Abstract
Weighted for 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.
1. Introduction
We will be working on a space of homogeneous type. Let be a set endowed with a positive Borel regular measure and a symmetric quasimetric satisfying that there exists a constant such that for all , , , The triple is said to be a space of homogeneous type in the sense of Coifman and Weiss [1] if satisfies the following doubling condition: there exists a constant such that for all and ,It is easy to see that the above doubling property implies the following strong homogeneity: there exist positive constants and such that for all , and ,Moreover, there also exist constants and such that for all , and ,
We remark that although all balls defined by satisfy the axioms of complete system of neighborhoods in , and therefore induce a (separated) topology in , the balls for and need not be open with respect to this topology. However, by a remarkable result of Macías and Segovia in [2], we know that there exists another quasimetric such that(i) there exists a constant such that for all , (ii) there exist constants and such that for all , The balls corresponding to are open in the topology induced by . Thus, throughout this paper, we always assume that there exist constants and such that for all ,and that the balls for all and are open.
Now let be a positive integer and , define the th-order commutator of the Hardy-Littlewood maximal operator with byfor all . For the case that is the Euclidean space, García-Cuerva et al. [3] proved that is bounded on for any , and Alphonse [4] proved that enjoys a weak-type estimate, that is, there exists a positive constant , depending on , such that for all suitable functions ,Li et al. [5] established a weighted estimate with any general weight for in . As it was shown in [3–5] for the setting of Euclidean spaces, the operator 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 in spaces of homogeneous type. To state our results, we first give some notation.
Let be a measurable set with . For any fixed , and suitable function , setThe maximal operator is defined by where the supremum is taken over all balls containing . In the following, we denote by for simplicity, and denote by the set of bounded functions with bounded support.
With the notation above, we now formulate our main results as follows. Theorem 1.1. Let be a positive integer, and . Then for any , there exists a positive constant , depending only on , and , such that for all nonnegative weights and ,Theorem 1.2. Let be a positive integer, and . There exists a positive constant such that for all nonnegative weights , and ,where, and in the following, if is even and if is odd.
As a corollary of Theorem 1.2, we establish a weighted endpoint estimate for the maximal commutator of singular integral operators with symbols. Let be a Calderón-Zygmund operator, that is, is a linear -bounded operator and satisfies that for all with bounded support and almost all ,where is a locally integrable function on and satisfies that for all ,and that for all with ,with positive constants and . For any , suitable function and , define the truncated operator byLet and let be a positive integer. Define the commutator byfor all and . The maximal operator associated with the commutator is defined byfor all . In [6], it was proved that if is a Calderón-Zygmund operator, then for any , there exists a positive constant such that for all and all nonnegative weights ,In [5], it was proved that in , enjoys the following weighted weak-type endpoint estimate: for any , there exists a positive constant , depending on , , and , such that
Using Theorem 1.2, we will prove the following result.Theorem 1.3. Let be a Calderón-Zygmund operator. Then for any , nonnegative integer and , there exists a positive constant , depending on , , and , such that for all , and nonnegative weights ,
We mention that Theorems 1.1, 1.2, and 1.3 are also new even when for all .
We now make some conventions. Throughout the paper, we always denote by a positive constant which is independent of main parameters, but it may vary from line to line. We denote and simply by and , respectively. If , we then write . Constant, with subscript such as , does not change in different occurrences. A weight always means a nonnegative locally integrable function. For a measurable set and a weight , denotes the characteristic function of , . Given and a ball , denotes the ball with the same center as and whose radius is times that of . For a fixed with denotes the dual exponent of , namely, . For any measurable set and any integrable function on , we denote by the mean value of over , that is, For any locally integrable function and , the Fefferman-Stein sharp maximal function is defined bywhere the supremum is taken over all balls containing . For any fixed , the sharp maximal function of the function is defined by .
A generalization of Hölder's inequality will be used in the proofs of our theorems. For any measurable set with , positive integer , and suitable function , setThen the following generalization of Hölder's inequality:holds for any suitable functions and ; see [7] for details.
2. Proof of Theorem 1.1
To prove Theorem 1.1, we need some technical lemmas. In what follows, we denote by the Hardy-Littlewood maximal function. Moreover, for any and suitable function , we set .Lemma 2.1 (see [8]). There exists a positive constant such that for all weights and all nonnegative functions satisfying for all , then(i) if ,(ii) if ,Lemma 2.2. For any , there exists a positive constant such that for all with and all , .
For the case that 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 and let be a positive integer.(a) There exists a positive constant , depending only on and , such that for all and all weights ,(b) For any , there exists a positive constant , depending only on , and , such that for all and all weights ,
For Euclidean spaces, Lemma 2.3(a) is just Corollary 1.8 in [10] and Lemma 2.3(b) is included in the proof of Theorem 2 in [11] together with (4.11) in [12]. For spaces of homogeneous type, Lemma 2.3(a) 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 , (see [12]), we can prove Lemma 2.3(b) 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 with bounded support on . We omit the details here.
For any and , set .Lemma 2.4. Let be as in
(1.5). Then
there exists an approximation of the identity of order with bounded
support on . Namely, is a sequence
of bounded linear integral operators on , and there exist constants , such that for
all and all , , , and the integral
kernel of is a measurable
function from into satisfying
(i) if and (ii) for all ;(iii) for (iv) for all and ;(v)
For any and , letObviously, satisfies (i) through (v) of Lemma 2.4 with replaced by . From (iii) and (iv) of Lemma 2.4, it follows that there exist constants and such that for all and all satisfying ,For a positive integer and a function , let be the operator defined byfor all and , where for ,If , we denote and simply by and , respectively. From (i) of Lemma 2.4 together with (1.1), it follows that Notice that if , then . Thus,On the other hand, for each fixed , by (2.6) and , we haveBy the definition of , we further obtain . Thus, there exists some constant such that for all and ,For the sharp function estimate of , we have the following estimate. Lemma 2.5. Let be a positive integer and . For any and with , there exists a positive constant such that for all and all ,
Proof. By (i), (ii), and (iii) of Lemma 2.4, we obtain that for all ,and that for all and all with ,
To verify (2.12), by homogeneity, we may assume that . For all , and balls containing , it suffices to prove thatWe consider the following three
cases. Case 1 (). Where and in what follows . In this case, we have that for all ,The Kolmogorov inequality (see
[15, page 102]), along
with the fact that (and so ) is bounded
from to and the
inequality (1.22) gives us thatwhere the last inequality
follows from the John-Nirenberg inequality, which states that for any ball
,On the other hand, if , an application of Hölder's inequality implies
thatWe then get (2.15).Case 2 ( and ).
In this case, decompose into recalling that denotes the
characteristic function of the set . Let be a point in such thatWith the aid of the formulawhere is the constant
from Newton's formula, we haveThus for any ,As in Case 1, we have that As for , by (2.14) and (1.22), it is easy to getwhere the last inequality
follows from (2.18) and This leads to
our desired estimate (2.15).Case 3 ( and ). In this case, we take such that , and . We then have thatWith the ball replaced by in Case 2,
we also obtain the result that for any ,which completes the proof of
Lemma 2.5.
Lemma 2.6. Let . There exists a positive constant , depending only on and , such that for all weights ,
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 and suitable function ,Using this, Lemma 2.6 can be proved by applying the ideas used in [16]. For details, see [6, Lemma 7].
Proof of Theorem 1.1. We assume again that . At first, we claim that when , for all and , In fact, for
any , let be large enough
such that for some . Notice that for all , It then follows that for ,This, together with the estimate
thatleads to our claim.
By (2.11), to prove Theorem 1.1, it suffices to prove that
for all weights ,We proceed our proof by an
inductive argument on . When , (2.33) is implied by the fact that for all and the
following known inequality:See [18, pages 150-151], for a proof
of the last inequality when . The same ideas also work for . Now we assume that is a positive
integer and (2.33)
holds for any integer with . Then () can extend to a
bounded operator on for and so for any and ,
We now prove (2.33). To begin with,
we prove that for any given and , and for all weights and all ,We first consider the case that . Choose , such that . By Lemma 2.5, we obtain that for any and any weight ,Therefore, applying Lemmas 2.1 and 2.2, and the
estimate (2.35),
we havewhich leads toRepeating the argument above times, we then
have that for all weights ,On the other hand, notice that
for all , and that, by
(2.12) and the
fact that for all (see [12, (4.11)]), we then have
that for all ,
From these
inequalities, it then follows thatwhich together with (i) of Lemma 2.1 gives
(2.36).
We turn our attention to (2.36) for the case of . For all ,Moreover, the Kolmogorov
inequality, together with Hölder's inequality, the inequalities (1.22), and (2.18), tells us that
for any , and ,Combining the above estimates,
we obtainLet , be as in the
case of . Another application of Kolmogorov inequality and the
fact that is bounded from to leads toAs in the case of , by Lemmas 2.1, 2.2, and 2.5, we have that for any ,Combining the two cases yields
(2.36).
For any fixed and , choose and such that . This, via a duality argument, (2.36), and Lemma 2.3, leads
towhere 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 be a space of homogeneous type and let be a nonnegative integrable function. Then for every if , there exist a sequence of pairwise disjoint balls and a constant such thatand for every ball centered at .Lemma 3.2. Let and be two nonnegative integers. Then for all ,
Proof. We may assume that , otherwise the conclusion holds obviously. Set , , and . Let . Denote by the inverse of , that is, It is well known that and (see [19]). On the other hand, it is easy to verify that when and when Therefore, for all , This via [7, Lemma 6, page 63] tells us that Our desired conclusion then follows directly.
Proof of Theorem 1.2. With the notation as in (2.7), by (2.11), it suffices to
prove that for and all and , where when is even and when is odd.
Recall that for all ,(See [18, page 151] for a proof when . The same idea also works for .) By Hölder's inequality, it follows that for all ,and so when is
odd,Thus, it suffices to prove
(3.3) for the
case that is even. We
employ some ideas from [20], and proceed our proof of (3.3) by an inductive
argument. When , (3.3) is implied by the fact that for all and (3.4). Now let be a positive
integer. We may assume that is finite
almost everywhere, otherwise there is nothing to be proved. For any fixed , we assume that for any nonnegative integer with , there exists a constant such that for
all ,where and in what follows, when is even and when is odd. If and , the inequality (3.3) is trivial. So it
remains to consider the case that . For each fixed bounded function with bounded
support and , applying Lemma 3.1 to at level , we obtain a sequence of balls with pairwise
disjoint interiors. As in the proof of Lemma 2.10 in [17], set and , it then follows that and . Define the functions and , respectively, by and with . Recall that is regular and
the set of continuous function is dense in for any . Lemma 3.1 implies that for any fixed ,with a constant
independent of and , which together with the Lebesgue differentiation
theorem and Lemma 3.1 again yields thatLet with . The doubling property of and (3.1) now state
thatFollowing an argument similar to
the case of Euclidean spaces (see [18, page 159]), we have that for any , there exists a positive constant , depending only on , such that for all ,Thus,For each fixed , choose and such that . From the last estimate, (2.11), Theorem 1.1, and
(3.9), it
follows that
Thus, our proof is now reduced to
provingwhere
For any , letWe now prove (3.14). With the aid of
the formula that for all ,since is even, for , we writeRecall that are mutually
disjoint. If we set , our inductive hypothesis (3.7) via (3.11) now tells us
thatAn application of Lemma 3.2 then
gives thatFor each fixed , notice that by (3.8) and Lemma
3.2,
It then follows
that
It remains to prove thatFor each fixed , let and be the center
and radius of , respectively. If , then by the vanishing moment of and the
estimate (2.14), we obtain thatThis in turn implies
thatwhere in the second to the last
inequality, we use the fact that for each fixed , and positive
integer , a standard argument involving the inequalities
(1.22) and
(2.18)
yieldssee 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 be a Calderón-Zygmund operator. Then, there exists a positive constant such that for all , and weights ,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 . Now let be a positive
integer. For any fixed , and any nonnegative integer with , we assume that for all and ,
We need only consider the case
that . For each fixed bounded function with bounded
support and , applying Lemma 3.1 to at level , with the same notation , , as in the proof
of Theorem 1.2, we decompose , where and with . Applying the estimate
(1.17), and a
similar argument to that used to deal with the term gives us
thatwhere and such that
We now turn to the term . For any and , set It then follows
thatNotice that for and , we have that . By the vanishing moment of and the
regularity condition (1.13), we haveand soOur inductive hypothesis (4.2),
via the argument for the term in the proof of
Theorem 1.2, leads toNotice that for and , we have that , where and with are two
positive constants. Therefore, for all ,This, along with Theorem 1.2 and
an argument for the term in the proof of
Theorem 1.2, leads to where when is even and when is odd.
Combining the estimates for the terms , , and gives us
thatwhich 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.