Boundedness of Toeplitz type operator associated to singular integral operator satisfying a variant of Hörmander’s condition on spaces with variable exponent
© Feng; licensee Springer. 2014
Received: 20 April 2014
Accepted: 27 August 2014
Published: 25 September 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
Remark We note that if and .
In this paper, we study some singular integral operator as follows (see ).
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.
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])
Lemma 2 ()
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 ()
Lemma 8 ()
This completes the proof of Theorem 1. □
This completes the proof of Theorem 2. □
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