-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
© Feng; licensee Springer. 2013
Received: 4 June 2013
Accepted: 15 November 2013
Published: 17 December 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.
KeywordsToeplitz-type operator singular integral operator sharp maximal function BMO Morrey space
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.
Remark We note that if and .
In this paper, we study some singular integral operators as follows (see ).
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 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])
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 )
This finishes the proof. □
The proof of the lemma is similar to that of Lemma 6 by Lemma 3, we omit the details.
4 Proofs of theorems
This completes the proof of Theorem 1. □
This completes the proof. □
This completes the proof. □
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