The Higher Integrability of Commutators of Calderón-Zygmund Singular Integral Operators on Differential Forms

Differential forms as the key tools are widely used in many fields including quasiconformal analysis, nonlinear elasticity, and differential geometry, due to their advantage of being coordinate system independent; see [1–5]. The integrability of various operators and the upper bound estimates for the norms of operators are very important and core topics while studying the Lp-theory of differential forms and investigating the qualitative and quantitative properties of the solutions of partial differential equations. In last few decades, a lot of related research has been done and many results on estimates for the Lp-norms of various operators applied to the differential form u in terms of the Lp-norms of u have been obtained; see [6–12]. In this paper, we define the commutators of Calderón-Zygmund singular integral operators [b, TΩ] on differential forms and give the strong type estimates for the commutators, which allows one to estimate ‖[b, TΩ]u‖p in terms of the norm ‖u‖p with p > 1. Meanwhile, we make a contribution to the estimates of the commutators ‖[b, TΩ]u‖s in terms of the norm ‖u‖p, where s > p, that is the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms. Then, we will establish the higher order Poincaré-type inequalities for the commutators applied to the solutions of Diracharmonic equations. The higher integrability and higher order inequalities in this paper can be used to study the regularity properties of the related operators. More results on the problem of higher order estimates and their applications in potential theory, quantum mechanics, and partial differential equations can be found in [13–17]. This paper is organised as follows. Section 2 contains, in addition to definitions and other preliminary material, the main lemmas. In Section 3, Theorems 12 and 13 show the local higher integrability of commutators of CalderónZygmund singular integral operators on differential forms. Based on these local results, the global higher integrability is presented inTheorems 14 and 15 by the well-known covering lemma. Especially, when the differential forms satisfy the Dirac-harmonic equations (in [18]), we establish the higher order Poincaré-type inequalities for the commutators in Section 4. The local higher order Poincaré-type inequalities are given in Theorems 16 and 17 and the global higher order Poincaré-type inequalities are obtained in Theorems 18 and 19. Finally, we demonstrate some applications of the main results by examples in Section 5. These results obtained in this paper will provide a further insight into the Lp-theory and regularity theory of the related operators and differential forms.


Introduction
Differential forms as the key tools are widely used in many fields including quasiconformal analysis, nonlinear elasticity, and differential geometry, due to their advantage of being coordinate system independent; see [1][2][3][4][5].The integrability of various operators and the upper bound estimates for the norms of operators are very important and core topics while studying the   -theory of differential forms and investigating the qualitative and quantitative properties of the solutions of partial differential equations.In last few decades, a lot of related research has been done and many results on estimates for the   -norms of various operators applied to the differential form  in terms of the   -norms of  have been obtained; see [6][7][8][9][10][11][12].In this paper, we define the commutators of Calderón-Zygmund singular integral operators [,  Ω ] on differential forms and give the strong type estimates for the commutators, which allows one to estimate ‖[,  Ω ]‖  in terms of the norm ‖‖  with  > 1.Meanwhile, we make a contribution to the estimates of the commutators ‖[,  Ω ]‖  in terms of the norm ‖‖  , where  > , that is the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms.Then, we will establish the higher order Poincaré-type inequalities for the commutators applied to the solutions of Diracharmonic equations.The higher integrability and higher order inequalities in this paper can be used to study the regularity properties of the related operators.More results on the problem of higher order estimates and their applications in potential theory, quantum mechanics, and partial differential equations can be found in [13][14][15][16][17].
This paper is organised as follows.Section 2 contains, in addition to definitions and other preliminary material, the main lemmas.In Section 3, Theorems 12 and 13 show the local higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms.Based on these local results, the global higher integrability is presented in Theorems 14 and 15 by the well-known covering lemma.Especially, when the differential forms satisfy the Dirac-harmonic equations (in [18]), we establish the higher order Poincaré-type inequalities for the commutators in Section 4. The local higher order Poincaré-type inequalities are given in Theorems 16 and 17 and the global higher order Poincaré-type inequalities are obtained in Theorems 18 and 19.Finally, we demonstrate some applications of the main results by examples in Section 5.These results obtained in this paper will provide a further insight into the   -theory and regularity theory of the related operators and differential forms.
The Calderón-Zygmund singular integral operator  Ω on differential forms is defined by where Ω() is defined on  −1 , has mean 0, and is sufficiently smooth.
If  ∈ (R  ), the commutator of Calderón-Zygmund singular integral operator on differential forms is of the form When taking () as a 0-form, the commutator in (4) reduces to the corresponding operator on function space as follows: For the degenerated operator and the related applications in partial differential equations, see [22][23][24].
In order to prove our conclusions, we need several lemmas.The following   -boundedness result for commutator [,  Ω ] on function spaces was proved in [25].
Lemma 1.Let  be a weight satisfying   condition: for all cubes , if there is a constant  such that where 1 <  < ∞ and 1/ + 1/  = 1. Ω is any Calderón-Zygmund singular integral operator.Then, given any function  ∈ (R  ), [,  Ω ] satisfies the following inequality: The following lemma was given by S. Ding and B. Liu in [18].Lemma 2. Let  ∈   (, Λ  ) be a solution of -harmonic equation (1) in ;  > 1 and 0 < ,  < ∞ are constants.Then, there exists a constant , independent of , such that for all cubes or balls  with  ⊂ .
In [26], T. Iwaniec and A. Lutoborski gave the following three lemmas which will be used repeatedly in this paper.
The following lemma appears in [27].
The covering lemma below belongs to [19].

Lemma 7. Each domain 𝑀 has a modified Whitney cover of cubes
and some  > 1, and if   ∩   ̸ = 0, then there exists a cube  (this cube need not be a member of V) in   ∩   such that   ∪   ⊂ .Moreover, if  is -John, then there is a distinguished cube  0 ∈ V which can be connected with every cube  ∈ V by a chain of cubes  0 ,  1 , . . .,   =  from V and such that  ⊂   ,  = 0, 1, 2, . . ., , for some  = (, ).

Higher Integrability
In this section, we show the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms.We first concentrate on the local higher integrability of the commutators.We need the following lemma appearing in [28]. where Proof.For any differential -form (), by the definition of commutator of Calderón-Zygmund singular integral operator on differential forms, we have Let ( − ) = Ω( − )/| − |  and then according to the definition of the exterior differential operator, we obtain Using the elementary inequality that is, Substituting ( 25) into (23) gives This completes the proof of Lemma 9.
Using Lemma 1 and the analogous method developed in Lemma 9, we have the following estimate for [,  Ω ].
We now present the local higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms.
for all balls  with  ⊂  for some  > 1.
Proof.For any ball  with  ⊂  for some  > Moreover, inequality (38) can be written as the following integral average inequality: Clearly, with  close to , the integral exponent  on the left hand side could be much larger than the integral exponent  on the right hand side since the condition 0 <  < /( − ).Hence the higher integrability of operator [,  Ω ] for the case that 1 <  <  is obtained.
Next, we consider the higher integrability of [,  Ω ] for the case  ≥ .
for all balls  with  ⊂  for some  > 1.
Then for any differential form , we have Replacing  by [,  Ω ] in (43), we get Taking into account the fact that /( − ) =  > , we see from the monotonic property of the   -space, (42), and ( 44) that We have completed the proof of Theorem 13.Now, we are ready to assert the global higher integrability of the commutator of Calderón-Zygmund singular integral operator on differential forms.Theorem 14.Let  Ω be the Calderón-Zygmund singular integral operator on differential forms and  be sufficiently smooth and bounded.If  ∈   (, Λ  ),  = 1, 2, . . ., , 1 <  < , then [,  Ω ] ∈   (, Λ  ) for any 0 <  < /( − ).Moreover, there exists a constant , independent of , such that for any bounded domain  ⊂ R  .
Using the similar method as we did in Theorem 14 and combining with Theorem 13, we can deduce the following global result for the case  ≥ .Theorem 15.Let  Ω be the Calderón-Zygmund singular integral operator on differential forms and  be sufficiently smooth and bounded.If  ∈   (, Λ  ),  = 1, 2, . . ., ,  ≥ , then [,  Ω ] ∈   (, Λ  ) for any  > 0.Moreover, there exists a constant C, independent of u, such that for any bounded domain  ⊂ R  .

Higher Order Poincaré-Type Inequalities
In this section, we shall state the higher order Poincaré-type inequalities for commutator of Calderón-Zygmund singular integral operator acting on the solutions of the Diracharmonic equations.
Next, we will prove the higher order Poincaré-type inequality still holds for the case that  ≥ .
Based on Theorems 16 and 17, we can obtain the following global higher order Poincaré-type inequalities for the commutator [,  Ω ] using the analogous method developed in Theorem 14.

Applications
In this section, we demonstrate some applications of our main results established in the previous sections.
Example 20.Let  > 0 and  > 0 be any constants and  1 = {( 1 ,  2 ,  3 ) : We should notice that the above example can be extended to the case of R  as follows.Remark.It is worth pointing out that the global results obtained in this paper can be extended to larger classes of domains, such as   -averaging domains and   ()-averaging domains; see [1,27].Also, the techniques developed in this paper provide an effective method to study the higher integrability of bilinear commutators of singular integrals on differential forms, which are defined in [25].We leave the statements and proofs to the interested readers.