Appl
.Mat
h.J.Chi
ne
s
eUni
v.Se
r
.B
2006,21(1):6978
A CLASSOFOSCI
LLATORY SI
NGULAR I
NTEGRALSON
TRI
EBELLI
ZORKI
N SPACES
J
i
a
ngLi
ya Che
nJ
i
e
c
he
ng
.Thebounde
Abs
t
r
ac
t
dne
s
sonTr
i
e
be
l
Li
z
or
ki
ns
pa
c
e
sofos
c
i
l
l
a
t
or
ys
i
ngul
a
ri
nt
e
gr
a
lope
r
a
t
or
i
|x|a
Ti
nt
hef
or
me
-n
1
n- 1
shomoge
ne
ous
Ω(x)|x| i
ss
t
udi
e
d,whe
r
ea∈R,a≠0,1a
ndΩ∈L (S )i
+
n- 1
)∈Ll
ofde
gr
e
ez
e
r
oa
nds
a
t
i
s
f
i
e
sc
e
r
t
a
i
nc
a
nc
e
l
l
a
t
i
onc
ondi
t
i
on.Whe
nke
r
ne
lΩ(x'
og L(S ),
α,q
n
1
·
p (
t
heF
R )bounde
dne
s
soft
hea
boveope
r
a
t
ori
sobt
a
i
ne
d.Me
a
nwhi
l
e,whe
nΩ(x)s
a
t
i
s
f
i
e
sL 0,1
n
·
1 (
R ).
Di
nic
ondi
t
i
on,t
hea
boveope
r
a
t
orT i
sbounde
donF
45 I
6t
r
789c
t
:
76a68;a:
6r
<
s
9=
t
s
n- 1
n
Le
tΩ(x)beaf
unc
t
i
onove
rt
heuni
ts
phe
r
eS ofR ands
at
i
s
f
y
> Ω(x')dx'? 0.
(1.1)
n- 1
S
n
Theos
c
i
l
l
at
or
ys
i
ngul
ari
nt
e
gr
alope
r
at
orT i
sde
f
i
ne
dont
het
e
s
tf
unc
t
i
ons
pac
eS(R )by
a
i
|A|
T@(x)? p.v.e
Ω(A)
B@(x),a∈ R,a≠ 0,1.
|A|n
(1.2)
p
l
l
o,Kur
i
ght
e
d
C
nD4EandD5E,Chani
t
sandSamps
ons
t
udi
e
dt
heL (F)(1GpGH)andwe
a
-1 i
|I |
we
akt
ype(1,1)bounde
dne
s
sofope
r
at
orT@(x)?p.v.(1+|I|) e
B@(x),whe
r
eF
∈Ap.Jss
howni
nD5E,t
hes
amer
e
s
ul
t
sar
eal
s
ot
r
uef
ort
heope
r
at
orde
f
i
ne
di
n(1.2)
.
wi
t
hs
t
andar
dCZke
r
ne
l
Fora?0i
n(1.2),t
heope
r
at
orT i
sj
us
tt
hes
i
ngul
ari
nt
e
gr
alope
r
at
orofc
onvol
ut
i
on
α
,q
n
r
n- 1
·
t
ype,andt
heFp (R )bounde
dne
s
si
st
r
uef
orr
oughke
r
ne
lΩ∈L (S )(1G rK H ),s
e
e
+
n- 1
α
,q
·
D2E.Me
anwhi
l
e,f
orΩ(x)∈ Ll
og L(S ),t
heFp (F)bounde
dne
s
swasc
ons
i
de
r
e
df
or
c
e
r
t
ai
nwe
i
ghtf
unc
t
i
onF(x)i
nD8E.
α
,q
n
·
Thef
i
r
s
tai
m oft
hi
spape
ri
st
oe
s
t
abl
i
s
ht
heF
R )bounde
dne
s
sofope
r
at
orT
p (
+
n- 1
gi
ve
ni
n(1.2)wi
t
hr
oughke
r
ne
lΩ(x)∈Ll
og L(S ),whe
r
eα∈R,1Gp,qGH.
.Choos
We r
e
c
al
lt
he de
f
i
ni
t
i
on oft
he homoge
ne
ous Tr
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e
be
l
Li
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ea
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on:42B20.
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,Tr
Ke
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c
i
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a
t
or
ys
i
ngul
a
ri
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e
gr
a
l
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be
l
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ns
pa
c
e.
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t
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d by 973pr
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t(G1999075109),NSFZJ(RC97017),RFDP (20030335019),NSFC
(10271107).
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;pp=
?.@.C?A
nB
C
BDnA
E.SB
r
70
∞
n
.21,No.1
Gol
j
n
∈C0 (R )s
≤1.Le
(2ξ)s
)⊂{x∈R :
f
unc
t
i
onφ
uc
ht
hat0≤φ
tφ
ξ)=φ
at
i
s
f
yt
hats
upp(φ
j(
+∞
1
3
5
≤|x|≤2}andφ
(x)>c
>0f
x)2 = 1f
or ≤|x|≤ .Weas
s
umeΣ φ
oral
lx≠0.
j(
2
5
3
j
=-∞
.Le
De
f
i
ni
t
i
on 1.1Suppos
et
hatφ
at
i
f
i
e
sal
lt
heabovec
ondi
t
i
ons
tΦj bede
f
i
ne
d by
js
^
(Rn)╲P(Rn),whe
Φj(ξ)= φ
ξ).Wr
i
t
eSjf= Φj*f.Forα∈R,1<p,q< ∞ andf∈S'
r
e
j(
n
n
P(R )de
not
e
st
hec
l
as
sofpol
ynomi
al
sonR ,t
hehomoge
ne
ousTr
i
e
be
l
Li
z
or
ki
ns
pac
e
α
,q
n
·
F (R )i
st
hes
e
tofal
lfs
at
i
s
f
yi
ng
p
+∞
1
‖f‖ F·pα,q(Rn) = ‖ (Σ 2- αjq|Sjf|q)q‖ p < ∞.
(1.3)
j
=-∞
*
Le
tSj bet
hedualope
r
at
orofSj,i
ti
se
as
yt
os
e
e
+∞
1
‖ (Σ 2- αjq|S*
|q)q‖ p ～ ‖f‖ F·pα,q(Rn).
jf
j
=-∞
.
Now,l
e
tuss
t
at
eourt
he
or
e
ms
The
or
e
m 1.1.Le
tα∈R,1<p,q<∞,andT bede
f
i
ne
dasi
n(1.2),whe
r
ea∈R,a≠0,1.
+
n- 1
α
,q
·
(Rn).
I
fΩ∈Ll
og L(S )ands
at
i
s
f
i
e
sc
ondi
t
i
on(1.1),t
he
nT i
sbounde
donF
p
0,1
n
·
Theanot
he
rpur
pos
ei
st
oc
ons
i
de
rt
heF1 (R )bounde
dne
s
sofT.Weknow t
hat
0,1
n
1
n
0,1
n
·
·
,t
oi
nor
de
rt
oobt
ai
nt
heF1 (R )bounde
dne
s
s
her
e
l
at
e
dke
r
ne
lΩ(x)
F1 (R )⊂H (R ),s
[6]
0,1
n
·
mus
ts
at
i
s
f
yc
e
r
t
ai
nc
ont
i
nui
t
y.I
n1994,Fan s
t
h
howe
dt
hatT i
sbounde
donF1 (R )wi
1
Ω∈ C .Butwhe
nT i
sas
i
ngul
ari
nt
e
gr
alope
r
at
orofc
onvol
ut
i
on t
ype,t
hati
sa= 0,
1
f
ol
l
owi
ngt
her
e
s
ul
t
si
n[7],weknow t
hati
fΩs
at
i
s
f
i
e
st
heL Di
nic
ondi
t
i
on,i
ti
sbounde
d
0,1
n
·
onF (R ).
1
r
n- 1
≤∞,and
De
f
i
ni
t
i
on1.2.Le
tΩ∈L(S ),1≤r
1
r
(∫
ωr(t
)= s
up
|R|< t
)
|Ω(Ry'
)- Ω(y'
)|rdy' ,
n- 1
S
n
r
whe
r
eR i
sar
ot
at
i
ononR ,and|R|= ‖ R- I‖.Wes
ayt
hatΩ s
at
i
s
f
i
e
st
heLDi
ni
c
ondi
t
i
on,i
f
1
dt
∫ω(t) t< ∞.
r
0
0,1
n
·
Thef
ol
l
owi
ngt
he
or
e
ms
howst
hatwhe
na≠ 0,T i
sal
s
obounde
donF
R )i
fΩ
1 (
1
s
at
i
s
f
i
e
st
heL Di
nic
ondi
t
i
on.
1
The
or
e
m 1.2.Le
tΩs
at
i
s
f
yt
heL Di
nic
ondi
t
i
onandT beasi
n(1.2),whe
r
ea∈R,a≠0,
0,1
n
·
1,t
he
nT i
sbounde
donF (R ).
1
42 5r
oofofThe
or
e
m 1.1
,wes
.
Fi
r
s
t
t
at
et
hef
ol
l
owi
ngl
e
mmas
.I
6e
mm72.1.Le
tT:S8S'beac
onvol
ut
i
onope
r
at
or
ff
ors
ome1<q< ∞,t
hei
ne
9ual
i
t
y
q
‖Tf‖ Lq(:)≤;‖f‖ Lq(:) hol
dsf
oral
l:∈;1 andf∈L (:),whe
r
e;1 i
st
he<uc
ke
nhoupt
A CLASSOFOSCI
LLATORY SI
NGULAR I
NTEGRALS
ON TRI
EBELLI
ZORKI
N SPACES
.
Ji
angLi
ya,e
tal
71
∈R,T i
we
i
ghtc
l
as
sandA i
si
nde
pe
nde
ntoffandw,t
he
nf
oral
l1<p<∞,s
sabounde
d
s
,q
n
·
ope
r
at
oronFp (R ),and
‖Tf‖ F·sp,q(Rn) ≤ A‖f‖ F·sp,q(Rn).
n
.Le
i
t
e
Pr
oof
tφ
et
hes
ameasi
nt
hei
nt
r
oduc
t
i
on,f
orf∈S(R ),wr
jb
Tf(x)=
Σ STSf(x).
k
k
k∈ Z
Wehave
1
*
k
Σ <TS,S g> ≤ ‖g‖
|<Tf,g>|=
k
‖ (Σ 2- skq|TSkf|q)q‖ p,
-s
,q'
·
F
p'
k∈ Z
k∈ Z
-s
,q'
n
·
f
org∈F
R ).He
nc
e
p' (
1
‖Tf‖ F·sp,q(Rn) ≤ ‖ (Σ 2- skq|TSkf|q)q‖ p.
k∈ Z
p
t
h‖u‖ L(q)'=1s
uc
ht
hat
I
fq≤p,wec
anc
hoos
eaf
unc
t
i
onu(x)∈L(q) wi
'
p
1
‖ (Σ 2- skq|TSkf|q)q‖ q
p=
k∈ Z
Σ∫2
-s
kq
n
k∈ Z R
|TSkf|q(x)u(x)dx.
1
p
n
r
'
,wehaveM(ur)(x)<∞ f
ora.e.x∈R .Thus(M(u))r∈A1.Byt
he
q
we
i
ght
e
de
s
t
i
mat
eofope
r
at
orT,i
tf
ol
l
owst
hat
()
<
Pi
c
ki
ng1<r
1
r
1
r
∫|TSf|(x)u(x)dx≤∫|TSf|(x)(M(u)) dx≤ A∫|Sf|(x)(M(u)) dx.
q
n
q
k
r
k
n
R
q
n
R
r
k
R
Thus
∫
-s
kq
Σ2
k∈ Z
R
k∈ Z
-s
kq
q
A‖ Σ 2 |Skf|‖
p
Lq
k∈ Z
r
1
∫
|TSkf|q(x)u(x)dx≤ AΣ 2- skq
n
1
r
|Skf|q(x)(M(ur))rdx≤
n
R
p
q
‖(M(u)) ‖ L(q)' ≤ A‖f‖ F·sp,q(Rn).
The
r
e
f
or
e‖Tf‖ F·sp,q(Rn)≤A‖f‖ F·sp,q(Rn) i
fq≤p.Bydual
i
t
ywec
anobt
ai
nt
hes
i
mi
l
arr
e
s
ul
t
i
fq≥p.Thepr
oofofLe
mma2.1i
sc
ompl
e
t
e
d.
[11]
)i
eφ(t
sr
e
al
val
ue
dands
moot
hi
n(a,b),and
Le
mma2.2.(Vande
rCor
put )Suppos
(k)
(t
)|≥1f
∈(a,b
),t
|φ
oral
lt
he
n
b
∫e
i
λ
φ
(t
)
a
1
k
dt≤ Ckλ
hol
dswhe
n:
(i
)k≥2,or
(i
)k=1andφ
'
(t
)i
.
i
smonot
oni
c
.
TheboundCk i
si
nde
pe
nde
ntofφandλ
.Le
Now ac
c
or
di
ngt
o[3],wede
c
ompos
eΩ(x)asf
ol
l
ows
t
θ
x'∈ Sn- 1:|Ω(x'
)|≤ 1},θ
x'∈ Sn- 1:2d- 1 ≤ |Ω(x'
)|≤ 2d}(d≥ 1),
0= {
d= {
～
Ωd(x)= Ω(x)xθd(x),
～
∫ Ω (x)dx.
～
Ωd(x)= Ωd(x)-
n- 1
d
S
ωn
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
72
.21,No.1
Vol
The
nwehave
Σ Ω (x)= Ω(x),∫
d
Ωd(x)dx= 0,
n- 1
S
d≥ 0
‖Ωd‖ ∞ ≤ C2d,‖Ωd‖ L1 ≤ C2d|θ
,Σ d2d|θ
‖Ω‖ Llog+ L.
d|
d|≤ C
d≥ 0
n
Forf∈S(R ),wr
i
t
e
a
i
|·|
ΣΣe
Tf(x)=
d≥ 0 k∈ Z
Ωd(·) k- 1
x2 < |·|≤ 2k*f(x)=
|·|n
d
k
Σ Σ T f(x).
d≥ 0 k∈ Z
Ωd(x) k- 1
x2 < |x|≤ 2k(x),k∈Z.Foral
l1<p<∞,t
he
r
ehol
ds
|x|n
‖|σd
*|f|‖ p ≤ ‖Ωd‖ L1‖f‖ p.
k|
d
a
i
|x|
Le
mma2.3.Le
tσk(x)=e
.Wenot
Pr
oof
et
hat
k
2
1
∫ |f(x- ry')| rdrdσ(y').
|σd
*|f|(x)≤ fSn- 1|Ωd(y'
)|
k|
k- 1
2
He
nc
e
k
∫∫
‖|σd
*|f|‖ p
k|
p≤
n
R
2
S
∫
|Ωd(y'
)|
n- 1
p
1
|f(x- ry'
)| drdσ(y'
) dx≤ ‖Ωd‖ L1‖f‖ p.
r
k- 1
2
Le
mma2.4.Forα∈R,1<p,q<∞,t
he
r
ehol
ds
‖ F·pα,q(Rn) ≤ ‖Ωd‖ L1‖f‖ F·pα,q(Rn).
‖Td
kf
-α
,q'
n
·
.Byt
Pr
oof
hes
i
mi
l
arpr
oofofLe
mma2.1,f
oranyg∈Fp' (R ),wehave
<Td
,g>=
kf
(2.1)
d
k j
+k
Σ <T S
f,S*
>≤
j
+ kg
j
1
1
‖ (Σ 2
‖
-α
q(j
+ k)
|Td
|q)q
kS
j
+ kf
i
‖ (Σ 2
‖
α
q(j
+ k)
p
|S*
|q')q'
j
+ kg
j
.
p'
I
tf
ol
l
owst
hat
1
‖Td
‖ F·pα,q(Rn) ≤
kf
‖ (Σ 2
‖.
-α
q(j
+ k)
|Td
|q)q
j
+ kf
kS
j
p
ByLe
mma2.3,
‖s
up2- α(j+ k)|Td
|‖ p ≤ ‖Ωd‖ L1‖ s
up2- α(j+ k)|Sj+ kf|‖ p.
kS
j
+ kf
j
j
(2.2)
p'
Si
nc
ep>1,t
he
r
ee
xi
s
t
saf
unc
t
i
ong∈L wi
t
h‖g‖ p'=1s
uc
ht
hat
‖ Σ 2- α(j+ k)|Td
|‖ p =
kS
j
+ kf
j
-α
(j
+ k)
|Td
|,g>≤
kS
j
+ kf
Σ <2
j
-α
(j
+ k)
Σ <2
|σd
*|Sj+ kf|,|g|>≤
k|
j
～d
‖ Σ 2- α(j+ k)|Sj+ kf|‖ p‖|σ
*|g|‖ p',
k|
j
d
～d
whe
r
e|σk|(x)=|σk|(-x).Us
i
ngLe
mma2.3,weobt
ai
n
‖ Σ 2- α(j+ k)|Td
|‖ p ≤ ‖Ωd‖ L1‖ Σ 2- α(j+ k)|Sj+ kf|‖ p.
j
+ kf
kS
j
(2.3)
j
Thus(2.1)f
ol
l
owsi
mme
di
at
e
l
ybyus
i
ngani
nt
e
r
pol
at
i
onbe
t
we
e
n(2.2)and(2.3).
:(Ⅰ)a>0;(Ⅱ)a<0.
Now l
e
tuspr
oveThe
or
e
m 1.1.The
r
ear
et
woc
as
e
s
A CLASSOFOSCI
LLATORY SI
NGULAR I
NTEGRALS
ON TRI
EBELLI
ZORKI
N SPACES
.
Ji
angLi
ya,e
tal
73
I
nt
hef
ol
l
owi
ng,wef
i
r
s
tde
alwi
t
ht
hec
as
e(Ⅰ)a>0.
,wr
Asus
ual
i
t
e
a
i
|·|
Tf(x)= e
a
Ω(·)
·)
i
|·| Ω(
(x)+ e
x|·|> 1*f(x):=
|·|≤ 1*f
nx
|·|
|·|n
T0f(x)+ T∞ f(x).
hal
lpr
ovet
hatunde
rt
hec
ondi
t
i
onsofThe
or
e
m 1.1,t
he
r
ehol
ds
ForT0f,wes
‖T0f‖ F·pα,q(Rn) ≤ C(1+ ‖Ω‖ Llog+ L)‖f‖ F·pα,q(Rn).
(2.4)
Re
wr
i
t
e
T0f(x)=
a
i
|·|
ΣΣe
d≥ 0 k≤ 0
Ωd(·) k- 1
x2 < |·|≤ 2k*f(x)=
|·|n
d
0,k
ΣΣT
f(x).
d≥ 0 k≤ 0
>0s
I
ft
he
r
ei
sac
ons
t
antε
uc
ht
hat
‖Td
‖ F·pα,q(Rn) ≤ C2εk2d‖f‖ F·pα,q(Rn),
0,kf
(2.5)
t
he
nt
he
r
ehol
ds
‖Td
‖ F·pα,q(Rn) ≤ CΣ 2- εdN2d‖f‖ F·pα,q(Rn) ≤ C‖f‖ F·pα,q(Rn)
0,kf
ΣΣ
d≥ 0k≤ - dN
(2.6)
d≥ 0
i
fwec
hoos
eN s
uf
f
i
c
i
e
ntl
ar
ge.Thus(2.4)f
ol
l
owsf
r
om (2.6)andLe
mma2.4.
∫
d
Topr
ove(2.5),l
e
tS0,kf(x)=
k- 1
2
k
< |x- y|≤ 2
Ωd(x- y)
f(y)dy,t
he
ns
i
mi
l
art
ot
he
|x- y|n
pr
oofofThe
or
e
m 1i
n[8],wehave
‖Σ
Σ
Sd
(x)‖ F·pα,q(Rn) ≤ C(1+ ‖Ω‖ Llog+ L)‖f‖ F·pα,q(Rn).
0,kf
d≥ 0k≤ - dN
He
nc
ei
ti
se
nought
oe
s
t
i
mat
e
～
Td
(x)- Sd
(x):=Td
(x).
0,kf
0,kf
0,kf
Not
et
hat
～
|Td
(x)|=
0,kf
∫
a
Ωd(x- y)
i
|x- y|
[e
- 1]
f(y)dy≤ 2ka‖Ωd‖ ∞ M(f)(x),
< |x- y|≤ 2
|x- y|n
k- 1
2
k
.Sof
whe
r
eM de
not
et
heHar
dyLi
t
t
l
e
woodmaxi
malope
r
at
or
orany1<q<∞ andω∈Aq,
t
he
r
ehol
ds
～
‖Td
‖ q,w ≤ 2ka‖Ωd‖ ∞ ‖f‖ q,w.
0,kf
(2.7)
The
r
e
f
or
e,byLe
mma2.1wege
t
～d
‖T0,kf‖ F·pα,q(Rn) ≤ 2ka‖Ωd‖ ∞ ‖f‖ F·pα,q(Rn).
d
=a.
Not
i
c
et
hat‖Ωd‖ ∞ ≤2 anda>0.Thust
hei
ne
qual
i
t
y(2.5)hol
dswi
t
hε
,wes
Ne
xt
hal
le
s
t
i
mat
eT∞ .Asbe
f
or
e,wr
i
t
e
T∞ f(x)=
a
i
|·|
ΣΣe
d≥ 0 k≥ 1
Ωd(·) k- 1
x2 < |·|≤ 2k*f(x)=
|·|n
d
∞ ,k
ΣΣT
d≥ 0 k≥ 1
whe
r
e
k
∫
Td
∞ ,kf=
S
2
a
dr
dθ
.
r
∫ e f(x- rθ)
Ω (θ
)
n- 1 d
ByLe
mma2.3,f
orany1<q<∞,wehave
i
r
k- 1
2
f(x),
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
74
.21,No.1
Vol
‖Td
‖ q ≤ C‖Ωd‖ L1‖f‖ q.
∞ ,kf
2
(2.8)
d
∞ ,k
f.
Now wee
s
t
i
mat
eL nor
m ofT
n- 1
.For
Fi
xθ∈ S ,l
e
tY bet
hehype
r
pl
anepas
s
i
ngt
hr
ought
heor
i
gi
nor
t
hogonalt
oθ
n
∈R.So
x∈R ,wr
i
t
ex=z+s
θ,z∈Y,s
k
2
k
2
a
dr
dθ
i
r
e
f(x- r
θ
)
=
k- 1
2
r
∫
∫
dr
a
∫ e f(z+ (s- r)θ) r =
i
r
k- 1
2
a
dt
i
(s
-t
)
= Nk(f(z+·θ
e
f(z+ t
θ
)
))(s
),
<s
-t
≤2
s- t
k- 1
2
k
2
*
.Now,t
sal
i
ne
arope
r
at
orde
f
i
ne
d on L (R),Nk i
si
t
sadj
oi
ntope
r
at
or
he
whe
r
eNk i
*
:
ope
r
at
orNk Nk hast
hef
ol
l
owi
ngke
r
ne
l
a
∫
Mk(u,v)=
k
k- 1
k
<r
- v≤ 2
2
∫
2
<r
- u≤ 2
k a
k
a
i
[(2 r
) - (2 r
+ v- u) ]
e
1
<r
≤1
2
k- 1
a
i
[(r
- v) - (r
- u) ]
e
k- 1
2
k
k
< 2r
+ v- u≤ 2
1
1
dr=
r- vr- u
1
1
dr
.
r 2kr+ v- u
-k
I
ti
se
as
yt
oc
he
c
kt
hat|Mk(u,v)|≤ 2 x{|u- v|≤ 2k- 1}. Wi
t
houtl
os
sofge
ne
r
al
i
t
y,wemay
k
a- 1
k
a- 1
as
s
umet
hatu- v≠ 0,t
he
nf
oranyf
i
xe
du,v,(2r) - (2r+ v- u) (0< a≠ 1)i
s
1
≤1,and
,f
monot
oni
ci
nr
or <r
2
|a2k[(2kr
)a- 1 - (2kr+ v- u)a- 1]|≥ C2k(a- 1)|u- v|.
- ka
-1
- k(1- δ) - kaδ
2 ×
ThusbyLe
mma2.2,|Mk(u,v)|≤ C2 |u- v| .So|Mk(u,v)|≤ C2
|u-v|- δx{|u- v|≤ 2k- 1},whe
r
e0<δ<1.He
nc
e
∫|M (u,v)|du+∫|M (u,v)|dv≤ C2
- kaδ
k
k
R
R
*
- kaδ
Thus‖Nk Nk‖ L2→ L2≤C2
‖Td
‖2=
∞ ,kf
.
1
- kaδ
tf
ol
l
owst
hat
and‖Nk‖ L2→ L2≤C2 2 .I
2
(∫∫∫
S
Y R
1
2
)≤
Ω (θ
)Nk(f(z+·θ
))(s
)dθ ds
dz
n- 1 d
(2.9)
1
C2- 2kaδ‖Ωd‖ ∞ ‖f‖ 2.
≤C,wege
Us
i
ngt
hei
nt
e
r
pol
at
i
ont
he
or
e
m be
t
we
e
n(2.8)and(2.9),andnot
i
ngt
hat|θ
t
d|
<1s
t
hatt
he
r
ee
xi
s
t
sac
ons
t
ant0<δ'
uc
ht
hat
1
‖Td
‖ p ≤ C2- 2kaδ'2d‖f‖ p.
∞ ,kf
(2.10)
Ont
heot
he
rhand,
∫
|Td
|≤
∞ ,kf
k- 1
2
k
< |x- y|≤ 2
Ωd(x- y)
|f(y)|dy≤ C‖Ωd‖ ∞ M(f)(x).
|x- y|n
d
Sof
orany1<q<∞ andw(x)∈Aq,wehave‖T∞ ,kf‖ p,w≤C‖Ωd‖ ∞ ‖f‖ p,w.
1+ ε
>0s
Foranyw∈Aq,t
he
r
ei
sanε
uc
ht
hatw
d
∞ ,k
‖T
∈Aq.Thus
,f
oranyw∈Aq,wehave
f‖ p,w1+ ε ≤ C‖Ωd‖ ∞ ‖f‖ p,w1+ ε.
(2.11)
The
r
e
f
or
e,us
i
ngt
hei
nt
e
r
pol
at
i
ont
he
or
e
m wi
t
hc
hangeofme
as
ur
e(s
e
e[1])be
t
we
e
n(2.
10)and(2.11),wege
t
A CLASSOFOSCI
LLATORY SI
NGULAR I
NTEGRALS
ON TRI
EBELLI
ZORKI
N SPACES
.
Ji
axgLi
ya,e
tal
‖Td
‖ p,w ≤ C2- kσ2d‖f‖ p,w
∞ ,kf
75
(0< σ< 1).
(2.12)
Fr
om Le
mma2.1wehave
‖Td
‖ F·sp,q ≤ C2- kσ2d‖f‖ F·sp,q.
∞ ,kf
(2.13)
Ac
c
or
di
ngt
oLe
mma2.4,weal
s
oge
t
‖Td
‖ F·sp,q ≤ C‖Ωd‖ L1‖f‖ F·sp,q.
∞ ,kf
So
Nd
Σ Σ ‖Td∞ ,kf‖ F·sp,q =
‖T∞ f‖ F·sp,q ≤
d≥ 0 k≥ 1
(Σ Σ
∞
+
d≥ 0 k= 1
d
∞ ,k
Σ Σ )‖T
f‖ F·sp,q = I+ I
I
.
d≥ 0k= Nd+ 1
Nd
I≤
d
Σ Σ C2|θ|‖f‖
d
s
,q
·
F
p
≤ CN‖Ω‖ Llog+ L‖f‖ F·sp,q,
d≥ 0 k= 1
∞
I
I≤
Σ Σ
C2- kσ2d‖f‖ F·sp,q ≤ C‖f‖ F·sp,q,
d≥ 0k= Nd+ 1
pr
ovi
de
dt
hatN i
ss
uf
f
i
c
i
e
ntl
ar
ge.Thust
hec
as
ea> 0ofThe
or
e
m 1.1i
sc
ompl
e
t
e
l
y
pr
ove
d.
)a<0.Toobt
,wee
Now wede
alwi
t
ht
hec
as
e(I
I
ai
nt
her
e
s
ul
t
s
xc
hanget
heme
t
hodt
o
),whi
.
t
r
e
atT0fandT∞ ff
orc
as
e(I
c
hi
se
as
yt
oc
he
c
k.He
r
eweomi
tt
hede
t
ai
l
s
gh ij
kklklmno
kj
o
p q.r
0,p
0,p
0,1
·
·
·
suet
oF
i
tonl
yne
e
dst
opr
ovet
hatT i
sbounde
dont1 .
p =tp ,
v
,q
·
(wx)(vyw,1≤p,q≤∞).
,l
Fi
r
s
t
e
t
u
ss
t
at
et
hede
f
i
ni
t
i
onoft
p
∞
x
zo
l
{
|{
}
{
k|h.q.Le
t~(x)bear
adi
alf
unc
t
i
oni
nC (w )whi
c
hs
at
i
s
f
i
e
s
^
~(0)= 0,
(3.1)
x
s
upp~⊂ {xy w :|x|≤ 1},
(3.2)
∞
ds
^
|~(x)|2 < ∞.
0
s
v
,q
·
Forvyw,1≤p,q≤∞,t
hehomoge
ne
ousBe
s
ovs
pac
et
sde
f
i
ne
dby
p i
∫
·
v
,q
t
wx)= fy S'
p (
(wx)╲P(wx):‖f‖ t·pv,q(wx) =
{
-x
1
∞
(∫
0
(3.3)
-v
q
t
‖~t*f‖ q
p
dt q
<∞ ,
t
)
}
x
(t).
whe
r
e~t(x)=t ~
zo
l
{
|{
}
{
k|h.r.Wes
ayt
hata(x)i
sa(1,q)at
om,i
fa(x)s
at
i
s
f
i
e
ss
uppa(x)⊂t(x0,ρ)
and
‖a‖ q ≤ ρ- x+ x/q,
J
(3.4)
- x- |J|+ x/q
‖D a‖ q ≤ ρ
,
∫xa(x)dx= 0
β
x
w
f
oral
lmul
t
i
i
ndi
c
e
sJandβwi
t
h|J|≤2,|β|≤1.
[6]
0,1
·
(wx),i
Lo
ppah.q. A f
unc
t
i
onfbe
l
ongst
ot
fandonl
yi
f
1
(3.5)
(3.6)
.Mat
.B
A{{l
h.J.Chi
ne
s
eUni
|.Se
r
76
f=
.21,No.1
Vol
Σ Ca,
k k
k
wi
t
h‖f‖ B·01,1(Rn) ～
Σ |C |,wherea isa(1,∞)atom.
k
k
k
Topr
oveThe
or
e
m 1.2,i
ti
se
nought
opr
ovet
hatf
orany(1,∞ )at
om a(x),t
he
r
e
hol
ds‖Ta‖ B·01,1(Rn)≤C,whe
r
eCi
sac
ons
t
anti
nde
pe
nde
ntofat
om.
[6]
Le
mma3.2. Le
ta(x)beadi
f
f
e
r
e
nt
i
abl
ef
unc
t
i
onwi
t
hs
uppa⊂B(0,r)ands
at
i
s
f
yt
he
c
anc
e
l
l
at
i
onc
ondi
t
i
on(3.6).Le
t
Aj(r
,a)=
j
s
upj+ 4
2r
≤ |y|≤ 2
a
∫e
i
|y- x|
n
R
r
a(x)dx ,
a+ n j
,a)≤Cr
2(a- 1)‖a‖ ∞ ,andt
=1,2,...,wehaveAj(r
he
r
ee
xi
s
t
sani
nt
e
ge
r
t
he
nf
oral
lj
≥N,t
N>0i
nde
pe
nde
ntofrs
uc
ht
hatwhe
nj
hef
ol
l
owi
nghol
ds
Δ
n
Aj(r
,a)≤ Cr
[(2jr
)- a+ 1‖
a‖ ∞ + (2jr
)- a‖a‖ ∞ ].
[6]
Le
mma3.3. Le
ta(x)be(1,2)at
om,t
he
nt
he
r
ee
xi
s
t
sc
ons
t
antCi
nde
pe
nde
ntofa(x)
s
uc
ht
hat
‖a‖ B·01,1(Rn) ≤ C.
,f
Re
mar
k3.1.I
nf
ac
t
ol
l
owi
ngt
hepr
oofofLe
mma2.5i
n[6],wec
ane
as
i
l
yf
i
ndt
heabove
l
e
mmai
sal
s
ot
r
uei
ft
hef
unc
t
i
onas
at
i
s
f
i
e
st
hec
ondi
t
i
ons(3.4),(3.5),(3.6)f
or|J|=1
and|β|=0onl
y.
,wes
Ne
xt
hal
lpr
ove‖Ta‖ B·01,1(Rn)≤C.
∞
n
Wi
t
houtl
os
sofge
ne
r
al
i
t
y,wemayas
s
umet
hats
uppa⊂ B(0,ρ).Choos
eaC (R )
f
unc
t
i
ons≥0whi
c
hs
at
i
s
f
i
e
ss
upps⊂
1
t2≤|x|≤2u,andΣ
∞
-j
-2
s(2- jy)= 1(xyy 0).
j
vw
N- 1
ρ|y|)= Σ j= - ∞ s(2- j- 2ρ|y|),
Le
tN beasi
nLe
mma3.2,andz(x)= 1- Σ j= Ns(2
t
he
n
a
i
|}|
Ta(x)= {.|.e
~(})
z(})*a(x)+
|}|n
∞
a
i
|}|
Σ {.|.e
j
=N
~(})
s(2- j- 2ρ|}|)*a(x)
|}|n
∞
:=T0a(x)+
Σ T a(x).
j
j
=N
Obvi
ous
l
y,t
he
r
ehol
ds
s
uppT0a(x)⊂ B(0,2N+ 5ρ),
s
uppTja(x)⊂ B(0,2j+ 4ρ),
∫xT a(x)dx=∫xT a(x)dx= 0
J
n
J
0
R
n
j
R
x|J|≤ 1.
ByThe
or
e
m 1.1wege
t
Δ
‖T0a‖ 2 ≤ C‖a‖ 2 ≤ Cρ- n and ‖
T0a‖ 2 ≤ C‖Δa‖ 2 ≤ Cρ- n- 1
ThusbyLe
mma3.3andRe
mar
k3.1,‖T0a‖ B·01,1(Rn)≤C,whe
r
eCi
si
nde
pe
nde
ntofa.
I
nt
hef
ol
l
owi
ngwewi
l
lpr
ove
t a[tuuvhvuaI
[[t]vwx uI
Nyz[tw I
N]{ywt[u
vN ]wI
{U{[|
[I
}vw~I
N uPta{u
.
op
anqLp
ya,r
tas
77
-j
+1
2
{
∫
‖Tja‖ L1 ≤ C 2- j + mi
n(2- j(a- 1)ρ- a,2j(a- 1)ρa)+
dt .
ω(t
)
t
}
-j
-3
2
(3.7)
j
j
+4
Not
et
hatwhe
n2ρ≤|x|≤2 ρ,Tja(x)≠0.I
tf
ol
l
owst
hat
a
∫
∫
‖Tja‖ L1 =
Ω(x- y)
∫e |x- y| Q(2 ρ |x- y|)a(y)dy dx≤
Ω(x- y)
∫e R|x- y| Q(2 ρ |x- y|)Ω(x)
Q(2 ρ |x|) a(y)dy dx+
S
|x|
Ω(x)
∫e |x| Q(2 ρ |x|a(y)dy dx
i
|x- y|
j
R
ρ
a
-j
-2 -1
i
|x- y|
j
j
+4
2ρ≤ |x|≤ 2
n
n
ρ
-j
-2 -1
n
n
j
+4
2ρ≤ |x|≤ 2
R
-j
-2 -1
n
a
∫
i
|x- y|
j
2ρ≤ |x|≤ 2
-j
-2 -1
n
n
j
+4
R
ρ
T=I
I
j+ I
j.
Ue
V
aWs
eai
s(1,X)at
om andYZ[e
mma3.2,weoYt
ai
n
∫
I
I
j≤ C
j
j
+4
2ρ≤ |x|≤ 2
ρ
|Ω(x)|
|\j(ρ,a)|dx≤
|x|n
C‖Ω‖ 1mi
n(2- j(a- 1)ρ- a,2j(a- 1)ρa).
]oe
s
t
i
mat
eI
w^
i
t
e
j,
Ω(x- y)
Ω(x)
∫
∫ |x- y| - |x| |a(y)|dydx+
|Ω(x- y)|
∫
∫ |x- y| |Q(2 ρ |x- y|)- Q(2
I
j≤
j
j
+4
2ρ≤ |x|≤ 2
n
n
ρ R
n
-j
-2 -1
j
j
+4
2ρ≤ |x|≤ 2
-j
-2 -1
ρ |x|)‖a(y)|dydx
n
n
ρ R
T=\j + _j.
-j
I
ti
se
as
Zt
o‘e
t_j≤C2 .
1
1
∫
C
∫
\j≤
j
∫ |x- y| - |x| |Ω(x)||a(y)|dydx+
|Ω(x- y)- Ω(x)|
∫ |x- y| |a(y)|dydx
j
+4
2ρ≤ |x|≤ 2
j
j
+4
2ρ≤ |x|≤ 2
n
n
ρ R
n
n
n
ρ R
T=\1j + \2j.
j
j
j
+4
bcb 1,
al
e
a^
l
Z,\1j≤C2.]oe
s
t
i
mat
e\2j,wenot
et
hatwhe
n2ρ≤|x|≤2 ρ,|y|≤ρ,j
2ρ
2|y|
d|x|e≤ωd|x|e,andsinVe‖a‖ ≤1wehafe
2ρ
ω
|x|e
dt
d
∫ |x| |a(y)|dydx≤ C∫ ω(t) t.
t
he
^
ehol
ds|Ω(x-y)-Ω(x)|≤ω
1
L
-j
+1
∫
\2j ≤ C
j
2
j
+4
2ρ≤ |x|≤ 2
n
ρ R
n
-j
-3
2
]he
^
e
f
o^
eweoYt
ai
ni
ne
gWal
i
t
Z(3.7).
hi
nal
l
Zwes
hal
ls
how
‖ i Tja‖ _j01,1(Rn) ≤ C.
j
bc
0,1
n
j
kelnow t
hatal
i
ne
a^ome
^
at
o^T ofV
onfol
Wt
i
ont
Zmei
sYoWnde
don_
R )i
fand
1 (
0,1
n
0,X
n
j
j
onl
Zi
fT i
sYoWnde
df
^
om _1 (R )t
o_1 (R ).ne
nV
eweonl
Zne
e
dt
om^
ofe
‖ i Tja‖ _j01,X (Rn) ≤
j
bc
i ‖T a‖
j
j
bc
0,X
n
j
U
R)
1 (
≤ C.
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
78
.21,No.1
Vol
0,∞
n
·
Byt
hede
f
i
ni
t
i
onofB1 (R ),
‖Tja‖ B·01,∞ (Rn) = s
up‖ΦtTja‖ 1 ≤ C‖Tja‖ 1.
t
>0
I
tf
ol
l
owst
hat
-j
+1
2
{
∫
‖Tja‖ B·01,∞ (Rn) ≤ C 2- j + mi
n(2- j(a- 1)ρ- a,2j(a- 1)ρa)+
dt .
ω(t
)
t
-j
-3
2
}
Thus
‖ Σ Tja‖ B·01,∞ (Rn) ≤ C.
j
≥N
He
nc
et
hepr
oofofThe
or
e
m 1.2i
snow c
ompl
e
t
e
d.
Re
f
e
r
e
nc
e
s
‥
1 Be
,Lo
.I
,AnI
r
ghJ
f
s
t
r
om J
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r
pol
a
t
i
onSpa
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s
nt
r
oduc
t
i
on,Be
r
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i
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w Yor
k:Spr
i
ngVe
r
l
a
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2 Che
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,JMa
,2002,276:
nJ
nD,Yi
ngY.Si
ngul
a
ri
nt
e
gr
a
lope
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a
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or
sonf
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i
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pa
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s
t
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lAppl
691708.
+
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3 Che
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nD,Yi
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r
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gr
a
l
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og L) ke
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l
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4 Cha
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,1983,21(2):
ni
l
l
oS,Kur
t
zD,Sa
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onG.We
i
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e
dL e
s
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i
ma
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sf
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a
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i
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ne
l
s
kMa
t
233257.
p
5 Cha
ni
l
l
oS,Kur
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zD,Sa
mps
on G.We
i
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dwe
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k (1,1)a
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e
dL e
s
t
i
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t
e
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c
i
l
l
a
t
i
ng
,Tr
,1986,295(1):127145.
ke
r
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l
s
a
nsAme
rMa
t
hSoc
0,1
·
6 Fa
,1994,187:9861002.
1 (
nD.Anos
c
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l
l
a
t
or
yi
nt
e
gr
a
li
nB
Rn),JMa
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or
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a
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i
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t
or
s
lMa
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orr
oughs
i
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a
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nt
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a
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eAnn
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sf
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gr
a
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swi
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hr
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,Re
218.
ke
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s
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t
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be
r1992,8(2):21010 Ri
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Va
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h.,Zhe
j
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