一类离散型奇异积分算子
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Fa
q 2 (1/2) 1 − ζ1 1/ 2 2 ≤ (1 − s2 )1/2 ≤ 2 1 − ζ1 1/ 2
.
(2.15)
≤ Cρ−1
"! ρ =C
Fa
q
c
≤ Cρ
d5
2(−1+1/q )
1−s2 ≤10000ρ2
|1 − s2 |−q/2 ds
1/q
.
≤ C ρ2 |ζ1 | + ρ|ζ2 |
ζ ∈ S n−1 .
5 ζ 7}E
2.1 ρ
7 ^$
n ≥ 3,
a
7 S n−1 wg S n−1 ∩ B (ζ, ρ) $AFj fP
S n−2
rk B (ζ, ρ) 7Y
Fa (s) = (1 − s2 )(n−3)/2 χ(−1,1) (s)
a(s, (1 − s2 )1/2 y )dσ (y),
∞ k S n−1
Ω(y )dσ (y ) = 0
TΩ f (x) =
k=−∞
S n−1
f (x − αk y )Ω(y )dσ (y )
and the associated maximal operator
∗ TΩ f (x) = sup N ∞ k=N S n−1
f (x − αk y )Ω(y )dσ (y )
": ~ 1996-10-18, U$ ~ a$ q-nB% NSF ;
1997-03-31
230
(
O
O
41
`
(Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh PA 15260, USA) (yibiao@tomato.math.pitt.edu) In Memory of Professor Rui-Lin Long
inf αk+1 /dk = α > 1, Ω(y ) n Besov 7 1 i {αk }∞ k=−∞ n~kcsvY k 0,1 VR B1 (S n−1) hFMk _; S n−1 n Rn (n ≥ 2) hFEob\ ?p ] d Ω(y )dσ (y ) = 0, |Wet`xPH S
−1+1/q
≤ Cr−1+1/q .
g! `j 2.2 v −ψ ∈ S , u /g`j 2.1 D`j 2.2 }$ r(ξ ), ψ (dk r(ξ )|ξ |), k ∈ Z. od <$ 5 2.3 J C' sup |ψk ∗ f | m Lp(Rn) 7dZ k vxR [5].
ψ (0) = 1,
β>0
p
$C'
p,
vxo
(1.2)
TΩ (f )
≤C f
∀p ∈ (1, ∞).
0,1 9 Tz$mgvx ! p = 2, Ω(y ) %g< $ Besov eQ B1 (S n−1 ) &dZ ∞ αk+1 /αk = α > 1. ?+\W.h IXt m {αk }k=−∞ 7u' ALp u inf k F;]H8 TΩ,N f (x) 7
{ S0 = ζ1 L r = |ξ|−1|Aρξ|, o (2.8) }$4V Xt m 2jdv (2.8) $(1 X " a D ρ∇a d@2$ tU m 6 (2.10) > m ?mTv (2.9)–(2.9 ) m 2 1/ 2 1 − ζ1 > 99ρ. c (2.13), d CDwK1S ?7v (2.9), zKRv (2.9) }$(V c Fa $ dx Fa ∞ ≤ Cρ−1 1−ζ12 −1/2 ≤ Cr−1 , M (2.9 ). 2jdv (2.9 ) m lc (2.8) # Fa q ≤ Cr−1+1/q . x! 1−ζ12 1/2 > 99ρ (2.9) m ! 1−ζ12 1/2 = |ζ2| ≤ 99ρ c (2.13) dx (1 − s2 )1/2 ≤ 100ρ. g! c Fa $ #
(2.3)
λj aj
j
|λj | < ∞,
0,1 1
(2.4)
} aj G!l6j G!AFj 6 Ω B ∼ inf |λj | , rk< Z!/V 8 j Y (2.4) 0 $ / V7q$ ξ = (ξ1, · · · , ξn) ∈ Rn , P ξ/|ξ| = ξ = (ξ1, · · · , ξn ) = (ζ1, · · · , ζn ) = ζ . >
Abstract Suppose that {αk }∞ k=−∞ is a Lacunary sequence of positive numbers satis0,1 fying inf αk+1 /αk = α > 1 and that Ω(y ) is a function in the Besov space B1 (S n−1 ) where S n−1 is the unit sphere on R n (n ≥ 2). We prove that if then the discrete singular integral operator
o m S0 ∈ R DV Xfg a $ ' C , u
Fa
∞
Supp Fa ⊂ (S0 − 2r, S0 + 2r), ≤ C/r
(2.5) (2.6) (2.7)
L
R
|Fa (s)|ds ≤ Cr−1 ,
} r = r(ξ ) = |ξ|−1|Aρ ξ|, S0 = ξ1 , 0 Aρ 7 n × n V_s diag (ρ2, ρ, · · · , ρ). vx | 2 [4], rk s /g n = 2 $ H a !wg S 1 ∩ B (ζ, ρ) $AFj P YL 5
Fa (s) = (1 − s2 )−1/2 χ(−1,1) (s){a(s, (1 − s2 )1/2 ) + a(s, −(1 − s2 )1/2 )} Fa (s) = (1 − s2 )−1/2 χ(−1,1) (s) 2.2 d {a(s, (1 − s2 )1/2 ) + a(s, −(1 − s2 )1/2 )}. ds Supp Fa ⊂ (S0 − 2r, S0 + 2r),
1
60 y 7 Rn } 8 w S n−1 $V* pbro<&h F$;]H8
TΩ (f )(x) =
k ∈Z S n−1 n−1 L1 ), β (S
dσ (y )
7 S n−1 $ Lebesgue , m [1] }
(1.1)
f (x − 2k y )Ω(y )dσ (y ),
} Ω(y ) ! Sobolev eQ
S n−1
E
∞
≤1
L
(2.1) (2.2)
a(ξ )dσ (ξ ) = 0, ≤ ρ−(n−1)
a
} ρ ∈ (0, 1]. 0,1 0,1 (S n−1 ) d <$8Y.t (R [2, 3]): Ω ∈ B1 (S n−1 ) ! \! Besov eQ B1
Ω=
j
∞
YL
∇a
∞
≤ ρ −n ,
n−1
(i) TΩ,N (f )
L2 (Rn )
≤C Ω
∗ (f ) L2 (R n ) TΩ
≤C Ω
0,1 B1 (S n−1 )
f
L2 (R n ) , L2 (R n ) ,
0,1 B1 (S n−1 )
f
} ' C Xfg f, N L Ω.
2
}
3
' Wgi G ^I9,
231
2
∇E
6 ∇ 7 S n−1 $/, V l6j E !y S n−1 $AFC' u AFj a !y S n−1 $AFC' u ∞ ≤ 1. V Supp a ⊂ x ∈ S n−1 : |x − x0 | < ρ /yV x0 ∈ S n−1 ,
n−1
∞
TΩ (f )(x) =
k=−∞
S n−1
f (x − αk y )Ω(y )dσ (y )
OrJFQCl
∗ TΩ (f )(x) = sup N
∞ k=N S n−1
f (x − αk y )Ω(y )dσ (y )
U{ L2(Rn) hzT hjSLmKX Duoandikoetxea O Rubio de Francia[1] { L2 auqFwIS L `xPH Besov VR BAN MR(1991) ) 42B20 :* O174.3
)41`)2} 1998|3k 3
ACTA MATHEMATICA SINICA
)
P
P
Vol.41, No. 2 March, 1998
&/$4 GDf
(Wisconsin-Milwaukee
(<
)
Q*QcQ
(
Zg} ] 4 Q*Q ] 100875) ^y@ (Pittsburgh Q*Qi3O ) -+ # %> ' =
A Discrete Singular Integral Operator
Fan Dashan (Department of Mathematics Sciences, University of Wisconsin-Milwaukee Milwaukee, WI 53201, USA) (fan@alphal.csd.uwm.edu) Lu Shanzhen (Department of Mathematics, Beijing Normal University, Beijing 100875, China) (lusz@sun.ihep.ac.cn) Pan Yibiao
> 99ρ,
232
(
O
wenku.baidu.comO !
2 1 − ζ1 1/ 2
41
`
|Fa (s)| ≤ Cr−1 ρ−1 (1 − s2 )−1/2 ,
R
> 99ρ,
(2.9 ) (2.10)
Fa (s)ds = 0,
2 2 1/ 2 } 1 < q < 2, S0 = ξ1 , r = r(ξ ) = |ξ|−1 ρ4ξ1 + ρ2 ξ 2 . 1 8! V I a wg S ∩ B (ζ, ρ) } ρ = C ζ ∈ S 1 . { ζ = ζ1 , 1 / 2 2 σ , } σ ∈ {±1}. Fa (s) = 0, o (s, (1 − s2 )1/2 δ ) ∈ B (ζ, ρ), } δ ∈ {±1}. _ ζ1 2 2 1/ 2 2 1/ 2 2 (s − ζ1 ) + δ (1 − s ) − σ 1 − ζ1 < ρ2 , " δ = σ G δ = −σ, Zvx 2 (s − ζ1 )2 + (1 − s2 )1/2 − 1 − ζ1 1/ 2 2
TΩ,N (f )(x) =
k ≥N
S n−1
f (x − αk y )Ω(y )dσ (y ),
(1.3)
0@a$J +
1 (ii)
∗ +\7 TΩ f (x) = supN |TΩ,N f (x)|. 9Tv#Y<$ 0,1 n−1 Ω ∈ B1 (S ), u Ω(y )dσ (y ) = 0, od S ∗ TΩ
are both bounded in the space L2 (R n ). The theorems in ths paper improve a result by Duoandikoetxea and Rubio de Francia[1] in the L2 case. Keywords Singular integral, Besov space, Rough kernel 1991 MR Subject Classification 42B20 Chinese Library Classification O174.3
{ ψ k 7 ψ k (ξ ) =
2
} 9" 2 0,1 Ω ∈ B1 (S n−1 ), o Ω =
1
3
' Wgi G ^I9, }vV aj 7AFj
≤C
j ∗ |λj | Ta (f ) j
233
3
j
1−
≤ ρ2 .
(2.11)
c d5 YL _ d
|s − ζ1 | ≤ ρ, (1 − s2 )1/2 − 1 −
2 1/ 2 ζ1
(2.12) ≤ ρ,
1/ 2
(2.13)
2 |s − ζ1 | ≤ ρ2 + 2ρ 1 − ζ1
.
(2.14)
|s − ζ1 | ≤ 2|ξ |−1 |Aρ ξ |.
Lq (R )
R
Fa (s)ds = 0,
m S0 ∈ R DV
Fa Fa
Lq (R ) ∞
Xfg a $ ' C , # ! L
F∇a 1−
2 1/ 2 ζ1
Supp Fa ⊂ (S0 − 2r, S0 + 2r); ≤ Cr−1+1/q ≤ Cr−1 ,
(2.8) (2.9) (2.9 )
≤ Cr−1+1/q ρ−1 ,
q 2 (1/2) 1 − ζ1 1/ 2 2 ≤ (1 − s2 )1/2 ≤ 2 1 − ζ1 1/ 2
.
(2.15)
≤ Cρ−1
"! ρ =C
Fa
q
c
≤ Cρ
d5
2(−1+1/q )
1−s2 ≤10000ρ2
|1 − s2 |−q/2 ds
1/q
.
≤ C ρ2 |ζ1 | + ρ|ζ2 |
ζ ∈ S n−1 .
5 ζ 7}E
2.1 ρ
7 ^$
n ≥ 3,
a
7 S n−1 wg S n−1 ∩ B (ζ, ρ) $AFj fP
S n−2
rk B (ζ, ρ) 7Y
Fa (s) = (1 − s2 )(n−3)/2 χ(−1,1) (s)
a(s, (1 − s2 )1/2 y )dσ (y),
∞ k S n−1
Ω(y )dσ (y ) = 0
TΩ f (x) =
k=−∞
S n−1
f (x − αk y )Ω(y )dσ (y )
and the associated maximal operator
∗ TΩ f (x) = sup N ∞ k=N S n−1
f (x − αk y )Ω(y )dσ (y )
": ~ 1996-10-18, U$ ~ a$ q-nB% NSF ;
1997-03-31
230
(
O
O
41
`
(Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh PA 15260, USA) (yibiao@tomato.math.pitt.edu) In Memory of Professor Rui-Lin Long
inf αk+1 /dk = α > 1, Ω(y ) n Besov 7 1 i {αk }∞ k=−∞ n~kcsvY k 0,1 VR B1 (S n−1) hFMk _; S n−1 n Rn (n ≥ 2) hFEob\ ?p ] d Ω(y )dσ (y ) = 0, |Wet`xPH S
−1+1/q
≤ Cr−1+1/q .
g! `j 2.2 v −ψ ∈ S , u /g`j 2.1 D`j 2.2 }$ r(ξ ), ψ (dk r(ξ )|ξ |), k ∈ Z. od <$ 5 2.3 J C' sup |ψk ∗ f | m Lp(Rn) 7dZ k vxR [5].
ψ (0) = 1,
β>0
p
$C'
p,
vxo
(1.2)
TΩ (f )
≤C f
∀p ∈ (1, ∞).
0,1 9 Tz$mgvx ! p = 2, Ω(y ) %g< $ Besov eQ B1 (S n−1 ) &dZ ∞ αk+1 /αk = α > 1. ?+\W.h IXt m {αk }k=−∞ 7u' ALp u inf k F;]H8 TΩ,N f (x) 7
{ S0 = ζ1 L r = |ξ|−1|Aρξ|, o (2.8) }$4V Xt m 2jdv (2.8) $(1 X " a D ρ∇a d@2$ tU m 6 (2.10) > m ?mTv (2.9)–(2.9 ) m 2 1/ 2 1 − ζ1 > 99ρ. c (2.13), d CDwK1S ?7v (2.9), zKRv (2.9) }$(V c Fa $ dx Fa ∞ ≤ Cρ−1 1−ζ12 −1/2 ≤ Cr−1 , M (2.9 ). 2jdv (2.9 ) m lc (2.8) # Fa q ≤ Cr−1+1/q . x! 1−ζ12 1/2 > 99ρ (2.9) m ! 1−ζ12 1/2 = |ζ2| ≤ 99ρ c (2.13) dx (1 − s2 )1/2 ≤ 100ρ. g! c Fa $ #
(2.3)
λj aj
j
|λj | < ∞,
0,1 1
(2.4)
} aj G!l6j G!AFj 6 Ω B ∼ inf |λj | , rk< Z!/V 8 j Y (2.4) 0 $ / V7q$ ξ = (ξ1, · · · , ξn) ∈ Rn , P ξ/|ξ| = ξ = (ξ1, · · · , ξn ) = (ζ1, · · · , ζn ) = ζ . >
Abstract Suppose that {αk }∞ k=−∞ is a Lacunary sequence of positive numbers satis0,1 fying inf αk+1 /αk = α > 1 and that Ω(y ) is a function in the Besov space B1 (S n−1 ) where S n−1 is the unit sphere on R n (n ≥ 2). We prove that if then the discrete singular integral operator
o m S0 ∈ R DV Xfg a $ ' C , u
Fa
∞
Supp Fa ⊂ (S0 − 2r, S0 + 2r), ≤ C/r
(2.5) (2.6) (2.7)
L
R
|Fa (s)|ds ≤ Cr−1 ,
} r = r(ξ ) = |ξ|−1|Aρ ξ|, S0 = ξ1 , 0 Aρ 7 n × n V_s diag (ρ2, ρ, · · · , ρ). vx | 2 [4], rk s /g n = 2 $ H a !wg S 1 ∩ B (ζ, ρ) $AFj P YL 5
Fa (s) = (1 − s2 )−1/2 χ(−1,1) (s){a(s, (1 − s2 )1/2 ) + a(s, −(1 − s2 )1/2 )} Fa (s) = (1 − s2 )−1/2 χ(−1,1) (s) 2.2 d {a(s, (1 − s2 )1/2 ) + a(s, −(1 − s2 )1/2 )}. ds Supp Fa ⊂ (S0 − 2r, S0 + 2r),
1
60 y 7 Rn } 8 w S n−1 $V* pbro<&h F$;]H8
TΩ (f )(x) =
k ∈Z S n−1 n−1 L1 ), β (S
dσ (y )
7 S n−1 $ Lebesgue , m [1] }
(1.1)
f (x − 2k y )Ω(y )dσ (y ),
} Ω(y ) ! Sobolev eQ
S n−1
E
∞
≤1
L
(2.1) (2.2)
a(ξ )dσ (ξ ) = 0, ≤ ρ−(n−1)
a
} ρ ∈ (0, 1]. 0,1 0,1 (S n−1 ) d <$8Y.t (R [2, 3]): Ω ∈ B1 (S n−1 ) ! \! Besov eQ B1
Ω=
j
∞
YL
∇a
∞
≤ ρ −n ,
n−1
(i) TΩ,N (f )
L2 (Rn )
≤C Ω
∗ (f ) L2 (R n ) TΩ
≤C Ω
0,1 B1 (S n−1 )
f
L2 (R n ) , L2 (R n ) ,
0,1 B1 (S n−1 )
f
} ' C Xfg f, N L Ω.
2
}
3
' Wgi G ^I9,
231
2
∇E
6 ∇ 7 S n−1 $/, V l6j E !y S n−1 $AFC' u AFj a !y S n−1 $AFC' u ∞ ≤ 1. V Supp a ⊂ x ∈ S n−1 : |x − x0 | < ρ /yV x0 ∈ S n−1 ,
n−1
∞
TΩ (f )(x) =
k=−∞
S n−1
f (x − αk y )Ω(y )dσ (y )
OrJFQCl
∗ TΩ (f )(x) = sup N
∞ k=N S n−1
f (x − αk y )Ω(y )dσ (y )
U{ L2(Rn) hzT hjSLmKX Duoandikoetxea O Rubio de Francia[1] { L2 auqFwIS L `xPH Besov VR BAN MR(1991) ) 42B20 :* O174.3
)41`)2} 1998|3k 3
ACTA MATHEMATICA SINICA
)
P
P
Vol.41, No. 2 March, 1998
&/$4 GDf
(Wisconsin-Milwaukee
(<
)
Q*QcQ
(
Zg} ] 4 Q*Q ] 100875) ^y@ (Pittsburgh Q*Qi3O ) -+ # %> ' =
A Discrete Singular Integral Operator
Fan Dashan (Department of Mathematics Sciences, University of Wisconsin-Milwaukee Milwaukee, WI 53201, USA) (fan@alphal.csd.uwm.edu) Lu Shanzhen (Department of Mathematics, Beijing Normal University, Beijing 100875, China) (lusz@sun.ihep.ac.cn) Pan Yibiao
> 99ρ,
232
(
O
wenku.baidu.comO !
2 1 − ζ1 1/ 2
41
`
|Fa (s)| ≤ Cr−1 ρ−1 (1 − s2 )−1/2 ,
R
> 99ρ,
(2.9 ) (2.10)
Fa (s)ds = 0,
2 2 1/ 2 } 1 < q < 2, S0 = ξ1 , r = r(ξ ) = |ξ|−1 ρ4ξ1 + ρ2 ξ 2 . 1 8! V I a wg S ∩ B (ζ, ρ) } ρ = C ζ ∈ S 1 . { ζ = ζ1 , 1 / 2 2 σ , } σ ∈ {±1}. Fa (s) = 0, o (s, (1 − s2 )1/2 δ ) ∈ B (ζ, ρ), } δ ∈ {±1}. _ ζ1 2 2 1/ 2 2 1/ 2 2 (s − ζ1 ) + δ (1 − s ) − σ 1 − ζ1 < ρ2 , " δ = σ G δ = −σ, Zvx 2 (s − ζ1 )2 + (1 − s2 )1/2 − 1 − ζ1 1/ 2 2
TΩ,N (f )(x) =
k ≥N
S n−1
f (x − αk y )Ω(y )dσ (y ),
(1.3)
0@a$J +
1 (ii)
∗ +\7 TΩ f (x) = supN |TΩ,N f (x)|. 9Tv#Y<$ 0,1 n−1 Ω ∈ B1 (S ), u Ω(y )dσ (y ) = 0, od S ∗ TΩ
are both bounded in the space L2 (R n ). The theorems in ths paper improve a result by Duoandikoetxea and Rubio de Francia[1] in the L2 case. Keywords Singular integral, Besov space, Rough kernel 1991 MR Subject Classification 42B20 Chinese Library Classification O174.3
{ ψ k 7 ψ k (ξ ) =
2
} 9" 2 0,1 Ω ∈ B1 (S n−1 ), o Ω =
1
3
' Wgi G ^I9, }vV aj 7AFj
≤C
j ∗ |λj | Ta (f ) j
233
3
j
1−
≤ ρ2 .
(2.11)
c d5 YL _ d
|s − ζ1 | ≤ ρ, (1 − s2 )1/2 − 1 −
2 1/ 2 ζ1
(2.12) ≤ ρ,
1/ 2
(2.13)
2 |s − ζ1 | ≤ ρ2 + 2ρ 1 − ζ1
.
(2.14)
|s − ζ1 | ≤ 2|ξ |−1 |Aρ ξ |.
Lq (R )
R
Fa (s)ds = 0,
m S0 ∈ R DV
Fa Fa
Lq (R ) ∞
Xfg a $ ' C , # ! L
F∇a 1−
2 1/ 2 ζ1
Supp Fa ⊂ (S0 − 2r, S0 + 2r); ≤ Cr−1+1/q ≤ Cr−1 ,
(2.8) (2.9) (2.9 )
≤ Cr−1+1/q ρ−1 ,