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Toeplitz-type operators in weighted Morrey spaces
Journal of Inequalities and Applications volume 2013, Article number: 253 (2013)
Abstract
Let and be singular integrals with non-smooth kernels, which are associated with an approximation of identity or ±I (I is the identity operator). Denote the Toeplitz-type operator by , where . In this paper, the estimates of the Toeplitz operator related to singular integral operators with non-smooth kernels and in weighted Morrey spaces is established.
MSC:47B35.
1 Introduction
The classical Morrey spaces were introduced by Morrey in [1] to investigate the local behavior of solutions to second-order elliptic partial differential equations. The boundedness of the Hardy-Littlewood maximal operator, the singular integral operator, the fractional integral operator and the commutator of these operators in Morrey spaces have been studied by many authors; see [2–5] and the references therein. In [6], Komori and Shirai studied the boundedness of these operators in weighted spaces.
It is well known that the commutator is defined by , where T is a Calderón-Zygmund operator and . The commutator generated by the Calderón-Zygmund operators and a locally integrable function b can be regarded as a special case of the Toeplitz operator , where and are the Calderón-Zygmund operators or ±I (I is the identity operator), . When , Krantz and Li discussed the boundedness of on the homogeneous space, see [7, 8]. In [9], the authors studied the boundedness of in Morrey spaces. In this paper, we study the boundedness of Toeplitz-type operators related to singular integral operators with non-smooth kernels in weighted Morrey spaces.
The singular integral operators with non-smooth kernels previously appeared in [10]. We say that T is a singular integral operator with non-smooth kernel if it satisfies the following conditions.
-
(i)
There exists a class of operators with kernels , which satisfy the condition (2.3) in Section 2, so that the kernels of the operators satisfy the condition
(1.1)
when for some .
-
(ii)
There exists a class of operators with kernels , which satisfy the condition (2.3), such that have associated kernels and there exist positive constants , such that
(1.2)
Note that the classes of operators and play the role of a generalized approximation to the identity. It is not difficult to check that conditions (1.1) and (1.2) are consequences of the standard Calderón-Zygmund operator. See Proposition 2 in [10].
The paper is organized as follows. In Section 2, we recall some important estimates on BMO functions, maximal functions and sharp maximal functions. In Section 3, we prove the main result.
2 Definitions and preliminary results
Let , and w be a weight. The weighted Morrey space is defined by
where
and the supremum is taken over all balls B in . If and with , then , the classical Morrey spaces.
The standard Hardy-Littlewood maximal function , , is defined by
where the sup is taken over all balls containing x. If , will be denoted by Mf. The Fefferman-Stein sharp maximal function of f, , is defined by
where . We will say if and . If , the BMO semi-norm of f is given by
A weight w is a non-negative locally integrable function. We say that , , if there exists a constant C such that for every ball ,
where . For , we say that if there is a constant C such that for every ball ,
or, equivalently, a.e. We denote . For the above definition, see [11].
A family of operators , , is said to be a ‘generalized approximation to the identity’ if, for every , can be represented by kernels in the following sense: For every function , , , and the following condition holds:
in which m is a positive constant and s is a positive, bounded, decreasing function satisfying
for some .
Note that (2.2) implies that
In [12], the sharp maximal function associated with a ‘generalized approximation to the identity’ is defined by
where , and for some .
The following results are proved in the context of spaces of homogeneous type in [13, 14] and [10].
Lemma 2.1
-
(i)
For every , there exists a constant C such that for every ,
-
(ii)
Assume that and . Then, for every ball , we have
-
(iii)
(John-Nirenberg lemma) Let and , then if and only if
Lemma 2.2 For , and , we have .
For the proof of this lemma, see [[2], Theorem 3.2].
Lemma 2.3 Let be a ‘generalized approximation to the identity’ and let . Then, for every function , , and , we have
where .
For the proof of this lemma, see Lemma 2.3 in [15].
Now, we have the following analogy of the classical Fefferman-Stein inequality [[11], Chapter IV] for the sharp maximal function . For the proof, see Proposition 4.1 in [12].
Lemma 2.4 Take , , and a ball such that there exists with . Then, for every , there exist (independent of λ, , f, ) and which only depend on w such that
where is a fixed constant which depends only on the ‘generalized approximation to the identity’ .
3 The main results
In this section, we consider the Toeplitz operator related to a singular integral with non-smooth kernel , where and are singular integrals with non-smooth kernels, which are associated with an approximation of identity or ±I. For , , we assume that if , then:
-
(a)
are bounded operators on .
-
(b)
There exist ‘generalized approximations of the identity’ such that have associated kernels and there exist positive constants , such that
-
(c)
There exists a ‘generalized approximation to the identity’ such that the kernels of the operators satisfy
(3.1)
when for some .
It is proved in [10] that if T is an operator satisfying (a) and (b) above, then T is of weak and of strong type for . In addition, if (c) is also satisfied, the operator T is bounded on for all . Moreover, if , then T is bounded on (see [12]).
In order to study the boundedness of in weighted Morrey spaces, we need the following result.
Lemma 3.1 Let , and . Then, for every with , there exists a constant , which only depends on w, such that
Proof Let B be a ball in . Set . Then from the Whitney decomposition theorem, we know that there exist mutually disjoint cubes such that and . Denote to be the ball with the same center as and . Let . Then there exists an , that is, . Let us use Lemma 2.4. There are ; and such that, if (to be chosen later), we can find in such a way that
Set and so since . Then
where we used the fact that weights are doubling measures and C is a constant that only depends on the weight. One can prove that
Let us choose η such that . The former inequality turns out to be
This implies that
The proof of this lemma is completed. □
The aim of this section is to prove the following theorem.
Theorem 3.2 Let be operators satisfying the above conditions (a), (b) and (c) or ±I. Let , and . Suppose that when . If , then there exists a constant C such that
for all .
Proof Without loss of generality, we may assume that
For , it is well known that there exists such that . Then we can choose two real numbers r and s larger than 1 such that and . We will prove that there exists a constant C such that
for all .
We now prove (3.4). For an arbitrary fixed , choose a ball which contains x. We have that , and so . Thus
and
where . Then
Let be the dual of r such that . By Lemma 2.1 and the boundedness of , we have
Similarly, by Lemma 2.1 and the boundedness of , we obtain
We now consider the term III. There are two cases:
-
(1)
Suppose that (), then using the assumption (c), we have
-
(2)
Suppose that there are i identity or −I operators in . Without loss of generality, we assume that are identity operators, then
It is obvious that .
By (2.4) and Lemma 2.1, we obtain
Moreover, from case (1) it follows that
So, .
Combining the above estimates of I, II and III, we obtain (3.4).
From (3.4), we know that if T is an operator satisfying (a), (b) and (c), then there exists such that and
For the proof of (3.5), one can also see [[12], Proposition 5.4]. Then combining (3.5), Lemmas 2.2 and 3.1, we have
Combining this, (3.4), Lemmas 2.2 and 3.1, we have
for all . The proof of this theorem is completed. □
Corollary 3.3 Let T be operators satisfying the above conditions (a), (b) and (c). Let , . If , then there exists a constant C such that
for all .
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Acknowledgements
The authors would like to thank the referee for carefully reading the manuscript and for making several useful suggestions. This research was supported by the National Natural Science Foundation of China (Grant No. 11271092), Natural Science Foundation of Guangdong Province (Grant No. s2011010005367), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20114410110001, 20124410120002) and SRF of Guangzhou Education Bureau (Grant No. 2012A088).
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Xie, P., Cao, G. Toeplitz-type operators in weighted Morrey spaces. J Inequal Appl 2013, 253 (2013). https://doi.org/10.1186/1029-242X-2013-253
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DOI: https://doi.org/10.1186/1029-242X-2013-253