^{1}

^{2}

^{1}

^{2}

We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

Let the Calderón-Zygmund singular integral operator

For a function

It is well known that commutators play a very important role in harmonic analysis and PDEs. Indeed, Coifman et al. [

During last three decades, the theory of variable function spaces has developed quickly; see [

Let

For

In this section, we give a compactness criterion in variable Morrey spaces. We remark here that a compactness criterion for variable exponent Lebesgue spaces was given in [

Let

Norm boundedness uniformly is

Translation continuity uniformly is

Uniformly convergence at infinity is

Then

To prove Theorem

A set

Let

There exists a constant

For every

Now there is a position to prove Theorem

Let

Finally, we verify that

By Lemma

To finish the proof, we only need to show that, for

Now, we choose

To show (

If

If

Here,

Therefore,

To consider the boundedness of singular integrals, a fundamental operator is the Hardy-Littlewood maximal operator. Given a function

Let

Let

Suppose

Let

Let

Firstly, we estimate

Next we turn to the boundedness of commutators in variable Morrey spaces. Many authors have studied it; see [

Suppose

For any

For

By the well-known fact that, for any

For

Therefore, as the argument as for

Finally, for

Let

Corollary

We remark here that the boundedness in variable Lebesgue spaces

Let

Here we say

Let

Suppose that

Given that

Now we obtain sufficient conditions for the commutator

Let

Let

Lemma

Suppose that

We will use the method in [

Notice that

Next we show that (

In fact, for any

Finally, we show that the translation continuity condition (

Since

As for

Regarding

Finally, by

From (

We remark that in Theorem

The authors declare that they have no conflicts of interest.

The first author was supported by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant no. 11426221) and the High Level Introduction of Talent Research Start-Up Fund by Central South University of Forestry and Technology (Grant no. 1040212). The second author is supported by the National Natural Science Foundation of China (Grant no. 11361020).