Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 237937, 21 pages
doi:10.1155/2008/237937
Research Article
Commutators of the Hardy-Littlewood
Maximal Operator with BMO Symbols on
Spaces of Homogeneous Type
Guoen Hu,1 Haibo Lin,2 and Dachun Yang2
1
Department of Applied Mathematics, University of Information Engineering,
P.O. Box 1001-747 Zhengzhou 450002, China
2
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and
Complex Systems, Ministry of Education, Beijing 100875, China
Correspondence should be addressed to Dachun Yang, dcyang@bnu.edu.cn
Received 20 August 2007; Revised 19 November 2007; Accepted 5 January 2008
Recommended by Yong Zhou
Weighted Lp for p ∈ 1, ∞ and weak-type endpoint estimates with general weights are established
for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal
operators associated with commutators of singular integral operators with BMO symbols on spaces
of homogeneous type. All results with no weight on spaces of homogeneous type are also new.
Copyright q 2008 Guoen Hu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
We will be working on a space of homogeneous type. Let X be a set endowed with a positive
Borel regular measure μ and a symmetric quasimetric d satisfying that there exists a constant
κ ≥ 1 such that for all x, y, z ∈ X, dx, y ≤ κdx, z dz, y. The triple X, d, μ is said to be
a space of homogeneous type in the sense of Coifman and Weiss 1 if μ satisfies the following
doubling condition: there exists a constant C ≥ 1 such that for all x ∈ X and r > 0,
μ Bx, 2r ≤ Cμ Bx, r .
1.1
It is easy to see that the above doubling property implies the following strong homogeneity:
there exist positive constants C and n such that for all λ ≥ 1, r > 0, and x ∈ X,
μ Bx, λr ≤ Cλn μ Bx, r .
1.2
2
Abstract and Applied Analysis
Moreover, there also exist constants C > 0 and N ∈ 0, n such that for all x, y ∈ X and r > 0,
dx, y N μ By, r ≤ C 1 μ Bx, r .
1.3
r
We remark that although all balls defined by d satisfy the axioms of complete system
of neighborhoods in X, and therefore induce a separated topology in X, the balls Bx, r for
x ∈ X and r > 0 need not be open with respect to this topology. However, by a remarkable
result of Macı́as and Segovia in 2, we know that there exists another quasimetric d such that
y ≤ dx, y ≤ Cdx,
y;
i there exists a constant C ≥ 1 such that for all x, y ∈ X, C−1 dx,
ii there exist constants C > 0 and γ ∈ 0, 1 such that for all x, x , y ∈ X,
y − d x , y ≤ C d x, x γ dx,
y d x , y 1−γ .
dx,
1.4
Thus, throughout this
The balls corresponding to d are open in the topology induced by d.
paper, we always assume that there exist constants C > 0 and γ ∈ 0, 1 such that for all
x, x , y ∈ X,
dx, y − d x , y ≤ C d x, x γ dx, y d x , y 1−γ ,
1.5
and that the balls Bx, r for all x ∈ X and r > 0 are open.
Now let k be a positive integer and b ∈ BMO X, define the kth-order commutator Mb,k
of the Hardy-Littlewood maximal operator with b by
1
bx − byk fydμy
1.6
Mb,k fx sup
B x μB B
for all x ∈ X. For the case that X, d, μ is the Euclidean space, Garcı́a-Cuerva et al. 3 proved
that Mb,k is bounded on Lp Rn for any p ∈ 1, ∞, and Alphonse 4 proved that Mb,1 enjoys a weak-type Llog L estimate, that is, there exists a positive constant C, depending on
bBMO Rn , such that for all suitable functions f,
fx fx
x ∈ Rn : Mb,1 fx > λ ≤ C
log e dx.
1.7
λ
λ
Rn
Li et al. 5 established a weighted estimate with any general weight for Mb,1 in Rn . As it was
shown in 3–5 for the setting of Euclidean spaces, the operator Mb,k plays an important role in
the study of commutators of singular integral operators with BMO symbols. In this paper, we
establish weighted estimates with general weights for Mb,k in spaces of homogeneous type. To
state our results, we first give some notation.
Let E be a measurable set with μE < ∞. For any fixed p ∈ 1, ∞, δ > 0, and suitable
function f, set
p
fx fx
1
δ
fLp log Lδ ,E inf λ > 0 :
log e dμx ≤ 1 .
1.8
μE E
λ
λ
The maximal operator MLp log Lδ is defined by MLp log Lδ fx supBx fLp log Lδ ,B , where
the supremum is taken over all balls containing x. In the following, we denote ML1 log Lδ by
MLlog Lδ for simplicity, and denote by L∞
X the set of bounded functions with bounded
b
support.
With the notation above, we now formulate our main results as follows.
Guoen Hu et al.
3
Theorem 1.1. Let k be a positive integer, p ∈ 1, ∞, and b ∈ BMO X. Then for any δ > 0, there
exists a positive constant C, depending only on p, k, and δ, such that for all nonnegative weights w and
f ∈ L∞
X,
b
X
p
kp
Mb,k fx wxdμx ≤ CbBMO X
X
fxp M
Llog Lkpδ wxdμx.
1.9
Theorem 1.2. Let k be a positive integer, b ∈ BMO X, and δ > 0. There exists a positive constant
C Ck,bBMO X such that for all nonnegative weights w, f ∈ L∞
μ and λ > 0,
b
w x ∈ X : Mb,k fx > λ ≤ C
fx fx
k
wxdμx,
log e MLlog Lkδ
λ
λ
X
1.10
where, and in the following, k k if k is even and k k 1 if k is odd.
As a corollary of Theorem 1.2, we establish a weighted endpoint estimate for the maximal commutator of singular integral operators with BMO X symbols. Let T be a CalderónZygmund operator, that is, T is a linear L2 X-bounded operator and satisfies that for all
f ∈ L2 X with bounded support and almost all x /
∈ suppf,
Kx, yfydμy,
1.11
T fx X
where K is a locally integrable function on X × X \ {x, y : x y} and satisfies that for all
x
/ y,
Kx, y ≤ C
,
μ B x, dx, y
1.12
and that for all x, y, y ∈ X with dx, y ≥ 2dy, y ,
τ
d y, y Kx, y − K x, y Ky, x − K y , x ≤ C τ
μ B x, dx, y dx, y
1.13
with positive constants C and τ ≤ 1. For any > 0, suitable function f and x ∈ X, define the
truncated operator T by
Kx, yfydμy.
1.14
T fx dx,y>
Let b ∈ BMO X and let k be a positive integer. Define the commutator T;b,k by
T;b,0 fx T fx, T;b,k fx bxT;b,k−1 fx − T;b,k−1 bfx
1.15
for all x ∈ X and f ∈ L∞
X. The maximal operator associated with the commutator Tb,k is
b
defined by
∗
Tb,k
fx supT;b,k fx
>0
1.16
4
Abstract and Applied Analysis
for all x ∈ X. In 6, it was proved that if T is a Calderón-Zygmund operator, then for any
p ∈ 1, ∞, there exists a positive constant C such that for all f ∈ L∞
X and all nonnegative
b
weights w,
∗
T fxp wxdμx ≤ Cbkp
fxp M
1.17
Llog Lk1pδ wxdμx.
b,k
BMO μ
X
X
∗
Tb,1
enjoys the following weighted weak-type endpoint estiIn 5, it was proved that in R ,
mate: for any δ > 0, there exists a positive constant C, depending on n, δ, and bBMO Rn , such
that
fx fx
n
∗
w x ∈ R : Tb,1 fx > λ ≤ C
log e MLlog2δ wxdx.
1.18
λ
λ
Rn
n
Using Theorem 1.2, we will prove the following result.
Theorem 1.3. Let T be a Calderón-Zygmund operator. Then for any b ∈ BMO X, nonnegative
integer k and δ > 0, there exists a positive constant C, depending on k, δ, and bBMO X , such that for
all λ > 0, f ∈ L∞
X and nonnegative weights w,
b
fx fx
k
MLlog Lk1δ wxdμx.
wxdμx ≤ C
log e 1.19
∗
λ
λ
{x∈X:Tb,k
fx>λ}
X
We mention that Theorems 1.1, 1.2, and 1.3 are also new even when wx ≡ 1 for all
x ∈ X.
We now make some conventions. Throughout the paper, we always denote by C a positive constant which is independent of main parameters, but it may vary from line to line. We
denote f ≤ Cg and f ≥ Cg simply by f g and f g, respectively. If f g f, we then
write f∼g. Constant, with subscript such as C1 , does not change in different occurrences. A
weight w always means a nonnegative locally integrable function.
For a measurable set E and
a weight w, χE denotes the characteristic function of E, wE E wxdμx. Given λ > 0 and
a ball B, λB denotes the ball with the same center as B and whose radius is λ times that of B.
For a fixed p with p ∈ 1, ∞, p denotes the dual exponent of p, namely, p p/p − 1. For
any measurable set E and any integrable
function f on E, we denote by mE f the mean value
of f over E, that is, mE f 1/μE E fxdμx. For any locally integrable function f and
x ∈ X, the Fefferman-Stein sharp maximal function M# fx is defined by
1
fy − mB fdμy,
M# fx sup
1.20
B x μB B
where the supremum is taken over all balls B containing x. For any fixed q ∈ 0, 1, the sharp
maximal function Mq# f of the function f is defined by Mq# f M# |f|q 1/q .
A generalization of Hölder’s inequality will be used in the proofs of our theorems. For
any measurable set E with μE < ∞, positive integer l, and suitable function f, set
1/l
fx
1
fexp L1/l ,E inf λ > 0 :
exp
dμx ≤ 2 .
1.21
μE E
λ
Then the following generalization of Hölder’s inequality:
1
fxhxdμx ≤ Cf
Llog Ll ,E hexp L1/l ,E
μE E
holds for any suitable functions f and h; see 7 for details.
1.22
Guoen Hu et al.
5
2. Proof of Theorem 1.1
To prove Theorem 1.1, we need some technical lemmas. In what follows, we denote by M the
Hardy-Littlewood maximal function. Moreover, for any s > 0 and suitable function f, we set
Ms f M|f|s 1/s .
Lemma 2.1 see 8. There exists a positive constant C such that for all weights w and all nonnegative
functions f satisfying μ{x ∈ X : fx > λ} < ∞ for all λ > 0, then
i if μX ∞,
X
fxwxdμx ≤ C
M# fxMwxdμx;
2.1
M# fxMwxdμx CwXmX f.
2.2
X
ii if μX < ∞,
X
fxwxdμx ≤ C
X
Lemma 2.2. For any q ∈ 0, 1, there exists a positive constant C such that for all f ∈ Lp X with
p ∈ 1, ∞ and all x ∈ X, Mq# Mfx ≤ CM# fx.
For the case that X, d, μ is the Euclidean space, this lemma was proved in 9. For
spaces of homogeneous type, the proof is similar to the case of Euclidean spaces; see 6.
Lemma 2.3. Let p ∈ 1, ∞ and let k be a positive integer.
a There exists a positive constant C, depending only on k and p, such that for all f ∈ L∞
X
b
and all weights w,
X
p
MLlog Lk fx wxdμx ≤ C
X
fxp Mwxdμx.
2.3
b For any δ1 > 0, there exists a positive constant C, depending only on k, p, and δ1 , such that
for all f ∈ L∞
X and all weights w,
b
X
p 1−p
dμx ≤ C
Mk fx MLlog Lkp −1δ1 wx
X
fxp wx 1−p dμx.
2.4
For Euclidean spaces, Lemma 2.3a is just Corollary 1.8 in 10 and Lemma 2.3b is
included in the proof of Theorem 2 in 11 together with 4.11 in 12. For spaces of homogeneous type, Lemma 2.3a is a simple corollary of Theorem 1.4 in 13. On the other hand, by
Theorem 1.4 in 13, and the estimate that for all weights w, MLlog Lk w ≈ Mk1 w see 12,
we can prove Lemma 2.3b by the ideas used in 11, page 751. For details, see 6.
By a similar argument that was used in the proof of Theorem 2.1 in 14, we can verify
the existence of the following approximation of the identity of order γ with bounded support
on X. We omit the details here.
For any x ∈ X and r > 0, set Vr x μ Bx, r .
6
Abstract and Applied Analysis
Lemma 2.4. Let γ be as in 1.5. Then there exists an approximation of the identity {Sk }k∈Z of order
γ with bounded support on X. Namely, {Sk }k∈Z is a sequence of bounded linear integral operators on
> 0 such that for all k ∈ Z and all x, x , y, and y ∈ X, Sk x, y,
L2 X, and there exist constants C0 , C
the integral kernel of Sk is a measurable function from X × X into C satisfying
−k and 0 ≤ Sk x, y ≤ C0 1/V2−k x V2−k y;
i Sk x, y 0 if dx, y ≥ C2
ii Sk x, y Sk y, x for all x, y ∈ X;
iii |Sk x, y − Sk x , y| ≤ C0 2kγ dx, x γ 1/V2−k x V2−k y for dx, x ≤ max{C/κ,
1−k
1/κ}2 ;
iv C0 V2−k xSk x, x > 1 for all x ∈ X and k ∈ Z;
v X Sk x, ydμy 1 X Sk x, ydμx.
For any > 0 and x, y ∈ X, let
S x, y ≡ Sk x, yχ2−k−1 , 2−k .
2.5
Obviously, S satisfies i through v of Lemma 2.4 with 2−k replaced by . From iii and iv
of Lemma 2.4, it follows that there exist constants C ∈ 0, min{C/κ,
1/κ, C0 −2/γ } and C > 1
such that for all > 0 and all x, y ∈ X satisfying dx, y < C ,
CV xS x, y > 1.
2.6
b,k be the operator defined by
For a positive integer k and a function b ∈ BMO X, let M
;b,k fx
b,k fx supM
M
2.7
>0
for all f ∈ L∞
X and x ∈ X, where for > 0,
b
k ;b,k fx S x, ybx − by fydμy.
M
X
2.8
b,k and M
;b,k simply by M
and M
, respectively. From i of Lemma 2.4
If k 0, we denote M
together with 1.1, it follows that S x, y 1/V x 1/V2C
x. Notice that if dx, y ≥ 2C,
then S x, y 0. Thus,
1
bx − byk fydμy Mb,k fx.
Mb,k fx supM;b,k fx sup
V
x
>0
>0 2C
Bx,2C
2.9
On the other hand, for each fixed > 0, by 2.6 and VC x∼V x, we have
k 1
bx − byk fydμy S x, ybx − by fydμy
VC x Bx,C X
b,k fx.
M
2.10
b,k fx. Thus, there exists some
By the definition of Mb,k , we further obtain Mb,k fx M
∞
constant C ≥ 1 such that for all x ∈ X and f ∈ Lb X,
b,k fx.
b,k fx ≤ Mb,k fx ≤ CM
C−1 M
b,k , we have the following estimate.
For the sharp function estimate of M
2.11
Guoen Hu et al.
7
Lemma 2.5. Let k be a positive integer and b ∈ BMO X. For any q and s with 0 < q < s < 1, there
exists a positive constant C such that for all f ∈ L∞
X and all x ∈ X,
b
k−1
b,k f x ≤ C bk−j
b,j f x Cbk
Mq# M
Ms M
BMO X MLlog Lk fx.
BMO X
2.12
j0
Proof. By i, ii, and iii of Lemma 2.4, we obtain that for all x, y ∈ X,
S x, y 1
,
μ B x, dx, y
2.13
and that for all > 0 and all x, y, y ∈ X with dx, y ≥ 2κdy, y ,
S x, y − S x, y S y, x − S y , x 1
μ B x, dx, y
γ
d y, y .
dx, y
2.14
To verify 2.12, by homogeneity, we may assume that bBMO X 1. For all f ∈ L∞
X,
b
x ∈ X, and balls B containing x, it suffices to prove that
inf
c∈C
1
μB
1/q
M
b,k fy − cq dμy
B
k−1
b,j f x M
Ms M
Llog Lk fx.
2.15
j0
We consider the following three cases.
Case 1 μX \ C1 B 0. Where and in what follows C1 κ4κ 1. In this case, we have that
for all x ∈ X,
b,k fx M
k−1 mB b − bxk−j M
b − mB b k f x.
b,j fx M
2.16
j0
is
The Kolmogorov inequality see 15, page 102, along with the fact that M and so M
bounded from L1 X to L1,∞ X and the inequality 1.22 gives us that
1
μB
1/q
q
b − mB b k f y dμy
M
1
by − mB bk fydμy
μB C1 B
k 1
1
fydμy
by − mC1 B b fy dμy mC1 B b − mB b
μB C1 B
μB C1 B
B
MLlog Lk fx,
2.17
where the last inequality follows from the John-Nirenberg inequality, which states that for any
ball Q,
mQ b − bk exp L1/k ,Q
bkBMO X .
2.18
8
Abstract and Applied Analysis
On the other hand, if 0 < q < s < 1, an application of Hölder’s inequality implies that
1
μB
1/q
b,k fx q dμx
M
B
k−1
b,j f x Ms M
j0
1
μB
2.19
1/q
q
b − mB b k f y dμy
M
.
B
We then get 2.15.
0 and μC1 B \B > 0. In this case, decompose f into f fχC1 B fχX\C1 B ≡
Case 2 μX\C1 B /
f1 f2 , recalling that χE denotes the characteristic function of the set E. Let y0 be a point in B
such that
k CB ≡ sup S y0 , z mB b − bz f2 zdz < ∞.
2.20
>0
X
With the aid of the formula
mB b − bz
k
k−1
k k−j
j j
by − bz Ck by − bz mB b − by
,
2.21
j0
j
where Ck is the constant from Newton’s formula, we have
k k by − bz fz − mB b − bz f2 z
k−1 by − bzj mB b − byk−j fz mB b − bzk f1 z.
2.22
j0
Thus for any y ∈ B,
M
b,k fy − CB supM
;b,k fy − supM
mB b − b k f2 y0 >0
>0
;b,k fy − M
mB b − b k f2 y0 ≤ supM
>0
k−1 mB b − b k f1 y mB b − byk−j M
b,j fy
M
2.23
j0
mB b − b k f2 y − M
mB b − b k f2 y0 supM
>0
≡ I y II y III y.
As in Case 1, we have that
1/q
1
Iyq dμy
MLlog Lk fx,
μB B
1/q k−1
1
IIyq dμy
b,j f x.
Ms M
μB B
j0
2.24
Guoen Hu et al.
9
As for III y, by 2.14 and 1.22, it is easy to get
III y ≤ sup
X
>0
S y, z − S y0 , z mB b − bzk f2 zdμz
γ
k d y, y0
γ mB b − bz fzdz
dy, z
X\C1 B μ B y, dy, z
∞
k 1
2−lγ l
l m2l C1 B b − bz fzdμz
μ 2 C1 B 2 C1 B
l1
∞
k
1
2−lγ m2l C1 B b − mB b l
l fzdμz
μ 2 C1 B 2 C1 B
l1
∞
k
2−lγ m2l C1 B b − b exp L1/k ,2l C1 B fLlog Lk ,2l C1 B
2.25
l1
∞
k
2−lγ m2l C1 B b − mB b Mfx MLlog Lk fx,
l1
where the last inequality follows from 2.18 and |m2l Q b−mQ b| l. This leads to our desired
estimate 2.15.
Case 3 μX\C1 B /
0 and μC1 B\B 0. In this case, we take B such that B ⊂ B , μB μB,
and μC1 B \ B > 0. We then have that
1
inf
c∈C μB
M
b,k fx − cq dμx ≤ inf 1 c∈C μ B B
B
M
b,k fx − cq dμx.
2.26
With the ball B replaced by B in Case 2, we also obtain the result that for any y ∈ B ,
sup
>0
X
S y, z − S y0 , z mB b − bzk f2 zdz ≤ CM
Llog Lk fx,
2.27
which completes the proof of Lemma 2.5.
Lemma 2.6. Let α, β ∈ 0, ∞. There exists a positive constant C, depending only on α and β, such
that for all weights w,
MLlog Lα MLlog Lβ w x ≤ CMLlog Lαβ1 wx.
2.28
For Euclidean spaces, a generalization of Lemma 2.6 was proved in 16. For spaces of
homogeneous type, by a standard argument involving a covering lemma in 17, page 138, we
have that for any λ > 0 and suitable function f,
μ x ∈ X : MLlog Lα fx > λ fx
X
λ
log
α
fx
e
dμx.
λ
2.29
Using this, Lemma 2.6 can be proved by applying the ideas used in 16. For details, see 6,
Lemma 7.
10
Abstract and Applied Analysis
Proof of Theorem 1.1. We assume again that bBMO X 1. At first, we claim that when μX b,k fx > λ} < ∞. In fact, for any f ∈ L∞ X,
X, μ{x ∈ X : M
∞, for all λ > 0 and f ∈ L∞
b
b
let R be large enough such that supp f ⊂ Bx0 , R for some x0 ∈ X. Notice that for all x ∈
X \ Bx0 , 3R,
fL1 X
Mfx .
μ B x, d x, x0
2.30
It then follows that for p ∈ 1, ∞,
μ
k
x ∈ X \ B x0 , 3R : bx − mBx0 ,R b Mfx > λ
bx − mBx ,R bkp Mfx p dμx
≤ λ−p
0
X\Bx0 ,3R
p
≤ λ−p fL1 X
bx − mBx ,dx,x bkp
0
0
p dμx < ∞.
μ B x, d x, x0
X\Bx0 ,3R
2.31
This, together with the estimate that
μ
k p
k x ∈ X : M b − mBx0 ,R b f x > λ λ−p b − mBx0 ,R b f Lp X < ∞,
2.32
leads to our claim.
By 2.11, to prove Theorem 1.1, it suffices to prove that for all weights w,
X
b,k fx p wxdμx M
X
fxp M
Llog Lkpδ wxdμx.
2.33
We proceed our proof by an inductive argument on k. When k 0, 2.33 is implied by the fact
that Mwx ≤ MLlog Lδ wx for all x ∈ X and the following known inequality:
X
p
wxdμx Mfx
X
fxp Mwxdμx.
2.34
See 18, pages 150-151, for a proof of the last inequality when X Rn . The same ideas also
work for X. Now we assume that k is a positive integer and 2.33 holds for any integer l with
b,l 0 ≤ l ≤ k − 1 can extend to a bounded operator on Lp X for p ∈ 1, ∞
0 ≤ l ≤ k − 1. Then M
and so for any λ > 0 and σ ∈ 0, 1,
μ
b,l f σ x > λ < ∞.
x∈X:M M
2.35
We now prove 2.33. To begin with, we prove that for any given q ∈ 0, 1 and k ∈ N,
and for all weights h and all f ∈ L∞
X,
b
X
b,k fx q hxdμx M
X
q
MLlog Lk fx Mk hxdμx.
2.36
Guoen Hu et al.
11
We first consider the case that μX ∞. Choose r1 , . . . , rk−1 , rk such that 0 < q r0 < r1 < · · · <
rk−1 < rk < 1. By Lemma 2.5, we obtain that for any 1 ≤ m ≤ k − 1 and any weight h,
X
b,m f x q hxdμx
Mr#j M
m−1
l0
X
b,l f x q hxdμx Mrj1 M
X
q
MLlog Lm fx hxdμx.
2.37
Therefore, applying Lemmas 2.1 and 2.2, and the estimate 2.35, we have
X
Mrj1
b,l f x q hxdμx M
X
X
X
X
b,l f rj1 x q/rj1 hxdμx
M M
#
b,l f rj1 x q/rj1 Mhxdμx
Mq/r
M M
j1
M#
b,l f rj1 x q/rj1 Mhxdμx
M
2.38
b,l f x q Mhxdμx,
Mr#j1 M
which leads to
X
b,m f x q hxdμx
Mr#j M
m−1
l0
X
Mr#j1
b,l f x q Mhxdμx M
X
q
2.39
MLlog L fx hxdμx.
m
Repeating the argument above k − 1 times, we then have that for all weights h,
X
b,k f x q hxdμx
Mq# M
k−1 q
q
#
Mr1 Mb,j f x Mhxdμx MLlog Lk fx hxdμx
j0 X
k−2 j0 X
X
X
#
b,j f x q M2 hxdμx Mr2 M
k−1 j0 X
q
MLlog Lj fx Mhxdμx
q
MLlog Lk fx hxdμx
1 k q
# b,j f x q Mk−1 hxdμx Mrk−1 M
MLlog Lj fx Mk−1 hxdμx.
j0 X
j0 X
2.40
b,0 fx M2 fx, and that, by 2.12
On the other hand, notice that for all x ∈ X, Mr#k−1 M
2
and the fact that M fx∼MLlog L fx for all x ∈ X see 12, 4.11, we then have that for all
12
Abstract and Applied Analysis
b,1 fx Mrk M
b,0 fx MLlog L fx M2 fx. From these inequalities,
x ∈ X, Mr#k−1 M
it then follows that
#
b,k f x q hxdμx
Mq M
X
X
X
2
q
M fx M
k−1
hxdμx k j0 X
q
MLlog Lj fx Mk−1 hxdμx
2.41
q
MLlog Lk fx Mk−1 hxdμx,
which together with i of Lemma 2.1 gives 2.36.
We turn our attention to 2.36 for the case of μX < ∞. For all x ∈ X,
l bx − mX bj M
mX b − b l−j f x.
b,l fx M
2.42
j0
Moreover, the Kolmogorov inequality, together with Hölder’s inequality, the inequalities
1.22, and 2.18, tells us that for any 0 ≤ j ≤ k, r ∈ 0, 1, and t ∈ r, 1,
r
1
bx − mX brj M
mX b − b l−j f x dμx
μX X
r/t
l−j t
1
2.43
M mX b − b
fx dμx
μX X
r
r
1
fxmX b − bxl−j dμx f
Llog Ll−j ,X .
μX X
Combining the above estimates, we obtain
r
b,l f r inf M
mX M
Llog Ll fx .
x∈X
2.44
Let q, r1 , r2 , . . . , rk be as in the case of μX ∞. Another application of Kolmogorov inequality
and the fact that M is bounded from L1 X to L1,∞ X leads to
rj
q/rj rj /q
b,l f rj
b,l f rj inf M
l fx
mX M M
mX M
.
2.45
Llog L
x∈X
As in the case of μX ∞, by Lemmas 2.1, 2.2, and 2.5, we have that for any q ∈ 0, 1,
#
b,k f q hxdμx b,k f x q Mhxdμx hXmX M
b,k f q
M
Mq M
X
X
k−1 j0 X
X
k−1
b,j f x q M2 hxdμx
Mr#1 M
q
MLlog Lk fx Mhxdμx
q/r1
b,j f r1
b,k f q
MhXmX M M
hXmX M
j0
X
q
MLlog Lk fx Mk hxdμx.
Combining the two cases yields2.36.
2.46
Guoen Hu et al.
13
For any fixed p ∈ 1, ∞ and δ > 0, choose q ∈ 0, 1 and δ1 > 0 such that kp/qδ1 < kpδ.
This, via a duality argument, 2.36, and Lemma 2.3, leads to
M
b,k f q Lp/q w
sup
h
Lp/q
w1−p/q
h
sup
h
sup
Lp/q
Lp/q
q
Mb,k fx hxdμx
X
≤1
w1−p/q
w1−p/q
≤1
X
X
X
X
q
MLlog Lk fx Mk hxdμx
≤1
×
Mk hx
X
q/p
p
MLlog Lk fx MLlog Lkp/q−1δ1 wxdμx
2.47
1/p/q
1−p/q
dμx
MLlog Lkp/q−1δ1 wx
p/q q/p
p
MLlog Lk fx MLlog Lkp/q−1δ1 wxdμx
fxp M
Llog L
kpδ
q/p
wxdμx
,
where in the last inequality we have used Lemma 2.6. This completes the proof of Theorem 1.1.
3. Proof of Theorem 1.2
We begin with some preliminary lemmas.
Lemma 3.1 see 17. Let X, d, μ be a space of homogeneous type and let f be a nonnegative integrable function. Then for every λ > mX fmX f 0 if μX ∞, there exist a sequence of pairwise
disjoint balls {Bj }j≥1 and a constant C4 ≥ 1 such that
mC4 Bj f ≤ λ < mBj f
3.1
and mB f ≤ λ for every ball B centered at x ∈ X \ ∪j C4 Bj .
Lemma 3.2. Let d and l be two nonnegative integers. Then for all t1 , t2 ≥ 0,
t1 td2 log e t1 td2 ≤ C t1 logdl e t1 exp t2 .
3.2
Proof. We may assume that d ≥ 1, otherwise the conclusion holds obviously. Set Φ1 t tlogl e t, Φ2 t tlogld e t, and Φ3 t expt1/d . Let j 1, 2, 3. Denote by Φ−1
j the inverse
−l
−1
of Φj , that is, Φ−1
j t inf{s > 0 : Φj s > t}. It is well known that Φ1 t ≈ tlog e t and
−dl
Φ−1
e t see 19. On the other hand, it is easy to verify that Φ−1
2 t ≈ tlog
3 t 0 when
−1
−1
−1
tΦ
t ∈ 0, 1 and Φ3 t logd t when t ∈ 1, ∞. Therefore, for all t ∈ 0, ∞, Φ−1
2
3 t Φ1 t.
d
d
This via 7, Lemma 6, page 63 tells us that Φ1 t1 t2 Φ2 t1 Φ3 t2 . Our desired conclusion
then follows directly.
14
Abstract and Applied Analysis
b,k as in 2.7, by 2.11, it suffices to prove that for
Proof of Theorem 1.2. With the notation M
bBMO X 1 and all f ∈ L∞
X
and
λ
> 0,
b
b,k fx > λ w x∈X:M
fx fx
k
wxdμx,
MLlog Lkδ
log e λ
λ
X
3.3
where k k when k is even and k k 1 when k is odd.
Recall that for all f ∈ L∞
X,
b
1
w{x ∈ X : Mfx
> λ} λ
X
|fx|Mwxdμx.
3.4
See 18, page 151 for a proof when X Rn . The same idea also works for X. By Hölder’s
inequality, it follows that for all x ∈ X,
1/k1
b,k fx ≤ M
b,k1 fx k/k1 Mfx
M
3.5
and so when k is odd,
w
b,k fx > λ ≤ w x ∈ X : M
b,k1 fx > λ w x ∈ X : Mfx
x∈X:M
>λ
fxMwxdμx.
b,k1 fx > λ 1
w x∈X:M
λ X
3.6
Thus, it suffices to prove 3.3 for the case that k is even. We employ some ideas from 20, and
proceed our proof of 3.3 by an inductive argument. When k 0, 3.3 is implied by the fact
that Mwx ≤ MLlog Lδ wx for all x ∈ X and 3.4. Now let k be a positive integer. We may
assume that MLlog Lkδ w is finite almost everywhere, otherwise there is nothing to be proved.
For any fixed δ > 0, we assume that for any nonnegative integer l with 0 ≤ l ≤ k − 1, there exists
a constant C Cl,δ such that for all λ > 0,
b,l fx > λ w x∈X:M
fx
X
λ
fx log e MLlog Llδ wxdμx,
λ
l
3.7
where and in what follows, l l when l is even and l l 1 when l is odd. If μX < ∞
and λ ≤ fL1 X μX−1 , the inequality 3.3 is trivial. So it remains to consider the case
that λ > fL1 X μX−1 . For each fixed bounded function f with bounded support and λ >
fL1 X μX−1 , applying Lemma 3.1 to |f| at level λ, we obtain a sequence of balls {Bj }j≥1
with pairwise disjoint interiors. As in the proof of Lemma 2.10 in 17, set V1 C4 B1 \∪n≥2 Bn j−1
and Vj C4 Bj \ ∪n1 Vn ∪ ∪l≥j1 Bl , it then follows that Bj ⊂ Vj ⊂ C4 Bj and ∪j Vj ∪j C4 Bj .
Define the functions g and h, respectively, by g ≡ |f|χX\∪j Vj j mVj |f|χVj and h ≡ j hj with
hj ≡ |f| − mVj |f|χVj . Recall that μ is regular and the set of continuous function is dense in
Lp X for any p ∈ 1, ∞. Lemma 3.1 implies that for any fixed j,
1
C6−1 λ ≤ μ Vj
Vj
fydμy ≤ C6 λ
3.8
Guoen Hu et al.
15
with C6 > 1 a constant independent of f and j, which together with the Lebesgue differentiation theorem and Lemma 3.1 again yields that
gL∞ X ≤ C6 λ.
3.9
Let Ω ∪j C7 Bj with C7 C1 C4 . The doubling property of μ and 3.1 now state that
w C7 Bj 1 1 wΩ inf Mwx fydμy fyMwydμy.
μ Bj λ j x∈Bj
λ X
Bj
j μ C7 Bj
3.10
Following an argument similar to the case of Euclidean spaces see 18, page 159, we have
that for any γ ≥ 0, there exists a positive constant C, depending only on γ, such that for all
x ∈ C4 Bj ,
MLlog Lγ wχX\Ω x inf MLlog Lγ wχX\Ω y.
y∈C4 Bj
3.11
Thus,
mV f M
j
Llog Lkδ wχX\Ω xdμx λμ Vj inf MLlog Lkδ wχX\Ω y
y∈Bj
Vj
λμ Bj inf MLlog Lkδ wχX\Ω y
y∈Bj
fxM
Llog Lkδ wχX\Ω xdμx.
3.12
Bj
For each fixed δ > 0, choose p0 ∈ 1, ∞ and δ1 > 0 such that kp0 δ1 < k δ. From the last
estimate, 2.11, Theorem 1.1, and 3.9, it follows that
−p0
gxp0 M
w x ∈ X \ Ω : Mb,k gx > λ/2 λ
Llog Lkp0 δ1 wχX\Ω xdμx
X
−1
gxM
λ
Llog Lkp0 δ1 wχX\Ω xdμx
X
−1
fxM
λ
Llog Lkδ wχX\Ω xdμx
X\∪j Vj
λ−1
mV f M
kδ wχX\Ω xdμx
j
Llog L
Vj
j
X
fxM
Llog Lkδ wxdμx.
3.13
Thus, our proof is now reduced to proving
w
∗ hx > λ
x ∈X\Ω:M
b,k
2
fx
fx
k
log e MLlog Lkδ wxdμx,
λ
λ
X
3.14
16
Abstract and Applied Analysis
where
∗ hx sup S x, ybx − byk hydμy.
M
b,k
X
>0
3.15
For any j ≥ 1, let
∗
M hj x sup S x, yhj ydμy.
>0
X
3.16
We now prove 3.14. With the aid of the formula that for all x, y ∈ X,
k
k l k−l
bx − by Ckl bx − mBj b mBj b − by
,
3.17
l0
since k is even, for x ∈ X \ Ω, we write
∗ hx
M
b,k
k−1
k ∗
k−l
bx − mB b M
hj x M
b,l
hj x
b· − mBj b
j
j
l0
j
3.18
≡ Gx Hx.
Recall that {Vj }j are mutually disjoint. If we set Φl t tlogl e t, our inductive hypothesis
3.7 via 3.11 now tells us that
λ
w
x ∈ X \ Ω : Hx >
4
k−l k−1
by − mBj b fy
Φl
dμy inf MLlog Llδ wχX\Ω z
3.19
z∈C4 Bj
λ
Vj
l0 j
k−1 by − mB bk−l mV f
j
j
Φl
dμy inf MLlog Llδ χX\Ω w z.
z∈B
λ
j
Vj
l0 j
An application of Lemma 3.2 then gives that
by − mB bk−l fy
j
Φl
dμy
λ
Vj
by − mC B bk−l fy
4 j
Φl
dμy
λ
Vj
fy
k−l Φl mC4 Bj b − mBj b
Φl
dμy
λ
Vj
by − mC B b
fx
4 j
exp
Φk
dμx dμx
C2
λ
Vj
Vj
fx
Φk
dμx.
λ
Vj
3.20
Guoen Hu et al.
17
For each fixed j, notice that by 3.8 and Lemma 3.2,
by − mB bk−l mV f
j
j
Φl
dμy μ Bj .
λ
Vj
3.21
It then follows that
fx
λ
Φk
w
x ∈ X \ Ω : Hx >
MLlog Lkδ wxdμx
4
λ
Vj
j
μ Bj inf MLlog Lkδ wy
y∈Bj
j
3.22
fx
Φk
MLlog Lkδ wxdμx.
λ
X
It remains to prove that
w
x ∈ X \ Ω : Gx >
λ
4
λ−1
X
fxM
Llog Lk wxdμx.
3.23
For each fixed j, let yj and rj be the center and radius of Bj , respectively. If x ∈ X \ Ω, then by
the vanishing moment of hj and the estimate 2.14, we obtain that
∗ hj x ≤ sup
M
>0
X
S x, y − S x, yj hj ydμy
−γ
d x, yj
γ
rj
hj ydμy.
μ B yj , d x, yj
X
3.24
This in turn implies that
w
λ
x ∈ X \ Ω : Gx >
4
bx − mB bk
wx
1 γ
j
rj
γ dμx hj ydμy
λ j
μ
B
y
,
d
x,
y
d x, yj
j
j
X\C7 Bj
X
∞ bx − mB bk wx
1 γ j
≤
rj
γ dμx hj ydμy
l
l−1
λ j
d x, yj
X
l1 2 C7 Bj \2 C7 Bj μ B yj , d x, yj
1 1 hj ydμy inf MLlog Lk wy fxMLlog Lk wxdμx,
y∈C7 Bj
λ j X
λ X
3.25
where in the second to the last inequality, we use the fact that for each fixed j, yj ∈ Vj
and positive integer l, a standard argument involving the inequalities 1.22 and 2.18
18
Abstract and Applied Analysis
yields
2l C7 Bj \2l−1 C7 Bj bx − mB bk wx
l −γ
j
inf MLlog Lk wy;
γ dμx lk 2 rj
y∈C7 Bj
μ B yj , d x, yj
d x, yj
3.26
see also the proof of 2.25. We then complete the proof of Theorem 1.2.
4. Proof of Theorem 1.3
This section is devoted to the proof of Theorem 1.3.
Lemma 4.1. Let T be a Calderón-Zygmund operator. Then, there exists a positive constant C such that
for all λ > 0, f ∈ L∞
X, and weights w,
b
fx
MLlogL1δ wxdμx.
wxdμx ≤ C
4.1
λ
{x∈X: T ∗ fx>λ}
X
Lemma 4.1 can be proved by a similar but more careful argument as that used in the
proof of Theorem 1.2 in 8. We omit the proof here for brevity.
Proof of Theorem 1.3. The argument here is similar to that used in the proof of Theorem 1.2,
and we will only give an outline. Also, we proceed our proof by an inductive argument. By
Lemma 4.1, it is obvious that 1.19 is true when k 0. Now let k be a positive integer. For any
fixed δ > 0, and any nonnegative integer l with 0 ≤ l ≤ k − 1, we assume that for all λ > 0 and
f ∈ L∞
X,
b
∗
w x ∈ X : Tb,l
fx > λ fx
X
λ
log
l
fx
e
MLlog Ll1δ wxdμx.
λ
4.2
We need only consider the case that λ > fL1 X μX−1 . For each fixed bounded function f
with bounded support and λ > fL1 X μX−1 , applying Lemma 3.1 to |f| at level λ, with the
same notation {Bj }j , {Vj }j , Ω as in the proof of Theorem 1.2, we decompose f ≡ g h, where
g ≡ fχX\∪j Vj j mVj fχVj and h ≡ j hj with hj ≡ f − mVj fχVj . Applying the estimate
b,k g gives us that
1.17, and a similar argument to that used to deal with the term M
w
p
λ
∗
x ∈ X \ Ω : Tb,k
gx >
λ−p0 gx 0 MLlog Lk1p0 δ1 wχX\Ω xdμx
2
X
4.3
k1δ wxdμx,
λ−1 fxM
X
Llog L
where p0 ∈ 1, ∞ and δ1 > 0 such that k 1p0 δ1 < k 1 δ.
∗
We now turn to the term Tb,k
h. For any x ∈ X \ Ω and > 0, set
I1 x, j
I2 x, j
I3 x, j
: ∀ y ∈ C4 Bj , dx, y ≤ ,
: ∀ y ∈ C4 Bj , dx, y > ,
: C4 Bj ∩ {y ∈ X : dx, y > / ∅, C4 Bj ∩ y ∈ X : dx, y ≤ /∅ .
4.4
Guoen Hu et al.
19
It then follows that
T;b,k hx ≤ T;b,k
hj x T;b,k
hj x
j∈I2 x,
j∈I3 x,
k−1 k
k−l
hj x
bx − mBj b T hj x T;b,l
b· − mBj b
j∈I x,
l0 j
2
k−1 k−l
hj x
b· − mBj b
T;b,l
l0
j∈I3 x,
T;b,k
hj x ≡ U x V x W x X x.
j∈I x,
3
4.5
Notice that for x ∈ X \ Ω and j ∈ I2 x, , we have that T hj x T hj x. By the vanishing
moment of hj and the regularity condition 1.13, we have
τ
k
d y, yj
bx − mBj b
4.6
supU x ≤
τ hj ydμy
dx, y
>0
X μ B y, dx, y
j
and so
λ
w
x ∈ X \ Ω : supU x >
6
>0
τ
−1
hj y d y, yj
λ
j
λ−1
X
X
X\C7 Bj
bx − mB bk wx
j
τ dμxdμy
μ B y, dx, y dx, y
4.7
fxM
Llog Lk wxdμx.
Our inductive hypothesis 4.2, via the argument for the term H in the proof of Theorem 1.2,
leads to
fx
fx
λ
k
log e MLlog Lk1δ wxdμx.
w
x ∈ X \ Ω : supV x >
4
λ
λ
>0
X
4.8
Notice that for x ∈ X \ Ω and j ∈ I3 x, , we have that C4 Bj ⊂ {Bx, C8 \ Bx, C9 }, where
C8 and C9 with C8 > C9 are two positive constants. Therefore, for all x ∈ X \ Ω,
k
k−l b − mB b
hj x.
4.9
sup W x X x Mb,l
j
>0
l0
j
This, along with Theorem 1.2 and an argument for the term H in the proof of Theorem 1.2,
leads to
λ
w
x ∈ X \ Ω : supW x X x >
6
>0
k−l k
hj y
by − mBj b
Φl
MLlog Llδ wχχ\Ω ydμy
4.10
λ
Vj
l0 j
fy
fy
k
wydμy,
log e MLlog Lkδ
λ
λ
X
20
Abstract and Applied Analysis
where l l when l is even and l l 1 when l is odd. Combining the estimates for the terms
sup>0 U , sup>0 V , and sup>0 W X gives us that
w
x ∈ X \ Ω :
∗
Tb,k
hx
λ
>
2
fy
|fy|
k
log e MLlog Lk1δ wydμy,
λ
λ
X
4.11
which completes the proof of Theorem 1.3.
Acknowledgments
The authors would like to thank the referees for their many helpful suggestions and corrections, which improve the presentation of this paper. The first author is supported by National
Natural Science Foundation of China Grant no. 10671210, and the third author is supported
by National Science Foundation for Distinguished Young Scholars Grant no. 10425106 and
NCET Grant no. 04-0142 of Ministry of Education of China.
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