Characterizations of the BMO and Lipschitz spaces via commutators on weak Lebesgue and Morrey spaces
Abstract.
We prove that the weak Morrey space is contained in the Morrey space for . As applications, we show that if the commutator is bounded from to for some , then , where is a CalderónZygmund operator. Also, for , if and only if is bounded from to . For belonging to Lipschitz class, we obtain similar results.
* Corresponding author, Email: .
1. Introduction
Let be a CalderónZygmund operator defined by
where the kernel satisfies the following conditions:

is homogeneous of degree zero on , i.e., for all and ;

and
A locally integrable function belongs to the space if satisfies
where and the supremum is taken over all cubes in . A well known result of Coifman, Rochberg and Weiss [2] states that the commutator
is bounded on some , , if and only if . An interesting question is raised. Is in if is of weak type for some ? We will give an affirmative answer in this paper.
For , we say that a function belongs to Morrey space if
a function belongs to weak Morrey space if
Morrey spaces describe local regularity more precisely than spaces and can be seen as a complement of . In fact, and for . In 1997, Ding [4] showed that is in if and only if the commutator of CalderónZygmund operator is bounded on Morrey spaces. We will demonstrate here that if and only if is weak bounded on Morrey spaces.
Another subject of this paper is to consider the characterizations of Lipschitz functions via commutators. For , the Lipschitz space is the set of functions such that
In 1978, Janson [5] proved that, for with , if and only if is bounded from to . In 1995, using SobolevBesov embedding, Paluszyński [8] obtained that, for and , if and only if is bounded from to the homogeneous TriebelLizorkin spaces . Paluszyński’s idea was novel for the study about the boundedness of commutators from to and shed new light on the characterization of Lipschitz space via commutators. Recently, Shi and Lu [10] showed that, for , ,
Throughout this paper, the letter denotes constants which are independent of main variables and may change from one occurrence to another.
2. Characterization of space via commutators
In this section, we characterize space via the boundedness of commutator on (weak) Lebesgue spaces or (weak) Morrey spaces. First of all, we compare with Morrey spaces and weak Morrey spaces.
It is clear that is contained in and if . However, for , one has the reverse inequality as follows.
Theorem 2.1.
If , then and .
Proof.
Let . Given a cube and ,
that is,
Choose
Thus,
which gives
Then
and the lemma follows. ∎
Remark 2.1.
Now we return to our first subject.
Theorem 2.2.
Let . The following statements are equivalent:

;

is a bounded operator from to ;

is a bounded operator from to .
Theorem 2.3.
Let . The following statements are equivalent:

;

is a bounded operator from to ;

is a bounded operator from to .
Proof of Theorem 2.2. The equivalence of and was proved in [3]. By the inequality , it is obvious that implies .
To show , we use Paluszyński idea given in [8]. For , let and denote the open cube centered at with side length . Then has an absolutely convergent Fourier series
with , where the exact form of the vectors is unrelated. Then, we have the expansion
Given cubes and , if and , then
Let . Then
Setting and , we have
Choose . By Remark 2.1,
(2.1) 
which yields and . Hence, the proof of Theorem 2.2 is completed. ∎
3. Characterization of via commutators
We give a lemma that can be used to prove a characterization of Lipschitz functions.
Lemma 3.1.
For and ,
where the supremum is taken over all cubes and means equivalence.
Proof.
The first result of this section is
Theorem 3.2.
Let , and . The following statements are equivalent:

;

is a bounded operator from to .
Proof.
: Let . Then
(3.1) 
which implies
Corollary 3.3.
Let , and . The following statements are equivalent:

;

is a bounded operator from to ;

is a bounded operator from to .
We give two remarkable results about the boundedness of fractional integral operator on Morrey spaces, where is defined by
Theorem 3.4.
(Peetre [9]) Let , and . If
Theorem 3.5.
(Adams [1]) Let , and . If
We will use the above two theorems to show other characterizations of Lipschitz functions.
Theorem 3.6.
Let , , ,

;

is a bounded operator from to .
Theorem 3.7.
Let , , , ,

;

is a bounded operator from to ;

is a bounded operator from to ;

is a bounded operator from to .
Proof of Theorem 3.6. : For , Theorem 3.4 and (3.1) yield
For , it follows from [11, Theorem 1.4] that
which gives
: A same argument as the proof of Theorem 2.2 except choosing and replacing (2.1) by
implies due to Lemma 3.1. ∎
: Since
and hence . Then .
It is obvious for .
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