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

Authors: Jinling Niu; Yuming Xing
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this article, the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms is derived. Also, the higher order Poincaré-type inequalities for the commutators acting on the solutions of Dirac-harmonic equations are obtained. Finally, some applications of the main results are demonstrated by examples.


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## Body

## 1. 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 [1–5]. 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,TΩu‖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 Dirac-harmonic 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. Section2 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 Lp-theory and regularity theory of the related operators and differential forms.


## 2. Preliminary

Before specifying the main results precisely, we introduce some notation. LetM⊂Rn be a bounded domain, n≥2, B and σB be the balls with the same center, and diam(σB)=σdiam(B) for any real number σ>1. We denote by μ(M) the n-dimensional Lebesgue measure of a set M⊆Rn. The set of all l-forms, denoted by Λl=Λl(Rn), is a l-vector, spanned by exterior products eI=ei1∧ei2⋯∧eil, for all ordered l-tuples I=(i1,i2,…,il), 1≤i1<⋯<il≤n. The l-form u(x)=∑IuI(x)dxI is called a differential l-form if its coefficients uI are differentiable functions. We shall denote by D′(M,Λl) the differential l-forms space on M and denote by Lp(M,Λl) the space of differential l-forms on M satisfying ∫M|uI|p<∞ and with norm ‖u‖p,M=(∫M∑IuIx2p/2dx)1/p. In particular, we know that a 0-form is a function. A differential l-form u∈D′(M,Λl) is called a closed form if du=0 in M. Similarly, a differential (l+1)-form v∈D′(M,Λl+1) is called a coclosed form if d⋆v=0. From the Poincaré lemma, ddu=0, we know that du is a closed form. The operator ⋆:Λl(Rn)→Λn-l(Rn) is the Hodge-star operator which is an isometric isomorphism and the linear operator d:D′(M,Λl)→D′(M,Λl+1), 0≤l≤n-1, is called the exterior differential. The Hodge codifferential operator d⋆:D′(M,Λl+1)→D′(M,Λl), the formal adjoint of d, is defined by d⋆=(-1)nl+1⋆d⋆; see [19] for more introduction. The following Dirac-harmonic equation for differential forms was initially introduced by S. Ding and B. Liu in [18]:(1)d⋆Ax,Du=0,for almost every x∈M, where D=d+d⋆ is the Dirac operator, M⊂Rn is a domain, and A:M×Λl(Rn)→Λl(Rn) satisfies the following conditions:(2)Ax,ξ≤aξp-1,Ax,ξ,ξ≥ξpfor almost every x∈M and all ξ∈Λl(Rn). Here, a>0 is a constant and 1<p<∞ is a fixed exponent associated with (1).We should point out that the Dirac-harmonic equation is a kind of general equation which includes many existing harmonic equations as special cases, such as theA-harmonic equation; see [18, 20, 21] for more information.The Calderón-Zygmund singular integral operatorTΩ on differential forms is defined by(3)TΩux=∑I∫RnΩx-yx-ynuIydydxI,where Ω(x) is defined on Sn-1, has mean 0, and is sufficiently smooth.Ifb∈BMO(Rn), the commutator of Calderón-Zygmund singular integral operator on differential forms is of the form(4)b,TΩux=bxTΩux-TΩbux=∑I∫Rnbx-byΩx-yx-ynuIydydxI.When takingu(x) as a 0-form, the commutator in (4) reduces to the corresponding operator on function space as follows:(5)b,TΩfx=∫Rnbx-byΩx-yx-ynfydy.For the degenerated operator and the related applications in partial differential equations, see [22–24].In order to prove our conclusions, we need several lemmas. The followingLp-boundedness result for commutator [b,TΩ] on function spaces was proved in [25].Lemma 1.
Letw be a weight satisfying Ap condition: for all cubes Q, if there is a constant C such that(6)1Q∫Qwxdx1Q∫Qwx1-p′dxp-1≤C<∞,where 1<p<∞ and 1/p+1/p′=1. TΩ is any Calderón-Zygmund singular integral operator. Then, given any function b∈BMO(Rn), [b,TΩ] satisfies the following inequality:(7)∫Rnb,TΩfxpwxdx1/p≤CbBMO∫Rnfxpwxdx1/p.The following lemma was given by S. Ding and B. Liu in [18].Lemma 2.
Letu∈D′(M,Λl) be a solution of Dirac-harmonic equation (1) in M; σ>1 and 0<s,t<∞ are constants. Then, there exists a constant C, independent of u, such that(8)us,B≤CμBt-s/stut,σB for all cubes or balls B with σB⊂M.In [26], T. Iwaniec and A. Lutoborski gave the following three lemmas which will be used repeatedly in this paper.Lemma 3.
Letu∈Llocp(M,Λl), l=1,2,…,n, 1<p<∞, be a differential form and T:Lp(M,Λl)→W1,p(M,Λl-1) be the homotopy operator. Then, we have the following decomposition:(9)u=dTu+Tduand the inequality(10)Tup,M≤Cn,pμMdiamMup,Mholds for any bounded domain M⊂Rn, where C(n,p) is a constant, independent of u.From [26], the l-form uM∈Lp(M,Λl) is defined by uM=1/μ(M)∫Mu(y)dy if l=0 and uM=d(Tu) if l=1,…,n, 1≤p≤∞.Lemma 4.
Letu∈Lp(M,Λl), l=1,2,…,n. Then, uM∈Lp(M,Λl) and (11)uMp,M≤Cn,pμMup,M,where C(n,p) is a constant, independent of u and 1<p<∞.Lemma 5.
Letu∈D′(M,Λl) and du∈Lp(M,Λl+1). Then u-uM is in Lnp/(n-p)(M,Λl) and(12)∫Mu-uMnp/n-pdxn-p/np≤Cpn∫Mdupdx1/p, where l=1,2,…,n, 1<p<n, and Cp(n) is a constant independent of u.The following lemma appears in [27].Lemma 6.
Letψ defined on [0,+∞) be a strictly increasing convex function, ψ(0)=0, and M⊂Rn be a domain. Assume that u(x)∈D′(M,Λl) satisfies ψ(k(|u|+|uM|))∈L1(M,μ) for any real number k>0, and μ(x∈M:|μ-μM|>0)>0 where μ is a Radon measure defined by dμ(x)=ω(x)dx with a weight ω(x), then for any a>0, we obtain (13)∫Mψ12u-uMdμ≤C1∫Mψaudμ≤C2∫Mψ2au-uMdμ, where C1>0 and C2>0 are constants.Chooseψ(t)=tp, p>1, and w(x)=1 and let M be a ball B in Lemma 6; we find that the norms ‖u‖p,B and ‖u-uB‖p,B are comparable; that is,(14)u-uBp,B≤C1up,B≤C2u-uBp,Bfor any ball B with |{x∈B:|u-uB|>0}|>0.The covering lemma below belongs to [19].Lemma 7.
Each domainM has a modified Whitney cover of cubes V={Qi} such that (15)∪iQi=M,∑Qi∈Vχ5/4Qi≤NχΩand some N>1, and if Qi∩Qj≠∅, then there exists a cube R (this cube need not be a member of V) in Qi∩Qj such that Qi∪Qj⊂NR. Moreover, if M is δ-John, then there is a distinguished cube Q0∈V which can be connected with every cube Q∈V by a chain of cubes Q0,Q1,…,Qk=Q from V and such that Q⊂ρQi, i=0,1,2,…,k, for some ρ=ρ(n,δ).


## 3. 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].Lemma 8.
Let(16)Afx=limϵ→0∫x-y>ϵkx-yfydy,where x, y are points in n-dimensional Euclidean space Rn and k(x) is a homogeneous function of degree -n with mean value zero on |x|=1, and let B(f)=b(x)f(x). If k and b are sufficiently smooth and b is bounded, then,(17)∂∂xjAB-BAfp≤cfpfor 1<p<∞, where c is a constant, independent of f.Lemma 9.
Letu∈Lp(Rn,Λl), l=1,2,…,n, 1<p<∞, TΩ be the Calderón-Zygmund singular integral operator on differential forms, and b be sufficiently smooth and bounded. Then, there exists a constant C, independent of u, such that(18)db,TΩup,Rn≤Cup,Rn.Proof.
For any differentiall-form u(x), by the definition of commutator of Calderón-Zygmund singular integral operator on differential forms, we have(19)b,TΩux=bxTΩux-TΩbux=∑I∫RnΩx-yx-ynbx-byuIydydxI.Let K(x-y)=Ω(x-y)/|x-y|n and then according to the definition of the exterior differential operator, we obtain(20)db,TΩux=∑I∑k=1n∂∫Rnbx-byKx-yuIydy∂xkdxk∧dxI.Using the elementary inequality |∑i=1Nti|s≤Ns-1∑i=1N|ti|s, for constants N,s>0, we deduce(21)db,TΩuxp,Rnp≤∫Rn∑I∑k=1n∂∫Rnbx-byKx-yuIydy∂xk2p/2dx≤C1∫Rn∑I∑k=1n∂∫Rnbx-byKx-yuIydy∂xkpdx=C1∑I∑k=1n∫Rn∂∫Rnbx-byKx-yuIydy∂xkpdx=C1∑I∑k=1n∂∫Rnbx-byKx-yuIydy∂xkp,Rnp.By Lemma 8, we obtain(22)∂∫Rnbx-byKx-yuIydy∂xkp,Rn≤C2uIxp,Rn.Substituting (22) into (21) and applying the fundamental inequality ‖a‖s+‖b‖s≤(‖a‖+‖b‖)s, s≥1 yields that(23)db,TΩuxp,Rnp≤C3∑IuIxp,Rnp=C3∑I∫RnuIxpdx=C3∫Rn∑IuIxpdx≤C4∫Rn∑IuIxpdx.Using the inequality |∑i=1Nti|s≤Ns-1∑i=1N|ti|s again, we easily have(24)∑IuIx2≤C5∑IuIx2, that is,(25)∑IuIx≤C5∑IuIx21/2.Substituting (25) into (23) gives(26)db,TΩup,Rnp≤C6∫Rn∑IuIx2p/2dx=C6∫Rnuxpdx=C6up,Rnp. This completes the proof of Lemma 9.In Lemma9, let u=u in bounded domain M and u=0 in Rn∖M; we can easily obtain the following lemma.Lemma 10.
Letu∈Lp(M,Λl), l=1,2,…,n, 1<p<∞, TΩ be the Calderón-Zygmund singular integral operator on differential forms, and b be sufficiently smooth and bounded. Then, there exists a constant C, independent of u, such that(27)db,TΩup,M≤Cup,M, where M⊂Rn is any bounded domain.Using Lemma1 and the analogous method developed in Lemma 9, we have the following estimate for [b,TΩ].Lemma 11.
Letu∈Lp(M,Λl), l=0,1,…,n, 1<p<∞, and TΩ be a Calderón-Zygmund singular integral operator on differential forms. Then, given any function b∈BMO(Rn), [b,TΩ] satisfies the strong (p,p) inequality(28)b,TΩup,M≤CbBMOup,M for any bounded domain M⊂Rn, where C is a constant independent of u.We now present the local higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms.Theorem 12.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Llocp(M,Λl), l=1,2,…,n, 1<p<n, then [b,TΩ]u∈Llocs(M,Λl) for any 0<s<np/(n-p). Moreover, there exists a constant C, independent of u, such that(29)b,TΩus,B≤CμB1/s+1/n-1/pup,σBfor all balls B with σB⊂M for some σ>1.Proof.
For any ballB with σB⊂M for some σ>1, in the case that the measure μ{x∈B:|[b,TΩ]u-([b,TΩ]u)B|>0}=0. We have obviously that [b,TΩ]u=([b,TΩ]u)B almost everywhere in B, which implies [b,T]u is a closed form and a solution of the Dirac-harmonic equation (1) in M. Applying Lemmas 4 and 2 to [b,TΩ]u with any s,p>0, we have(30)b,TΩus,B=b,TΩuBs,B≤C1μBb,TΩus,B≤C2μB1+1/s-1/pb,TΩup,σBfor all balls B with σB⊂M for some σ>1. We note that b is bounded; thus b∈BMO(Rn). By Lemma 11, it follows that(31)b,TΩus,B≤C2μB1+1/s-1/pb,TΩup,σB≤C3μB1+1/s-1/pbBMOup,σB≤C4μB1+1/s-1/pup,σB.
On the other hand, if the measureμ{x∈B:|[b,TΩ]u-([b,TΩ]u)B|>0}>0. Applying Lemma 5 to [b,TΩ]u, we get(32)∫Bb,TΩu-b,TΩuBnp/n-pdxn-p/np≤C5n,p∫Bdb,TΩupdx1/p. Then by Lemma 10, it follows that(33)∫Bb,TΩu-b,TΩuBnp/n-pdxn-p/np≤C6n,p∫Bupdx1/p.Noticing that the measure |{x∈B:|[b,TΩ]u-([b,TΩ]u)B|>0}|>0, we could use Lemma 6. Taking ψ(t)=tnp/(n-p) in Lemma 6, we have(34)∫Bvnp/n-pdxn-p/np≤C7∫Bv-vBnp/n-pdxn-p/npfor any differential form v and any ball B with |{x∈B:|v-vB|>0}|>0. Replacing v by [b,TΩ]u in (34), we obtain(35)∫Bb,TΩunp/n-pdxn-p/np≤C8∫Bb,TΩu-b,TΩuBnp/n-pdxn-p/np.Combining (33) and (35) yields that(36)∫Bb,TΩunp/n-pdxn-p/np≤C9∫Bupdx1/p.In view of the monotonic property of the Lp-space with 0<s<np/(n-p), we have(37)1μB∫Bb,TΩusdx1/s≤1μB∫Bb,TΩunp/n-pdxn-p/np.Substituting (36) and (37) gives(38)∫Bb,TΩusdx1/s≤C10μB1/s-1/p+1/n∫Bupdx1/p.Therefore, inequality (29) follows in both cases, which indicates that if u∈Llocp(M,Λl), then [b,T]u∈Llocs(M,Λl). We have completed the proof of Theorem 12.Moreover, inequality (38) can be written as the following integral average inequality:(39)1μB∫Bb,TΩusdx1/s≤C11μB1/n1μB∫σBupdx1/p.Clearly, withp close to n, the integral exponent s on the left hand side could be much larger than the integral exponent p on the right hand side since the condition 0<s<np/(n-p). Hence the higher integrability of operator [b,TΩ] for the case that 1<p<n is obtained.Next, we consider the higher integrability of[b,TΩ] for the case p≥n.Theorem 13.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Llocp(M,Λl), l=1,2,…,n, p≥n, then [b,TΩ]u∈Llocs(M,Λl) for any s>0. Moreover, there exists a constant C, independent of u, such that(40)b,TΩus,B≤CμB1/s+1/n-1/pup,σBfor all balls B with σB⊂M for some σ>1.Proof.
It is immediate by the same method developed in the proof of Theorem12 that (40) holds for μ{x∈B:|[b,TΩ]u-([b,TΩ]u)B|>0}=0. Let us consider the remain case that μ{x∈B:|[b,TΩ]u-([b,TΩ]u)B|>0}>0. Select α=max{1,s/p} and q=αnp/(n+αp). Then it is easy to check that(41)q-p=pαn-p-nn+αp<0, since n-p≤0. Notice that 1<q=αnp/(n+αp)<n. Using Lemma 5 to [b,TΩ]u and combining Lemma 10 and the monotonic property of the Lp-space, we have(42)∫Bb,TΩu-b,TΩuBnq/n-qdxn-q/nq≤C1∫Bdb,TΩuqdx1/q≤C2∫Buqdx1/q≤C3μB1/q-1/p∫Bupdx1/p.Notice that the measure |{x∈B:|[b,TΩ]u-([b,TΩ]u)B|>0}|>0. Therefore, we can select ψ(t)=tnq/(n-q) in Lemma 6. Then for any differential form w, we have(43)∫Bwnq/n-qdxn-q/nq≤C4∫Bw-wBnq/n-qdxn-q/nq.Replacing w by [b,TΩ]u in (43), we get(44)∫Bb,TΩunq/n-qdxn-q/nq≤C5∫Bb,TΩu-b,TΩuBnq/n-qdxn-q/nq.Taking into account the fact that nq/(n-q)=αp>s, we see from the monotonic property of the Lp-space, (42), and (44) that(45)∫Bb,TΩusdx1/s≤C6μB1/s-1/αp∫Bb,TΩunq/n-qdxn-q/nq≤C7μB1/s-1/αp∫Bb,TΩu-b,TΩuBnq/n-qdxn-q/nq≤C8μB1/q-1/p+1/s-1/αp∫Bupdx1/p=C8μB1/n+1/s-1/p∫Bupdx1/p,which means that if u∈Llocp(M,Λl), then [b,TΩ]u∈Llocs(M,Λl). 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.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Lp(M,Λl), l=1,2,…,n, 1<p<n, then [b,TΩ]u∈Ls(M,Λl) for any 0<s<np/(n-p). Moreover, there exists a constant C, independent of u, such that(46)b,TΩus,M≤CμM1/s+1/n-1/pup,Mfor any bounded domain M⊂Rn.Proof.
Notice that1/s+1/n-1/p>0 since 0<s<np/(n-p). By Lemma 7 and Theorem 12, we have(47)b,Tus,M≤∑B∈νb,Tus,B≤∑B∈νC1μB1/s+1/n-1/pup,σB≤∑B∈νC1μM1/s+1/n-1/pup,σB≤C2μM1/s+1/n-1/pNup,M≤C3up,M,which finishes the proof of Theorem 14.Using the similar method as we did in Theorem14 and combining with Theorem 13, we can deduce the following global result for the case p≥n.Theorem 15.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Lp(M,Λl), l=1,2,…,n, p≥n, then [b,TΩ]u∈Ls(M,Λl) for any s>0. Moreover, there exists a constant C, independent of u, such that(48)b,TΩus,M≤CμM1/s+1/n-1/pup,Mfor any bounded domain M⊂Rn.


## 4. 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 Dirac-harmonic equations.Theorem 16.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Llocp(M,Λl) is a solution of the Dirac-harmonic equation (1), l=1,2,…,n, 1<p<n. Then, for any 0<s<np/(n-p), there exists a constant C>0, independent of u, such that(49)b,TΩu-b,TΩuBs,B≤CμB1+1/s-1/p+1/nup,σBfor all balls B with σB⊂M for some σ>1.Proof.
For anyu∈Llocp(M,Λl), using the Lp-decomposition formula (9) to [b,TΩ]u, we obtain(50)b,TΩu=dTb,TΩu+Tdb,TΩu.Noticing that dT([b,TΩ]u)=([b,TΩ]u)B for any differential form u, by (50), (10) and Lemma 10, we get(51)b,TΩu-b,TΩuBnp/n-p,B=Tdb,TΩunp/n-p,B≤C1n,pμBdiamBdb,TΩunp/n-p,B≤C2n,pμB1+1/nunp/n-p,B.Since u is a solution of the Dirac-harmonic equation (1), by Lemma 2, we have(52)unp/n-p,B≤μB-1/nup,σB,where σ>1 is a constant. Substituting (52) into (51) gives that(53)b,TΩu-b,TΩuBnp/n-p,B≤C3n,pμBup,σB.Moreover, for any 0<s<np/n-p, applying the monotonic properties of Lp-space, it follows that(54)b,TΩu-b,TΩuBs,B≤μB1/s-1/p+1/nb,TΩu-b,TΩuBnp/n-p,B.Combined with (53), the above inequality becomes(55)b,TΩu-b,TΩuBs,B≤C4n,pμB1+1/s-1/p+1/nup,σB,which finishes the proof of Theorem 16.Next, we will prove the higher order Poincaré-type inequality still holds for the case thatp≥n.Theorem 17.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Llocp(M,Λl) is a solution of the Dirac-harmonic equation (1), l=1,2,…,n, p≥n. Then, for any s>0, there exists a constant C, independent of u, such that(56)b,TΩu-b,TΩuBs,B≤CμB1+1/s-1/p+1/nup,σBfor all balls B with σB⊂M for some σ>1.Proof.
Selectα=max⁡{1,s/p} and q=αnp/(n+αp). Then, we easily obtain(57)q-p=pαn-p-nn+αp<0,that is, q<p, since n-p≤0. We can also find 1<q=αnp/(n+αp)<n. Using the same technique as in the proof of Theorem 16, we have(58)b,TΩu-b,TΩuBnq/n-q,B=Tdb,TΩunq/n-q,B≤C1μBdiamBdb,TΩunq/n-q,B≤C2μB1+1/nunq/n-q,B≤C3μB1+1/nμB1/q-1/n-1/pup,σB≤C4μB1+1/q-1/pup,σB.Noticing that nq/(n-q)=αp>s and combining the monotonic property of the Lp-space, we obtain(59)b,TΩu-b,TΩuBs,B≤μB1/s-1/q+1/nb,TΩu-b,TΩuBnq/n-q,B.Combining (58) and (59), we finally have(60)b,TΩu-b,TΩuBs,B≤C5μB1+1/s-1/p+1/nup,σB.The proof of Theorem 17 is completed.Based on Theorems16 and 17, we can obtain the following global higher order Poincaré-type inequalities for the commutator [b,TΩ] using the analogous method developed in Theorem 14.Theorem 18.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Lp(M,Λl) is a solution of the Dirac-harmonic equation (1), l=1,2,…,n, 1<p<n. Then, for any 0<s<np/(n-p), there exists a constant C>0, independent of u, such that(61)b,TΩu-b,TΩuMs,M≤CμM1/n+1/s-1/pup,Mfor any bounded domain M⊂Rn.Theorem 19.
LetTΩ be the Calderón-Zygmund singular integral operator on differential forms and b be sufficiently smooth and bounded. If u∈Lp(M,Λl) is a solution of the Dirac-harmonic equation (1), l=1,2,…,n, p≥n. Then, for any s>0, there exists a constant C, independent of u, such that(62)b,TΩu-b,TΩuMs,M≤CμM1/n+1/s-1/pup,Mfor any bounded domain M⊂Rn.


## 5. Applications

In this section, we demonstrate some applications of our main results established in the previous sections.Example 20.
Letr>0 and k>0 be any constants and M1={(x1,x2,x3):x12+x22+x32≤r2}⊂R3. Consider the 1-form (63)ux1,x2,x3=x1k+x12+x22+x32dx1+x2k+x12+x22+x32dx2+x3k+x12+x22+x32dx3defined in M such that M1⊂M. It is easy to check that du=0. Hence, u is a solution of the Dirac-harmonic equation (1) for any operators A satisfying (2). Also, it can be calculated that(64)u=x1k+x12+x22+x322+x2k+x12+x22+x322+x3k+x12+x22+x3221/2=x12+x22+x32k+x12+x22+x3221/2<1for any x=(x1,x2,x3)∈M1. Thus,(65)up,M1=∫M1updx1/p=∫M1x12+x22+x32k+x12+x22+x322p/2dx1/p≤∫M11dx1/p=4π3r31/p.Applying Theorem 14 for 1<p<3, we have [b,TΩ]u∈Ls(M,Λl) for any 0<s<3p/(3-p). Analogously, we have [b,TΩ]u∈Ls(M,Λl) for any s>0 when p≥3 by Theorem 15. In the meantime, we have the higher estimate for [b,TΩ]u; that is,(66)b,TΩus,M1≤C4π3r31/p.In addition, we can evaluate the following integrals by Theorem 18 and Theorem 19 that(67)b,TΩu-b,TΩuM1s,M1≤C4π3r31/p.We should notice that the above example can be extended to the case ofRn as follows.Example 21.
We can check that the 1-form defined inRn(68)ux1,…,xn=∑i=1nxik+x12+⋯+xn2dxi,k>0,is a solution of the Dirac-harmonic equation (1) for any operators A satisfying (2). Hence, Theorems 14, 15, 18, and 19 are also applicable to u(x1,…,xn).Remark. It is worth pointing out that the global results obtained in this paper can be extended to larger classes of domains, such as Lp-averaging domains and Lφ(μ)-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.



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