Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

1Department of Mathematics, School of Mathematics and Statistics, Hengyang Normal University, Hengyang 421008, China 2School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China 3Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China 4School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Here, the inner product notation Therefore,  ⋅  ≡ ∑ ,  , ()    .Associated with such a matrix , we define a second-order divergence form operator: Theorem 1.Let 1 <  < 2, 0 <  ≤ 1,  ∈   (), and  be an integer greater than .Let  and  be two closed subsets of R  with a Euclidean distance (, ) between each other, and let {  } >0 be a family of sublinear operators acting on  2 (R  ).
Assume that, for  > 0 and 2 < We recall a square function, which is representative of larger classes of square functions associated with , given as follows [3]: In 2007, Aushcer [3] proved that ‖  ‖   ∼ ‖‖   for   <  < q .The interval (  , q ) is the maximal open interval required for the semigroup ( √ ∇ − ) >0 to be   bounded.We recall that (  , p ) is the maximal open interval required for the semigroup ( − ) >0 to be   bounded.In [3], the author has shown in general that Many researchers have contributed to the commutators associated with the second-order elliptic operator, and among the numerous studies, some related to development and applications have been cited herein [4][5][6][7][8][9].In particular, commutators with fractional differentiations associated with  play an important role in the theory of linear partial differential equations and harmonic analysis [10][11][12][13][14]. Naturally, the case of the commutators of square functions being related to fractional differentials associated with  is worth studying.
In this paper, we define a square function related to the fractional differential operator associated with  as follows: Moreover, for 0 <  < 1 and  ∈   (), the commutator of    can be defined by In this paper, we also establish the   boundedness for   ; .
Theorem 2. Let  be a second-order elliptic operator in divergence form defined by ( ), 0 <  < The remainder of this paper is organised as follows: in Section 2, we present some lemmas that play an important role in the proof of the main results; in Section 3, we prove Theorem 1; in Section 4, we prove Theorem 2. For  ≥ 1,   denotes the dual exponent of , i.e.,   = /( − 1).Throughout this paper, the letter "" will stand for a positive constant that is independent of the essential variables but will not necessarily have the same value for each occurrence.

Preliminary Lemmas
The second-order elliptic operator  in divergence form is defined by (5) and has the following off-diagonal estimates (see [3,4,9] and references therein).
Another very useful and well-known lemma for offdiagonal estimates is introduced here, which could be proved by using a similar argument for the proof of a previous lemma [9, lemma 2.3].
The following two lemmas are about the   −   offdiagonal estimates related to some commutators of the Lipschitz function and semigroups for second-order elliptic operators.

Proof of Theorem 1
For any fixed  > 0, without loss of generality, we may assume that  ∈   (R  ) is nonnegative.Let us write  for the Hardy-Littlewood maximal function.We use the Calderón-Zygmund decomposition for ()  at height   .Then, there exists a collection of pairwise disjoint cubes {  }  such that and they satisfy the following property: Then, we write  =  + ℎ =  + ∑  ℎ  , where After estimating (24),  > 1 and the standard arguments yield 0 ≤ () ≤  for almost every  ∈ R  .Then, and We estimate every term separately.For , we use (8) and the properties of  to obtain Now, we proceed with .Let us fix an integer  ≥ 1.We write   = ℓ(  ) 2 , where ℓ(  ) stands for the side length of the cube   .We use the notation  *  = 2  , where, in general, we write  for the -dilated , i.e., for the cube with the same centre as  and with the side length ℓ().

Proof of Theorem 2
First, we introduce a lemma that will be used to prove Theorem 2.