Precompact Sets , Boundedness , and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

Here Sn−1 fl {x ∈ R: |x| = 1} is the unit sphere in R and dσ is the area measure on it. For a function b ∈ L loc(Rn) (the set of all locally integrable functions onR), let Mb be the correspondingmultiplication operator defined by Mbf = bf for a measurable function f. Then the commutator between T and Mb is denoted by [b, T] fl MbT − TMb = p.v. ∫ R Ω (x − y) 󵄨󵄨󵄨󵄨x − y󵄨󵄨󵄨󵄨n (b (x) − b (y)) f (y) dy, (4) for suitable functions f. Denote the bounded mean oscillation function space by BMO (Rn) fl {b ∈ L loc (Rn) : ‖b‖∗ fl sup cube:Q⊂R Mb,Q < ∞} , (5)


Introduction
Let the Calderón-Zygmund singular integral operator  be defined by where Ω is a measurable function on R  and satisfies the following conditions: (i) Ω is a homogeneous function of degree zero on R  \ {0}; that is, (ii) Ω has mean zero on  −1 ; that is, Here  −1 fl { ∈ R  : || = 1} is the unit sphere in R  and d is the area measure on it.
For a function  ∈  loc (R  ) (the set of all locally integrable functions on R  ), let   be the corresponding multiplication operator defined by    =  for a measurable function .Then the commutator between  and   is denoted by It is well known that commutators play a very important role in harmonic analysis and PDEs.Indeed, Coifman et al. [1] characterized the   -boundedness of [,   ], where   ( = 1, . . ., ) are the Reisz transforms and  ∈ BMO(R  ).Using this characterization, the authors of [1] obtained a decomposition theorem of the real Hardy spaces  1 (R  ).Uchiyama [2] and Janson [3] showed that the Riesz transform   may be replaced by the Calderón-Zygmund singular integral operator  as in (1).Coifman et al. generalized the boundedness results of [, ] to Hardy spaces and gave important applications to some nonlinear PDEs in [4].The characterization of   -compactness of [, ] was obtained by Uchiyama [2].We remark that the interest in the compactness of [, ] in complex analysis is from the connection between the commutators and the Hankel-type operators; see [5].In recent years, Chen et al. have considered the compactness of commutators in [6][7][8].Specially, the results in [2] were generalized to Morrey spaces in [8].The Morrey space  , (R  ) was introduced by Morrey in 1938 and it is connected to certain problems in elliptic PDE [9].After that the Morrey spaces were found to have many important applications to the Navier-Stokes equations (see [10]), the Schrödinger equations (see [11]), and potential theory (see [12][13][14]).
During last three decades, the theory of variable function spaces has developed quickly; see .We claim that the list is not exhaust.The boundedness in variable function spaces of many classical operators from harmonic analysis has been obtained; see [18,19,[41][42][43][44]. Motivated by these works, we will consider analogous results in [8] to variable exponent situation.The structure of this paper is as follows.In Section 2, we give sufficient conditions for a set to be a precompact set in a variable Morrey space.In Section 3, we obtain the boundedness of singular integrals and their commutator in variable Morrey spaces.In Section 4, we discuss compactness of commutators in variable Morrey spaces.The remainder of this section is some notions.
Let  be a measurable subset in R  with || > 0, where as usual || is the Lebesgue measure of .Let (⋅) be a measurable function on  with range in [1, ∞).The variable exponent modulus is defined for measurable functions  on  by (⋅) () denotes the set of measurable functions  on  such that  (⋅) () < ∞ for some  > 0. The set becomes a Banach function space when equipped with the norm These spaces are the so-called variable Lebesgue spaces.Denote by P 1 () the set of measurable functions (⋅) on  with range in [1, ∞) such that For (⋅) ∈ P 1 (R  ) and 0 < () <  for  ∈ R  , the variable Morrey space  (⋅),(⋅) (R  ) is defined as the set of integrable functions  on R  with the finite norm (,)     (⋅) (R  ) , (10) where (, ) denotes a ball centered at  with radius  and V 1 is the volume of the unit ball in R  .

Precompact Sets in Variable Morrey Spaces
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 [45].
To prove Theorem 1, we need the following two lemmas, which are well known; for example, see [46] Now there is a position to prove Theorem 1.

Boundedness of Singular Integrals and Their Commutators
To consider the boundedness of singular integrals, a fundamental operator is the Hardy-Littlewood maximal operator.Given a function , the maximal function  is defined by where the supremum is taken over all cubes containing .
It is well known that  is bounded on   , 1 <  < ∞.However, for any (⋅) ∈ P 1 (R  ),  need not be bounded in For the set B(R  ), we refer the reader to [19,41] for details.If (⋅) ∈ B(R  ), we will use the following results.
Lemma 6 (see [23,24]).Suppose (⋅) ∈ B(R  ), and then there exists a positive constant  such that for each  ∈ (R  ) If (⋅) ∈ B(R  ),  + <  − , where  is as in Lemma 4 and the operator  is bounded on  (⋅) (R  ), then  is also bounded on where the constant  is independent of .
Next we turn to the boundedness of commutators in variable Morrey spaces.Many authors have studied it; see [44], but they considered that it restricts the underlying space with finite measure.Theorem 8. Suppose  is a linear operator satisfying where Ω is a bounded measurable function.Let 0 < () < , and  ∈ (R  ) and (⋅) ∈ B(R  ).If the commutator [, ] is bounded on  (⋅) (R  ), then [, ] is also bounded on  (⋅),(⋅) (R  ).
Proof.For any  ∈ R  and  > 0, let  = (, ) and write  as in the proof of Theorem 7. By the  (⋅) -boundedness of [, ], we obtain For  > 0 and  ∈ , we write Hence, for  > 0, using Lemmas 4 and 5 we have For  3 (), by Hölder's inequality and Lemma 6, we have Therefore, as the argument as for  2 , we have Finally, for  1 () by Hölder's inequality, we have Thus, as the argument as before, we obtain that From ( 45), (47), and (49), we get Hence, This finishes the proof of Theorem 8.
Proof.Corollary 9 is the result of the following lemmas.Indeed, the boundedness of  on  (⋅),(⋅) (R  ) is the direct result of Theorem for bounded functions  and compactly supported functions .Finally, using the similar argument for Theorem 7.5.6 in [47], we obtain that the last inequality holds for any  ∈ BMO(R  ).Thus, by Lemma 13, we obtain that [, ] is bounded on   (⋅)(R  ).Consequently, by Theorem 8, [, ] is bounded on  (⋅),(⋅) (R  ).
We remark here that the boundedness in variable Lebesgue spaces  (⋅) (R  ) of commutator [, ] has been proved in [44] by another method when Ω is an infinitely differentiable function on  −1 .
Lemma 10 (see Lemma 2.1 in [48]).Let 1 <  < ∞ and  ∈   (Muckenhoupt weight); then there exists a positive constant for functions  such that the left-hand side is finite.
Here we say  ∈   , 1 <  < ∞ if for every cube For properties of   , we refer the reader to [47].

Compactness of Commutators
Now we obtain sufficient conditions for the commutator [, ] to be a compact operator on  (⋅),(⋅) (R  ).