




p
0
dz
!
1
p
0
=C2
−
jnρ
p
X
|α|≤3n
Z
R
n




Z
R
n
e
iz·ξ

2
jρ|α|
∂
α
ξ
a
j
(x,ξ)

dξ




p
0
dz
!
1
p
0
≤C2
−
jnρ
p
X
|α|≤3n

Z
R
n
|2
jρ|α|
∂
α
ξ
a
j
(x,ξ)|
p
dξ

1
p
≤C2
−
jnρ
p
X
|α|≤3n

Z
|ξ|<2
1+j
2
jmp
dξ

1
p
≤C2
−
jnρ
p
2
j(m+
n
p
)
= C2
j(m−m
0
)
.(3.3)
ù‡Øªép>2™7¤á,Ï•·‚I‡^Hausdorff-YoungØª.,5¿ùpm
0
‹pk
',~êC‹xÃ'.
Ïd,Šâ(3.3),·‚Œ±
|T
a
∆
j
f(x)|=




Z
R
n
k
j
(x,x−y)f(y)dy




≤
Z
R
n
|
k
j
(x,x−y)
σ
1
p
j
(x−y)
|σ
1
p
j
(x−y)|f(y)|dy
≤


Z
R
n
|
k
j
(x,x−y)
σ
1
p
j
(x−y)
|
p
0
dy


1
p
0

Z
R
n
σ
j
(x−y)|f(y)|
p
dy

1
p
≤C2
j(m−m
0
)
(σ
j
∗|f|
p
)
1
p
.(3.4)
y3,|^(3.4)ÚYoungØªŒ
kT
a
∆
j
fk
p
≤C2
j(m−m
0
)
kσ
j
∗|f|
p
k
1
p
1
≤C2
j(m−m
0
)
kσ
j
k
1
p
1
kfk
p
= C2
j(m−m
0
)
kfk
p
.(3.5)
5¿p≥2žm
0
‹äNpÃ',é?¿p>2d(3.4)·‚k
|T
a
∆
j
f(x)|≤C2
j(m−m
0
)
(σ
j
∗|f|
2
)
1
2
.
DOI:10.12677/aam.2023.123086842A^êÆ?Ð
é¬k
|^aq(3.5)OŽ,p>2ž,·‚Œ±y²
kT
a
∆
j
fk
p
≤C2
j(m−m
0
)
k(σ
j
∗|f|
2
)
1
2
k
p
=C2
j(m−m
0
)
kσ
j
∗|f|
2
k
1
2
p
2
≤C2
j(m−m
0
)
kσ
j
k
1
2
1
kf
2
k
1
2
p
2
=C2
j(m−m
0
)
kfk
p
.
Ïdé¤k1 ≤p≤∞·‚y²ù‡Ún. 2
Ún3jÚL´ü‡šKê,b•3λ
j
¦é?¿õ-•Iα,|α|≤LÑk
k∂
α
x
F
j
k
p
≤C2
j|α|
λ
j
,Ké?¿êk·‚k
k∆
k
F
j
k
p
≤C2
(j−k)L
λ
j
.(3.6)
y²ηXc½Â,él= 1,2,...,n½Â
b
0
(ξ) = η(ξ),b
l
(ξ) = (1−η(ξ))
ξ
l
|ξ|
2
,
Kw,b
0
,b
1
,...,b
n
∈S
−1
1,0
…
b
0
(ξ)+
n
X
l=1
b
l
(ξ)ξ
l
= 1, ∀ξ∈R
n
.
Ïdé?¿šKêLk
∆
k
F
j
(x)=
Z
R
n
e
ix·ξ
cϕ
k
(ξ)(b
0
(ξ)+
n
X
l=1
b
l
(ξ)ξ
l
)
L
c
F
j
(ξ)dξ
=
X
|α|≤L
Z
R
n
e
ix·ξ
b
α
(ξ)cϕ
k
(ξ)ξ
α
c
F
j
(ξ)dξ
=
X
|α|≤L
T
b
α
∆
k
(∂
α
F
j
)(x)
Ù¥b
α
= b
L−|α|
0
n
Q
l=1
b
α
l
l
∈S
−L
1,0
.3Ún2¥m= −L,ρ= 1(džm
0
(1,p) = 0)·‚
k∆
k
F
j
k
p
≤
X
|α|≤L
kT
b
α
∆
k
(∂
α
F
j
)k
p
≤C2
−kL
X
|α|≤L
k∂
α
F
j
k
p
≤C2
−kL
X
|α|≤L
2
j|α|
λ
j
≤C2
(j−k)L
λ
j
.
Ún3y..2
e¡·‚y²½n1.
éf∈B
s
p,q
,j≥1,Pf
0
j
= (∆
j−1
+∆
j
+∆
j+1
)f,Kw,k∆
j
f= ∆
j
f
0
j
.
DOI:10.12677/aam.2023.123086843A^êÆ?Ð
é¬k
Šâa∈S
m
ρ,1
,ØJwÑ
∂
α
x
T
a
f(x) = ∂
α
x
Z
R
n
e
ix·ξ
a(x,ξ)
b
f(ξ)dξ=
X
β+γ=α
Z
R
n
e
ix·ξ
ξ
β
∂
γ
x
a(x,ξ)
b
f(ξ)dξ= T
a
α
f(x)
Ù¥a
α
(x,ξ) = ξ
β
∂
γ
x
a(x,ξ) ∈S
m+|α|
ρ,1
.(ù‡ªféo÷H¨ormanderaL
∞
S
m
ρ
ؤá.)
y3,éj≥1Ú?¿õ-•Iα,dÚn2
k∂
α
x
T
a
∆
j
fk
p
= kT
a
α
∆
j
f
0
j
k
p
≤C2
j(|α|+m−m
0
)
kf
0
j
k
p
.(3.7)
-λ
j
= 2
js
kf
0
j
k
p
,j≥1.k≤jž,3Ún3¥L= 0
2
k(s−m+m
0
)
k∆
k
T
a
∆
j
fk
p
≤C2
k(s−m+m
0
)
2
j(m−m
0
)
kf
0
j
k
p
= C2
(k−j)(s−m+m
0
)
λ
j
.
k>jž,PL
0
•Œus−m+m
0
•ê,3Ún3¥L= L
0

2
k(s−m+m
0
)
k∆
k
T
a
∆
j
fk
p
≤C2
k(s−m+m
0
)
2
(j−k)L
0
2
j(m−m
0
)
kf
0
j
k
p
= C2
(j−k)(L
0
−s+m−m
0
)
λ
j
.
-δ= min{s−m+m
0
,L
0
−s+m−m
0
},KdsbÚL
0
½Â•δ>0.nÜþ¡ü‡ªf,é
¤kj,k≥1·‚k
2
k(s−m+m
0
)
k∆
k
T
a
∆
j
fk
p
≤C2
−|j−k|δ
λ
j
.(3.8)
|^ÓOŽÚÚn1k
2
k(s−m+m
0
)
k∆
k
T
a
S
0
fk
p
≤C2
−kδ
kS
0
fk
p
.(3.9)
••Bå„,j≤0žØ”-λ
j
= 0,Šâ(3.8),(3.9)ÚMinkovskyØª·‚k
kT
a
fk
B
s−m+m
0
p,q
=kS
0
T
a
fk
p
+
∞
X
k=1
(2
k(s−m+m
0
)
k∆
k
T
a
fk
p
)
q
!
1
q
≤kS
0
fk
p
+
∞
X
k=1
(2
k(s−m+m
0
)
k∆
k
T
a
S
0
fk
p
+
∞
X
j=1
2
k(s−m+m
0
)
k∆
k
T
a
∆
j
fk
p
)
q
!
1
q
≤kS
0
fk
p
+C
∞
X
k=1
(2
−kδ
kS
0
fk
p
+
∞
X
j=1
2
−|j−k|δ
λ
j
)
q
!
1
q
≤C


kS
0
fk
p
+
∞
X
k=1
(
∞
X
j=−∞
2
−|j|δ
λ
k−j
)
q
!
1
q


≤C
kS
0
fk
p
+
∞
X
j=−∞
2
−|j|δ
(
∞
X
k=1
λ
q
k−j
)
1
q
!
= C
kS
0
fk
p
+
∞
X
j=−∞
2
−|j|δ
(
∞
X
k=1
λ
q
k
)
1
q
!
DOI:10.12677/aam.2023.123086844A^êÆ?Ð
é¬k
≤C
kS
0
fk
p
+
∞
X
j=−∞
2
−|j|δ
(
∞
X
k=1
(2
ks
k(∆
k−1
+∆
k
+∆
k+1
)fk
p
)
q
)
1
q
!
≤C
kS
0
fk
p
+(
∞
X
k=1
(2
ks
k∆
k
fk
p
)
q
)
1
q
!
≤Ckfk
B
s
p,q
.
ù‡½ny..2
4.o(
lStein[8]P.253,Proposition6¥¼éu,©±Littlewood-Paley©)•óä,ïÄ[‡
©Žf3˜½^‡eBesov˜mþk.5.8,·‚ïÄSN´ò©(Jí2Fp“È©
Žf3Besov˜mþk.5.
ë•©z
[1]H¨ormander,L.(1965)Pseudo-DifferentialOperators.CommunicationsonPureandApplied
Mathematics,18,501-517.https://doi.org/10.1002/cpa.3160180307
[2]Kohn, J.J.andNirenberg, L.(1965) AnAlgebraofPseudo-DifferentialOperators.Communica-
tionsonPureand AppliedMathematics, 18, 269-305. https://doi.org/10.1002/cpa.3160180121
[3]H¨ormander,L.(1967)Pseudo-DifferentialOperatorsandHypoellipticEquations.In:Calder´on,
A.P.,Ed.,SingularIntegrals.ProceedingsofSymposiainPureMathematics,Vol.10,AMS,
Providence,RI,138-183.
[4]H¨ormander,L.(1971)Onthe L
2
Continuity ofPseudo-DifferentialOperators.Communications
onPureandAppliedMathematics,24,529-535.https://doi.org/10.1002/cpa.3160240406
[5]Hounie,J.(1986)On theL
2
-ContinuityofPseudo-Differential Operators.Communicationsin
PartialDifferentialEquations,11,765-778.https://doi.org/10.1080/03605308608820444
[6]Rodino,L.(1976)OntheBoundednessofPseudoDifferentialOperatorsintheClassL
m
ρ,1
.
ProceedingsoftheAMS,58,211-215.https://doi.org/10.2307/2041387
[7]Ching,C.H.(1972)Pseudo-DifferentialOperatorswithNonregularSymbols.JournalofDif-
ferentialEquations,11,436-447.https://doi.org/10.1016/0022-0396(72)90057-5
[8]Stein, E.M. (1993) Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory
Integrals(PMS-43).PrincetonUniversityPress,Princeton,NJ.
[9]
´
Alvarez,J.andHounie,J.(1990)EstimatesfortheKernelandContinuityPropertiesof
Pseudo-DifferentialOperators.ArkivMatematik,28,1-22.
https://doi.org/10.1007/BF02387364
[10]Guo,J.andZhu,X.(2022)SomeNotesonEndpointEstimatesforPseudo-DifferentialOper-
ators.https://arxiv.org/abs/2201.10724
DOI:10.12677/aam.2023.123086845A^êÆ?Ð
é¬k
[11]Kenig,C.E. andStaubach,W. (2007)Ψ-Pseudodifferential OperatorsandEstimates forMax-
imalOscillatoryIntegrals.StudiaMathematica,183,249-258.
https://doi.org/10.4064/sm183-3-3
DOI:10.12677/aam.2023.123086846A^êÆ?Ð

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