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3
SOME HARDY SPACES ESTIMATES FOR MULTILINEAR
LITTLEWOOD-PALEY S-OPERATORS
L. Lanzhe
Department of Applied Mathematics,
Hunan University, Changsha, 410082, P.R.of China,
E-mail: lanzheliu@263.net
Abstract
In this paper, we prove the boundedness for the multilinear
Littlewood-Paley S-operators on certain Hardy and Herz-Hardy spaces.
Key words: Littlewood-Paley S-operator, Multilinear operator, BMO(Rn ),
Hardy space, Herz-Hardy space.
2000 MR Subject Classification 42B25, 42B20.
1. Introduction
Let ψR be a function on Rn which satisfies the following properties:
(1)
ψ(x)dx = 0,
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1) ,
(3) |ψ(x + y) − ψ(x)| ≤ C|y|(1 + |x|)−(n+2) when 2|y| < |x|;
Let m be a positive integer and let A be a function on Rn . We denote
n+1
Γ(x) = {(y, t) ∈ R+
: |x − y| < t}. The multilinear Littlewood-Paley
operator is defined by


SψA (f )(x) = 
1/2
Z
|FtA (f )(x, y)|2
dydt 

tn+1
,
Γ(x)
where
Z
FtA (f )(x, y)
=
Rn
f (z)ψt (y − z)
Rm+1 (A; x, z)dz,
|x − z|m
Rm+1 (A; x, y) = A(x) −
X 1
Dα A(y)(x − y)α ,
α!
|α|≤m
and ψt (x) = t−n ψ(x/t) for t > 0. We also define that


Sψ (f )(x) = 
1/2
Z
|f ∗ ψt (y)|2
dydt 

tn+1
,
Γ(x)
which is the Littlewood-Paley operator (see [15]).
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It is well known that multilinear operators are of great interest in harmonic
analysis and have been widely studied by many authors (see [2-6]). The main
purpose of this paper is to consider the continuity of the multilinear LittlewoodPaley operators on certain Hardy and Herz-Hardy spaces. Let us first introduce
some definitions (see [7] [8] [9] [10] [11] [12]).
Definition 1. Let A be a function on Rn and let m be a positive integer
and 0 < p ≤ 1. A bounded measurable function a on Rn is said to be a
(p, Dm A) atom if
i) suppa ⊂ B = B(x0 , r),
−1/p
ii) ||a||
R L∞ ≤ |B|R , α
iii)
a(y)dy = a(y)D A(y)dy = 0, |α| = m;
p
n
A temperate distribution f is said to belong to HD
m A (R ), if, in the Schwartz
distributional sense, it can be written as
f (x) =
∞
X
λj aj (x),
j=0
where aj ’s are (p, Dm A) atoms, λj ∈ C and
³P
´1/p
∞
p
||f ||HDp m A ∼
|λ
|
.
j
j=0
P∞
j=0
|λj |p < ∞. Moreover,
Let Bk = {x ∈ Rn : |x| ≤ 2k }, Ck = Bk \ Bk−1 , k ∈ Z, mk (λ, f ) = |{x ∈
Ck : |f (x)| > λ}|; for k ∈ N , let m̃k (λ, f ) = mk (λ, f ) and m̃0 (λ, f ) = |{x ∈
B0 : |f (x)| > λ}|.
Definition 2. Let 0 < p, q < ∞, α ∈ R.
(1) The homogeneous Herz space is defined by
K̇qα,p = {f ∈ Lqloc (Rn \ {0}) : ||f ||K̇qα,p (Rn ) < ∞},
where
"
||f ||K̇qα,p =
∞
X
#1/p
2kαp ||f χk ||pLq
;
k=−∞
(2) The nonhomogeneous Herz space is defined by
Kqα,p (Rn ) = {f ∈ Lqloc (Rn ) : ||f ||Kqα,p (Rn ) < ∞},
where
"
||f ||Kqα,p =
∞
X
#1/p
2kαp ||f χk ||pLq
+
||f χB0 ||pLq
.
k=1
Definition 3. Let m be a positive integer and let A be a function on
Rn , α ∈ R, 0 < p < ∞, 1 < q ≤ ∞, A function a(x) on Rn is called a central
(α, q, Dm A)-atom (or a central (a, q, Dm A)-atom of restrict type), if
1) suppa ⊂ B(0, r) for some r > 0 (or for some r ≥ 1),
2) ||a||Lq ≤ |B(0, r)|−α/n ,
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R
R
3)
a(x)dx = a(x)Dβ A(x)dx = 0, |β| = m;
α,p
α,p
n
n
A temperate distribution fP
is said to belong to HP
K̇q,D
m A (R ) (or HKq,D m A (R )),
∞
0
n
if it can be written as f = ∞
j=−∞ λj aj (or f =
j=0 λj aj ) in the S (R ) sense,
m
where aj is a central (α, q, Dm A)-atomP(or a central (α, q, DP
A)-atom of re∞
j
p
p
strict type) supported on B(0, 2 ) and j=−∞ |λj | < ∞ (or ∞
j=0 |λj | < ∞),
´1/p
³P
p
α,p
|λ
|
.
moreover, ||f ||H K̇ α,p m (or ||f ||HKq,D
)
∼
j
j
mA
q,D A
Now we can state our main theorems.
Theorem 1. Let 1 ≥ p > n/(n + 1),Dβ A ∈ BM O(Rn ) for |β| = m. Then
p
n
p
n
A
Sψ is bounded from HD
m A (R ) to L (R ).
Theorem 2. Let 0 < p < ∞, 1 < q < ∞, n(1 − 1/q) ≤ α < n(1 − 1/q) + 1
α,p
n
and Dβ A ∈ BM O(Rn ) for |β| = m. Then SψA is bounded from H K̇q,D
m A (R )
to K̇qα,p (Rn ).
Theorem 3. Let Dβ A ∈ BM O(Rn ) for |β| = m and 0 < p ≤ 1 ≤ q < ∞,
α,p
n
α = n(1 − 1/q) + 1. Then, for any λ > 0 and f ∈ HKq,D
m A (R ), we have
#1/p
"∞
X
2kαp m̃k (λ, SψA (f ))p/q
k=0
³
´
+ −1
α,p
α,p
≤ Cλ−1 ||f ||HKq,D
1
+
log
(λ
||f
||
)
.
n
n
HKq,Dm A (R )
m A (R )
Remark. Note that when m = 0, SψA is just the commutator of LittlewoodPaley operator (see [1]). Theorem 1-3 are an extension of the results in [1].
2. Proof of Theorems
We begin with some preliminary lemmas.
Lemma 1.[4] Let A be a function on Rn and Dα A ∈ Lq (Rn ) for |α| = m
and some q > n. Then

1/q
Z
X 
1

|Rm (A; x, y)| ≤ C|x − y|m
|Dα A(z)|q dz  ,

|Q̃(x, y)|
|α|=m
Q̃(x,y)
√
where Q̃(x, y) is the cube centered at x and having side length 5 n|x − y|.
Lemma 2. Let 1 < p < ∞ and Dα A ∈ Lr (Rn ), |α| = m, 1 < r ≤ ∞,
1/q = 1/p + 1/r. Then SψA is bounded from Lp (Rn ) to Lq (Rn ), that is
X
||Dα A||Lr ||f ||Lp .
||SψA (f )||Lq ≤ C
|α|=m
Proof.
By Minkowski inequality and the condition of ψ, we have

1/2
Z
Z
|f (z)| |Rm+1 (A; x, z)| 
2 dydt 
|ψ
(y
−
z)|
SψA (f )(x) ≤

 dz
t
|x − z|m
t1+n
Rn
Γ(x)
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Z
≤ C
Rn
Z
≤ C
Rn
Z
≤ C
Rn

|f (z)||Rm+1 (A; x, z)| 

|x − z|m

|f (z)||Rm+1 (A; x, z)| 

|x − z|m

|f (z)||Rm+1 (A; x, z)| 

|x − z|m
Z∞ Z
1/2
t−2n
dydt 

2n+4
(1 + |y − z|/t)
t1+n
0 |x−y|≤t
Z∞ Z
1/2
−2n
dydt 
t

(1 + |y − z|/t)2n+4 tn+1
0 |x−y|≤t
Z∞ Z
dz
dz,
1/2
22n+4 · t1−n

dydt
2n+2
(2t + |y − z|)
dz,
0 |x−y|≤t
noting that 2t + |y − z| ≥ 2t + |x − z| − |x − y| ≥ t + |x − z| when |x − y| ≤ t
and
Z∞
tdt
= C|x − z|−2n ,
2n+2
(t + |x − z|)
0
we obtain
Z
SψA (f )(x) ≤ C
Rn
Z
= C
Rn

|f (z)|
|Rm+1 (A; x, z)| 
|x − z|m
Z∞
1/2
tdt

(t + |x − z|)2n+2
dz
0
|f (z)|
|Rm+1 (A; x, z)|dz,
|x − z|m+n
thus, the lemma follows from [5] and [6].
Now let us turn to the proof of Theorems in this paper.
Proof of Theorem 1. It suffices to show that there exists a constant
c > 0 such that for every (p, Dm A) atom a,
||SψA (a)||Lp ≤ C.
Let a be a (p, Dm A) atom supported on a ball B = B(x0 , r). We write
Z
Z
Z
A
p
A
p
[Sψ (a)(x)] dx =
[Sψ (a)(x)] dx +
[SψA (a)(x)]p dx = I + II.
Rn
|x−x0 |>2r
|x−x0 |≤2r
For I, taking q > 1, by Hölder’s inequality and the Lq -boundedness of SψA (see
Lemma 2), we see that
I ≤ C||SψA (a)||pLq · |B(x0 , 2r)|1−p/q ≤ C||a||pLq |B|1−p/q ≤ C.
To obtain the estimate of II, we need to estimate SψA (a)(x) for x ∈ (2B)c . Let
P 1 α
√
(D A)B̃ · xα , where (A)B are the mean
B̃ = 5 nB and Ã(x) = A(x) −
α!
|α|=m
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values of A on B. Then Rm (A; x, y) = Rm (Ã; x, y). We write, by the vanishing
moment of a,
Z ·
FtA (a)(x, y)
=
B
¸
ψt (y − z) ψt (y − x0 )
−
Rm (Ã; x, z)a(z)dz
|x − z|m
|y − x0 |m
Z
ψt (y − x0 )
[Rm (Ã; x, z) − Rm (Ã; x, x0 )]a(y)dy
|y − x0 |m
B
X 1 Z ψt (y − z)(x − z)α
(Dα A(z) − (Dα A)B )a(z)dz,
−
m
α!
|x − z|
+
|α|=m
B
thus,
SψA (a)(x)


Z
Z


≤ 
Γ(x)


+ 
B

Z
Z

Γ(x)

B
1/2
2
¯
¯
¯ ψt (y − z) ψt (y − x0 ) ¯
¯
¯ |Rm (Ã; x, z)||a(z)|dz  dydt 
−

¯ |x − z|m
|x − x0 |m ¯
tn+1
2
1/2
dydt 
|ψt (y − x0 )|
|Rm (Ã; x, z) − Rm (Ã; x, x0 )||a(z)|dz  n+1 
m
|x − x0 |
t
1/2
¯
¯2
¯
Z ¯X
Z
¯
¯ dydt 
ψt (y − z)(x − z)α α
1

α
¯
¯
+ 
(D
A(z)
−
(D
A)
)a(z)dz
B
¯
¯ tn+1 
m
α!
|x
−
z|
¯
¯
Γ(x)
|α|=m
B
≡ II1 + II2 + II3 .
By Lemma 1, for z ∈ B and x ∈ 2k+1 B \ 2k B, we know
X
|Dα A(x) − (Dα A)2k B |;
|Rm (Ã; x, z)| ≤ C|x − z|m
|α|=m
By the condition of ψ and Minkowski’s inequality, similar to the proof of
Lemma 2, noting that |x − z| ∼ |x − x0 | for z ∈ B and x ∈ Rn \ B, we obtain
II1 ≤
¯
¶
· Z µZ µ¯
¯
¯
|ψt (y − z) − ψt (y − x0 )|
1
1
¯
¯
¯ |x − z|m − |x − x0 |m ¯ |ψt (y − x0 )| +
|x − z|m
Γ(x)
B
|Rm (Ã; x, z)||a(z)|dz
¶2
dydt
tn+1
¸1/2
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
Z
|x0 − z| 

|x − x0 |m+1
≤ C
B

Z
1


m
|x − x0 |
+ C
B
1/2
Z
|ψt (y − x0 )|2
dydt 

tn+1
|Rm (Ã; x, z)||a(z)|dz
Γ(x)
1/2
Z
|ψt (y − z) − ψt (y − x0 )|2
dydt 

tn+1
Γ(x)
×|Rm (Ã; x, z)||a(z)|dz

1/2 

Z
Z
1−n
1/n−1/p
t dydt
|B|

 
|Rm (Ã; x, z)|dz 
≤ C


m+1
|x − x0 |
(2t + |y − x0 |)2n+2
B
Γ(x)

+C
|B|1/n−1/p
|x − x0 |m
1/2 

Z
Z
1−n
t dydt

 
|Rm (Ã; x, z)|dz 


2n+4
(2t + |y − x0 |)
Γ(x)

B
Z
≤ C|x − x0 |−(m+n+1) |B|1/n−1/p 

|Rm (A; x, z)|dz 
B
≤ Ck|x − x0 |
−n−1
|B|
1/n−1/p+1
X
|Dα A(x) − (Dα A)2k+1 B |;
|α|=m
On the other hand, by the formula (see [4]):
X 1
Rm (Ã; x, z) − Rm (Ã; x, x0 ) =
Rm−|β| (Dβ Ã; z, x0 )(x − x0 )β
β!
|β|<m
and Lemma 1, we get
|Rm (Ã; x, z) − Rm (Ã; x, x0 )| ≤ C
X X
|x0 − z|m−|β| |x − x0 ||β| ||Dα A||BM O ,
|β|<m |α|=m
so that
Z
II2 ≤ C
Z
|x − x0 |
|β|<m
B
|x − x0 |−(n+m)
≤C
X
Z
||Dα A||BM O
|α|=m
≤C
X
X
|x0 − z|m−|β| |x − x0 ||β|
|β|<m
B
≤C
¯
X ¯¯
¯
β
¯Rm−|β| (D Ã; z, x0 )¯ |x − x0 ||β| |a(z)|dz
−(n+m)
B
X
||Dα A||BM O |a(z)|dz
|α|=m
|x0 − z|
|a(z)|dz
|x − x0 |n+1
α
||D A||BM O |x − x0 |−n−1 |B|1/n−1/p+1 ;
|α|=m
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For II3 , we write
Z
ψt (y − z)(x − z)α α
(D A(z) − (Dα A)B )a(z)dz
m
|x − y|
B
¸
Z ·
ψt (y − z)(x − z)α ψt (y − x0 )(y − x0 )α
=
−
|x − z|m
|x − x0 |m
B
× [Dα A(z) − (Dα A)B ]a(z)dz.
Similar to the estimate of II1 , we obtain
X
II3 ≤ C
Z
|x − x0 |
−(n+1)
|α|=m
X
≤ C
|x0 − z||Dα A(z) − (Dα A)B ||a(z)|dz
B
α
||D A||BM O |B|1/n−1/p+1 |x − x0 |−n−1 .
|α|=m
Therefore, recall that p > n/(n + 1) ,
II ≤
Z
∞
X
[SψA (a)(x)]p dx
k=1
≤ C
2k+1 B\2k B
Z
∞
X
k=1

×
k p |x − x0 |−p(n+1) |B|p(1+1/n−1/p)
2k+1 B\2k B
X
p
|Dα A(x) − (Dα A)2k+1 B | dx
|α|=m

+C 
X
p
||D A||BM O 
α
k=1
|α|=m
p

≤ C
X
∞
X
||Dα A||BM O 
∞
X
Z
|x − x0 |−p(n+1) |B|p(1+1/n−1/p) dx
2k+1 B\2k B
k p 2k(n−p−pn) ≤ C 
k=1
|α|=m

X
p
||Dα A||BM O  ,
|α|=m
which together with the estimate for I yields the desired result. This finishes
the proof of Theorem 1.
P∞
α,p
n
Proof of Theorem 2. Let f ∈ H K̇q,D
m A (R ) and f (x) =
j=−∞ λj aj (x)
be the atomic decomposition for f as in Definition 3. We write
"
||SψA (f )||K̇qα,p (Rn ) ≤ C
∞
X
k=−∞
Ã
2kαp
k−3
X
!p #1/p
|λj | ||SψA (aj )χk ||Lq
j=−∞
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"
∞
X
+C
Ã
2kαp
k=−∞
!p #1/p
∞
X
|λj | ||SψA (aj )χk ||Lq
= I + II.
j=k−2
For II, by the boundedness of SψA on Lq (Rn ) (see Lemma 2), we have
"
II ≤ C
∞
X
Ã
2kαp
≤ C
∞
X
|λj | ||aj ||Lq
j=k−2
k=−∞
"
!p #1/p
∞
X
Ã
2kαp
k=−∞
∞
X
!p #1/p
|λj |2−jα
j=k−2
 "
#1/p
∞
∞

X
X



2kαp
|λj |p 2−jαp
, 0<p≤1


 k=−∞
j=k−2

≤ C
!p/p0 1/p
!Ã ∞
à ∞
∞

X
X
X

0


 , p>1

2−jαp /2
|λj |p 2−jαp/2
2kαp



j=k−2
j=k−2
k=−∞
 "
Ã
!
#1/p
j+2
∞

X
X



|λj |p
2(k−j)αp |λj |p 2−jαp
, 0<p≤1


j=−∞
k=−∞
≤ C
!#1/p
" ∞
à j+2

X
X


(k−j)αp/2
p

2
, p>1
|λj |


Ã
≤ C
j=−∞
∞
X
!1/p
|λj |p
k=−∞
≤ C||f ||H K̇ α,p m .
q,D
j=−∞
A
For I, similar to the proof of Theorem 1, we have, for x ∈ Ck , j ≤ k − 3,


SψA (aj )(x) ≤ C|x − x0 |−n−m−1 |Bj |1/n 
Z


|aj (y)||Rm (Ã; x, y)|dy 
Bj
+C
X
|β|=m
||D A||BM O (k − j)|x − x0 |
+C
−n−1
|Bj |
1/n
|a(y)|dy
Bj

≤ C2−k(n+1) 2j(1+n(1−1/q)−α) 
X
Z
β
X

|Dβ A(x) − (Dβ A)Bk |
|β|=m
||Dβ A||BM O (k − j)2−k(n+1) 2j(1+n(1−1/q)−α) ,
|β|=m
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thus,
I ≤ C
·X
∞
kαp
2
k−3
³X
j=−∞
k=−∞

X
+C
|D A(x) − (D A)Bk | dx
β
β
q
´p ¸1/p
Bk
·X
∞
2
kαp
k−3
³X
|λj |(k − j)2−k(n+1)+j(1+n(1−1/q)−α) 2kn/q
j=−∞
k=−∞
X
1/q
Z

|β|=m
|λj |2−k(n+1)+j(1+n(1−1/q)−α)
β
||D A||BM O
´p ¸1/p
= I1 + I2 ;
|β|=m
To estimate I1 and I2 , we consider two cases.
Case 1 0 < p ≤ 1.
I1 ≤

C
∞
X
k−3
X
2kαp
= C
|λj |p 2[−k(n+1)+j(1+n(1−1/q)−α)]p 2knp/q (
j=−∞
k=−∞
X
≤ C
"
I2 ≤ C
||Dβ A||BM O
≤ C
∞
X
|λj |p
j=−∞
Ã
"
∞
X
∞
X
β
||D A||BM O
2(j−k)(1+n(1−1/q)−α)p
k=j+3
!1/p
|λj |p
|λj |
Ã
∞
X
∞
X
p
j=−∞
||Dβ A||BM O
≤ C||f ||H K̇ α,p m ,
q,D
A
#1/p
p (j−k)(1+n(1−1/q)−α)p
(k − j) 2
k=j+3
!1/p
|λj |p
≤ C||f ||H K̇ α,p m ;
q,D
j=−∞
|β|=m
||Dβ A||BM O )p 
#1/p
j=−∞
|β|=m
X
∞
X
||Dβ A||BM O
|β|=m
X
1/p
|β|=m
|β|=m
X
X
A
Case 2 p > 1. By Hölder’s inequality, we deduce
I1 ≤ C
X
|β|=m
||Dβ A||BM O
∞
hX
j=−∞
Ã
k−3
X
!
|λj |p 2(j−k)p(1(1−1/q)−α)/2
j=−∞
http://www.viam.hepi.edu.ge/others/ticmi/blt/vol7/aliu.pdf
12
Bulletin of TICMI
Ã
k−3
X
×
!p/p0
(j−k)p0 (1+n(1−1/q)−α)/2
2
j=−∞
Ã
≤ C
!1/p
∞
X
|λj |
p
≤ C||f ||H K̇ α,p m ,
q,D
j=−∞
"
X
I2 ≤ C
||Dβ A||BM O
×
k−3
X
Ã
∞
X
j=−∞
|β|=m
Ã
i1/p
A
!
k−3
X
|λj |p (k − j)p 2(j−k)p(1+n(1−1/q)−α)/2
j=−∞
!p/p0 1/p
0

2(j−k)p (1+n(1−1/q)−α)/2
j=−∞
≤ C
"
X
||Dβ A||BM O
|β|=m
Ã
≤ C
∞
X
∞
X
|λj |p
j=−∞
!1/p
|λj |p
∞
X
#1/p
(k − j)p 2(j−k)p(1+n(1−1/q)−α)/2
k=j+3
≤ C||f ||H K̇ α,p m .
q,D
j=−∞
A
This finishes the proof of Theorem 2.
Remark. Theorem 2 also holds for nonhomogeneous Herz-type
P∞ space.
α,p
n
Proof of Theorem 3. Let f ∈ HKq,Dm A (R ) and let f (x) = j=0 λj aj (x)
be the atomic decomposition for f as in Definition 3. We write
"∞
X
#1/p
2kαp m̃k (λ, SψA (f ))p/q
≤C
k=0

+C 

+C 
" 3
X
#1/p
2kαp m̃k (λ, SψA (f ))p/q
k=0
∞
X
Ã
2kαp m̃k
λ/2,
!p/q 1/p

|λj |SψA (aj )
j=0
k=4
∞
X
k−3
X
Ã
Ã
λ/2, SψA
2kαp m̃k
k=4
∞
X
!!p/q 1/p
λj aj

≡ I1 + I2 + I3 .
j=k−2
For I1 , I3 , by the weak (q, q) type boundedness of SψA and 0 < p ≤ 1, we have
"
I1 ≤ Cλ
−1
3
X
#1/p
kαp
2
||f ||pLq
k=0
Ã
≤ Cλ−1
∞
X
≤ Cλ
!1/p
−1
̰
X
!1/p
|λj |
p
||aj ||pLq
j=0
|λj |p · 2−jαp
j=0
http://www.viam.hepi.edu.ge/others/ticmi/blt/vol7/aliu.pdf
Volume 7, 2003
Ã
≤ Cλ−1
∞
X
13
!1/p
|λj |p
α,p
≤ Cλ−1 ||f ||HKq,D
,
mA
j=0

I3 ≤ Cλ−1 
"
≤ Cλ−1
"
≤ Cλ−1
∞
X
k=4
∞
X
¯¯
¯¯p 1/p
∞
¯¯ X
¯¯
¯¯
kαp ¯¯
2 ¯¯
λj aj ¯¯ 
¯¯
¯¯
j=k−2
∞
X
2kαp
k=4
j=k−2
∞
X
j+2
X
|λj |p
j=0
Lq
#1/p
|λj |p 2−jαp
#1/p
2(k−j)αp
≤ Cλ−1
̰
X
!1/p
|λj |p
j=0
k=0
−1
α,p
≤ Cλ ||f ||HKq,D
.
mA
For I2 , by the same argument as the proof of Theorem 1 and 2, we have


X
X
|Dβ A(x) − (Dβ A)Bk | + k
||Dβ A||BM O  ,
SψA (aj )(x) ≤ C2−k(n+1) 
|β|=m
|β|=m
therefore,


p/q 1/p
∞
∞
X
X
X kαp 

|Dβ A(x) − (Dβ A)Bk |
2 m̃k λ/4, C2−k(n+1)
|λj | 
I2 ≤ C 
k=4


+C 
j=0
|β|=m
∞
X

2kαp m̃k λ/4, C2−k(n+ε) k
k=4
(1)
X
|β|=m
||Dβ A||BM O
∞
X
p/q 1/p

|λj | 
j=0
(2)
≡ I2 + I2 .
(1)
For I2 , by using John-Nirenberg inequality (see [15]), we gain

!
!p/q 1/p
Ã
Ã
∞
k(n+1)
X
C2
λ
(1)

P∞
I2
≤ C
2kn
2kαp exp − P
β
|λ
|
||D
A||
j
BM O
j=0
|β|=m
k=4
"∞
Ã
!#1/p
k(n+1)
X
Cλ2
P∞
≤ C
2k(n+1)p exp − P
β A||
||D
BM
O
j=0 |λj |
|β|=m
k=0
∞
Ã
! 1/p
Z
cλx
P∞
≤ C  xp−1 exp − P
dx
β
j=0 |λj |
|β|=m ||D A||BM O
0
http://www.viam.hepi.edu.ge/others/ticmi/blt/vol7/aliu.pdf
14
Bulletin of TICMI
= Cλ−1 ||Dβ A||BM O
∞
X

|λj | 
j=0
≤ Cλ−1
̰
X
1/p
Z∞
tp−1 e−t dt
0
!1/p
|λj |p
j=0
−1
α,p
≤ Cλ ||f ||HKq,D
;
mA
(2)
For I2 , by using the following fact: If there exists u > 1, such that 2x /x ≤ u
for x ≥ 3, then 2x ≤ cu log+ u. We have, if
∞
¯n
o¯
X
X
¯
¯
−k(n+1)
β
k
||D A||BM O
|λj | > λ/4 ¯ 6= 0,
¯ x ∈ Ck : C2
j=0
|β|=m
then
k(n+1)/k(n+1)
1<2
< Cλ
X
−1
β
||D A||BM O
2k(n+1) ≤ Cλ−1
∞
X
|λj |,
j=0
|β|=m
thus,
∞
X
Ã
|λj | log+
∞
X
λ−1
j=0
!
|λj | .
j=0
Let Kλ be the maximal integer k which satisfies this estimate, then
(2)
I2
≤ C
ÃK
λ
X
≤ Cλ
!1/p
2kαp 2knp/q
k=4
∞
X
−1
Ã
≤ Cλ−1
∞
X
≤ Cλ ||f ||
|λj |

log+ λ−1
|λj |p
α,p
HKq,D
mA
!
j=0
!1/p
j=0
−1
∞
X
|λj | log+ λ−1
j=0
Ã
≤ C2Kλ (n+1)
log
+
̰
X
!1/p 
|λj |p
j=0
³
−1
λ ||f ||
α,p
HKq,D
mA

´
.
Now, summing up the above estimates, we have
"∞
X
#1/p
2kαp m̃k (λ, SψA (f ))p/q
k=0
³
³
´´
+
−1
α,p
α,p
≤ Cλ−1 ||f ||Hq,D
1
+
log
λ
||f
||
.
HK
mA
q,D m A
This completes the proof of Theorem 3.
http://www.viam.hepi.edu.ge/others/ticmi/blt/vol7/aliu.pdf
Volume 7, 2003
15
References
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Received June 16, 2003; accepted October 2, 2003.
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