CHEN Dazhao

School of Science,Shaoyang University,Shaoyang 422000,China.

Abstract. In this paper,the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the fractional and pseudo-differential operators is obtained.

Key Words: Toeplitz type operator;pseudo-differential operator;BMO space;Orlicz space.

1 Introduction

As the development of singular integral operators (see [1,2]), their commutators have been well studied. In [3-5], the authors proved that the commutators generated by the singular integral operators andBMOfunctions are bounded onLp(Rn) for 1

First,let us introduce some notations.Throughout this paper,Qwill denote a cube ofRnwith sides parallel to the axes.For any locally integrable functionf,the sharp function offis defined by

We say thatfbelongs toBMO(Rn)iff#belongs toL∞(Rn)and||f||BMO=||f#||L∞. More generally, letρbe a non-decreasing positive function on [0,+∞) and defineBMOρ(Rn)as the space of all functionsfsuch that

2 The main Theorem

whereTk,1areTor±I(the identity operator),Tk,2andTk,4are the bounded linear operators onLp(Rn)for 1

Note that the commutator [b,T](f)=bT(f)−T(b f) is a particular operator of the Toeplitz type operatorTb. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [3,17,18]). The main purpose of this paper is to prove the boundedness properties for the Toeplitz type operatorTbfrom Lebesgue spaces to Orlicz spaces.

We shall prove the following theorem in Section 4.

Remark 2.1. Ifϕ(t)≡1 andψ(t)=tpfor 1

(b).Ifψ(t)=tqandϕ(t)=tn(1/p−1/q)for 1

3 A key lemma

We begin with the following preliminary lemmas.

Proof.For suppf⊂(2Q)candx,˜x∈Q,note that|x−y|∼|x0−y|forx∈Qandy∈Rn2Q.

(I).We consider the following two cases:

Case 1.Whend≤1,let ˜Qbe the cube concentric withQof side lengthd1−θ. We have

Let 12, we have,by Lemma 3.2,

We then complete the proof.

4 Proof of Theorem 2.1

Now we are in position to prove our main theorem.

Proof.Without loss of generality, we may assumeTk,1areT(k=1,...,l). We prove the theorem in several steps.First,we prove,ifb∈BMO(Rn),

where

We now put these estimates together,and taking the supremum over allQsuch that ˜x∈Q,we obtain

Thus,takingr,usuch that 2

Secondly,we prove that,ifb∈Lipβ(Rn),

for any 2

Now we verify thatTbsatisfies the conditions of Lemma 3.8. In fact,for any 1

Acknowledgement

The author are very grateful to the anonymous referees for their constructive suggestions.This research was supported by the National Natural Science Foundation of China(Grant No.11901126),the Scientific Research Funds of Hunan Provincial Education Department.(Grant No.19B509).