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Boundedness of Toeplitz type operator associated to singular integral operator satisfying a variant of Hörmander’s condition on spaces with variable exponent
Journal of Inequalities and Applicationsvolume 2014, Article number: 369 (2014)
In this paper, the boundedness for some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition on spaces with variable exponent is obtained by using a sharp estimate of the operator.
As the development of singular integral operators (see [1, 2]), their commutators have been well studied (see [3, 4]). In [5, 6], some singular integral operators satisfying a variant of Hörmander’s condition and the boundedness for the operators and their commutators are obtained (see ). In [7–9], some Toeplitz type operators related to singular integral operators and strongly singular integral operators are introduced and the boundedness for the operators generated by and Lipschitz functions are obtained. In the last years, the theory of spaces with variable exponent has been developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity (see [10–13] and their references). Karlovich and Lerner study the boundedness of the commutators of singular integral operators on spaces with variable exponent (see ). Motivated by these papers, the main purpose of this paper is to introduce some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition and prove the boundedness for the operator on spaces with variable exponent by using a sharp estimate of the operator.
2 Preliminaries and results
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 and , the sharp function of f is defined by
We write that if . We say that f belongs to if belongs to and define . Let M be the Hardy-Littlewood maximal operator defined by
For , we denote by the operator M iterated k times, i.e., and
Let Ψ be a Young function and be the complementary associated to Ψ. We denote the Ψ-average by, for a function f,
and the maximal function associated to Ψ by
The Young functions to be used in this paper are and , the corresponding average and maximal functions are denoted by , and , . Following , we know the generalized Hölder inequality,
and the following inequality: for , with and any , ,
The non-increasing rearrangement of a measurable function f on is defined by
For and a measurable function f on , the local sharp maximal function of f is defined by
Let be a measurable function. Denote by the sets of all Lebesgue measurable functions f on such that for some , where
The sets become Banach spaces with respect to the following norm:
Denote by the sets of all measurable functions such that the Hardy-Littlewood maximal operator M is bounded on and the following holds:
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
In this paper, we study some singular integral operator as follows (see ).
Definition 3 Let and satisfy
There exist functions and such that , and for a fixed and any ,
For , we define a singular integral operator related to the kernel K by
Let b be a locally integrable function on and T be a singular integral operator with variable Calderón-Zygmund kernels. The Toeplitz type operator associated to T is defined by
where are the singular integral operators T with variable Calderón-Zygmund kernels or ±I (the identity operator), are the linear operators for and .
We shall prove the following theorems.
Theorem 1 Let T be a singular integral operator as in Definition 3, and . If for any (), then there exists a constant such that for any and ,
Theorem 2 Let T be a singular integral operator as in Definition 3, and . If for any () and are the bounded linear operators on for , then is bounded on , that is,
Corollary Let be a commutator generated by the singular integral operators T and b. Then Theorems 1 and 2 hold for .
3 Proofs of theorems
To prove the theorems, we need the following lemmas.
Lemma 1 ([, p.485])
Let . We define that for any function and ,
where the sup is taken for all measurable sets E with . Then
Lemma 2 ()
Let for , we denote that . Then
Lemma 3 (see )
Let T be a singular integral operator as in Definition 3. Then T is weak bounded of .
Lemma 4 ()
Let be a measurable function satisfying (1). Then is dense in .
Lemma 5 ()
Let and g be a measurable function satisfying
Let , , and such that . Then
Let be a measurable function satisfying (1). If and with , then fg is integrable on and
Lemma 8 ()
Let be a measurable function satisfying (1). Set
Proof of Theorem 1 It suffices to prove for and some constant that the following inequality holds:
where Q is any cube centered at , and . Without loss of generality, we may assume that 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 Lemmas 4-7, we get, for and ,
thus, by Lemma 8,
This completes the proof of Theorem 2. □
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The author declares that they have no competing interests.