Integral Estimates for the Potential Operator on Differential Forms

Abstract

We develop the local inequalities with new weights for the potential operator applied to differential forms. We also prove the global weighted norm inequalities for the potential operator in averaging domains and explore applications of our new results.

1. Introduction

This paper deals with the weighted estimates for the potential operator applied to differential forms. Throughout this paper, Ω will denote an open subset of ℝ^𝑛, 𝑛≥2, and ℝ=ℝ¹. Let 𝑒₁=(1,0,…,0), 𝑒₂=(0,1,…,0),…, 𝑒_𝑛=(0,0,…,1) be the standard unit basis of ℝ^𝑛. For 𝑙=0,1,…,𝑛, the linear space of 𝑙-vectors, spanned by the exterior products 𝑒_𝐼=𝑒_𝑖1∧𝑒_𝑖2∧⋯∧𝑒_𝑖𝑙, corresponding to all ordered 𝑙-tuples 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑙), 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑙≤𝑛, is denoted by ∧^𝑙=∧^𝑙(ℝ^𝑛). The Grassman algebra ∧=∧ℝ^𝑛=⊕^𝑛𝑙_𝑙=0∧ℝ^𝑛 (1) is a graded algebra with respect to the exterior products. For 𝛼=∑𝛼^𝐼𝑒_𝐼∈∧ and 𝛽=∑𝛽^𝐼𝑒_𝐼∈∧, the inner product in ∧ is given by 𝛼,𝛽=𝛼^𝐼𝛽^𝐼 (2) with summation over all 𝑙-tuples 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑙) and all integers 𝑙=0,1,…,𝑛. We should also notice that 𝑑𝑥_𝑖∧𝑑𝑥_𝑗=−𝑑𝑥_𝑗∧𝑑𝑥_𝑖,𝑖≠𝑗, and 𝑑𝑥_𝑖∧𝑑𝑥_𝑖=0.

Assume that 𝐵⊂ℝ^𝑛 is a ball and 𝜎𝐵 is the ball with the same center as 𝐵 and with diam(𝜎𝐵)=𝜎diam(𝐵). Differential forms are extensions of functions defined in ℝ^𝑛. A function 𝑓(𝑥₁,…,𝑥_𝑛) in ℝ^𝑛 is called a 0-form. A differential 𝑙-form is of the form 𝑤(𝑥)=_𝐼𝑤_𝐼(𝑥)𝑑𝑥_𝐼=𝑤_{𝑖1𝑖2⋯𝑖𝑙}(𝑥)𝑑𝑥_𝑖1∧𝑑𝑥_𝑖2∧⋯∧𝑑𝑥_𝑖𝑙 (3) in ℝ^𝑛. 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. Many interesting and useful results about the differential forms have been obtained during recent years; particularly, for the differential forms satisfying some version of 𝐴-hrmonic equation, see [1–8]. The 𝑛-dimensional Lebesgue measure of a set 𝐸⊆ℝ^𝑛 is denoted by |𝐸|. We call 𝑤 a weight if 𝑤∈𝐿¹_loc(ℝ^𝑛) and 𝑤>0 a.e. For 0<𝑝<∞, we denote the weighted 𝐿^𝑝 norm of a measurable function 𝑓 over 𝐸 by 𝑓_{𝑝,𝐸,𝑤𝛼}=_𝐸𝑓(𝑥)^𝑝𝑤^𝛼𝑑𝑥^1/𝑝 (4) if the above integral exists. Here 𝛼 is a real number. It should be noticed that the Hodge star operator can be defined equivalently as follows.

Definition 1. If 𝜔=𝛼_{𝑖1,𝑖2,…,𝑖𝑘}(𝑥₁,𝑥₂,…,𝑥_𝑛)𝑑𝑥_𝑖1∧𝑑𝑥_𝑖2∧⋯∧𝑑𝑥_𝑖𝑘=𝛼_𝐼𝑑𝑥_𝐼,𝑖₁<𝑖₂<⋯<𝑖_𝑘, is a differential 𝑘-form, then ⋆𝜔=⋆𝛼_{𝑖1𝑖2⋯𝑖𝑘}𝑑𝑥_𝑖1∧𝑑𝑥_𝑖2∧⋯∧𝑑𝑥_𝑖𝑘=(−1)^∑(𝐼)𝛼_𝐼𝑑𝑥_𝐽, (5) where 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑘), 𝐽={1,2,…,𝑛}−𝐼, and (𝐼)=𝑘(𝑘+1)2+^𝑘_𝑗=1𝑖_𝑗. (6)

The following 𝐴(𝛼,𝛽,𝛾;𝐸)-weights were introduced in [8].

Definition 2. One says that a measurable function 𝑔(𝑥) defined on a subset 𝐸⊂ℝ^𝑛 satisfies the 𝐴(𝛼,𝛽,𝛾;𝐸)-condition for some positive constants 𝛼,𝛽,𝛾, writes 𝑔(𝑥)∈𝐴(𝛼,𝛽,𝛾;𝐸) if 𝑔(𝑥)>0 a.e., and writes sup_𝐵1|𝐵|_𝐵𝑔^𝛼1𝑑𝑥|𝐵|_𝐵𝑔^−𝛽𝑑𝑥^𝛾/𝛽<∞, (7) where the supremum is over all balls 𝐵⊂𝐸. One says that 𝑔(𝑥) satisfies the 𝐴(𝛼,𝛽;𝐸)-condition if (7) holds for 𝛾=1 and write 𝑔(𝑥)∈𝐴(𝛼,𝛽;𝐸)=𝐴(𝛼,𝛽,1;𝐸).
Notice that there are three parameters in the definition of the 𝐴(𝛼,𝛽,𝛾;𝐸)-class. We obtain some existing weighted classes if we choose some particular values for these parameters. For example, it is easy to see that the 𝐴(𝛼,𝛽,𝛾;𝐸)-class reduces to the usual 𝐴_𝑟(𝐸)-class if 𝛼=𝛾=1 and 𝛽=1/(𝑟−1).

Recently, Bi extended the definition of the potential operator to the case of differential forms; see [2]. For any differential 𝑘-form 𝑤(𝑥), the potential operator 𝑃 is defined by 𝑃𝑤(𝑥)=_𝐼𝐸𝐾𝑥,𝑦𝑤_𝐼𝑦𝑑𝑦𝑑𝑥_𝐼, (8) where the kernel 𝐾(𝑥,𝑦) is a nonnegative measurable function defined for 𝑥≠𝑦 and the summation is over all ordered 𝑘-tuples 𝐼. The 𝑘=0 case reduces to the usual potential operator, 𝑃𝑓(𝑥)=_𝐸𝐾𝑥,𝑦𝑓𝑦𝑑𝑦, (9) where 𝑓(𝑥) is a function defined on 𝐸⊂ℝ^𝑛. A kernel 𝐾 on ℝ^𝑛×ℝ^𝑛 is said to satisfy the standard estimates if there are 𝛿, 0<𝛿≤1, and constant 𝐶 such that, for all distinct points 𝑥 and 𝑦 in ℝ^𝑛 and all 𝑧 with |𝑥−𝑧|<(1/2)|𝑥−𝑦|, the kernel 𝐾 satisfies (i) 𝐾(𝑥,𝑦)≤𝐶|𝑥−𝑦|^−𝑛, (ii) |𝐾(𝑥,𝑦)−𝐾(𝑧,𝑦)|≤𝐶|𝑥−𝑧|^𝛿|𝑥−𝑦|^{−𝑛−𝛿}, and (iii) |𝐾(𝑦,𝑥)−𝐾(𝑦,𝑧)|≤𝐶|𝑥−𝑧|^𝛿|𝑥−𝑦|^{−𝑛−𝛿}. In this paper, we always assume that 𝑃 is the potential operator with the kernel 𝐾(𝑥,𝑦) satisfying the above condition (i) in the standard estimates. In [2], Bi proved the following inequality for the potential operator: 𝑃(𝑢)−(𝑃(𝑢))_{𝐸𝑝,𝐸}≤𝐶|𝐸|diam(𝐸)‖𝑢‖_𝑝,𝐸, (10) where 𝑢∈𝐷^′(𝐸,∧^𝑘),𝑘=0,1,…,𝑛−1, is a differential form defined in a bounded and convex domain 𝐸.

Definition 3. One says that a differential form 𝑢∈∧^𝑙(𝐸) belongs to the WRH(∧^𝑙,𝐸)-class and writes 𝑢∈WRH(∧^𝑙,𝐸), 𝑙=0,1,2,…,𝑛, if for any constants 0<𝑠,𝑡<∞, the inequality ‖𝑢‖_𝑠,𝐵≤𝐶|𝐵|^{(𝑡−𝑠)/𝑠𝑡}‖𝑢‖_{𝑡,𝜎𝐵} (11) holds for any ball 𝐵 with 𝜎𝐵⊂Ω, where 𝜎>1 and 𝐶>0 are constants.
From [1], we know that any solution of 𝐴-harmonic equations satisfies (11). Hence, the WRH(∧^𝑙,Ω)-class is a large class of differential forms. We will use the following Hölder inequality repeatedly in this paper.

Lemma 4. Let both 𝑓 and 𝑔 be measurable functions in ℝ^𝑛 and 𝑝,𝑞>0 and 1/𝑠=(1/𝑝)+(1/𝑞). Then _𝐸𝑓𝑔^𝑠𝑑𝑥^1/𝑠≤_𝐸𝑓^𝑝𝑑𝑥^1/𝑝_𝐸𝑔^𝑞𝑑𝑥^1/𝑞 (12) for any 𝐸⊂ℝ^𝑛.

2. Local Inequalities

In this section, we will prove some local weighted norm inequalities for the potential operator.

Theorem 5. Let 𝑃 be the potential operator applied to a differential form 𝑢∈WRH(∧^𝑙,Ω), where Ω⊂ℝ^𝑛 is a domain. Assume that 𝑤(𝑥)∈𝐴(𝛼,𝛽;Ω) with 𝛼,𝛽>0. Then, there exists a constant 𝐶, independent of 𝑢, such that 𝑃(𝑢)−(𝑃(𝑢))_{𝐵𝑠,𝐵,𝑤}≤𝐶|𝐵|diam(𝐵)‖𝑢‖_{𝑠,𝐵,𝑤} (13) for all balls 𝐵 with 𝐵⊂Ω, where 𝑠>1 is a constant.

Proof. Let 𝑡=𝑠𝛼/(𝛼−1), then 𝑡>𝑠. Using Lemma 4 with 1/𝑠=(1/𝑡)+((𝑡−𝑠)/𝑠𝑡) yields 𝑃(𝑢)−(𝑃(𝑢))_{𝐵𝑠,𝑏,𝑤}=_𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵^𝑠𝑤𝑑𝑥^1/𝑠=_𝐵|𝑃(𝑢)−(𝑃(𝑢))|_𝐵𝑤^1/𝑠𝑠𝑑𝑥^1/𝑠≤_𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵^𝑡𝑑𝑥^1/𝑡×_𝐵𝑤^{1/𝑠𝑠𝑡/(𝑡−𝑠)}𝑑𝑥^{(𝑡−𝑠)/𝑠𝑡}≤𝐶₁‖𝐵‖diam(𝐵)‖𝑢‖_{𝑡,𝐵𝐵}𝑤^𝛼(𝑥)𝑑𝑥^1/𝛼𝑠. (14) Set 𝑘=𝛽𝑠/(1+𝛽), then 0<𝑘<𝑠. Since 𝑢 is in the WRH(Ω) class, ‖𝑢‖_𝑡,𝐵≤𝐶₂‖𝐵‖^{(𝑘−𝑡)/𝑘𝑡}‖𝑢‖_{𝑘,𝜎1𝐵}, (15) where 𝜎₁>1 in a constraint. Since 𝑢 is in the WRH(Ω)-class again (note that 1/𝑘=(1/𝑠)+(𝑠−𝑘)/𝑠𝑘), ‖𝑢‖_{𝑘,𝜎1𝐵}=_𝜎1𝐵|𝑢|𝑤^1/𝑠𝑠^{−1/𝑠𝑘}𝑑𝑥^1/𝑘≤_𝜎1𝐵|𝑢|𝑤^1/𝑠𝑤^{−1/𝑠𝑘}𝑑𝑥^1/𝑘×_𝜎1𝐵𝑤^{−1/𝑠𝑘𝑠/(𝑠−𝑘)}𝑑𝑥^{(𝑠−𝑘)/𝑘𝑠}=_𝜎1𝐵|𝑢|^𝑠𝑤𝑑𝑥^1/𝑠×_𝜎1𝐵𝑤^{−𝑘/(𝑠−𝑘)}𝑑𝑥^{(𝑠−𝑘)/𝑘𝑠}=_𝜎1𝐵|𝑢|^𝑠𝑤𝑑𝑥^1/𝑠_𝜎1𝐵𝑤^−𝛽𝑑𝑥^1/𝛽𝑠, (16) where we have used the following calculation: 𝑘=𝑠−𝑘𝛽𝑠/1+𝛽=𝑠−𝛽𝑠/1+𝛽𝛽/1+𝛽=𝑎−𝛽/1+𝛽𝛽/1+𝛽1+𝛽−𝛽/1+𝛽=𝛽. (17)
Combining (14), (15) and, (16) gives 𝑃(𝑢)−(𝑃(𝑢))_{𝐵𝑠,𝐵,𝑤}≤𝐶₃|𝐵|diam(𝐵)|𝐵|^{(𝑘−𝑡)/𝑘𝑡}_𝜎1𝐵|𝑢|^𝑠𝑤𝑑𝑥^1/𝑠×_𝐵𝑤^𝛼𝑑𝑥^1/𝛼𝑠_𝜎1𝐵𝑤^−𝛽𝑑𝑥^1/𝛽𝑠. (18) Note that since 𝑤∈𝐴(𝛼,𝛽,𝛼;Ω), it follows that _𝐵𝑤^𝛼𝑑𝑥^1/𝛼𝑠_𝜎1𝐵𝑤^−𝛽𝑑𝑥^1/𝛽𝑠=_𝜎1𝐵𝑤^𝛼𝑑𝑥_𝜎1𝐵𝑤^−𝛽𝑑𝑥^{𝛼/𝛽1/𝛼𝑠}=𝜎₁𝐵^{1+(𝛼/𝛽)}1𝛽_𝜎1𝐵𝑤^𝛼1𝑑𝑥𝛽_𝜎1𝐵𝑤^−𝛽𝑑𝑥^{𝛼/𝛽1/𝛼𝑠}=𝐶₄|𝐵|^{(1/𝛼𝑠)+(1/𝛽𝑠)}. (19) Plugging (19) into (18), we have 𝑃(𝑈)−(𝑃(𝑢))_{𝐵𝑠,𝐵,𝑤}≤𝐶₅diam(𝐵)|𝐵|^{1+(1/𝑡)−(1/𝑘)}×_𝜎1𝐵|𝑢|^𝑠𝑤𝑑𝑥^1/𝑠𝐶₄|𝐵|^{(1/𝛼𝑠)+(1/𝛽𝑠)}≤𝐶₆diam(𝐵)|𝐵|^{1+(1/𝑡)−(1/𝑘)+(1/𝛼𝑠)+(1/𝛽𝑠)}×_𝜎1𝐵|𝑢|^𝑠𝑤𝑑𝑥^1/𝑠. (20) We should notice that 11+𝑡−1𝑘+1+1𝛼𝑠𝛽𝑠=1+𝛼−1−𝑠𝛼1+𝛽+1𝑠𝛽+1𝛼𝑠𝛽𝑠=1. (21) Combining (20) and (21) gives us 𝑃(𝑢)−(𝑃(𝑢))_{𝐵𝑠,𝐵,𝑤}≤𝐶|𝐵|diam(𝐵)_𝜎1𝐵|𝑢|^𝑠𝑤𝑑𝑥^1/𝑠. (22) The proof of Theorem 5 has been completed.

A continuously increasing function 𝜑∶[0,∞)→[0,∞) with 𝜑(0)=0 is called an Orlicz function. A convex Orlicz function 𝜑 is often called a Young function. The Orlicz space 𝐿^𝜑(Ω,𝜇) consists of all measurable functions 𝑓 on Ω such that ∫_Ω𝜑(|𝑓|/𝜆)𝑑𝜇<∞ for some 𝜆=𝜆(𝑓)>0. If 𝜑 is a Young function, then 𝑓_{𝜑(Ω,𝜇)}=inf𝜆>0∶_Ω𝜑𝑓𝜆𝑑𝜇≤1 (23) defines a norm in 𝐿^𝜑(Ω,𝜇), which is called the Orlicz norm or the Luxemburg norm.

Definition 6 (see [9]). One says that a Young function 𝜑 lies in the class 𝐺(𝑝,𝑞,𝐶), 1≤𝑝<𝑞<∞, 𝐶≥1, if (i) 1/𝐶≤𝜑(𝑡^1/𝑝)/𝑗(𝑡)≤𝐶 and (ii) 1/𝐶≤𝜑(𝑡^1/𝑞)/ℎ(𝑡)≤𝐶 for all 𝑡>0, where 𝑗 is a convex increasing function and ℎ is a concave increasing function on [0,∞).
From [9], each of 𝜑,𝑗, 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⁻¹𝜑(𝑡)≤𝐶₂𝑡^𝑞,𝐶₁𝑡^𝑝≤𝑗⁻¹𝜑(𝑡)≤𝐶₂𝑡^𝑝, (24) where 𝐶₁ and 𝐶₂ are constants. Also, for all 1≤𝑝₁<𝑝<𝑝₂ and 𝛼∈ℝ, the function 𝜑(𝑡)=𝑡^𝑝log^𝛼₊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^𝑝 lies in 𝐺(𝑝₁,𝑝₂,𝐶), 1≤𝑝₁<𝑝<𝑝₂.

Theorem 7. Let 𝑃 be the potential operator applied to a differential form 𝑢∈WRH(∧^𝑙,Ω) and 𝜑 a Young function in the class 𝐺(𝑝,𝑞,𝐶),1≤𝑝<𝑞<∞,𝐶≤1, and where Ω is a bounded domain. Assume that 𝜑(|𝑢|)∈𝐿¹_loc(Ω,𝜇). Then, there exists a constant 𝑀>0, independent of 𝑢, such that _𝐵𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘𝑑𝑥≤𝑀_𝜎𝐵𝜑|𝑢|𝑘𝑑𝑥 (25) for all balls 𝐵 with 𝐵⊂Ω, where 𝜎>1 is a constant.

Proof. Using Jensen's inequality for ℎ⁻¹ and (24), we have _𝐵𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘ℎ𝑑𝑥=ℎ⁻¹_𝐵𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘𝑑𝑥≤ℎ_𝐵ℎ⁻¹𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘𝐶𝑑𝑥≤ℎ_1𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘^𝑞𝑑𝑥≤𝐶₂𝜑𝐶_1𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘^𝑞𝑑𝑥^1/𝑞≤𝐶₂𝜑1𝑘𝐶_1𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵^𝑞𝑑𝑥^1/𝑞≤𝐶₃𝜑1𝑘_𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵^𝑞𝑑𝑥^1/𝑞. (26)
Since 𝑢∈WRH(Ω), by (11) we obtain _𝐵|𝑢|^𝑞𝑑𝑥^1/𝑞≤𝐶₄|𝐵|^{(𝑝−𝑞)/𝑝𝑞}_𝜎𝐵|𝑢|^𝑝𝑑𝑥^1/𝑝, (27) where 𝜎>1 is a constant. Using (13), (24), and Jensen's inequality, 𝜑1𝑘_𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵^𝑞𝑑𝑥^1/𝑞1≤𝜑𝑘|𝐵|^1+(1/𝑛)_𝐵|𝑢|^𝑞𝑑𝑥^1/𝑞1≤𝜑𝑘|𝐵|^1+(1/𝑛)𝐶₄|𝐵|^{(1/𝑞)−(1/𝑝)}_𝜎𝐵|𝑢|^𝑝𝑑𝑥^1/𝑝1=𝜑𝑘^𝑝𝐶^𝑝₄|𝐵|^{𝑝(1+(1/𝑛)+(1/𝑞)−(1/𝑝))}_𝜎𝐵|𝑢|^𝑝𝑑𝑥^1/𝑝≤𝐶₅𝑗1𝑘^𝑝𝐶^𝑝₄|𝐵|^{𝑝(1+(1/𝑛)+(1/𝑞)−(1/𝑝))}_𝜎𝐵|𝑢|^𝑝𝑑𝑥≤𝐶₅𝑗_𝜎𝐵1𝑘^𝑝𝐶^𝑝₄|𝐵|^{𝑝(1+(1/𝑛)+(1/𝑞)−(1/𝑝))}|𝑢|^𝑝𝑑𝑥≤𝐶_5𝜎𝐵𝑗1𝑘^𝑝𝐶^𝑝₄|𝐵|^{𝑝(1+(1/𝑛)+(1/𝑞)−(1/𝑝))}|𝑢|^𝑝𝑑𝑥. (28) Note that 𝑝≥1, then 1+(1/𝑛)+(1/𝑞)−(1/𝑝)>0. Thus, |𝐵|^{1+(1/𝑛)+(1/𝑞)−(1/𝑝)}≤|Ω|^{1+(1/𝑛)+(1/𝑞)−(1/𝑝)}≤𝐶₆. (29) Using the above inequality and (i) in Definition 6, we find that 𝑗(𝑡)≤𝐶₇𝜑(𝑡^1/𝑝). Therefore, _𝜎𝐵𝑗1𝑘^𝑝𝐶^𝑝₄|𝐵|^{𝑝(1+(1/𝑛)+(1/𝑞)−(1/𝑝))}|𝑢|^𝑝𝑑𝑥≤𝐶_7𝜎𝐵𝜑1𝑘𝐶₄|𝐵|^{1+(1/𝑛)+(1/𝑞)−(1/𝑝)}|𝑢|𝑑𝑥≤𝐶_7𝜎𝐵𝜑𝐶₈|𝑢|𝑘𝑑𝑥≤𝐶_9𝜎𝐵𝜑|𝑢|𝑘𝑑𝑥. (30) Combining (28) and (30) yields 𝜑1𝑘_𝐵𝑃(𝑢)−(𝑃(𝑢))_𝐵^𝑞𝑑𝑥^1/𝑞≤𝐶_10𝜎𝐵𝜑|𝑢|𝑘𝑑𝑥. (31) Finally, substituting (31) into (26), we obtain _𝐵𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑘𝑑𝑥≤𝑀_𝜎𝐵𝜑|𝑢|𝑘𝑑𝑥. (32) The proof of Theorem 7 has been completed.

3. Global Inequalities

In 1989, Staples introduced the following 𝐿^𝑠-averaging domains in [10]. A proper subdomain Ω⊂ℝ^𝑛 is called an 𝐿^𝑠-averaging domain, 𝑠≥1, if there exists a constant 𝐶 such that 1|Ω|_Ω𝑢−𝑢_Ω^𝑠𝑑𝑥^1/𝑠≤𝐶sup_𝐵⊂Ω1|Ω|_𝐵𝑢−𝑢_𝐵^𝑠𝑑𝑥^1/𝑠 (33) for all 𝑢∈𝐿^𝑠_loc(Ω). Here the supremum is over all balls 𝐵⊂Ω. The 𝐿^𝑠-averaging domains were extended into the 𝐿^𝑠(𝜇)-averaging domains recently in [11]. We call a proper subdomain Ω⊂ℝ^𝑛 an 𝐿^𝑠(𝜇)-averaging domain, 𝑠≥1, if 𝜇(Ω)<∞ and there exists a constant 𝐶 such that 1𝜇(Ω)_Ω𝑢−𝑢_𝐵0^𝑠𝑑𝜇^1/𝑠≤𝐶sup_4𝐵⊂Ω1𝜇(𝐵)_𝐵𝑢−𝑢_𝐵^𝑠𝑑𝜇^1/𝑠 (34) for some ball 𝐵₀⊂Ω and all 𝑢∈𝐿^𝑠_loc(Ω;𝜇). The 𝐿^𝑠(𝜇)-averaging domain was genralized into the following 𝐿^𝜑(𝜇)-averaging domain in [12].

Definition 8. Let 𝜑 be a continuous increasing convex function on [0,∞) with 𝜑(0)=0. One calls a proper subdomain Ω⊂ℝ^𝑛 an 𝐿^𝜑(𝜇)-averaging domain, if 𝜇(Ω)<∞ and there exists a constant 𝐶 such that 1𝜇(Ω)_Ω𝜑𝜏𝑢−𝑢_𝐵0𝑑𝜇≤𝐶sup_4𝐵⊂Ω1𝜇(𝐵)_𝐵𝜑𝜎𝑢−𝑢_𝐵𝑑𝜇 (35) for some ball 𝐵₀⊂Ω and all 𝑢 such that 𝜑(|𝑢|)∈𝐿¹_𝑙𝑜𝑐(Ω;𝜇), where the measure 𝜇 is defined by 𝑑𝜇=𝑤(𝑥)𝑑𝑥,𝑤(𝑥) is a weight, and 𝜏,𝜎 are constants with 0<𝜏<1, 0<𝜎<1, and the supremum is over all balls 𝐵⊂Ω with 4𝐵⊂Ω.
From the above definition, we see that 𝐿^𝑠(𝜇)-averaging domains are special 𝐿^𝜑(𝜇)-averaging domains when 𝜑(𝑡)=𝑡^𝑠 in Definition 8.

Theorem 9. Let 𝜑 be a Young function in the class 𝐺(𝑝,𝑞,𝐶), 1≤𝑝<𝑞<∞, 𝐶≥1; and Ω⊂ℝ^𝑛 any bounded 𝐿^𝜑(𝜇)-averaging domain, and 𝑃 the potential operator applied to a differential form 𝑢∈WRH(∧^𝑙,Ω), 𝑙=1,2…,𝑛. Assume that 𝜑(|𝑢|)∈𝐿¹(Ω,𝜇). Then, there exists a constant 𝐶, independent of 𝑢, such that _Ω𝜑1𝑘𝑃(𝑢)−(𝑃(𝑢))_𝐵0𝑑𝜇≤𝐶_Ω𝜑|𝑢|𝑘𝑑𝜇, (36) where 𝐵₀⊂Ω is some fixed ball.

Proof. From Definition 8, (25), and noticing that 𝜑 is doubling, we have _Ω𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵0𝑑𝜇≤𝐶₁sup_{𝐵⊂Ω𝐵}𝜑𝑃(𝑢)−(𝑃(𝑢))_𝐵𝑑𝜇≤𝐶₁sup_𝐵⊂Ω𝐶_2𝜎𝐵𝜑(|𝑢|)𝑑𝜇≤𝐶₁sup_𝐵⊂Ω𝐶_2Ω𝜑(|𝑢|)𝑑𝜇≤𝐶_3Ω𝜑(|𝑢|)𝑑𝜇. (37)
We have completed the proof of Theorem 9.

Choosing 𝜑(𝑡)=𝑡^𝑠log^𝛼₊𝑡 in Theorem 9, we obtain the following Poincaré inequalities with the 𝐿^𝑠(log^𝛼₊𝐿)-norms.

Corollary 10. Let 𝜑(𝑡)=𝑡^𝑠log^𝛼₊𝑡, 𝑠≥1, 𝛼∈ℝ, and 𝑃 the potential operator applied to a differential form 𝑢∈WRH(∧^𝑙,Ω), 𝑙=1,2…,𝑛. Assume that 𝜑(|𝑢|)∈𝐿¹(Ω,𝜇). Then, there exists a constant 𝐶, independent of 𝑢, such that _Ω𝑃(𝑢)−(𝑃(𝑢))_𝐵0^𝑠log^𝛼₊𝑃(𝑢)−(𝑃(𝑢))_𝐵0𝑑𝜇≤𝐶_Ω|𝑢|^𝑠log^𝛼₊(|𝑢|)𝑑𝜇 (38) for any bounded 𝐿^𝜑(𝜇)-averaging domain Ω and 𝐵₀⊂Ω is some fixed ball.

Note that (38) can be written as the following version with the Luxemburg norm: 𝑃(𝑢)−(𝑃(𝑢))_{𝐵0𝐿𝑠(log𝛼+𝐿)(Ω)}≤𝐶‖𝑢‖_𝐿^𝑠_(log^𝛼_+𝐿)(Ω) (39) provided that the conditions in Corollary 10 are satisfied.

4. Applications

We have established the local and global weighted estimates for the potential operator applied to the differential forms in the WRH(∧^𝑙,Ω)-class. It is well known that any solution to 𝐴-harmonic equations belongs to the WRH(∧^𝑙,Ω)-class. Hence, our inequalities can be used to estimate solutions of 𝐴-harmonic equations. Next, as applications of the main theorems, we develop some estimates for the Jacobian 𝐽(𝑥,𝑓) of a mapping 𝑓∶Ω→ℝ^𝑛, 𝑓=(𝑓¹,…,𝑓^𝑛). We know that the Jacobian 𝐽(𝑥,𝑓) of a mapping 𝑓 is an 𝑛-form, specifically, 𝐽(𝑥,𝑓)𝑑𝑥=𝑑𝑓¹∧⋯∧𝑑𝑓^𝑛, where 𝑑𝑥=𝑑𝑥₁∧𝑑𝑥₂∧⋯∧𝑑𝑥_𝑛. For example, let 𝑓=(𝑓¹,𝑓²) be a differential mapping in ℝ². Then, 𝐽𝑓𝑥,𝑓𝑑𝑥∧𝑑𝑦=^1𝑥𝑓¹_𝑦𝑓²_𝑥𝑓²_𝑦=𝑓𝑑𝑥∧𝑑𝑦_1𝑥𝑓²_𝑦−𝑓¹_𝑦𝑓²_𝑥𝑑𝑥∧𝑑𝑦,𝑑𝑓¹∧𝑑𝑓²=𝑓¹_𝑥𝑑𝑥+𝑓¹_𝑦𝑑𝑦∧𝑓²_𝑥𝑑𝑥+𝑓²_𝑦𝑑𝑦=𝑓¹_𝑦𝑓²_𝑥𝑑𝑦∧𝑑𝑥+𝑓¹_𝑥𝑓²_𝑦=𝑓𝑑𝑥∧𝑑𝑦_1𝑥𝑓²_𝑦−𝑓¹_𝑦𝑓²_𝑥𝑑𝑥∧𝑑𝑦, (40) where we have used the property 𝑑𝑥_𝑖∧𝑑𝑥_𝑗=−𝑑𝑥_𝑗∧𝑑𝑥_𝑖 if 𝑖≠𝑗, and 𝑑𝑥_𝑖∧𝑑𝑥_𝑗=0 if 𝑖=𝑗. Clearly, 𝐽(𝑥,𝑓)𝑑𝑥∧𝑑𝑦=𝑑𝑓¹∧𝑑𝑓².

Let 𝑓∶Ω→ℝ^𝑛, 𝑓=(𝑓¹,…,𝑓^𝑛) be a mapping, whose distributional differential Df=[𝜕𝑓^𝑖/𝜕𝑥_𝑗]∶Ω→GL(𝑛) is a locally integrable function on 𝑀 with values in the space GL(𝑛) of all 𝑛×𝑛-matrices. We use 𝐽(𝑥)=𝑓𝑥,𝑓=detDf^1𝑥1𝑓¹_𝑥2𝑓¹_𝑥3⋯𝑓¹_𝑥𝑛𝑓²_𝑥1𝑓²_𝑥2𝑓²_𝑥3⋯𝑓²_𝑥𝑛𝑓⋮⋮⋮⋱⋮_𝑛𝑥1𝑓^𝑛_𝑥2𝑓^𝑛_𝑥3⋯𝑓^𝑛_𝑥𝑛 (41) to denote the Jacobian determinant of 𝑓. Assume that 𝑢 is the subdeterminant of Jacobian 𝐽(𝑥,𝑓), which is obtained by deleting the 𝑘 rows and 𝑘 columns, 𝑘=0,1,…,𝑛−1; that is, 𝑢=𝐽𝑥_𝑗1,𝑥_𝑗2,…,𝑥_{𝑗𝑛−𝑘};𝑓^𝑖1,𝑓^𝑖2,…,𝑓^{𝑖𝑛−𝑘}=𝑓^𝑖1_𝑥𝑗1𝑓^𝑖1_𝑥𝑗2𝑓^𝑖1_𝑥𝑗3⋯𝑓^𝑖1_{𝑥𝑗𝑛−𝑘}𝑓^𝑖2_𝑥𝑗1𝑓^𝑖2_𝑥𝑗2𝑓^𝑖2_𝑥𝑗3⋯𝑓^𝑖2_{𝑥𝑗𝑛−𝑘}𝑓⋮⋮⋮⋱⋮_{𝑖𝑛−𝑘𝑥𝑗1}𝑓^{𝑖𝑛−𝑘}_𝑥𝑗2𝑓^{𝑖𝑛−𝑘}_𝑥𝑗3⋯𝑓^{𝑖𝑛−𝑘}_{𝑥𝑗𝑛−𝑘}, (42) which is an (𝑛−𝑘)×(𝑛−𝑘) subdeterminant of 𝐽(𝑥,𝑓), {𝑖₁,𝑖₂,…,𝑖_𝑛−𝑘}⊂{1,2,…,𝑛} and {𝑗₁,𝑗₂,…,𝑗_𝑛−𝑘}⊂{1,2,…,𝑛}. Note that 𝐽(𝑥_𝑗1,𝑥_𝑗2,…,𝑥_{𝑗𝑛−𝑘};𝑓^𝑖1,𝑓^𝑖2,…,𝑓^{𝑖𝑛−𝑘})𝑑𝑥_𝑗1∧𝑑𝑥_𝑗2∧⋯∧𝑑𝑥_{𝑗𝑛−𝑘} is an (𝑛−𝑘)-form. Thus, all estimates for differential forms are applicable to the (𝑛−𝑘)-form 𝐽(𝑥_𝑗1,𝑥_𝑗2,…,𝑥_{𝑗𝑛−𝑘};𝑓^𝑖1,𝑓^𝑖2,…,𝑓^{𝑖𝑛−𝑘})𝑑𝑥_𝑗1∧𝑑𝑥_𝑗2∧⋯∧𝑑𝑥_{𝑗𝑛−𝑘}. For example, choosing 𝑢=𝐽(𝑥,𝑓)𝑑𝑥 and applying Theorems 7 and 9 to 𝑢, respectively, we have the following theorems.

Theorem 11. Let 𝜑 be a Young function in the class 𝐺(𝑝,𝑞,𝐶), 1≤𝑝<𝑞<∞,𝐶≤1. Let 𝑓=(𝑓¹,…,𝑓^𝑛)∶Ω→ℝ^𝑛 be a mapping such that 𝐽(𝑥,𝑓)𝑑𝑥∈WRH(∧^𝑛,Ω), where 𝐽(𝑥,𝑓) is the Jacobian of the mapping 𝑓 and Ω⊂ℝ^𝑛 is a bounded domain in ℝ^𝑛. Assume that 𝜑(|𝐽(𝑥,𝑓)|)∈𝐿¹_loc(Ω,𝜇). Then, there exists a constant 𝐶, independent of 𝐽(𝑥,𝑓), such that _𝐵𝜑𝑃𝐽𝑥,𝑓−𝑃𝐽𝑥,𝑓_𝐵𝑘𝑑𝑢≤𝐶_𝜎𝐵𝜑𝐽𝑥,𝑓𝑘𝑑𝑢 (43) for all balls 𝐵⊂Ω and some constant 𝜎>1.

Theorem 12. Let 𝜑 be a Young function in the class 𝐺(𝑝,𝑞,𝐶), 1≤𝑝<𝑞<∞,𝐶≤1. Let 𝑓=(𝑓¹,…,𝑓^𝑛)∶Ω→ℝ^𝑛 be a mapping such that 𝐽(𝑥,𝑓)𝑑𝑥∈WRH(∧^𝑛,Ω), where 𝐽(𝑥,𝑓) is the Jacobian of the mapping 𝑓 and Ω⊂ℝ^𝑛 is a bounded 𝐿^𝜑(𝜇)-averaging domain in ℝ^𝑛. Assume that 𝜑(|𝐽(𝑥,𝑓)|)∈𝐿¹(Ω,𝜇). Then, there exists a constant 𝐶, independent of 𝐽(𝑥,𝑓), such that _Ω𝜑𝑃𝐽𝑥,𝑓−𝑃𝐽𝑥,𝑓_𝐵0𝑘𝑑𝑢≤𝐶_Ω𝜑𝐽𝑥,𝑓𝑘𝑑𝑢 (44) for some ball 𝐵₀⊂Ω.