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-Type sharp estimates and boundedness on a Morrey space for Toeplitz-type operators associated to singular integral operators satisfying a variant of Hörmander’s condition
Journal of Inequalities and Applications volume 2013, Article number: 589 (2013)
In this paper, we prove the -type sharp maximal function estimates for the Toeplitz-type operators associated to certain singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the operators on the Lebesgue and Morrey spaces.
As the development of singular integral operators (see [1, 2]), their commutators have been well studied. In [3, 4], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on for . Chanillo (see ) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [6, 7], some Toeplitz-type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions is obtained. In , some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators is obtained (see , ). In this paper, we prove the sharp maximal function inequalities for the Toeplitz-type operator related to some singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the Toeplitz-type operator on Lebesgue and Morrey spaces.
First, let us introduce some notations. Throughout this paper, Q will denote a cube of with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by
where, and in what follows, . We say that f belongs to if belongs to and define . It has been known that (see )
Let M be the Hardy-Littlewood maximal operator defined by
For , let . For , we denote by the operator M iterated k times, i.e., and
Let Φ be a Young function and be the complementary associated to Φ. We denote the Φ-average by, for a function f,
and the maximal function associated to Φ by
The Young functions to be used in this paper are and , the corresponding average and maximal functions denoted by , and , . Following , we know that the generalized Hölder inequality and the following inequalities hold:
The weight is defined by (see )
Given a weight function w, for , the weighted Lebesgue space is the space of functions f such that
Definition 1 Let be a finite family of bounded functions in . For any locally integrable function f, the Φ sharp maximal function of f is defined by
where the infimum is taken over all m-tuples of complex numbers and is the center of Q. For , let
Remark We note that if and .
Definition 2 Given a positive and locally integrable function f in , we say that f satisfies the reverse Hölder condition (write this as ) if for any cube Q centered at the origin, we have
Definition 3 Let φ be a positive, increasing function on , and there exists a constant such that
Let w be a weight function and f be a locally integrable function on . Set, for ,
where . The generalized Morrey space is defined by
In this paper, we study some singular integral operators as follows (see ).
Definition 4 Let and satisfy
there exist functions and such that , and for a fixed and any ,
For , we define the singular integral operator related to the kernel K by
Moreover, let b be a locally integrable function on . The Toeplitz-type operator related to T is defined by
where are T or ±I (the identity operator), are the bounded linear operators on for and , , .
Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 4 (see , ). Also note that the commutator is a particular operator of the Toeplitz-type operators . The Toeplitz-type operators are the non-trivial generalizations of the commutator. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [12, 14]). The main purpose of this paper is to prove the sharp maximal inequalities for the Toeplitz-type operator . As the application, we obtain the weighted -norm inequality and Morrey space boundedness for the Toeplitz-type operators .
3 Theorems and lemmas
We shall prove the following theorems.
Theorem 1 Let T be the singular integral operator as Definition 4, and . If for any (), then there exists a constant such that, for any and ,
Theorem 2 Let T be the singular integral operator as Definition 4, , and . If for any (), then is bounded on .
Theorem 3 Let T be the singular integral operator as Definition 4, , , and . If for any (), then is bounded on .
To prove the theorems, we need the following lemmas.
Lemma 1 ([, p.485])
Let and for any function . We define that for ,
where the sup is taken for all measurable sets E with . Then
Lemma 2 (see )
Lemma 3 (see )
Let T be the singular integral operator as Definition 4. Then T is bounded on for , and weak bounded.
Lemma 4 (see )
Let , , and such that . Then, for any smooth function f for which the left-hand side is finite,
Let , and . Then, for any smooth function f for which the left-hand side is finite,
Lemma 6 Let , , , and such that . Then, for any smooth function f for which the left-hand side is finite,
Proof For any cube in , we know for any cube by . By Lemma 4, we have, for ,
This finishes the proof. □
Lemma 7 Let T be the singular integral operator as Definition 3 or the bounded linear operator on for any and , , and . Then
The proof of the lemma is similar to that of Lemma 6 by Lemma 3, we omit the details.
4 Proofs of theorems
Proof of Theorem 1 It suffices to prove that for and some constant , the following inequality holds:
where Q is any cube centered at , and . Without loss of generality, we may assume are T (). Let . Fix a cube and . Write
For I, by Lemmas 1, 2 and 3, we obtain
For II, we get, for ,
This completes the proof of Theorem 1. □
Proof of Theorem 2 By Theorem 1 and Lemmas 3-4, we have
This completes the proof. □
Proof of Theorem 3 By Theorem 1 and Lemmas 5-7, we have
This completes the proof. □
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The author declares that they have no competing interests.
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Feng, Q. -Type sharp estimates and boundedness on a Morrey space for Toeplitz-type operators associated to singular integral operators satisfying a variant of Hörmander’s condition. J Inequal Appl 2013, 589 (2013). https://doi.org/10.1186/1029-242X-2013-589
- Toeplitz-type operator
- singular integral operator
- sharp maximal function
- Morrey space