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Boundedness of sublinear operators and their commutators on generalized central Morrey spaces
Journal of Inequalities and Applications volume 2013, Article number: 411 (2013)
Abstract
In this paper, we introduce the generalized central Morrey spaces and get the boundedness of a large class of rough operators on them. We also consider the CBMO estimates of their commutators on generalized central Morrey spaces. As applications, we obtain the boundedness characterizations of rough Hardy-Littlewood maximal function, rough Calderón-Zygmund singular integral, rough fractional integral, etc. on generalized central Morrey spaces.
MSC:42B20, 42B25.
1 Introduction
Let be homogeneous of degree zero on , where denotes the unit sphere of and . We define for any . Suppose that represents a sublinear operator, which satisfies that for any with compact support and ,
where is an absolute constant. Similarly, for any , we assume that represents a sublinear operator, which satisfies that
for any with compact support and .
Let T be a linear operator. For a locally integrable function b on , we define the commutator by
for any suitable function f.
To study the local behavior of solutions to second-order elliptic partial differential equations, Morrey [1] introduced the classical Morrey spaces . The readers can find more details in [2].
Let and . denotes a ball centered at of radius t. Morrey spaces are defined by
where
When , and ; when , .
Many authors have studied the mapping properties of many operators on Morrey spaces; see [3–5] and [6]. Alvarez et al. [7], in order to study the relationship between central BMO spaces and Morrey spaces, introduced λ-central bounded mean oscillation spaces and central Morrey spaces.
Let and . A function belongs to the λ-central bounded mean oscillation spaces if
where . If two functions which differ by a constant are regarded as functions in the spaces , then spaces become Banach spaces. spaces become the spaces of constants when and they coincide with modulo constants when .
Let and . The central Morrey spaces are defined by
It follows that spaces are Banach spaces continuously included in spaces. spaces reduce to when , and it is true that , .
Recently, Guliyev [8] introduced the generalized Morrey spaces , where is a positive measurable function on and . For all functions , the generalized Morrey spaces are defined by
Obviously, if , .
When , Guliyev obtained the sufficient condition on and
for the boundedness of satisfying (1.1) from to in [9] and gave the condition on the pair of
for the boundedness of satisfying (1.2) from to in [10], where .
Inspired by the above, we consider the boundedness of sublinear operators on the following generalized central Morrey spaces and give the λ-central bounded mean oscillation estimates for linear operator commutators.
Definition 1.1 Let be a positive measurable function on and . We denote by the generalized central Morrey spaces, the spaces of all with finite quasinorm
We can recover the spaces under the choice .
Recall that in 1994 the doctoral thesis [11] by Guliyev (see also [12–15]) introduced the local Morrey-type space given by
where ω is a positive measurable function defined on . The main purpose of [11] (also of [12–15]) is to give some sufficient conditions for the boundedness of fractional integral operators and singular integral operators defined on homogeneous Lie groups in the local Morrey-type space . In a series of papers by Burenkov, H Guliyev and V Guliyev, etc. (see [16–21]), some necessary and sufficient conditions for the boundedness of fractional maximal operators, fractional integral operators and singular integral operators in local Morrey-type spaces were given.
Particularly, if , , then the generalized central Morrey spaces are the same spaces as the local Morrey spaces with .
The following statements were proved in [11] (see also [14]).
Theorem A Let and satisfy the condition
where C does not depend on r. Then the Calderón-Zygmund operator T is bounded from to .
Theorem B Let , , and satisfy the condition
where C does not depend on r. Then the Riesz potential is bounded from to .
From Lemmas 4.4 and 5.3 in [9] we get the following for the generalized central (local) Morrey spaces .
Theorem C Let , T be a sublinear operator satisfying that for any with compact support and ,
and bounded on . Let also the pair satisfy the condition
where C does not depend on r. Then the operator T is bounded from to .
Theorem D Let , , , be a sublinear operator satisfying that for any with compact support and ,
and bounded from to . Let also the pair satisfy the condition
where C does not depend on r. Then the operator is bounded from to .
2 Sublinear operator with rough kernel
Theorem E Let ω be a positive weight function on . The inequality
holds for all non-negative and non-increasing g on if and only if
and , where the is the weighted Hardy operator
Note that Theorem E can be proved analogously to Theorem 1 in [22]; particularly, when , it was proved in [23].
In this section we are going to discuss the boundedness of and on generalized central Morrey spaces.
Lemma 2.1 Let , be a sublinear operator and satisfy (1.1) with .
When and is bounded on for , then the inequality
holds for any ball and for all ; or and is bounded on for , then the inequality
holds for any ball and for all .
Proof Let . For any , set and . We write
and have
Since is bounded on , it follows that
where the constant is independent of f.
It is known that , , which implies . Thus
-
(i)
When and by Fubini’s theorem, we have
Hence, for all , the inequality
holds.
-
(ii)
When , by Fubini’s theorem and the Minkowski inequality, we get
On the other hand, for any , we have
Combining the above estimates, we complete the proof of Lemma 2.1. □
Theorem 2.2 Let and . Let be a sublinear operator satisfying (1.1) and bounded on for . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then the operator is bounded from to .
Proof When , by Lemma 2.1 and Theorem E, for , , we have
For the case of , we can use the same method to prove the desirable conclusion. □
The Calderón-Zygmund operator with rough kernel has the following integral expression:
for any test function f and . The kernel is a locally integral function defined away from the diagonal satisfying the size condition
for all and .
, the rough Hardy-Littlewood maximal function is defined by
Then we can get the following corollary.
Corollary 2.3 Let and . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then and are both bounded from to .
In the following statements, the boundedness of satisfying (1.2) in generalized central Morrey spaces is proved.
Lemma 2.4 Let , and , be a sublinear operator and satisfy (1.2) with .
When and is bounded from to , then the inequality
holds for any ball and for all ; or and is bounded from to , then the inequality
holds for any ball and for all .
Proof Let , and . For any , set and . We write
and have
Since is bounded from to , it follows that
where the constant is independent of f.
Since , , thus
-
(i)
When and by Fubini’s theorem, we have
Hence, for all , the inequality
holds.
-
(ii)
When , by Fubini’s theorem and the Minkowski inequality, we get
Similarly, combining the above estimates, we finish this proof. □
Theorem 2.5 Let , , and . Let be a sublinear operator satisfying (1.2) and bounded from to . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then the operator is bounded from to .
Proof When , by Lemma 2.1 and Theorem E, for and , we have
For the case of , we can also use the same method to prove the desirable conclusion. □
, the rough fractional maximal function and the rough fractional integral are defined by
for .
Corollary 2.6 Let , , and . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then and are both bounded from to .
Remark 1 When , the comments in Theorem 2.2 and in Theorem 2.5 can be obtained from Lemmas 4.4 and 5.3 in [9].
3 The commutators of a linear operator with rough kernel
Let be a Calderón-Zygmund singular integral operator and . The commutator operator is defined by
A well-known result of Coifman et al. [24] states that the commutator is bounded on for if and only if .
Since when , if we only assume , then may not be a bounded operator on . However, it has some boundedness properties on other spaces. As a matter of fact, in [25] and [26], they considered the commutators with . Here we also obtain some boundedness of the commutators with on generalized central Morrey spaces.
We need the following statement on the boundedness of the Hardy-type operator
Theorem F [27]
The inequality
holds for all non-negative and non-increasing g on if and only if
and .
Lemma 3.1 Let , , and , is a sublinear operator and satisfies (1.1) with .
When and is bounded on for , then the inequality
holds for any ball and for all ; or and is bounded on for , then the inequality
holds for any ball and for all .
Proof Let , and . For any , set and . We can write
and
Hence, we have
Since is bounded on , it follows that
where the constant is independent of f.
For , we have
For , it is known that , , which implies .
-
(i)
When and by Fubini’s theorem, we have
thus
-
(ii)
When , by Fubini’s theorem and the Minkowski inequality, we get
On the other hand, for , by Fubini’s theorem, we have
-
(i)
When , we obtain
then
Moreover,
then
By estimating and , we obtain
-
(ii)
When , by the Minkowski inequality, we get
and
Hence, we have
Moreover, for any , we have
Now combining all the above estimates, we end the proof. □
Then we have the following conclusions.
Theorem 3.2 Let , , , and . Let be a linear operator satisfying (1.1) and bounded on for . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then the operator is bounded from to .
Corollary 3.3 Let , , , and . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then the operator is bounded from to .
About the commutator of linear operator satisfying (1.2), we get the following corresponding results.
Lemma 3.4 Let , , , , and let
is a sublinear operator and satisfies (1.2) with .
When , is bounded from to for any and , then the inequality
holds for any ball and all ; or , is bounded from to for any and , then the inequality
holds for any ball and all .
Proof Let , , , . For any , set and . We also write
and
Hence, we have
Since is bounded from to and , it follows that
where the constant is independent of f.
For , and ,
For ,
-
(i)
when , by Fubini’s theorem, since , we have
thus
-
(ii)
when , by Fubini’s theorem and the Minkowski inequality, we get
On the other hand, for , by Fubini’s theorem, we have
-
(i)
When , we obtain
then
For , we have
then
Then, by estimating and , we obtain
-
(ii)
When , by the Minkowski inequality, we get
and
Hence, we have
Then we end this proof. □
Theorem 3.5 Let , , , , and
Let be a linear operator satisfying (1.2) with , which is bounded from to for any , . If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then the operator is bounded from to .
Corollary 3.6 Let , , , , , and
If either of the two conditions
-
(i)
when , satisfies the condition
-
(ii)
when , satisfies the condition
is satisfied, then the operator is bounded from to .
Remark 2 In our main results, if we let and , then by calculating we can recover some known results in [7] and [25].
Author’s contributions
The author contributed to all the main results in this paper.
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Acknowledgements
This work was supported by NSF of Zhejiang Province (No. LQ13A010005). The author thanks the referees for their valuable suggestions.
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Fan, Y. Boundedness of sublinear operators and their commutators on generalized central Morrey spaces. J Inequal Appl 2013, 411 (2013). https://doi.org/10.1186/1029-242X-2013-411
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DOI: https://doi.org/10.1186/1029-242X-2013-411