CONTINUITY FOR MAXIMAL COMMUTATOR
OF BOCHNER-RIESZ OPERATORS ON SOME
WEIGHTED HARDY SPACES
LIU LANZHE AND TONG QINGSHAN
Received 17 May 2004 and in revised form 3 November 2004
We show the boundedness for the commutator of Bochner-Riesz operator on some
weighted H 1 space.
1. Introduction
Let b be a locally integrable function. The maximal operator B∗δ ,b associated with the
commutator generated by the Bochner-Riesz operator is defined by
δ
B∗δ ,b ( f )(x) = sup Br,b
( f )(x),
(1.1)
r>0
where
δ
Br,b
( f )(x) =
Rn
Brδ (x − y) f (y) b(x) − b(y) d y
(1.2)
and (Brδ ( fˆ ))(ξ) = (1 − r 2 |ξ |2 )δ+ fˆ (ξ). We also define that
B∗δ ( f )(x) = sup Brδ ( f )(x),
(1.3)
r>0
which is the Bochner-Riesz operator (see [8]). Let E be the space E = {h : h =
supr>0 |h(r)| < ∞}, then, for each fixed x ∈ Rn , Brδ ( f )(x) may be viewed as a mapping
from [0,+∞) to E, and it is clear that B∗δ ( f )(x) = Brδ ( f )(x) and B∗δ ,b ( f )(x) =
b(x)Brδ ( f )(x) − Brδ (b f )(x).
As well known, a classical result of Coifman et al. [4] proved that the commutator
[b,T] generated by BMO(Rn ) functions and the Calderón-Zygmund operator is bounded
on L p (Rn ) (1 < p < ∞). However, it was observed that [b,T] is not bounded, in general,
from H p (Rn ) to L p (Rn ) and from L1 (Rn ) to L1,∞ (Rn ) for p ≤ 1. But, if H p (Rn ) is replaced
p
by some suitable atomic space Hb (Rn ) and HB1 (Rn ) (see [1, 6, 7, 9]), then [b,T] maps
p n
p
n
continuously Hb (R ) into L (R ) and HB1 (Rn ) into weak L1 (Rn ) for p ∈ (n/(n + 1),1]. The
main purpose of this paper is to establish the weighted boundedness of the commutators
Copyright © 2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:2 (2005) 195–201
DOI: 10.1155/IJMMS.2005.195
196
Maximal commutator of Bochner-Riesz operator
related to Bochner-Riesz operator and BMO(Rn ) function on some weighted H 1 space.
We first introduce some definitions (see [1, 6, 7, 9]).
Definition 1.1. Let b, w be locally integrable functions and w ∈ A1 (i.e., Mw(x) ≤ cw(x)
a.e.). A bounded measurable function a on Rn is said to be (w,b)-atom if
(i) suppa ⊂ B = B(x0 ,r),
−1
,
(ii) aL∞ ≤ w(B)
(iii) a(y)d y = a(y)b(y)d y = 0.
A temperate distribution f is said to belong to Hb1 (w) if, in the Schwartz distributional
sense, it can be written as
f (x) =
∞
λ j a j (x),
(1.4)
j =1
where a j ’s are (w,b)-atoms, λ j ∈ C, and
∞
j =1 |λ j | < ∞.
Moreover, f Hb1 (w) ∼
∞
j =1 |λ j |.
Definition 1.2. Let w ∈ A1 . A function f is said to belong to weighted Block H 1 space
as (1.4), where a j ’s are w-atoms (i.e., a j ’s satisfy Definition
HB1 (w) if f can be written
1.1(i), (ii), and (iii) a(y)d y = 0) and λ j ∈ C with
∞
λ j 1 + log+ 1 < ∞.
λ j (1.5)
j =1
Moreover, f HB1 (w) ∼
∞
+ j =1 |λ j |(1 + log (( i |λi |)/ |λ j |)).
Now, we formulate our results as follows.
Theorem 1.3. Let b ∈ BMO(Rn ) and w ∈ A1 . Then the maximal commutator B∗δ ,b is
bounded from Hb1 (w) to L1w (Rn ) when δ > (n − 1)/2.
Theorem 1.4. Let b ∈ BMO(Rn ) and w ∈ A1 . Then the maximal commutator B∗δ ,b is
∞
n
bounded from HB1 (w) to L1,
w (R ) when δ > (n − 1)/2.
Theorem 1.5. Let b ∈ BMO(Rn ) and w ∈ A1 . Then the maximal commutator B∗δ ,b is
∞
n
bounded from H 1 (w) to L1,
w (R ) when δ > (n − 1)/2.
2. Proof of theorems
Proof of Theorem 1.3. It suffices to show that there exists a constant C > 0 such that for
every (w,b)-atom a,
δ
B (a)
1 ≤ C.
∗,b
Lw
(2.1)
Let a be a (w,b)-atom supported on a ball B = B(x0 ,R). We write
Rn
B∗δ ,b (a)(x) w(x)dx
=
|x−x0 |≤2R
B∗δ ,b (a)(x) w(x)dx +
|x−x0 |>2R
B∗δ ,b (a)(x) w(x)dx ≡ I + II.
(2.2)
L. Lanzhe and T. Qingshan 197
For I, taking q > 1, by Hölder’s inequality and the Lq -boundedness of B∗δ ,b (see [2]), we
see that
I ≤ C B∗δ ,b (a)
Lwq · w(2B)1−1/q ≤ C aLwq w(B)1−1/q ≤ C.
For II, let b0 = |B(x0 ,R)|−1
II ≤
B(x0 ,R) b(y)d y,
∞ k+1
k
k=1 2 R≥|x−x0 |>2 R
+
(2.3)
then
b(x) − b0 B δ (a)(x)w(x)dx
∗
∞ δ
k+1
k
k=1 2 R≥|x−x0 |>2 R
(2.4)
B∗ b − b0 a (x)w(x)dx = II1 + II2 .
For II1 , we choose δ0 such that
n+1
n−1
< δ0 < min δ,
2
2
(2.5)
and consider the following two cases.
Case 1 (0 < r ≤ R). In this case, note that (see [8])
δ B (z) ≤ C 1 + |z| −(δ+(n+1)/2) ,
(2.6)
we have, for |x − x0 | > 2| y − x0 |,
a(y)
δ+(n+1)/2 d y
B(x0 ,R) 1 + |x − y |/r
−(δ0 +(n+1)/2)/n
≤ C |B |(δ0 +(n+1)/2)/n 2k+1 B w(B)−1 .
δ
B (a)(x) ≤ Cr −n
r
(2.7)
Case 2 (r > R). In this case, note that
β δ ∇ B (z) ≤ C 1 + |z| −(δ+(n+1)/2)
(2.8)
for any β = (β1 ,...,βn ) ∈ (N ∪ {0})n and |x − x0 | > 2| y − x0 |, where
∂
∇ =
∂x1
β
β1
∂
···
∂xn
βn
,
(2.9)
by the vanishing condition of a, we gain
a(y) y − x0 δ+(n+1)/2 d y
B(x0 ,R) 1 + x − x0 /r
−(δ0 +(n+1)/2)/n
≤ C |B |(δ0 +(n+1)/2)/n 2k+1 B w(B)−1 .
δ
B (a)(x) ≤ Cr −(n+1)
r
(2.10)
198
Maximal commutator of Bochner-Riesz operator
Combining Case 1 with Case 2, we obtain
II1 ≤ C
∞ 2k+1 R≥|x−x
k =1
≤C
∞
0
|>2k R
b(x) − b0 |B |(δ0 +(n+1)/2)/n
−(δ0 +(n+1)/2)/n
× 2k+1 B w(B)−1 w(x)dx
2−k(δ0 +(n+1)/2) w(B)−1
2k+1 R≥|x−x
k =1
0
|>2k R
(2.11)
b(x) − b0 w(x)dx.
Since w ∈ A1 , w satisfies the reverse of Hölder’s inequality as follows:
1
|B |
1/ p
p
B
w(x) dx
C
≤
|B |
B
w(x)dx
(2.12)
for any ball B and some 1 < p < ∞ (see[10]). Using the properties of BMO(Rn ) functions
(see [10]), and noting w ∈ A1 , then
B w B2
· 1 ≤C
B2 w B1
(2.13)
for all balls B1 , B2 with B1 ⊂ B2 . Thus, by Hölder’s and reverse of Hölder’s inequalities for
w ∈ A1 , we get, for 1/ p + 1/ p = 1,
II1 ≤ C
∞
k =1
2
−k(δ0 +(n+1)/2)
w(B)
1
× 2k+1 B ≤ C bBMO
∞
k2
−1 k+1
2
1/ p
p
2k+1 B
(2.14)
w(x) dx
−k(δ0 −(n−1)/2)
k =1
1/ p
p
1
B k+1 b(x) − b0 dx
2 B 2k+1 B
k w 2 B |B |
2k B w(B) ≤ C.
For II2 , similar to the estimate of II1 , we obtain
−(δ0 +(n+1)/2)
Brδ b − b0 a (x) ≤ C bBMO w(B)−1 |B |(δ0 +(n+1)/2)/n x − x0 ,
(2.15)
thus
II2 ≤ C bBMO
∞
k =1
≤ C bBMO
∞
−(δ0 +(n+1)/2)/n w(B)−1 |B |(δ0 +(n+1)/2)/n 2k B 2
−k(δ0 −(n−1)/2)
k =1
w 2k B
k w 2 B |B |
2k B w(B) ≤ C.
This finishes the proof of Theorem 1.3.
To prove Theorem 1.4, we recall the following lemma (see [5, 10]).
(2.16)
L. Lanzhe and T. Qingshan 199
Lemma 2.1. Let w ≥ 0 and {gk } be a sequence of measurable functions satisfying
g k ∞
L1,
w
≤ 1.
(2.17)
Then, for every numerical sequence {λk },
λk gk k
∞
L1,
w
≤C
λk + log
λj
λk
.
(2.18)
j
k
Proof of Theorem 1.4. By Lemma 2.1, it is enough to show that there exists a constant C
such that
δ
B (a)
1,∞ ≤ C
∗,b
Lw
for each w-atom a.
(2.19)
Let a be a w-atom supported on a ball B = B(x0 ,r). We write
w x ∈ Rn : B∗δ ,b (a)(x) > λ
≤w
x ∈ 2B : B∗δ ,b (a)(x) > λ
+ w x ∈ (2B)c : B∗δ ,b (a)(x) > λ
= I + II.
(2.20)
For I, by the Lq -boundedness of B∗δ ,b for q > 1, we gain
I ≤ λ−1 B∗δ ,b (a)χ2B L1w ≤ Cλ−1 B∗δ ,b (a)
Lwq · w(B)1−1/q
≤ Cλ−1 aLwq · w(B)1−1/q ≤ Cλ−1 .
For II, let b0 = |B |−1
B b(x)dx,
(2.21)
notice that
B∗δ ,b (a)(x) = b(x)Brδ (a)(x) − Brδ (ba)(x)
= b(x) − b0 Brδ (a)(x) − Brδ b − b0 a (x)
≤ b(x) − b0 Brδ (a)(x)
+ Brδ b − b0 a (x)
δ
≤ b(x) − b0 B∗
(a)(x) + B∗δ b − b0 a (x),
(2.22)
we have
II ≤ w
+w
x ∈ (2B)c : b(x) − b0 gµ∗ (a)(x) >
x ∈ (2B)c : gµ∗ b − b0 a (x) >
λ
2
λ
2
(2.23)
= II1 + II2 .
Similar to the proof of Theorem 1.3, we get
II1 ≤ Cλ
−1
= Cλ−1
(2B)c
∗
∞ k =1
II2 ≤ Cλ−1
b(x) − b0 B δ (a)(x)w(x)dx
2k+1 B \2k B
(2B)c
b(x) − b0 B δ (a)(x)w(x)dx ≤ Cλ−1 bBMO ,
∗
B∗δ b − b0 a (x)w(x)dx ≤ Cλ−1 bBMO .
(2.24)
200
Maximal commutator of Bochner-Riesz operator
Combining the estimate of I, II1 , and II2 , we gain
w x ∈ Rn : B∗δ ,b (a)(x) > λ
≤ Cλ−1 bBMO .
(2.25)
This completes the proof of Theorem 1.4.
Proof of Theorem 1.5. . Given f ∈ H 1 (w), let f = j λ j a j be the atomic decomposition
for
f . By a limiting
argument, it suffices to show Theorem 1.5 for a finite sum of f =
Q λQ aQ with
Q |λQ | ≤ C f H 1 (w) . We may assume that each Q (the supporting cube
of aQ ) is dyadic. For λ > 0 by [3, Lemma 4.1], there exists a collection of pairwise disjoint
dyadic cubes {S} such that
λQ ≤ Cλ|S|,
Q⊂S
|S| ≤ λ
−1
−1
λ
|Q| χQ Q
λ Q ,
S
∀S,
Q⊂S
Q
(2.26)
≤ Cλ.
L∞
Let E = S S, where for a fixed cube Q, Q denotes the cube with the same center as
√
Q but with the side-length 4 n times that of Q. Then, |E| ≤ Cλ−1 f H 1 . Set M(x) =
δ
2
S Q⊂S λQ aQ , N(x) = f (x) − M(x). By the L boundedness of B∗,b and the well-known
argument, it suffices to show that
w x ∈ Ec : B∗δ ,b (M)(x) > λ
Because B∗δ ,b (M)(x) ≤
S
δ
Q⊂S |λQ |B∗,b (aQ )(x),
w x ∈ Ec : B∗δ ,b (M)(x) > λ
≤ Cλ−1
≤ Cλ−1 f H 1 (w) .
(2.27)
we have
Ec
B∗δ ,b (M)(x)w(x)dx
∞ λQ ≤ Cλ−1
k+1
k
k=1 2 Q\2 Q
S Q ⊂S
(2.28)
B∗δ ,b aQ (x)w(x)dx,
similar to the estimate of Theorem 1.3, we get, when x ∈ Ec ,
−(δ0 +(n+1)/2)
B∗δ ,b aQ (x) ≤ C bBMO w(B)−1 |Q|(δ0 +(n+1)/2)/n x − x0 + C b(x) − b0 w(B)−1 2−k(δ0 +(n+1)/2) ,
(2.29)
L. Lanzhe and T. Qingshan 201
thus, by Hölder’s and reverse of Hölder’s inequalities for w ∈ A1 , we obtain
w x ∈ Ec : B∗δ ,b (M)(x) > λ
≤ Cλ−1 w(B)−1
∞
λ Q k2−k(δ0 +(n+1)/2) w 2k Q
S Q⊂S
k =1
∞
λ Q ≤ Cλ−1
k2−k(δ0 −(n−1)/2)
S Q⊂S
≤ Cλ
−1
(2.30)
k =1
λQ ≤ Cλ−1 f H 1 (w) .
S Q⊂S
This finishes the proof of Theorem 1.5.
Acknowledgment
The author would like to express his gratitude to the referee for his very valuable comments and suggestions.
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Liu Lanzhe: College of Mathematics and Computer, Changsha University of Science and Technology, Changsha 410077, China
E-mail address: lanzheliu@263.net
Tong Qingshan: College of Mathematics and Computer, Changsha University of Science and Technology, Changsha 410077, China