Toeplitz-type operators in weighted Morrey spaces
© Xie and Cao; licensee Springer. 2013
Received: 4 July 2012
Accepted: 30 April 2013
Published: 17 May 2013
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.
The classical Morrey spaces were introduced by Morrey in  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 , 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 , 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.
- (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)
- (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 .
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
and the supremum is taken over all balls B in . If and with , then , the classical Morrey spaces.
or, equivalently, a.e. We denote . For the above definition, see .
for some .
where , and for some .
- (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 [, Theorem 3.2].
For the proof of this lemma, see Lemma 2.3 in .
where is a fixed constant which depends only on the ‘generalized approximation to the identity’ .
3 The main results
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  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 ).
In order to study the boundedness of in weighted Morrey spaces, we need the following result.
The proof of this lemma is completed. □
The aim of this section is to prove the following theorem.
for all .
for all .
- (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 .
Combining the above estimates of I, II and III, we obtain (3.4).
for all . The proof of this theorem is completed. □
for all .
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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