Appl
.Mat
h.J.Chi
ne
s
eUni
v.Se
r
.B
2005,20(4):455461
1
1
MATEFOR COMMUTATOR OF
WEAK TYPE(H ,L)ESTI
MARCI
NKI
EWI
CZI
NTEGRAL
Zha
ngPu WuHuoxi
ong
.Le
Abs
t
r
ac
t
tµΩ,b bet
hec
ommut
a
t
orge
ne
r
a
l
i
z
e
dbyt
hendi
me
ns
i
ona
lMa
r
c
i
nki
e
wi
c
zi
nt
e
gr
a
l
1
n
∈BMO(Rn).I
µΩ a
ndaf
unc
t
i
onb
ti
spr
ove
dt
ha
tµΩ,bi
sbounde
df
r
om t
heHa
r
dys
pa
c
eH (R )
1
n
pa
c
e.
i
nt
ot
hewe
a
kL (R )s
01 I
2t
r
345c
t
6
32a24s
t
at
7
87
2t39r
7
s
5:
t
n< 1
;e
not
ebyS
n
t
heuni
ts
phe
r
ei
nR (n= 2)e
>ui
ppe
dwi
t
ht
henor
mal
i
z
e
dLe
be
s
gue
1
n< 1
AdB(?@
).Le
me
as
ur
ed?@
tΩ∈L (S )behomoge
ne
ousofde
gr
e
ez
e
r
oands
at
i
s
f
y
C Ω(?@)d?@A 0.
(1D1)
n< 1
S
Thendi
me
ns
i
onalMar
c
i
nki
e
wi
c
zi
nt
e
gr
alc
or
r
e
s
pondi
ngt
ot
heLi
t
t
l
e
woodPal
e
yEF1G
f
unc
t
i
oni
nt
r
oduc
e
dbySt
e
i
n i
sde
f
i
ne
dby
I
µΩ(H)(?)A
LC
0
JKΩ,t(H)(?)J2
dt 1N2
,
3
t
M
whe
r
e
C
KΩ,t(H)(?)A
J?< OJP t
Ω(?< O)
H(O)dO.
J?< OJn< 1
F1G
n< 1
I
n1Q58,St
e
i
n pr
ove
dt
hati
fΩ∈Li
pR(S )(0SRP1),t
he
nµΩ i
soft
ype(p,p)f
or1
S pP 2and ofwe
ak t
ype(1,1).Si
nc
et
he
n many aut
hor
ss
t
udi
e
dt
hebounde
dne
s
s
,wer
10Gf
pr
ope
r
t
i
e
sofµΩ on var
i
ousf
unc
t
i
on s
pac
e
s
e
f
e
rt
her
e
ade
r
st
os
e
eF2ori
t
s
F10G
.Re
de
ve
l
opme
nt
sandappl
i
c
at
i
ons
c
e
nt
l
y,FanandSat
o
e
s
t
abl
i
s
he
dt
hef
ol
l
owi
ngwe
ak
.
t
ype(1,1)i
ne
>ual
i
t
yf
orµΩ wi
t
hr
oughke
r
ne
l
F10G
n< 1
TT7
3r
7
8 1. I
fΩ∈ L l
ogL(S )ands
at
i
s
f
i
e
s(1D1),t
he
nµΩ i
sofwe
akt
ype(1,1),
t
hati
s
1218.
Re
c
e
i
ve
d:2003MR Subj
e
c
tCl
a
s
s
i
f
i
c
a
t
i
on:42B30,42B25.
:Ma
,c
,Ha
Ke
ywor
ds
r
c
i
nki
e
wi
c
zi
nt
e
gr
a
l
ommut
a
t
or
r
dys
pa
c
e,a
t
om.
Suppor
t
e
d byt
heRe
s
e
a
r
c
h Founda
t
i
on ofZhe
j
i
a
ngSc
i
Te
c
h Uni
ve
r
s
i
t
y (0313055Y)a
ndNSFZJ
(Y604563).
.<=t
.E
App;
>.?.C>@
nA
B
ACn@
D.SA
r
45:
.20,Go.4
Fol
s
upλ
|{x∈ Rn:µΩ(f)(x)> λ
}|≤ C‖f‖ L1(Rn),
λ
>0
whe
r
eCi
sapos
i
t
i
vec
ons
t
anti
nde
pe
nde
ntoff.
n
∈BMO(R ),t
hec
ommut
at
orge
ne
r
at
e
dbyµΩ andbi
sde
f
i
ne
dby
Forb
∞
µΩ,b(f)(x)=
2
[b
(x)- b
(y)]Ω(x- y)
dt
f(y)dy 3
n- 1
|x- y|≤ t
|x- y|
t
(∫∫
0
1/2
).
[11]
p
ove
dt
hatµΩ,b i
sbounde
df
r
om L (ω)i
nt
oi
t
s
e
l
f
I
n1990,Tor
c
hi
ns
kyandWang pr
n- 1
[12]
wi
t
hω∈Ap and1<p<∞ whe
nΩ∈Li
pγ(S )(0<γ≤1).I
n2003,Di
ng,LuandYabut
a
p
s
t
udi
e
dt
hewe
i
ght
e
dL bounde
dne
s
soft
hehi
ghe
ror
de
rc
ommut
at
orofµΩ.Wewi
l
lus
e
.
t
hes
pe
c
i
alc
as
eoft
he
i
rr
e
s
ul
t
sasf
ol
l
ows
[12]
The
or
e
m 2.
r
n- 1
n
)(r> 1)s
at
i
s
f
yi
ng(1.1)andb∈ BMO(R ).
Suppos
et
hatΩ∈ L (S
The
n,f
or1<p<∞,t
he
r
ei
sapos
i
t
i
vec
ons
t
antC,i
nde
pe
nde
ntoff,s
uc
ht
hat
‖µΩ,b(f)‖ Lp(Rn) ≤ C‖f‖ Lp(Rn).
[13]
Re
c
e
nt
l
y,Di
ng,LuandXue
[14]
andChe
n,ZhangandChe
n
s
t
udi
e
dt
hebounde
dne
s
s
1
n
pr
ope
r
t
i
e
soft
hehi
ghe
ror
de
rc
ommut
at
or
sofµΩ ont
heHar
dyt
ypes
pac
e
sHb(R )andt
he
α
,p
n
[15]
·
m
He
r
t
ypeHar
dys
pac
e
sHKq,b (R ),r
e
s
pe
c
t
i
ve
l
y.Di
ng,Lu andZhang e
z
s
t
abl
i
s
he
da
n- 1
we
i
ght
e
dwe
ak(Ll
ogL)t
ypee
s
t
i
mat
ef
orµΩ,bwhe
nΩ∈Li
pγ(S )(0<γ≤1)andω∈A1.
1
n
,wewi
I
nt
hi
spape
r
l
ls
t
udyt
hebounde
dne
s
sofµΩ,b ont
heHar
dys
pac
eH (R )and
1
n
1
n
sbounde
df
r
om H (R )i
nt
ot
hewe
akL (R )s
pac
e.Be
f
or
es
t
at
i
ngour
pr
ovet
hatµΩ,b i
r
t
he
or
e
m,wef
i
r
s
tr
e
c
al
lt
hede
f
i
ni
t
i
onoft
heLDi
nic
ondi
t
i
on.
r
r
n
De
f
i
ni
t
i
on1.Le
tr≥ 1,wes
ayt
hatΩ s
at
i
s
f
i
e
st
heLDi
nic
ondi
t
i
oni
fΩ∈ L (R )i
s
n
homoge
ne
ousofde
gr
e
ez
e
r
oi
nR and
1
ωr(δ)
dδ< ∞,
δ
∫
0
whe
r
eωr(δ)de
not
e
st
hei
nt
e
gr
almodul
usofc
ont
i
nui
t
yofor
de
rrofΩ de
f
i
ne
dby
1/r
(∫ |Ω(ρx')- Ω(x')|dx') ,
ωr(δ)= s
up
|ρ|< δ
r
n- 1
S
n
-x'
|.
andρi
sar
ot
at
i
oni
nR wi
t
h|ρ|=s
upx'∈ Sn- 1|ρx'
r
n- 1
n
>1)behomoge
The
or
e
m 3.Le
tΩ∈L (S )(r
ne
ousofde
gr
e
ez
e
r
oi
nR s
at
i
s
f
yi
ng(1.1)
and
1
ωr(δ)
1
l
og
dδ< ∞.
δ
δ
∫
0
(
)
(1.2)
1
n
1
n
1
n
>
The
nµΩ,bi
sbounde
df
r
om H (R )i
nt
owe
akL (R ),s
ay,f
oranyf∈H (R )andanyλ
0,
-1
|{x∈ Rn:µΩ,b(f)(x)> λ
}|≤ Cλ
‖f‖ H1(Rn),
(1.3)
.
whe
r
eCi
sapos
i
t
i
vec
ons
t
anti
nde
pe
nde
ntoffandλ
32 4r
e
5
i
mi
n6r
75
e
mm68
[1:]
r
>1,andΩ s
9e
mm61. Suppos
et
hat0<α<n,r
at
i
s
f
i
e
st
heLDi
nic
ondi
t
i
on.I
ft
he
r
ei
s
7@AA TB0@(H1,L1)@CTI
DAT@FEF
CEDDGTATEF EFDAFCI
NAI
@7I
CHI
NT@IFAL
.
;4ang<=,>
6a?
:97
ac
ons
t
anta0 wi
t
h0<a0<1/2s
uc
ht
hat|y|<a0R,t
he
n
(∫
R< |x|< 2R
Ω(x- y)
Ω(x) r
dx
n- α |x- y|
|x|n- α
1/r
)
|y|
{R +∫
≤ CRn/r- n+ α
|y|/2R< δ< |y|/R
1
ωr(δ)
dδ .
δ
}
n
Le
tusr
e
c
al
lt
heat
omi
cde
c
ompos
i
t
i
onoft
heHar
dys
pac
eH (R )(s
e
e[17]f
or
).
de
t
ai
l
s
De
f
i
ni
t
i
on2.A f
unc
t
i
ona(x)i
sc
al
l
e
da(1,∞)at
om,i
f
∫
) a(x)dx= 0.
(i
)a(x)i
)‖a‖ L∞ ≤|B|- 1;(i
i
i
ss
uppor
t
e
di
nabal
lB;(i
i
1
n
1
n
1
aii
Le
mma2.A f
unc
t
i
onf∈L (R )be
l
ongst
oH (R )i
fandonl
yi
ff= Σ iλ
nH nor
m
i
1
,λ
∈Cwi
|< ∞.Fur
orL nor
m,whe
r
eai'
sar
e(1,∞)at
oms
t
hΣ i|λ
t
he
r
mor
e,
i
i
‖f‖ H1(Rn) ～ i
nf
{Σ i|λi|},
"i
whe
r
e"i
nf
st
ake
nove
ral
lt
heaboveat
omi
cde
c
ompos
i
t
i
onsoff.
yz {|
oofof}~e
o|
e
mz
1
n
Foragi
ve
nf∈H (R ),i
tf
ol
l
owsf
r
om t
heat
omi
cde
c
ompos
i
t
i
ont
hatf=
Σ λa,
ii i
.Topr
whe
r
eai'
sar
e(1,∞)at
oms
oveThe
or
e
m 3,i
ts
uf
f
i
c
e
st
opr
ovet
hat(1.3)hol
dsf
or
≤ 2‖f‖ H1(Rn)ande
fbe
i
ngaf
i
ni
t
es
umf= Σ jλ
t
hΣ j|λ
ac
haji
sa(1,∞)at
om.
ja
jwi
j|
I
nde
e
d,onc
e(1.3)i
spr
ove
nf
ors
uc
hf,f
ort
hege
ne
r
alf=
1
n
Σ λa ∈ H (R ),wecan
ii i
c
hoos
eas
e
que
nc
eof{fk}wi
t
hfkbe
i
ngaf
i
ni
t
es
um asaboves
uc
ht
hat{fk}c
onve
r
ge
st
of
1
,The
nor
m oral
mos
te
ve
r
ywhe
r
es
e
ns
ewhe
nk→∞.The
n,byal
i
mi
tar
gume
nt
or
e
m
i
nH 2
3f
ol
l
owsf
r
om t
heL bounde
dne
s
sofµΩ,b.
‖f‖ H1(Rn) ,whe
Now,weas
s
umet
hatf= Σ jλ
saf
i
ni
t
es
um wi
t
hΣ j|λ
r
e
ja
ji
j|≤ 2
e
nt
e
r
e
datxjwi
e
ac
haji
sa(1,∞)at
om s
uppor
t
e
di
nabal
lBj=B(xj,r
t
hr
adi
usr
j)c
j.As
1
g(x)dx.
|B| B
Forc
onve
ni
e
nc
e,wet
aket
hepoi
ntofvi
e
w oft
heve
c
t
or
val
ue
ds
i
ngul
ari
nt
e
gr
alof
,f
us
ual
oral
oc
al
l
yi
nt
e
gr
abl
ef
unc
t
i
ongandabal
lB,wede
not
ebygB =
∫
[12]
Be
ne
de
k,Cal
de
r
/nand0an1
one .Le
t3 bet
heHi
l
be
r
ts
pac
ede
f
i
ne
dby
∞
{
3 = 45‖4‖ 3 =
(∫
0
1/2
|4(6
)|2
d6
3
6
)
}
<∞ .
7r
i
t
e
∫
8b
(f)(x)=
Ω,6
|x- y|≤ 6
[b
(x)- b
(y)]Ω(x- y)
f(y)dy.
|x- y|n- 1
b
7emayvi
e
w 8Ω,6(f)(x)and8Ω,6(f)(x)asmappi
ngsf
r
om [0,∞)i
nt
o3.I
ti
sc
l
e
art
hat
µΩ(f)(x)= ‖8Ω,6(f)(x)‖ 3 andµΩ,b(f)(x)= ‖8b
(f)(x)‖ 3 .
Ω,6
I
ti
snotdi
f
f
i
c
ul
tt
os
e
et
hat
µΩ,b(f)(x)≤‖ Σ λ
b
(x)- b
8Ω,6(aj)(x)‖ 3 + ‖8Ω,6(Σ λ
b- b
aj)(x)‖ 3 ≤
j(
Bj)
j(
Bj)
j
j
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
458
.20,No.4
Vol
Σ |λ‖b(x)- b |µ (a)(x)+ µ (Σ λ(b- b )a)(x).
j
Bj
Ω
j
Ω
j
j
Bj
j
j
>0,t
The
n,f
oranyf
i
xe
dλ
he
r
ehas
}|≤|{x∈ Rn:Σ |λ
(x)- b
µΩ(aj)(x)> λ
/2}|+
|{x∈ Rn:µΩ,b(f)(x)> λ
j‖b
Bj|
j
n
|{x∈ R :µΩ(Σ λ
b- b
aj)(x)> λ
/2}|:=
j(
Bj)
j
I+ II.
(3.1)
-1
n
Not
i
ngt
hat‖aj‖ L∞ (R )≤ |Bj| ,t
hewe
akt
ype(1,1)i
ne
qual
i
t
yofµΩ(The
or
e
m 1)
gi
ve
s
∫Σ |λ‖b(y)- b ‖a(y)|dy≤
Cλ Σ |λ‖ B |
∫|b(y)- b |dy≤
-1
II≤Cλ
j
n
R
Bj
j
j
-1
-1
j
j
Bj
Bj
j
-1
C‖b
‖* λ
Σ |λj|≤ Cλ- 1‖f‖ H1(Rn),
(3.2)
j
‖ * de
.
whe
r
e‖b
not
e
st
heBMO nor
m ofb
Ont
heot
he
rhand,wehave
∫Σ |λ‖b(x)- b |µ (a)(x)dx≤
Cλ Σ |λ| |b
∫ (x)- b |µ (a)(x)dx+
Cλ Σ |λ|
∫ |b(x)- b |µ (a)(x)dx:=
-1
I≤Cλ
j
n
R
Bj
Ω
j
j
-1
j
Bj
4Bj
j
Ω
j
-1
j
j
Bj
n
R ╲4Bj
Ω
j
I1 + I2.
(3.3)
n
Si
nc
eb∈ BMO(R ),t
he
nf
oranynonne
gat
i
vei
nt
e
ge
rkandanybal
lB,t
he
r
ehas
k+ 1
|b
≤C(k+1)‖b
‖ * (s
),andt
e
e[17],p.141f
orde
t
ai
l
s
he
n
2
B-b
B|
1/p
1
p
|b
(x)- b
x
≤ C(k+ 1)‖b
‖* .
(3.4)
B|d
k+ 1
|2 B| 2 B
- 1/2
2
‥
ForI1,not
i
ngt
hat‖ aj‖ L2(Rn)≤ |B| ,byt
heHo
l
de
ri
ne
qual
i
t
y,(3.4)andL -
(
∫
k+ 1
)
bounde
dne
s
sofµΩ wehave
(∫
-1
I1 ≤Cλ
Σ |λj|‖µΩ(aj)‖ L2(Rn)
j
1/2
2
|b
(x)- b
x
Bj|d
4Bj
)
≤
-1
Cλ
Σ |λj|‖aj‖ L2(Rn)|4Bj|1/2‖b‖ * ≤
j
-1
Cλ
-1
Σ |λ|≤ Cλ ‖f‖
j
1
n
.
H (R )
(3.5)
j
∫
Se
tJj=
n
-1
|b
(x)- b
µΩ(aj)(x)dx,t
Jj.Toe
he
nI2= Cλ Σ j|λ
s
t
i
mat
eI2,we
Bj|
j|
R ╲4Bj
.Wes
f
i
r
s
tpr
ovet
hatJj≤ C wi
t
hapos
i
t
i
vec
ons
t
antC i
nde
pe
nde
ntofj
pl
i
tJj i
nt
ot
wo
,
par
t
s
WEAK TYPE(H1,L1)ESTI
MATEFOR
COMMUTATOR OFMARCI
NKI
EWI
CZI
NTEGRAL
.
ZhangPu,e
tal
dt 1/2
dx≤
3
t
∞
(∫
∫ |b(x)- b |(∫
∫ |b(x)- b |(∫
∫
Jj =
|b
(x)- b
Bj|
n
R ╲4Bj
0
Bj
n
R ╲4Bj
0
Bj
R ╲4Bj
dt 1/2
dx+
3
t
)
dt
(a)(x)|
dx:=
t)
|FΩ,t(aj)(x)|2
1/2
∞
n
)
|FΩ,t(aj)(x)|2
|x- xj|+ 2r
j
459
2
|FΩ,t
|x- xj|+ 2r
j
j
3
G+ H.
(3.6)
n
Si
nc
e|x-y|～|x-xj|～|x-xj|+2r
e
ne
ve
rx∈R ╲4Bjandy∈Bj,t
he
n
jwh
Cr
1
1
j
,f
orx∈ Rn╲4Bj,y∈ Bj.
2 ≤
|x- y|2
(|x- xj|+ 2r
|x- y|3
j)
Byt
heMi
nkows
ki
'
si
ne
qual
i
t
y,t
hes
i
z
ec
ondi
t
i
onofajand(3.4),t
he
r
ehas
∫
|b
(x)- b
Bj|
∫
|b
(x)- b
Bj|
G≤
n
∫
R ╲4Bj
n
n
R
∫Σ∫
k+ 1
Bjk= 1 2
∞
=
<
Σ
Bjk= 1
=
k
Bj╲2 Bj
1/2
dydx≤
|Ω(x- y)|
|b
(x)- b
dxdy≤
Bj|
|x- y|n+ 1/2
|Ω(x- y)|r
dx
k+ 1
k
2
Bj╲2 Bj |
x- y|n+ 1/2
(∫
∫
r
'
)1/r')
|b
(x)- b
Bj|
>dy≤
d
x
n+ 1/2
k+ 1
k
= 2 Bj╲2 Bj |x- y|
) )
1/r
=
)∫
∞
|Ω(x- y)|r
dx
k+ 1
k
2
Bj╲2 Bj |
x- y|n+ 1/2
(∫
∫
1/2- n
Cr
j
)
|Ω(x- y)||aj(y)|
1
1
2
|x- y|2
(|x- xj|+ 2r
Bj
|x- y|n- 1
j)
∞
1/2- n
Cr
j
(∫
∫
R ╲4Bj
1/2- n
Cr
j
dt 1/2
dydx≤
3
|x- y|≤ t
≤ |x- xj|+ 2r
j t
|Ω(x- y)||aj(y)|
|x- y|n- 1
Σ (k+ 1)(2k+ 1rj)- 1/2r'
Bjk= 1
1/r
) dy.
(3.7)
k- 1
k+ 2
k+ 1
k
Si
nc
e2 r
x-y|≤2 r
e
ne
ve
ry∈Bjandx∈2 Bj╲2Bj,t
he
n
j≤|
jwh
∫
|Ω(x- y)|r
dx≤
Bj╲2 Bj |
x- y|n+ 1/2
k+ 1
2
∫
k
k- 1
2
|Ω(x- y)|r
dx≤
r
x- y|n+ 1/2
j |
k+ 2
r
x- y|≤ 2
j≤ |
- 1/2
C‖Ω‖ Lr(Sn- 1)(2kr
.
j)
,t
Thi
s
oge
t
he
rwi
t
h(3.7)gi
ve
s
∞
∞
∫
1/2- n
G≤ Cr
j
Σ (k+ 1)(2k+ 1rj)- 1/2r'- 1/2rdy≤ CΣ (k+ 1)2- k/2 ≤ C.
Bjk= 1
(3.8)
k= 1
Ω(x-xj)
Ω(x-y)
f
or
|x-xj|n- 1
|x-y|n- 1
,byt
s
i
mpl
i
c
i
t
y.Not
i
ngt
hat|x-y|≤|x-xj|+r
whe
ny∈Bjand|x-xj|+2r
he
j<t
j≤t
Now,l
e
tuse
s
t
i
mat
e H. We wr
i
t
e K (x,y,xj)=
c
anc
e
l
l
at
i
onc
ondi
t
i
onofaj,wehave
∞
∫
H=
(∫
|b
(x)- b
Bj|
n
R ╲4Bj
∫
|x- xj|+ 2r
j
2
K(x,y,xj)aj(y)dy
|x- y|≤ t
∞
∫
∫
C
(∫
∫
|b
(x)- b
K(x,y,xj)‖aj(y)|
Bj| n|
n
R ╲4Bj
n
R
|aj(y)|
dt
3
t
∫|x- x||K(x,y,x)|dydx≤
|b
(x)- b
Bj|
R ╲4Bj
|x- xj|+ 2r
j
j
Bj
j
dt 1/2
dx≤
3
t
)
1/2
) dydx≤
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
460
.20,No.4
Vol
∞
∫Σ (2r)∫
C|Bj|- 1
k
-1
j
k+ 1
Bjk= 1
2
k
|b
(x)- b
|K(x,y,xj)|dxdy≤
Bj|
Bj╲2 Bj
∞
(∫
∫
- n- 1
Cr
j
Σ (k+ 1)2- k|2k+ 1Bj|1/r'
Bjk= 1
1/r
k+ 1
2
|K(x,y,xj)|rdx
k
) dy,
Bj╲2 Bj
‥
whe
r
et
hel
as
ti
ne
qual
i
t
yf
ol
l
owsf
r
om t
heHo
l
de
r
'
si
ne
qual
i
t
yand(3.4).
I
tf
ol
l
owsf
r
om Le
mma2.1t
hat
(∫
1/r
k+ 1
2
k
|K(x,y,xj)|rdx
)
Bj╲2 Bj
≤
k
{
|y- xj|/2 r
j
∫
n/r
- n+ 1
C(2kr
2- k +
j)
k+ 1
|y- xj|/2
r
j
ωr(δ)
dδ .
δ
}
The
n,by(1.2)t
he
r
ehas
k
∞
|y- xj|/2 r
j
∫Σ (k+ 1){2 +∫
-n
j
H ≤Cr
-k
Bjk= 1
k
∞
|y- xj|/2 r
j
∫Σ {k2 +∫
-n
Cr
j
-k
Bjk= 1
∫{Σ k2
Cr
-k
Bj
k= 1
k+ 1
|y- xj|/2
∞
-n
j
k+ 1
|y- xj|/2
1
0
ωr(δ)
1
l
og
dδ dy≤
δ
δ
ωr(δ)
1
l
og
dδ dy≤ C.
δ
δ
∫
+
r
j
ωr(δ)
dδ dy≤
δ
}
( )}
r
j
(
)}
,t
oral
ljwi
t
hapos
i
t
i
vec
ons
t
ant
Thi
s
oge
t
he
rwi
t
h(3.6)and(3.8),gi
ve
sust
hatJi≤Cf
.The
Ci
nde
pe
nde
ntofj
n
-1
I2 = Cλ
Σ |λj|Jj≤ Cλ- 1Σ |λj|≤ Cλ- 1‖f‖ H1(Rn).
j
(3.9)
j
-1
Fr
om (3.3),(3.5)and(3.9),wes
e
et
hatI≤Cλ ‖f‖ H1(Rn).(3.1),(3.2)andt
he
e
s
t
i
mat
ef
orIi
mpl
yt
hede
s
i
r
e
di
ne
qual
i
t
y.Thi
sc
ompl
e
t
e
st
hepr
oofofThe
or
e
m 3.
Re
f
e
r
e
nc
e
s
1 St
,Tr
e
i
n E M.On t
hef
unc
t
i
onsofLi
t
t
l
e
woodPa
l
e
y,Lus
i
na
ndMa
r
c
i
nki
e
wi
c
z
a
ns
a
c
t
i
onsoft
he
466.
Ame
r
i
c
a
nMa
t
he
ma
t
i
c
a
lSoc
i
e
t
y,1958,88:430p
‥
2 Hor
,Ac
ma
nde
rL.Es
t
i
ma
t
e
sf
ort
r
a
ns
l
a
t
i
oni
nva
r
i
a
ntope
r
a
t
or
si
nL s
pa
c
e
s
t
aMa
t
h,1960,104:93139.
,Duke
3 Cha
ni
l
l
oS,Whe
e
de
nR L.I
ne
qua
l
i
t
yf
orPe
a
noma
xi
ma
lf
unc
t
i
onsa
ndMa
r
c
i
nki
e
wi
c
zi
nt
e
gr
a
l
s
,1983,50(3):573603.
Ma
t
he
ma
t
i
c
a
lJ
our
na
l
4 Sa
,St
ka
mot
oM,Ya
but
aK.Bounde
dne
s
sofMa
r
c
i
nki
e
wi
c
zf
unc
t
i
ons
udi
aMa
t
he
ma
t
i
c
a,1999,135:103142.
5
,Ma
Che
n J C,Fa
n D S,Pa
n Y B. A not
e on a Ma
r
c
i
nki
e
wi
c
zi
nt
e
gr
a
lope
r
a
t
or
t
he
ma
t
i
s
c
he
42.
Na
c
hr
i
c
ht
e
n,2001,227(1):33-
p
6 Di
ngY,Fa
nD S,Pa
nY B.L bounde
dne
s
sofMa
r
c
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nki
e
wi
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zi
nt
e
gr
a
l
swi
t
hHa
r
dys
pa
c
ef
unc
t
i
on
,Ac
,2000,16(4):593600.
ke
r
ne
l
t
aMa
t
he
ma
t
i
c
aSi
ni
c
a,Engl
i
s
hSe
r
i
e
s
7 Di
,I
ngY,LuSZ,XueQ Y.Ma
r
c
i
nki
e
wi
c
zi
nt
e
gr
a
lonHa
r
dys
pa
c
e
s
nt
e
gr
a
lEqua
t
i
onsa
ndOpe
r
a
t
or
182.
The
or
y,2002,42:174-
.
fghijkl,m
nho
WEAK TYPE(p1,q1)ESTI
MATEFOR
COMMUTATOR OFMARCI
NKI
EWI
CZI
NTErRAL
461
8 Che
nD X,Zha
ngP.TheMa
r
c
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e
wi
c
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nt
e
gr
a
lwi
t
hhomoge
ne
ouske
r
ne
l
sont
heHe
r
z
t
ypeHa
r
dy
,Chi
372.
s
pa
c
e
s
ne
s
eAnnMa
t
hSe
rA,2004,25(3):3679 WuH X,Zha
,Appl
.Ma
.
ngP.Onpa
r
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me
t
r
i
cMa
r
c
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c
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nt
e
gr
a
l
sr
e
l
a
t
e
dt
obl
oc
ks
pa
c
e
s
t
h.J
.B,2003,18(3):258266.
Chi
ne
s
eUni
v.Se
r
10 Fa
,Tohoku
nDS,Sa
t
oS.We
a
kt
ype(1,1)e
s
t
i
ma
t
e
sf
orMa
r
c
i
nki
e
wi
c
zi
nt
e
gr
a
l
swi
t
hr
oughke
r
ne
l
s
,2001,53(2):265284.
Ma
t
he
ma
t
i
c
a
lJ
our
na
l
11 Tor
,Col
c
hi
ns
kyA,Wa
ngSL.A not
eont
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c
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nki
e
wi
c
zi
nt
e
gr
a
l
l
oqui
um Ma
t
he
ma
t
i
c
um,1990,6061:235243.
12 Di
,J
ngY,LuSZ,Ya
but
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or
sofMa
r
c
i
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e
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c
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nt
e
gr
a
l
swi
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hr
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r
ne
l
our
na
lof
,2002,275(1):6068.
Ma
t
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ma
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i
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a
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sa
ndAppl
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a
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ons
13 Di
ngY,LuSZ,XueQ Y.Bounde
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ort
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t
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c
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c
hHot
Li
ne,2001,5(9):4714 Che
nD X,Zha
ngP,Che
nJC.Bounde
dne
s
sofMa
r
c
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c
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e
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a
lonhomoge
ne
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sont
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r
z
Ha
r
dys
pa
c
e
s
t
hJChi
ne
s
eUni
vSe
rA,2004,19(1):
109117.
15 Di
ngY,LuS Z,Zha
ngP.We
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95.
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nc
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naSe
rA,2004,47(1):8316 Di
,TohokuMa
,
ngY,LuSZ.Homoge
ne
ousf
r
a
c
t
i
ona
li
nt
e
gr
a
l
sonHa
r
dys
pa
c
e
s
t
he
ma
t
i
c
a
lJ
our
na
l
2000,52:153162.
17
:Re
,Or
,
St
e
i
n E M.Ha
r
moni
cAna
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s
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Ve
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l
i
t
y,a
nd Os
c
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l
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a
t
or
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nt
e
gr
a
l
s
,1993.
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i
nc
e
t
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wJ
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r
s
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t
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vPr
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s
s
18 Be
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bnA P,Pa
nz
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or
sonBa
na
c
hs
pa
c
eva
l
ue
df
unc
t
i
ons
oc
365.
Na
tAc
a
dSc
iUSA,1962,48:356-
.ofMa
:puz
De
pt
t
h.,Zhe
c
i
a
ngSc
i
Te
c
hUni
v.,Ha
ngz
hou310018,Chi
na.Ema
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ha
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ohu.c
om
.ofMa
:huoewudemu.e
De
pt
t
h.,Xi
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v.,Xi
a
me
n361005,Chi
na.Ema
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du.c
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