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 ℝn, n≥2, and ℝ=ℝ1. Let e1=(1,0,…,0), e2=(0,1,…,0),…, en=(0,0,…,1) be the standard unit basis of ℝn. For l=0,1,…,n, the linear space of l-vectors, spanned by the exterior products eI=ei1∧ei2∧⋯∧eil, corresponding to all ordered l-tuples I=(i1,i2,…,il), 1≤i1<i2<⋯<il≤n, is denoted by ∧l=∧l(ℝn). The Grassman algebra
(1)∧=∧(ℝn)=⊕l=0n∧l(ℝn)
is a graded algebra with respect to the exterior products. For α=∑αIeI∈∧ and β=∑βIeI∈∧, the inner product in ∧ is given by
(2)〈α,β〉=∑αIβI
with summation over all l-tuples I=(i1,i2,…,il) and all integers l=0,1,…,n. We should also notice that dxi∧dxj=-dxj∧dxi, i≠j, and dxi∧dxi=0.

Assume that B⊂ℝn is a ball and σB is the ball with the same center as B and with diam (σB)=σ diam (B). Differential forms are extensions of functions defined in ℝn. A function f(x1,…,xn) in ℝn is called a 0-form. A differential l-form is of the form
(3)w(x)=∑IwI(x)dxI=∑wi1i2⋯il(x)dxi1∧dxi2∧⋯∧dxil
in ℝn. 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⊆ ℝn is denoted by |E|. We call w a weight if w∈Lloc1(ℝn) and w>0 a.e. For 0<p<∞, we denote the weighted Lp norm of a measurable function f over E by
(4)∥f∥p,E,wα=(∫E|f(x)|pwαdx)1/p
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=αi1,i2,…,ik(x1,x2,…,xn)dxi1∧dxi2∧⋯∧dxik=αIdxI, i1<i2<⋯<ik, is a differential k-form, then
(5)⋆ω=⋆αi1i2⋯ikdxi1∧dxi2∧⋯∧dxik=(-1)∑(I)αIdxJ,
where I=(i1,i2,…,ik), J={1,2,…,n}-I, and
(6)∑(I)=k(k+1)2+∑j=1kij.

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⊂ ℝn 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
(7)supB(1|B|∫Bgαdx)(1|B|∫Bg-βdx)γ/β<∞,
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 Ar(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
(8)Pw(x)=∑I∫EK(x,y)wI(y)dy dxI,
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,
(9)Pf(x)=∫EK(x,y)f(y)dy,
where f(x) is a function defined on E⊂ℝn. A kernel K on ℝn× ℝn 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 ℝn 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:
(10)∥P(u)-(P(u))E∥p,E≤C|E|diam(E)∥u∥p,E,
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
(11)∥u∥s,B≤C|B|(t-s)/st∥u∥t,σB
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 ℝn and p,q>0 and 1/s=(1/p)+(1/q). Then
(12)(∫E|fg|sdx)1/s≤(∫E|f|pdx)1/p(∫E|g|qdx)1/q
for any E⊂ℝn.

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⊂ℝn 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
(13)∥P(u)-(P(u))B∥s,B,w≤C|B|diam(B)∥u∥s,B,w
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
(14)∥P(u)-(P(u))B∥s,b,w =(∫B|P(u)-(P(u))B|sw dx)1/s =(∫B(|P(u)-(P(u))|Bw1/s)s dx)1/s ≤(∫B|P(u)-(P(u))B|tdx)1/t ×(∫B(w1/s)st/(t-s) dx)(t-s)/st ≤C1∥B∥diam(B)∥u∥t,B(∫Bwα(x)dx)1/αs.
Set k=βs/(1+β), then 0<k<s. Since u is in the WRH(Ω) class,
(15)∥u∥t,B≤C2∥B∥(k-t)/kt∥u∥k,σ1B,
where σ1>1 in a constraint. Since u is in the WRH(Ω)-class again (note that 1/k=(1/s)+(s-k)/sk),
(16)∥u∥k,σ1B=(∫σ1B(|u|w1/ss-1/s)k dx)1/k≤(∫σ1B(|u|w1/sw-1/s)kdx)1/k×(∫σ1B(w-1/s)ks/(s-k)dx)(s-k)/ks=(∫σ1B|u|sw dx)1/s×(∫σ1Bw-k/(s-k)dx)(s-k)/ks=(∫σ1B|u|sw dx)1/s(∫σ1Bw-βdx)1/βs,
where we have used the following calculation:
(17)ks-k=βs/(1+β)s-(βs/(1+β))=β/(1+β)a-(β/(1+β))=β/(1+β)(1+β-β)/(1+β)=β.

Combining (14), (15) and, (16) gives
(18)∥P(u)-(P(u))B∥s,B,w ≤C3|B|diam(B)|B|(k-t)/kt(∫σ1B|u|sw dx)1/s ×(∫Bwαdx)1/αs(∫σ1Bw-βdx)1/βs.
Note that since w∈A(α,β,α;Ω), it follows that
(19)(∫Bwαdx)1/αs(∫σ1Bw-βdx)1/βs=((∫σ1Bwαdx)(∫σ1Bw-βdx)α/β)1/αs=(|σ1B|1+(α/β)(1|β|∫σ1Bwαdx)(1|β|∫σ1Bw-βdx)α/β)1/αs=C4|B|(1/αs)+(1/βs).
Plugging (19) into (18), we have
(20)∥P(U)-(P(u))B∥s,B,w ≤C5diam(B)|B|1+(1/t)-(1/k) ×(∫σ1B|u|sw dx)1/sC4|B|(1/αs)+(1/βs) ≤C6diam(B)|B|1+(1/t)-(1/k)+(1/αs)+(1/βs) ×(∫σ1B|u|sw dx)1/s.
We should notice that
(21)1+1t-1k+1αs+1βs=1+α-1sα-1+βsβ+1αs+1βs=1.
Combining (20) and (21) gives us
(22)∥P(u)-(P(u))B∥s,B,w≤C|B|diam(B)(∫σ1B|u|sw dx)1/s.
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 Lφ(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
(23)∥f∥φ(Ω,μ)=inf{λ>0:∫Ωφ(|f|λ)dμ≤1}
defines a norm in Lφ(Ω,μ), which is called the Orlicz norm or the Luxemburg norm.

Definition 6 (see [<xref ref-type="bibr" rid="B4">9</xref>]).
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(t1/p)/j(t)≤C and (ii) 1/C≤φ(t1/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
(24)C1tq≤h-1(φ(t))≤C2tq, C1tp≤j-1(φ(t))≤C2tp,
where C1 and C2 are constants. Also, for all 1≤p1<p<p2 and α∈ ℝ , the function φ(t)=tplog+αt 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)=tp 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|)∈Lloc1(Ω,μ). Then, there exists a constant M>0, independent of u, such that
(25)∫Bφ(|P(u)-(P(u))B|k)dx≤M∫σBφ(|u|k)dx
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
(26)∫Bφ(|P(u)-(P(u))B|k)dx =h(h-1(∫Bφ(|P(u)-(P(u))B|k)dx)) ≤h(∫Bh-1(φ(|P(u)-(P(u))B|k))dx) ≤h(C1∫B(|P(u)-(P(u))B|k)qdx) ≤C2φ((C1∫B(|P(u)-(P(u))B|k)qdx)1/q) ≤C2φ(1k(C1∫B(|P(u)-(P(u))B|)qdx)1/q) ≤C3φ(1k(∫B(|P(u)-(P(u))B|)qdx)1/q).

Since u∈WRH(Ω), by (11) we obtain
(27)(∫B|u|qdx)1/q≤C4|B|(p-q)/pq(∫σB|u|pdx)1/p,
where σ>1 is a constant. Using (13), (24), and Jensen's inequality,
(28)φ(1k(∫B|P(u)-(P(u))B|qdx)1/q) ≤φ(1k|B|1+(1/n)(∫B|u|qdx)1/q) ≤φ(1k|B|1+(1/n)C4|B|(1/q)-(1/p)(∫σB|u|pdx)1/p) =φ((1kpC4p|B|p(1+(1/n)+(1/q)-(1/p))∫σB|u|pdx)1/p) ≤C5j(1kpC4p|B|p(1+(1/n)+(1/q)-(1/p))∫σB|u|pdx) ≤C5j(∫σB1kpC4p|B|p(1+(1/n)+(1/q)-(1/p))|u|pdx) ≤C5∫σBj(1kpC4p|B|p(1+(1/n)+(1/q)-(1/p))|u|p)dx.
Note that p≥1, then 1+(1/n)+(1/q)-(1/p)>0. Thus,
(29)|B|1+(1/n)+(1/q)-(1/p)≤|Ω|1+(1/n)+(1/q)-(1/p)≤C6.
Using the above inequality and (i) in Definition 6, we find that j(t)≤C7φ(t1/p). Therefore,
(30)∫σBj(1kpC4p|B|p(1+(1/n)+(1/q)-(1/p))|u|p)dx ≤C7∫σBφ(1kC4|B|1+(1/n)+(1/q)-(1/p)|u|)dx ≤C7∫σBφ(C8|u|k)dx ≤C9∫σBφ(|u|k)dx.
Combining (28) and (30) yields
(31)φ(1k(∫B|P(u)-(P(u))B|qdx)1/q)≤C10∫σBφ(|u|k)dx.
Finally, substituting (31) into (26), we obtain
(32)∫Bφ(|P(u)-(P(u))B|k)dx≤M∫σBφ(|u|k)dx.
The proof of Theorem 7 has been completed.

3. Global Inequalities
In 1989, Staples introduced the following Ls-averaging domains in [10]. A proper subdomain Ω⊂ℝn is called an Ls-averaging domain, s≥1, if there exists a constant C such that
(33)(1|Ω|∫Ω|u-uΩ|sdx)1/s≤C supB⊂Ω(1|Ω|∫B|u-uB|sdx)1/s
for all u∈Llocs(Ω). Here the supremum is over all balls B⊂Ω. The Ls-averaging domains were extended into the Ls(μ)-averaging domains recently in [11]. We call a proper subdomain Ω⊂ℝn an Ls(μ)-averaging domain, s≥1, if μ(Ω)<∞ and there exists a constant C such that
(34)(1μ(Ω)∫Ω|u-uB0|sdμ)1/s ≤Csup4B⊂Ω(1μ(B)∫B|u-uB|sdμ)1/s
for some ball B0⊂Ω and all u∈L
loc
s(Ω;μ). The Ls(μ)-averaging domain was genralized into the following Lφ(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⊂ℝn an Lφ(μ)-averaging domain, if μ(Ω)<∞ and there exists a constant C such that
(35)1μ(Ω)∫Ωφ(τ|u-uB0|)dμ ≤Csup4B⊂Ω1μ(B)∫Bφ(σ|u-uB|)dμ
for some ball B0⊂Ω and all u such that φ(|u|)∈Lloc1(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 Ls(μ)-averaging domains are special Lφ(μ)-averaging domains when φ(t)=ts 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⊂ℝn any bounded Lφ(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
(36)∫Ωφ(1k|P(u)-(P(u))B0|)dμ≤C∫Ωφ(|u|k)dμ,
where B0⊂Ω is some fixed ball.

Proof.
From Definition 8, (25), and noticing that φ is doubling, we have
(37)∫Ωφ(|P(u)-(P(u))B0|)dμ ≤C1supB⊂Ω∫Bφ(|P(u)-(P(u))B|)dμ ≤C1supB⊂Ω(C2∫σBφ(|u|)dμ) ≤C1supB⊂Ω(C2∫Ωφ(|u|)dμ) ≤C3∫Ωφ(|u|)dμ.

We have completed the proof of Theorem 9.

Choosing φ(t)=tslog+αt in Theorem 9, we obtain the following Poincaré inequalities with the Ls(log+αL)-norms.

Corollary 10.
Let φ(t)=tslog+αt, 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
(38)∫Ω|P(u)-(P(u))B0|slog+α(|P(u)-(P(u))B0|)dμ ≤C∫Ω|u|slog+α(|u|)dμ
for any bounded Lφ(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:
(39)∥P(u)-(P(u))B0∥Ls(log+αL)(Ω)≤C∥u∥Ls(log+αL)(Ω)
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→ℝn, f=(f1,…,fn). We know that the Jacobian J(x,f) of a mapping f is an n-form, specifically, J(x,f)dx=df1∧⋯∧dfn, where dx=dx1∧dx2∧⋯∧dxn. For example, let f=(f1,f2) be a differential mapping in ℝ2. Then,
(40)J(x,f)dx∧dy=|fx1fy1fx2fy2|dx∧dy=(fx1fy2-fy1fx2)dx∧dy,df1∧df2=(fx1dx+fy1dy)∧(fx2dx+fy2dy)=fy1fx2dy∧dx+fx1fy2dx∧dy=(fx1fy2-fy1fx2)dx∧dy,
where we have used the property dxi∧dxj=-dxj∧dxi if i≠j, and dxi∧dxj=0 if i=j. Clearly, J(x,f)dx∧dy=df1∧df2.

Let f:Ω→ ℝn, f=(f1,…,fn) be a mapping, whose distributional differential Df=[∂fi/∂xj]:Ω→GL(n) is a locally integrable function on M with values in the space GL(n) of all n×n-matrices. We use
(41)J(x,f)= det Df(x)=|fx11fx21fx31⋯fxn1fx12fx22fx32⋯fxn2⋮⋮⋮⋱⋮fx1nfx2nfx3n⋯fxnn|
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,
(42)u=J(xj1,xj2,…,xjn-k; fi1,fi2,…,fin-k)=|fxj1i1fxj2i1fxj3i1⋯fxjn-ki1fxj1i2fxj2i2fxj3i2⋯fxjn-ki2⋮⋮⋮⋱⋮fxj1in-kfxj2in-kfxj3in-k⋯fxjn-kin-k|,
which is an (n-k)×(n-k) subdeterminant of J(x,f), {i1,i2,…,in-k}⊂{1,2,…,n} and {j1,j2,…,jn-k}⊂{1,2,…,n}. Note that J(xj1,xj2,…,xjn-k;fi1,fi2,…,fin-k)dxj1∧dxj2∧⋯∧dxjn-k is an (n-k)-form. Thus, all estimates for differential forms are applicable to the (n-k)-form J(xj1,xj2,…,xjn-k;fi1,fi2,…,fin-k)dxj1∧dxj2∧⋯∧dxjn-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,…,fn):Ω→ℝn 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⊂ℝn is a bounded domain in ℝn. Assume that φ(|J(x,f)|)∈Lloc1(Ω,μ). Then, there exists a constant C, independent of J(x,f), such that
(43)∫Bφ(|P(J(x,f))-(P(J(x,f)))B|kdu) ≤C∫σBφ(|J(x,f)|k)du
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,…,fn):Ω→ℝn 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⊂ℝn is a bounded Lφ(μ)-averaging domain in ℝn. Assume that φ(|J(x,f)|)∈L1(Ω,μ). Then, there exists a constant C, independent of J(x,f), such that
(44)∫Ωφ(|P(J(x,f))-(P(J(x,f)))B0|kdu) ≤C∫Ωφ(|J(x,f)|k)du
for some ball B0⊂Ω.