Recent Advances in L p-Theory of Homotopy Operator on Differential Forms

and Applied Analysis 3 From [19], each of φ, g, and h in the above definition is doubling in the sense that its values at t and 2t are uniformly comparable for all t > 0, and the consequent fact that C1t q ≤ h −1 (φ (t)) ≤ C2t q , C1t p ≤ g −1 (φ (t)) ≤ C2t p , (14) where C1 and C2 are constants. Also, for all 1 ≤ p1 < p < p2 and α ∈ R, the function φ(t) = t log + t belongs to G(p1, p2, C) for some constant C = C(p, α, p1, p2). Here log + (t) is defined by log + (t) = 1 for t ≤ e, and log + (t) = log(t) for t > e. Particularly, if α = 0, we see that φ(t) = t lies in G(p1, p2, C), 1 ≤ p1 < p < p2. Lemma 3 (see [1]). Let u ∈ D(M,Λ) be a solution to the nonhomogeneous A-harmonic (7) on M and σ > 1 be a constant.Then there exists a constantC, independent of u, such that ‖du‖p,B ≤ C diam (B) −1 ‖u − c‖p,σB (15) for all balls or cubes B with σB ⊂ M and all closed forms c. Here 1 < p < ∞. Lemma 4 (see [1]). Suppose that u is a solution to the nonhomogeneous A-harmonic (7) on M, σ > 1 and q > 0. There exists a constant C, depending only on σ, n, p, a, b, and q, such that ‖du‖p,Q ≤ C|Q| (q−p)/pq ‖du‖q,σQ (16) for all Q with σQ ⊂ M. The followingHölder inequality will be used in this paper. Lemma 5. Let 0 < α < ∞, 0 < β < ∞, and s = α + β. If f and g are measurable functions on R, then 󵄩󵄩󵄩󵄩fg 󵄩󵄩󵄩󵄩s,E ≤ 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩α,E ⋅ 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩β,E (17)


Introduction
The homotopy operator has been playing an important role in the study of   -theory of differential forms.We all know that any differential form  can be decomposed as  = () + (), where  is the differential operator and  is the homotopy operator.Hence, the homotopy operator provides an effective tool to study various properties of different norms and the related operators.As extensions of functions, differential forms have become invaluable tools for many fields of sciences and engineering, including theoretical physics, general relativity, potential theory, and electromagnetism.They can be used to describe various systems of PDEs and to express different geometrical structures on manifolds.In recent years, much progress has been made in the investigation of differential forms and the related operators; see [1][2][3][4][5][6][7].The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of   -theory of the homotopy operator and its compositions applied to differential forms.We will first discuss the   -norm and   -norm inequalities in Sections 2 and 3, respectively.Then, we present Lipschitz and BMO norm inequalities in Sections 4 and 5.We also give some global   -inequalities in Section 6.Finally, we discuss the compositions of homotopy operator with the projection operator, potential operator, and Green's operator in Sections 7, 8, and 9.We will keep using the traditional symbols and notations in this survey paper.Specifically, we always assume that Ω is a bounded domain in R  ,  ≥ 2,  and  are the balls with the same center and diam() =  diam() throughout this paper.We use || to denote the dimensional Lebesgue measure of a set ⊆ R  .For a function , the average of  over  is defined by   = (1/||) ∫  .All integrals involved in this paper are the Lebesgue integrals.We call  a weight if  ∈  1 loc ( R  ) and  > 0 a.e.. Differential forms are extensions of differentiable functions in R  .For instance, the function ( 1 ,  2 , . . .,   ) is called a 0-form.A differential 1-form () in R  can be written as () = ∑  =1   ( 1 ,  2 , . . .,   )  , where the coefficient functions   ( 1 ,  2 , . . .,   ),  = 1, 2, . . ., , are differentiable.Similarly, a differential -form () can be expressed as where  = ( 1 ,  2 , . . .,   ), 1 ≤  1 <  2 < ⋅ ⋅ ⋅ <   ≤ .Let ∧  = ∧  ( R  ) be the set of all -forms in R  , let   (Ω, ∧  ) be the space of all differential -forms in Ω, and let   (Ω, ∧  ) be the -forms () = ∑    ()  in Ω satisfying ∫ Ω |  ()|   < ∞ for all ordered -tuples ,  = 1, 2, . . ., .We denote the exterior derivative by  and the Hodge star operator by ⋆.The A continuously increasing function  : [0, ∞) → [0, ∞) with (0) = 0 is called an Orlicz function.The Orlicz space   (Ω) consists of all measurable functions  on Ω such that ∫ Ω (||/) < ∞ for some  = () > 0.   (Ω) is equipped with the nonlinear Luxemburg functional A convex Orlicz function  is often called a Young function.
If  is a Young function, then ‖ ⋅ ‖  defines a norm in   (Ω), which is called the Luxemburg norm or Orlicz norm.
The following Hölder inequality will be used in this paper.
Definition 8.A weight  is called a doubling weight and write  ∈ (Ω) if there exists a constant  such that for all balls  with 2 ⊂ Ω.Here the measure  is defined by  = ().If this condition holds only for all balls  with 4 ⊂ Ω, then  is weak doubling and we write  ∈ (Ω).
Definition 9. Let  > 1.It is said that  satisfies a weak reverse Hölder inequality and write  ∈ (Ω) when there exist constants  > 1 and  > 0 such that for all balls  with  ⊂ Ω.We say that  satisfies a reverse Hölder inequality when (22) holds with  = 1, and write  ∈ (Ω).In fact the space (Ω) is independent of  > 1.

𝑑𝑥)
−1 < ∞. (23) for all balls  ⊂ R  .It is clear that   (1) is the usual   -class; see [1] for more properties of   -weights.We prove some properties of the   ()-weights.The following theorem says that   () is an increasing class with respect to .
The following result shows that   ()-weights have the property similar to the strong doubling property of  weights: if  ∈   (),  ≥ 1, and the measure  is defined by  = (), then where  is a ball in R  and  is a measurable subset of .
If we put  = 1 (24), then we have which is called the strong doubling property of   -weights.
It is well known that an   -weight  satisfies the following reverse Hölder inequality.The definitions of the following several weight classes can be found in [1] and these weight classes have been widely used recently in the study of the integral properties of differential forms.
Definition 11.We say that the weight () > 0 satisfies the    ()-condition,  > 1 and  > 0, and write for any ball  ⊂ .Here  is a subset of R  .
for any ball  ⊂ .
Using the basic Poincaré-type estimate for the homotopy operator  established in Theorem 6, we have the following   (Ω)-weighted inequality.
The above   -norm inequality can also be written in the integral form as Also, using the procedure developed to extend the local inequalities into the John domains, we have the following global Poincaré-type inequality.
So far, we have presented the   (Ω)-weighted Poincarétype estimates for the homotopy operator .Now, we state other estimates with different weights, such as   (, Ω)weights and    (Ω)-weights.
The above inequalities have integral representations; for example, inequality (38) can be written as The above estimates can be extended into the following twoweight case.
Letting  = 1 in Corollary 25, we find the following symmetric two-weighted inequality.
Finally, when  =  in Theorem 23, we have the following two-weighted inequality.

𝐿 𝜑 -Norm Inequalities
The following local Poincaré-type inequality with the  norm was proved in [13], which can be used to establish the global inequality.
Since each of , , and ℎ in Definition 2 is doubling, from the proof of Theorem 28 or directly from (48), we have for all balls  with  ⊂ Ω and any constant  > 0.
From ( 13) and (54), the following Poincaré inequality with the Luxemburg norm holds under the conditions described in Theorem 28.
Similar to (55), from ( 18) and ( 56), the following Orlicz norm inequality holds if all conditions of Theorem 29 are satisfied.

Lipschitz and BMO Norm Inequalities
In this section, we will present Lipschitz and BMO norm inequalities for the homotopy operator.All results presented in this section and next section can be found in [14].Let us recall the definitions of Lipschitz and BMO norms first.Let  ∈  1 loc (, ∧  ),  = 0, 1, . . ., .We write for some  ≥ 1.Further, we write lip  (, ∧  ) for those forms whose coefficients are in the usual Lipschitz space with exponent  and write ‖‖ lip  , for this norm.Similarly, for  ∈  1 loc (, ∧  ),  = 0, 1, . . ., , we write  ∈ BMO(, ∧  ) if for some  ≥ 1.When  is a 0-form, (68) reduces to the classical definition of BMO().The definitions of the above Lipschitz and BMO norms can be found in [1].
The following Theorem 30 indicates that we can use the   -norm of  to estimate the Lipschitz norm of ().
Using the similar method involved in the proof of Theorem 30, we have the following Lipschitz norm inequalities for Green's operator  and the projection operator ; see [1] for more properties about Green's operator  and the projection operator .
We have discussed some estimates for the Lipschitz norm ‖ ⋅ ‖ locLip  ,Ω above.Next, we will focus on the estimates for the BMO norm ‖ ⋅ ‖ ⋆,Ω .For this, let  ∈ locLip  (Ω, ∧  ),  = 0, 1, . . ., , 0 ≤  ≤ 1, and let Ω be a bounded domain.Then, from the definitions of the Lipschitz and BMO norms, we have where  1 is a positive constant.Hence, we have proved the following inequality between the Lipschitz norm and the BMO norm.
where  is a constant.
Using Theorems 32 and 30, we obtain the following inequality between the BMO norm and the   norm.
As in the proof of Theorem 33, using inequality (75) and Theorem 31, we obtain the following result immediately.
Proof.First, we note that () = ∫     ≥ ∫     = where  1 is a positive constant.Hence, we have obtained the following theorem.

Global 𝐿 𝜑 -Inequalities
In this section, we discuss the global inequalities in the following   ()-averaging domains.See [13] for detailed proofs.
From the above definition, we see that   -averaging domains and   ()-averaging domains are special   ()averaging domains when () =   in Definition 38.Also, uniform domains and John domains are very special   ()averaging domains; see [20,21] for more results about domains.
Proof.From Definition 38, (48), and noticing that  is doubling, we have We have completed the proof of Theorem 39.
Similar to the local case, the following global inequality with the Orlicz norm holds if all conditions in Theorem 39 are satisfied.Also, by the same way, we can extend Theorem 28 into the following global result in   ()-averaging domains.
Then, there exists a constant , independent of , such that where  0 ⊂ Ω is some fixed ball.
Note that (95) can be written as It has been proved that any John domain is a special   ()averaging domain.Hence, we have the following results.
Note that (98) can be written as the following version with the Luxemburg norm provided the conditions in Corollary 42 are satisfied.

Composition of Homotopy and Projection Operators
In this section, we present the norm estimates for the composition of the homotopy operator and projection operator.
The results presented in this section can be found in [15,16].We assume that  is a domain in an oriented, compact,  ∞ smooth Riemannian manifold of dimension  ≥ 2.
Proof.Let  be the homotopy operator and let  be locally   integrable  form.Then, there exists a constant  1 (, , Ω), independent of , such that which ends the proof of Lemma 47.
In applications, such as in calculating electric or magnetic fields, we often face the fact that the integrand contains a singular factor.So, the above result was extended into the following singular weighted case.
Theorem 51.Let  ∈   loc (Ω, ∧  ),  = 1, 2, . . ., , 1 <  < ∞, be a solution of the nonhomogeneous -harmonic equation in a bounded domain Ω, let H be the projection operator, and let  be the homotopy operator.Then, there exists a constant , independent of , such that where ] > 1 is a constant.We may assume that   = 0. Otherwise, we can move the center to the origin by a simple transformation.Then, for any By using the polar coordinate substitution, we have Choose  = /( +  − ), then 0 <  < .By the reverse Hölder inequality, we find that where which does not contain a singular factor in the integral on the right side of the inequality.
Theorem 53.Let  ∈   (Ω, ∧ 1 ) be a solution of the nonhomogeneous -harmonic equation, let  be the projection operator, and let  be the homotopy operator.Assume that  is a fixed exponent associated with the nonhomogeneous -harmonic equation.Then, there exists a constant , independent of , such that for any bounded and convex   ()-averaging domain Ω ⊂ R  .
Proof.Let   be the radius of a ball  ⊂ Ω.We may assume the center of  is 0.Then, (, Ω) ≥   − || for any  ∈ .
We recall the following definition of -John domains with  > 0.
Definition 54.A proper subdomain Ω ⊂ R  is called a -John domain,  > 0, if there exists a point  0 ∈ Ω which can be joined with any other point  ∈ Ω by a continuous curve  ⊂ Ω so that for each  ∈ .Here (, Ω) is the Euclidean distance between  and Ω.
Theorem 55.Let  ∈   (Ω, ∧ 1 ) be a solution of the nonhomogeneous -harmonic (7), let  be the projection operator, and let  be the homotopy operator.Assume that  is a fixed exponent associated with the nonhomogeneous -harmonic equation.Then, there exists a constant (, , , , ,  0 , Ω), independent of , such that for any bounded and convex -John domain Ω ⊂ R  , where Here  and  are constants with 0 ≤  <  < min{,  +  + ( − 1)}, and the fixed cube  0 ⊂ Ω, the cubes   ⊂ Ω, and the constant  > 1 appeared in Lemma 46.
The following   -imbedding inequality with a singular factor in the John domain was also proved in [12].
Remark 58.Since the usual -harmonic equation div(∇|∇| −2 ) = 0 for functions is the special case of the nonhomogeneous -harmonic equation for differential forms, all results proved in Theorems 55, 56, and 57 are still true for -harmonic functions.

Composition of Homotopy and Potential Operators
Recently, Bi extended the definition of the potential operator to the case of differential forms; see [3].For any differential -form (), the potential operator  is defined by where the kernel (, ) is a nonnegative measurable function defined for  ̸ =  and the summation is over all ordered tuples .The  = 0 case reduces to the usual potential operator: where () is a function defined on  ⊂ R  .See [3,25] for more results about the potential operator.We say a kernel  on R  × R  satisfies the standard estimates if there exist , 0 <  ≤ 1, and constant  such that for all distinct points  and  in R In this paper, we always assume that  is the potential operator defined in (143) with the kernel (, ) satisfying condition (i) of the standard estimates.Recently, Bi in [3] proved the following inequality for the potential operator: where  ∈   (, ∧  ),  = 0, 1, . . .,  − 1, is a differential form defined in a bounded and convex domain  and  > 1 is a constant.
In this section, we prove the local   imbedding inequalities for  ∘  applied to solutions of the nonhomogeneous -harmonic equation in a bounded domain.For any subset  ⊂ R  , we use  1, (, ∧  ) to denote the Orlicz-Sobolev space of -forms which equals   (, ∧  )∩  1 (, ∧  ) with norm for all balls  with  ⊂ Ω for some  > 1.
The following local   -imbedding theorem was also obtained in [18].
Theorem 62.Let  be a Young function in the class (, , ), 1 ≤  <  < ∞,  ≥ 1, Ω be a bounded domain,  be the homotopy operator defined in (2), and let  be the potential operator defined in (143) with the kernel (, ) satisfying condition (i) of the standard estimates.Assume that (||) ∈  1 loc (Ω) and  is a solution of the nonhomogeneous harmonic (7) in Ω.Then, there exists a constant , independent of , such that for all balls  with  ⊂ Ω for some  > 1.
The following version of local imbedding will be used to establish a global imbedding theorem which indicates that the operator  ∘  is bounded.
Theorem 63.Let  be a Young function in the class (, , ), 1 ≤  <  < ∞,  ≥ 1, Ω be a bounded domain,  be the homotopy operator defined in (2), and let  be the potential operator defined in (143) with the kernel (, ) satisfying condition (i) of the standard estimates.Assume that (||) ∈  1 loc (Ω) and  is a solution of the nonhomogeneous harmonic (7) in Ω.Then, there exists a constant , independent of , such that for all balls  with  ⊂ Ω for some  > 1.
Proof.Applying (6) to (), then using (145), we find that for any differential form  and all balls  with  ⊂ Ω, where  > 1 is a constant.Starting with (152) and using the similar method developed in the proof of Theorem 61, we obtain where  = max{ 1 ,  2 }.The proof of Theorem 63 has been completed.
Selecting () =   in Theorem 62, we obtain the usual imbedding inequalities  ∘  with the   -norms.
It is well known that any John domain is a special  averaging domain; see [1].Hence, we have the following global   -imbedding theorem for John domains.
Next, let  be the set of all solutions of the nonhomogeneous -harmonic equation in Ω.We have the following version of imbedding theorem with   norm for any bounded domain, which says that the composite operator  ∘  maps  1, (Ω, ∧ 1 )∩ continuously into   (Ω).See [18] for the proof of Theorem 67.
Selecting () =   in Theorems 65, we have the following version of the imbedding inequality with   -norms.
Remark 69.(i) We know that the   -averaging domains are the special   -averaging domains.Thus, Theorem 65 also holds for the   -averaging domain; (ii) Theorem 67 holds for any bounded domain in R  .

Composition of Homotopy and Green's Operators
In this section, we estimate the Lipschitz norm ‖ ⋅ ‖ locLip  , or BMO norm ‖ ⋅ ‖ ⋆, of composition  ∘  in terms of the   norm.First, we present the following   norm inequality for the composition ∘ of the homotopy operator  and Green's operator .
Using Theorem 70, we obtain the following inequality with Lipschitz norm.
Theorem 72.Let  ∈   loc (, ∧ 1 ), 1 <  < ∞, be a solution of the nonhomogeneous -harmonic (7) in a bounded, convex domain .Let  be Green's operator and let  be the homotopy operator defined in (2).Then, there exists a constant , independent of , such that where  is a constant with 0 ≤  ≤ 1.
The following theorem gives an estimate for BMO norm ‖ ⋅ ‖ ⋆, of composition  ∘  in terms of   norm.