# Advances in Study of Poincaré Inequalities and Related Operators

Authors: Yuming Xing; Shusen Ding
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We will present an up-to-date account of the recent advances made in the study of Poincaré inequalities for differential forms and related operators.


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## Body

## 1. Introduction

The Poincaré inequalities have been playing an important role in analysis and related fields during the last several decades. The study and applications of Poincaré inequalities are now ubiquitous in different areas, including PDEs and potential analysis. Some versions of the Poincaré inequality with different conditions for various families of functions or differential forms have been developed in recent years. For example, in 1989, Staples in [1] proved the following Poincaré inequality for Sobolev functions in Ls-averaging domains. If D is an Ls-averaging domain, s≥n, then there exists a constant C, such that
(1)(1m(D)∫D‍|u-uD|sdm)1/s≤C(m(D))1/n(1m(D)∫D‍|∇u|sdm)1/s
for each Sobolev function u defined in D, where the integral is the Lebesgue integral, and m(D) is the Lebesgue measure of D; see [2–13] for more versions of the Poincaré inequality.Throughout this paper, we assume thatΩ is a domain in ℝn, n≥2, B and σB are the balls with the same center, and diam⁡(σB)=σdiam⁡(B), σ>0. We do not distinguish the balls from cubes, throughout this paper. We use |E| to denote the Lebesgue measure of the set E⊂ℝn. Differential forms are extensions of functions in ℝn. For example, the function u(x1,x2,…,xn) is called a 0-form. Moreover, if u(x1,x2,…,xn) is differentiable, then it is called a differential 0-form. The 1-form u(x) in ℝn can be written as u(x)=∑i=1n‍ui(x1,x2,…,xn)dxi. If the coefficient functions ui(x1,x2,…,xn), i=1,2,…,n, are differentiable, then u(x) is called a differential 1-form. Similarly, a differential k-form u(x) is generated by {dxi1∧dxi2∧⋯∧dxik}, k=1,2,…,n, that is, u(x)=∑I‍uI(x)dxI=∑‍ui1i2⋯ik(x)dxi1∧dxi2∧⋯∧dxik, where I=(i1,i2,…,ik), 1≤i1<i2<⋯<ik≤n. Let ∧l=∧l(ℝn) be the set of all l-forms in ℝn, D′(Ω,∧l) the space of all differential l-forms on Ω, and Lp(Ω,∧l) the l-forms u(x)=∑I‍uI(x)dxI on Ω satisfying ∫Ω‍|uI|p<∞ for all ordered l-tuples I, l=1,2,…,n. We denote the exterior derivative by d: D′(Ω,∧l)→D′(Ω,∧l+1) for l=0,1,…,n-1, and define the Hodge star operator ⋆: ∧k→∧n-k as follows. If u=ui1i2⋯ik(x1,x2,…,xn)dxi1∧dxi2∧⋯∧dxik=uIdxI, i1<i2<⋯<ik, is a differential k-form, then ⋆u=⋆(ui1i2⋯ikdxi1∧dxi2∧⋯∧dxik)=(-1)∑‍(I)uIdxJ, where I=(i1,i2,…,ik), J={1,2,…,n}-I, and ∑‍(I)=k(k+1)/2+∑j=1k‍ij. The Hodge codifferential operator d⋆: D′(Ω,∧l+1)→D′(Ω,∧l) is given by d⋆=(-1)nl+1⋆d⋆ on D′(Ω,∧l+1), l=0,1,…,n-1. We write ∥u∥s,E=(∫E‍|u|s)1/s and ∥u∥s,E,μ=(∫E‍|u|sdμ(x))1/s, where E⊂Ω, and μ(x) is the Radon measure. We use W1,p(Ω,∧k) to denote the Sobolev space of k-forms. For 0<p<∞ and the Radon measure μ(x), the Sobolev norm with Radon measure of u∈W1,p(Ω,∧k) over E⊂Ω is denoted by
(2)∥u∥W1,p(E,μ)=(diam⁡(E))-1∥u∥p,E,μ+∥∇u∥p,E,μ.
We consider here the solutions to the nonlinear partial differential equation
(3)d⋆A(x,du)=B(x,du),
which is called nonhomogeneous A-harmonic equation, where A: Ω×∧l(ℝn)→∧l(ℝn) and B: Ω×∧l(ℝn)→∧l-1(ℝn) satisfy the conditions |A(x,ξ)|≤a|ξ|p-1, A(x,ξ)·ξ≥|ξ|p and |B(x,ξ)|≤b|ξ|p-1, for almost every x∈Ω and all ξ∈∧l(ℝn). Here a,b>0 are constants and 1<p<∞ is a fixed exponent associated with (3). A solution to (3) is an element of the Sobolev space Wloc⁡1,p(Ω,∧l-1) such that ∫Ω‍A(x,du)·dφ+B(x,du)·φ=0 for all φ∈Wloc⁡1,p(Ω,∧l-1) with compact support. If u is a function (0-form) in ℝn, (3) reduces to
(4)div⁡A(x,∇u)=B(x,∇u).
If the operator B=0, (3) becomes d⋆A(x,du)=0, which is called the (homogeneous) A-harmonic equation. Let A:Ω×∧l(ℝn)→∧l(ℝn) be defined by A(x,ξ)=ξ|ξ|p-2 with p>1. Then, A satisfies the required conditions and d⋆A(x,du)=0 becomes the p-harmonic equation d⋆(du|du|p-2)=0 for differential forms. If u is a function (0-form), the above equation reduces to the usual p-harmonic equation div⁡(∇u|∇u|p-2)=0 for functions. See [8, 12–18] for recent results on the solutions to the different versions of the A-harmonic equation.LetD⊂ℝn be a bounded and convex domain. The linear operator Ky: C∞(D,∧l)→C∞(D,∧l-1) was first introduced in [19], and then it was generalized to the following version in [20]. For any y∈D, there exists a linear operator Ky:C∞(D,∧l)→C∞(D,∧l-1) defined by (Kyω)(x;ξ1,…,ξl-1)=∫01‍tl-1ω(tx+y-ty;x-y,ξ1,…,ξl-1)dt and the decomposition ω=d(Tω)+T(dω) holds. The homotopy operator T: C∞(D,∧l)→C∞(D,∧l-1) is defined by Tω=∫D‍φ(y)Kyωdy, averaging Ky over all points y in D, where φ∈C0∞(D) is normalized by ∫D‍φ(y)dy=1. The l-form ωD∈D′(D,∧l) is defined by ωD=|D|-1∫D‍ω(y)dy, l=0, and ωD=d(Tω),  l=1,2,…,n, for all ω∈Lp(D,∧l), 1≤p<∞. From [20], we know that, for any bounded and convex domain D, we have ∥∇(Tu)∥s,D≤C|D|∥u∥s,D and ∥Tu∥s,D≤C|D|diam⁡(D)∥u∥s,D. From [21], any open subset Q in ℝn is the union of a sequence of cubes Qk, whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances from F. More explicitly (i) Q=∪k=1∞Qk, (ii) Qj0∩Qk0=∅ if j≠k, (iii) there exist two constants c1,c2>0 (we can take c1=1 and c2=4), so that c1diam⁡(Qk)≤dist⁡(Qk,F)≤c2diam⁡(Qk). Hence, the definition of the homotopy operator T can be extended to any domain Q in ℝn. For any x∈Q, x∈Qk for some k, let TQk be the homotopy operator defined on Qk (each cube is bounded and convex). Thus, we can define the homotopy operator TQ on any domain Q by TQ=∑k=1∞‍TQkχQk. Hence, for any bounded domain Q and any differential form u∈Lloc⁡s(Q,∧l), we have
(5)∥∇(Tu)∥s,Q≤C|Q|∥u∥s,Q,∥Tu∥s,Q≤C|Q|diam⁡(Q)∥u∥s,Q,
where C is a constant, independent of u, and l=1,2,…,n, 1<s<∞.We begin the discussion with the following definitions and weak reverse Hölder inequality in [22], which will be used repeatedly later.Definition 1 (see [2]).
A weightw satisfies Ar(Ω)-condition in a subset Ω⊆ℝn, where r>1, and write w∈Ar(Ω) when
(6)sup⁡B(1|B|∫B‍wdx)(1|B|∫B‍w1/(1-r)dx)r-1<∞,
where supremum is over all B⊂Ω.Definition 2 (see [10]).
A pair of weights(w1,w2) satisfy the Arλ(E)-condition in a set E⊂ℝn, and write (w1,w2)∈Arλ(E) for some r>1 and λ>0, if
(7)sup⁡B(1|B|∫B‍w1dx)(1|B|∫B‍(1w2)1/(1-r)dx)r-1<∞
for any ball B⊂E.Lemma 3.
Letu be a solution of the nonhomogeneous A-harmonic equation (3) in a domain Ω and 0<s,t<∞. Then, there exists a constant C, independent of u, such that
(8)∥u∥s,B≤C|B|(t-s)/st∥u∥t,σB
for all balls or cubes B with σB⊂Ω for some σ>1.


## 2. Poincaré Inequalities for Differential Forms

We first discuss the Poincaré inequality for some differential forms. These forms are not necessary to be the solutions of any version of theA-harmonic equation.Definition 4 (see [7]).
We call a proper subdomainΩ⊂ℝn an Ls(μ,0)-averaging domain, s≥1, if μ(Ω)<∞ and there exists a constant C such that
(9)(1μ(B0)∫Ω‍|u-uB0|sdμ)1/s≤Csup⁡2B⊂Ω(1μ(B)∫B‍|u-uB|sdμ)1/s
for some ball B0⊂Ω and all u∈Lloc⁡s(Ω;∧0,μ). Here the measure μ is defined by dμ=w(x)dx, where w(x) is a weight and w(x)>0 a.e., and the supremum is over all balls B with 2B⊂Ω.
In 1993, the following Poincaré-Sobolev inequality was proved in [20], which can be used to generalize the theory of Sobolev functions to that of differential forms.Theorem 5.
Letu∈D′(Q,∧l) and du∈Lp(Q,∧l+1). Then, u-uQ is in Lnp/(n-p)(Q,∧l) and
(10)(∫Q‍|u-uQ|np/(n-p)dx)(n-p)/np≤Cp(n)(∫Q‍|du|pdx)1/p
for Q a cube or a ball in ℝn, l=0,1,…,n-1, and 1<p<n.From Corollary 4.1 in [20], we have the following version of Poincaré inequality for differential forms.Theorem 6.
Letu∈D′(Q,∧l) and du∈Lp(Q,∧l+1). Then, u-uQ is in W1,p(Q,∧l) with 1<p<∞ and
(11)∥u-uQ∥p,Q≤C(n,p)|Q|1/n∥du∥p,Q
for Q a cube or a ball in ℝn, l=0,1,…,n-1.The above Poincaré-Sobolev inequalities are about differential forms. We know that theA-harmonic tensors are differential forms that satisfy the A-harmonic equation. Then naturally, one would ask whether the Poincaré-Sobolev inequalities for A-harmonic tensors are sharper than those for differential forms. The answer is “yes”. In [8], Ding and Nolder proved the following symmetric Poincaré-Sobolev inequalities for solutions of the nonhomogeneous A-harmonic equation (3).Theorem 7.
Letu∈D′(Ω,∧l) be a solution of the nonhomogeneous A-harmonic equation (3) in a domain Ω⊂ℝn and du∈Ls(Ω,∧l+1), l=0,1,…,n-1. Assume that σ>1, 0<s<∞, and w∈Ar for some r>1. Then
(12)∥u-uB∥s,B,w≤C|B|1/n∥du∥s,σB,w
for all balls B with σB⊂Ω. Here C is a constant independent of u and du.Note that (12) is equivalent to
(12)′(1μ(B)∫B‍|u-uB|sdμ)1/s≤C|B|1/n(1μ(B)∫σB‍|du|sdμ)1/s.Theorem 8.
Letu∈D′(Ω,∧l) be a solution of the nonhomogeneous A-harmonic equation (3) in a domain Ω⊂ℝn and du∈Ls(Ω,∧l+1), l=0,1,…,n-1. Assume that σ>1, 0<α≤1, 1+α(r-1)<s<∞, and w∈Ar for some r>1. Then
(13)∥u-uB∥s,B,wα≤C|B|1/n∥du∥s,σB,wα
for all balls B with σB⊂Ω. Here C is a constant independent of u and du.Note that (13) can be written as
(13)′(1|B|∫B‍|u-uB|swαdx)1/s≤C|B|1/n(1|B|∫σB‍|du|swαdx)1/s.Next, we will prove the following global weighted Poincaré-Sobolev inequality inLs(μ)-averaging domains.Theorem 9.
Letw∈Ar with w≥τ>0, r>1, where τ is a constant. Assume that u∈D′(Ω,∧0) is an A-harmonic tensor and du∈Ls(Ω,∧1); then
(14)(1μ(Ω)∫Ω‍|u-uB0|sdμ)1/s≤C(∫Ω‍|du|sdμ)1/s
for any Ls(μ)-averaging domain Ω and some ball B0 with 2B0⊂Ω. Here the measure μ is defined by dμ=w(x)dx and C is a constant independent of u.Clearly, we can write (14) as
(15)∥u-uB0∥s,Ω,w≤Cμ(Ω)1/s∥du∥s,Ω,w.In [13], we have obtained Poincaré inequalities in which the integral on one side is about Lebesgue measure, but on the other side, the integral is about general measure induced by a weight w(x). We state these results in the following.Theorem 10.
Letu∈D′(Ω,∧l) be an A-harmonic tensor in a domain Ω⊂ℝn and du∈Llocs(Ω,∧l+1), l=0,1,…,n-1. Assume that σ>1, 1<s<∞, and w∈Ar for some r>1. Then, there exists a constant C, independent of u, such that
(16)(1μ(B)∫B‍|u-uB|sdμ)1/s≤C|B|1/n(1|B|∫σB‍|du|sdx)1/s
for all balls B with σB⊂Ω. Here, the measure μ is defined by dμ=w(x)dx.Theorem 11.
Letu∈D′(Ω,∧l) be an A-harmonic tensor in a domain Ω⊂ℝn and du∈Llocs(Ω,∧l+1), l=0,1,…,n-1. Assume that σ>1, 1<s<∞, and w∈Ar for some r>1. Then
(17)(1|B|∫B‍|u-uB|sdx)1/s≤C|B|1/n(1μ(σB)∫σB‍|du|sdμ)1/s
for all balls B with σB⊂Ω. Here, the measure μ is defined by dμ=w(x)dx and C is a constant independent of u and du.Theorem 12.
Letw∈Ar for some r>1, u∈D′(Ω,∧0), and du∈Ls(Ω,∧1). If s≥n, then
(18)(1μ(Ω)∫Ω‍|u-uB0|sdμ)1/s≤C|Ω|1/n(1|Ω|∫Ω‍|du|sdx)1/s
for any Ls(μ,0)-averaging domain Ω with μ(Ω)<∞ and some ball B0 with 2B0⊂Ω. Here, the measure μ is defined by dμ=w(x)dx and C is a constant independent of u and du.Theorem 13.
Letw∈Ar with w(x)≥α>0, r>1, u∈D′(Ω,∧0), and du∈Ls(Ω,∧1). If s≥n, then
(19)(1|Ω|∫Ω‍|u-uB0|sdx)1/s≤C|Ω|1/n(1|Ω|∫Ω‍|du|sdμ)1/s
for any Ls-averaging domain Ω and some ball B0 with 2B0⊂Ω. Here, the measure μ is defined by dμ=w(x)dx and C is a constant independent of u.In recent years, several versions of the two weight Poincaré inequalities have been developed; see [13, 23–25] for example.Theorem 14.
Letu∈D′(B,∧l) and du∈Lt(B,∧l+1), l=0,1,…,n-1. Then, there exists a constant β>1 such that if w1∈Ar1 and (w1,w2)∈At/ss/t, where 1<s<n, t=sβ, and r>1, we have
(20)(1|B|∫B‍|u-uB|sw1dx)1/s≤C(∫B‍|du|tw2dx)1/t
for all balls B⊂ℝn. Here C is a constant independent of u and du.Theorem 15.
Letu∈D′(B,∧l) and du∈Ln(B,∧l+1), l=0,1,…,n-1. If 1<s<n and (w1,w2)∈An/s1, then there exists a constant C, independent of u and du, such that
(21)(1|B|∫B‍|u-uB|sw1s/ndx)1/s≤C(∫B‍|du|nw2dx)1/n
for any ball or cube B⊂ℝn.We remark that the exponentst and n on the right hand sides of (20) and (21) can be improved. In fact, the following result is with the sharper right-hand side.Theorem 16.
Letu∈D′(B,∧l) and du∈Lt(B,∧l+1), l=0,1,…,n-1. Then, there exists a constant β>1 such that if w1∈Ar1 and (w1,w2)∈Arαs/t, where 1<s<n, t=s+αs(r-1) and r>1, we have
(22)(1|B|β∫B‍|u-uB|sw1dx)1/s≤C|B|α(r-1)/t(∫B‍|du|tw2αdx)1/t
for all balls B⊂ℝn and any constant α>0. Here C is a constant independent of u and du.Clearly, in this result ifα→0, then t→s.Theorem 17.
Letu∈D′(Ω,∧0) and du∈Lt(Ω,∧1). Then there exists a constant β>1 such that if w1∈Ar1 and (w1,w2)∈At/ss/t, where 1<s<n, t=sβ, r>1, and w1>w2≥η>0, we have
(23)(1μ1(Ω)∫Ω‍|u-uB0|sdμ1)1/s≤C(∫Ω‍|du|tdμ2)1/t
for any Ls(μ1)-averaging domain Ω and some ball B0 with 2B0⊂Ω. Here the measures μ1 and μ2 are defined by dμ1=w1(x)dx, dμ2=w2(x)dx and C is a constant independent of u and du.Remark 18.
(1) Theorems15 and 16 can be extended to the global versions. (2) From [26], we know that John domains are Ls(μ)-averaging domains. Thus, the global results and Theorem 17 also hold if Ω is a John domain.
Next, we discuss the following version of two-weight Poincaré inequality for differential forms.Theorem 19.
Letu∈D′(Ω,∧l) be a differential form satisfying the A-harmonic equation (3) in a domain Ω⊂ℝn and du∈Ls(Ω,∧l+1), l=0,1,…,n-1. Suppose that (w1,w2)∈Arλ(Ω) for some r>1 and λ>0. If 0<α<1,σ>1 and s>αλ(r-1)+1, then there exists a constant C, independent of u, such that
(24)(∫B‍|u-uB|sw1αdx)1/s≤C|B|1/n(∫σB‍|du|sw2αλdx)1/s
for all balls B with σB⊂Ω. Here uB is a closed form.If we chooseλ=1/α in Theorem 19, we get the following version of the Arλ-weighted Poincaré inequality.Corollary 20.
Letu∈D′(Ω,∧l) be a differential form satisfying the A-harmonic equation (3) in a domain Ω⊂ℝn and du∈Ls(Ω,∧l+1), l=0,1,…,n-1. Suppose that (w1,w2)∈Ar1/α(Ω) for some r>1 and 0<α<1. If s>r, then there exists a constant C, independent of u, such that
(25)(∫B‍|u-uB|sw1αdx)1/s≤C|B|1/n(∫σB‍|du|sw2dx)1/s
for all balls B with σB⊂Ω. Here uB is a closed form.Selectingα=1/s in Theorem 19, we have the following two-weighted Poincaré inequality.Corollary 21.
Letu∈D′(Ω,∧l) be a differential form satisfying the A-harmonic equation (3) in a domain Ω⊂ℝn and du∈Ls(Ω,∧l+1), l=0,1,…,n-1. Suppose that (w1,w2)∈Arλ(Ω) for some r>1 and λ>0. If σ>1 and s>λ(r-1)/s+1, then there exists a constant C, independent of u, such that
(26)(∫B‍|u-uB|sw11/sdx)1/s≤C|B|1/n(∫σB‍|du|sw2λ/sdx)1/s
for all balls B with σB⊂Ω. Here uB is a closed form.Whenλ=1 in Corollary 21, we obtain the following symmetric two-weighted inequality.Corollary 22.
Letu∈D′(Ω,∧l) be a differential form satisfying the A-harmonic equation (3) in a domain Ω⊂ℝn and du∈Ls(Ω,∧l+1), l=0,1,…,n-1. Suppose that (w1,w2)∈Ar(Ω) for some r>1. If σ>1 and s>(r-1)/s+1, then there exists a constant C, independent of u, such that
(27)(∫B‍|u-uB|sw11/sdx)1/s≤C|B|1/n(∫σB‍|du|sw21/sdx)1/s
for all balls B with σB⊂Ω. Here uB is a closed form.


## 3. Poincaré Inequalities with the Radon Measure

Normally, most of these inequalities are developed with the Lebesgue measure. It is noticeable that the following results from [27] established the Poincaré inequalities with Radon measure. The Radon measure μ(x) is induced by dμ=g(x)dx, where g(x) may be an unbounded function. For example, it is allowed that g(x) contains a singular factor 1/|x-x0|α; here α>0 is a constant and x0 is some fixed point in the integral domain. We are interested in the singular factor case because normally we have to deal with the singular factor in applications, such as in the estimating of the Cauchy operator.We first introduce the following lemmas that will be used to prove the local Poincaré inequality with the Radon measure.Lemma 23.
Let0<α<∞, 0<β<∞, and s-1=α-1+β-1. If f and g are measurable functions on ℝn, then
(28)∥fg∥s,E≤∥f∥α,E·∥g∥β,E
for any E⊂ℝn.Now, we prove the following local Poincaré inequality with the Radon measure which will be used to establish the global inequality.Theorem 24.
Letu∈Llocs(M,∧l) be a solution of the nonhomogeneous A-harmonic equation (3) in a bounded domain M, du∈Llocs(M,∧l+1), l=0,1,…,n-1, and 1<s<∞. Then, for any ball B with σB⊂M, there exists a constant C, independent of u, such that
(29)(∫B‍|u-uB|sdμ)1/s≤C|B|γ(∫σB‍|du|sdν)1/s,
where the Radon measures μ and ν are induced by dμ=g(x)dx and dν=h(x)dx, respectively, with g,h∈Lloc1(M), 0<g(x)≤K1/|x-xB|α, and K2/|x-xB|λ≤h(x). Here K1,K2,α,λ, and σ are some constants with K1>0, K2>0, n>α>λ, σ>1, and γ=1+1/n-(α-λ)/ns; xB is the center of ball B.Proof.
Assume thatε∈(0,1) is small enough so that εn<α-λ and B⊂M is any ball with center xB and radius rB. Also, let δ>0 be small enough, Bδ={x∈B:|x-xB|≤δ} and Dδ=B/Bδ. For any differential forms u, we have u=d(Tu)+T(du)=uB+T(du), where d is the exterior differential operator and T is the homotopy operator. From (5), we obtain
(30)∥u-uB∥s,B=∥T(du)∥s,B≤C1|B|diam⁡(B)∥du∥s,B.
Since 0<g(x)≤K1/|x-xB|α, it follows that
(31)(∫Dδ‍|u-uB|sdμ)1/s=(∫Dδ‍|u-uB|sg(x)dx)1/s≤(∫Dδ‍|u-uB|sK1|x-xB|αdx)1/s.
Choose t=s/(1-ε); then t>s. Select β=t/(t-s). By the Hölder inequality, (30) and (31), we obtain
(32)(∫Dδ‍|u-uB|sdμ)1/s≤(∫Dδ‍|u-uB|sK1|x-xB|αdx)1/s=(∫Dδ‍(|u-uB|C2|x-xB|α/s)sdx)1/s≤∥u-uB∥t,Dδ(∫Dδ‍(C2|x-xB|)tα/(t-s)dx)(t-s)/st=∥u-uB∥t,Dδ(∫Dδ‍C3|x-xB|-αβdx)1/βs≤∥u-uB∥t,B·∥|x-xB|-α∥β,Dδ1/s≤C4|B|diam⁡(B)∥du∥t,B·∥|x-xB|-α∥β,Dδ1/s.
We may suppose that xB=0. Otherwise, we can move the center to the origin by a simple transformation. Thus, for any x∈B, |x-xB|≥|x|-|xB|=|x|. Using the polar coordinate substitution, we find that
(33)∥|x-xB|-α∥β,Dδ1/s=(∫Dδ‍|x-xB|-αβdx)1/sβ≤(C5∫δrB‍ρ-αβρn-1dρ)1/sβ=|C5n-αβ(rBn-αβ-δn-αβ)|1/sβ≤C6|rBn-αβ-δn-αβ|1/sβ.
Set m=nst/(ns+αt-λt); then 0<m<s. From Lemma 3, we have
(34)∥du∥t,B≤C7|B|(m-t)/mt∥du∥m,σB,
where σ>1 is a constant. Using the Hölder inequality again, we obtain
(35)∥du∥m,σB=(∫σB‍(|du|(h(x))1/s(h(x))-1/s)mdx)1/m≤(∫σB‍|du|sh(x)dx)1/s×(∫σB‍((1h(x))1/s)ms/(s-m)dx)(s-m)/ms≤(∫σB‍|du|sh(x)dx)1/s(∫σB‍(1h(x))m/(s-m)dx)(s-m)/ms≤(∫σB‍|du|sh(x)dx)1/s×(∫σB‍(|x-xB|λK2)m/(s-m)dx)(s-m)/ms≤(∫σB‍|du|sh(x)dx)1/sC8(σrB)λ/s+n(s-m)/ms≤C9(∫σB‍|du|sdν)1/s(rB)λ/s+n(s-m)/ms.
By a simple calculation, we find that n-αβ+λβ+nβ(s-m)/m=0. Substituting (33), (34), and (35) into (32) yields
(36)(∫Dδ‍|u-uB|sdμ)1/s≤C10|B|1+1/n+(m-t)/mt(∫σB‍|du|sdν)1/s×rBλ/s+n(s-m)/ms|rBn-αβ-δn-αβ|1/sβ=C10|B|1+1/n+(m-t)/mt(∫σB‍|du|sdν)1/s×[rB(λ/s+n(s-m)/ms)sβ|rBn-αβ-δn-αβ|]1/sβ=C10|B|1+1/n+(m-t)/mt(∫σB‍|du|sdν)1/s×|C11rBn-αβ+λβ+nβ(s-m)/m-δn-αβrBλβ+nβ(s-m)/m|1/sβ≤C10|B|1+1/n+(m-t)/mt(∫σB‍|du|sdν)1/s×[C11rBn-αβ+λβ+nβ(s-m)/m-δn-αβδλβ+nβ(s-m)/m]1/sβ≤C10|B|1+1/n+(m-t)/mt(∫σB‍|du|sdν)1/s×[C11rBn-αβ+λβ+nβ(s-m)/m+δn-αβ+λβ+nβ(s-m)/m]1/sβ≤C12|B|1+1/n-(α-λ)/ns(∫σB‍|du|sdν)1/s≤C12|B|γ(∫σB‍|du|sdν)1/s,
that is,
(37)(∫Dδ‍|u-uB|sdμ)1/s≤C12|B|γ(∫σB‍|du|sdν)1/s.
Notice that lim⁡δ→0⁡(∫Dδ‍|u-uB|sdμ)1/s=(∫B‍|u-uB|sdμ)1/s. Letting δ→0 in (37), we obtain (29). The proof of Theorem 24 has been completed.Letg(x)=1/|x-xB|α and h(x)=1/|x-xB|λ in Theorem 24, where α and λ are constants with α>λ. We have the following version of the Poincaré inequality with the Radon measures.Corollary 25.
Letu∈Llocs(M,∧l) be a solution of the nonhomogeneous A-harmonic equation (3) in a bounded domain M, du∈Llocs(M,∧l+1), l=0,1,…,n-1, and 1<s<∞. Then, there exists a constant C, independent of u, such that
(38)∥u-uB∥s,B,μ≤C|B|γ∥du∥s,σB,ν
for all balls B with σB⊂M, σ>1, where the Radon measures μ and ν are induced by dμ=g(x)dx and dν=h(x)dx, respectively, with g(x)=1/|x-xB|α and h(x)=1/|x-xB|λ. Here α and λ are some constants with α>λ, and γ=1+1/n-(α-λ)/ns; xB is the center of ball B.Letu∈Lloc⁡s(M,∧0) be a solution of (4). From (2), we have
(39)∥u-uB∥W1,s(B,μ)=(diam⁡(B))-1∥u-uB∥s,B,μ+∥∇(u-uB)∥s,B,μ
for any ball B⊂M. Note that ∥∇u∥s,B,μ=∥du∥s,B,μ and ∥d(uB)∥s,B,μ=0. Hence,
(40)∥∇(u-uB)∥s,B,μ=∥d(u-uB)∥s,B,μ≤∥du∥s,B,μ+∥duB∥s,B,μ=∥du∥s,B,μ.
Substituting (40) into (39) and using (29) and the fact that diam⁡(B)=C1|B|1/n for some constant C1>0, we have
(41)∥u-uB∥W1,s(B,μ)=C2|B|-1/n∥u-uB∥s,B,μ+∥du∥s,B,μ≤C3|B|-1/n|B|1+1/n-(α-λ)/ns∥du∥s,σ1B,ν+∥du∥s,B,μ≤C3|B|1-(α-λ)/ns∥du∥s,σ1B,ν+∥du∥s,B,μ.
Using the same method as we developed in the proof of Theorem 24, we have
(42)∥du∥s,B,μ≤C4|B|(λ-α)/ns∥du∥s,σ2B,ν.
Combining (41) and (42) and noticing that M is bounded and B⊂M, we find that
(43)∥u-uB∥W1,s(B,μ)≤(C3|B|+C4)|B|(λ-α)/ns∥du∥s,σB,ν≤C5|B|(λ-α)/ns∥du∥s,σB,ν,
where σ=max⁡{σ1,σ2}. Hence, we obtain the following Sobolev-Poincaré imbedding inequality with the Radon measure.Corollary 26.
Letu∈Llocs(M,∧0) be a solution of (4) and all other conditions in Theorem 24 are satisfied. Then, there exists a constant C, independent of u, such that
(44)∥u-uB∥W1,s(B,μ)≤C|B|(λ-α)/ns∥du∥s,σB,ν
for all balls B with σB⊂M, σ>1.Then, we will prove the global Poincaré inequalities with the Radon measures in the following statement. We firstly introduce the definition of John domains and the Lemma.Definition 27.
A proper subdomainΩ⊂ℝn is called a δ-John domain, δ>0, if there exists a point x0∈Ω, which can be joined with any other point x∈Ω by a continuous curve γ⊂Ω, so that
(45)d(ξ,∂Ω)≥δ|x-ξ|
for each ξ∈γ. Here d(ξ,∂Ω) is the Euclidean distance between ξ and ∂Ω.Lemma 28 (see [16] (Covering Lemma)).
EachΩ has a modified Whitney cover of cubes 𝒱={Qi} such that ∪iQi=Ω,∑Qi∈𝒱‍χ(5/4)Qi≤NχΩ and 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 Ω is δ-John, then there is a distinguished cube Q0∈𝒱 which can be connected with every cube Qm∈𝒱 by a chain of cubes Q0,Q1,…,Qk=Qm from 𝒱 and such that Qm⊂ρQi, i=0,1,2,…,k, for some ρ=ρ(n,δ).Theorem 29.
LetΩ be any bounded and convex δ-John domain Ω⊂ℝn and let u∈Ls(Ω,∧0) be a solution of the nonhomogeneous A-harmonic equation (4), du∈Ls(Ω,∧1), and 1<s<∞. Then, there exists a constant C, independent of u, such that
(46)(∫Ω‍|u-uQ0|sdμ)1/s≤C|Ω|γ(∫Ω‍|du|sdν)1/s,
where the Radon measures μ and ν are induced by dμ=g(x)dx and dν=h(x)dx, respectively, with g,h∈L1(Ω), 0<g(x)≤K1/dα(x,∂Ω), and ∑i‍χQiK2/|x-xQi|λ≤h(x), xQi is the center of Qi with Ω=∪iQi, and γ=1+1/n-(α-λ)/ns. Here K1,K2,α,λ are some constants with K1,K2>0, λ<α<min⁡{n,s+λ+n(s-1)}, and the fixed cube Q0⊂Ω, the constant N>1, and the cubes Qi appeared in Lemma 28.Proof.
We may assumeg(x)≥1 a.e. Otherwise, let Ω1=Ω∩{x∈Ω:0<g(x)<1} and Ω2=Ω∩{x∈Ω:g(x)≥1}; then Ω=Ω1∪Ω2. We define the new function G(x) by
(47)G(x)={1,x∈Ω1g(x),x∈Ω2.
Also, we choose the constant K1≥diam⁡α(Ω); then K1/dα(x,∂Ω)≥1 for any x∈Ω. Therefore, G(x)≥g(x), and G(x) satisfies all conditions required for g(x), particularly, 0<G(x)≤K1/dα(x,∂Ω) and
(48)(∫Ω‍|u-uQ0|sdμ)1/s=(∫Ω‍|u-uQ0|sg(x)dx)1/s≤(∫Ω‍|u-uQ0|sG(x)dx)1/s
with G(x)≥1. Hence, we may suppose that g(x)≥1 a.e. and have
(49)μ(Q)=∫Q‍dμ=∫Q‍g(x)dx≥∫Q‍dx=|Q|.
We use the notation appearing in Lemma 28. There is a modified Whitney cover of cubes 𝒱={Qi} for Ω such that Ω=∪Qi, and ∑Qi∈𝒱‍χ(5/4)Qi≤NχΩ for some N>1. Since Ω=∪Qi, for any x∈Ω, it follows that x∈Qi for some i. It is easy to check that all conditions in Theorem 24 are satisfied. Applying Theorem 24 to Qi, we obtain
(50)(∫Qi‍|u-uQi|sdμ)1/s≤C1|Qi|γ(∫σQi‍|du|sdν)1/s,
where σ>1 is a constant. Using the elementary inequality (a+b)s≤2s(|a|s+|b|s), s≥0, we have
(51)(∫Ω‍|u-uQ0|sdμ)1/s=(∫∪Qi‍|u-uQ0|sdμ)1/s≤(∑Qi∈𝒱‍(2s∫Qi‍|u-uQi|sdμ+2s∫Qi‍|uQi-uQ0|sdμ))1/s≤C2(s)((∑Qi∈𝒱‍∫Qi‍|u-uQi|sdμ)1/s+(∑Qi∈𝒱‍∫Qi‍|uQi-uQ0|sdμ)1/s)
for a fixed Q0⊂Ω. The first sum in (51) can be estimated by using Theorem 24 and the Covering Lemma
(52)(∑Qi∈𝒱‍∫Qi‍|u-uQi|sdμ)1/s≤(∑Qi∈𝒱‍C3|Qi|sγ∫σQi‍|du|sdν)1/s≤(C3|Ω|sγ∑Qi∈𝒱‍∫σQi‍|du|sdν)1/s≤C4|Ω|γ(∫Ω‍|du|sdν)1/s.
We use the properties of δ-John domain to estimate the second sum in (51). Fix a cube Qm∈𝒱 and let Q0,Q1,…,Qk=Qm be the chain in Lemma 28. Consider
(53)|uQm-uQ0|≤∑i=0k-1‍|uQi-uQi+1|.
The chain {Qi} also has property that, for each i, i=0,1,…,k-1, with Qi∩Qi+1≠∅, there exists a cube Di such that Di⊂Qi∩Qi+1 and Qi∪Qi+1⊂NDi, N>1. Consider
(54)max⁡{|Qi|,|Qi+1|}|Qi∩Qi+1|≤max⁡{|Qi|,|Qi+1|}|Di|≤C5.
For such Dj, j=0,1,…,k-1, Let |D|=min⁡{|D0|,|D1|,…,|Dk-1|}; then
(55)max⁡{|Qi|,|Qi+1|}|Qi∩Qi+1|≤max⁡{|Q0|,|Q1|,…,|Qk|}|D|≤C6.
By (49), (53), (55), and (50), we have
(56)|uQi-uQi+1|s=1μ(Qi∩Qi+1)∫Qi∩Qi+1‍|uQi-uQi+1|sdμ≤1|Qi∩Qi+1|∫Qi∩Qi+1‍|uQi-uQi+1|sdμ≤C7max⁡{|Qi|,|Qi+1|}∫Qi∩Qi+1‍|uQi-uQi+1|sdμ≤C8∑j=ii+1‍1|Qj|∫Qj‍|u-uQj|sdμ≤C9∑j=ii+1‍|Qj|γs|Qj|∫σQj‍|du|sdν=C9∑j=ii+1‍|Qj|γs-1∫σQj‍|du|sdν.
Since Qm⊂ρQj for j=i, i+1, 0≤i≤k-1, and γs-1>0 when λ<α<min⁡{n,s+λ+n(s-1)}, using (56), we find that
(57)|uQi-uQi+1|sχQm(x)≤C10∑j=ii+1‍χρQj(x)|Qj|γs-1∫σQj‍|du|sdν≤C11∑j=ii+1‍χρQj(x)|Ω|γs-1∫σQj‍|du|sdν≤C11|Ω|γs-1∑j=ii+1‍χρQj(x)∫σQj‍|du|sdν.
Taking the sth root both sides in (57) and using (a+b)1/s≤21/s(|a|1/s+|b|1/s), (53), and (55), we obtain
(58)|uQm-uQ0|χQm(x)≤C12|Ω|γ-1/s∑Di∈𝒱‍(∫σDi‍|du|sdν)1/s·χρDi(x)
for every x∈ℝn. Raising both sides of inequality (58) to s powers and then integrating over ℝn both sides, we have
(59)∑Qm∈𝒱‍∫Qm‍|uQm-uQ0|sdμ≤C13|Ω|γs-1∫ℝn‍|∑Di∈𝒱‍(∫σDi‍|du|sdν)1/sχρDi(x)|sdμ.
Notice that
(60)∑Di∈𝒱‍χρDi(x)≤∑Di∈𝒱‍χρNDi(x)≤NχΩ(x).
Using elementary inequality |∑i=1M‍ti|s≤Ms-1∑i=1M‍|ti|s for s>1, we finally have
(61)(∑Qm∈𝒱‍∫Qm‍|uQm-uQ0|sdμ)1/s≤C14|Ω|γ-1/s×(∫ℝn‍(∑Di∈𝒱‍(∫σDi‍|du|sdν)χρDi(x))dμ)1/s=C15|Ω|γ-1/s(∑Di∈𝒱‍(∫σDi‍|du|sdν))1/s≤C16|Ω|γ(∫Ω‍|du|sdν)1/s.
Substituting (52) and (61) into (51), we obtain (46). The proof of Theorem 29 has been completed.Similarly, chooseg(x)=1/dα(x,∂Ω) and h(x)=∑i‍χQi(K/|x-xQi|λ) in Theorem 29, where xQi is the center of Qi with Ω=∪iQi, α>λ, and K>0. We have the following version of the Poincaré inequality with the Radon measures.Corollary 30.
Letu∈Ls(Ω,∧0) be a solution of the nonhomogeneous A-harmonic equation (4), du∈Ls(Ω,∧1), 1<s<∞. Then, there exists a constant C, independent of u, such that
(62)∥u-uQ0∥s,Ω,μ≤C|Ω|γ∥du∥s,Ω,ν
for any bounded and convex δ-John domain Ω⊂ℝn and γ=1+1/n-(α-λ)/ns, where the Radon measures μ and ν are induced by g(x)=1/dα(x,∂Ω) and h(x)=∑i‍χQi(K/|x-xQi|λ), respectively, where xQi is the center of Qi with Ω=∪iQi. Here K,α,λ are constants with K>0, λ<α<min⁡{n,s+λ+n(s-1)}, and the fixed cube Q0⊂Ω, the constant N>1, and the cubes Qi appeared in Lemma 28.Using (2) and (46), and noticing the fact that |Ω|<∞ since Ω is bounded, we have
(63)∥u-uQ0∥W1,s(Ω,μ)=(diam⁡(Ω))-1∥u-uQ0∥s,Ω,μ+∥∇(u-uQ0)∥s,Ω,μ≤(diam⁡(Ω))-1C1|Ω|γ∥du∥s,Ω,ν+∥d(u-uQ0)∥s,Ω,μ≤C1|Ω|1-(α-λ)/ns∥du∥s,Ω,ν+∥du∥s,Ω,μ≤C2∥du∥s,Ω,ν+∥du∥s,Ω,μ.
Thus, we have the following global Sobolev-Poincaré imbedding inequality with the Radon measure.Corollary 31.
Assume that all conditions in Theorem29 are satisfied. Then, there exists a constant C, independent of u, such that
(64)∥u-uQ0∥W1,s(Ω,μ)≤C∥du∥s,Ω,ν+∥du∥s,Ω,μ
for any bounded and convex δ-John domain Ω⊂ℝn.


## 4. Poincaré Inequalities with Luxemburg Norms

In this section, we establish the local Poincaré inequalities for the differential forms in any bounded domain. A continuously increasing functionφ: [0,∞)→[0,∞) with φ(0)=0 is called an Orlicz function. The Orlicz space Lφ(Ω) consists of all measurable functions f on Ω such that ∫Ω‍φ(|f|/λ)dx<∞ for some λ=λ(f)>0. Lφ(Ω) is equipped with the nonlinear Luxemburg functional
(65)∥f∥φ(Ω)=inf⁡{λ>0:∫Ω‍φ(|f|λ)dx≤1}.
A convex Orlicz function φ is often called a Young function. If φ is a Young function, then ∥·∥φ defines a norm in Lφ(Ω), which is called the Luxemburg norm.Definition 32 (see [28]).
We say a Young functionφ lies in the class G(p,q,C), 1≤p<q<∞, C≥1, if (i) 1/C≤φ(t1/p)/g(t)≤C and (ii) 1/C≤φ(t1/q)/h(t)≤C for all t>0, where g is a convex increasing function and h is a concave increasing function on [0,∞).
From [28], each of φ,g, and h in 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 is that
(66)C1tq≤h-1(φ(t))≤C2tq,C1tp≤g-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.
The following results were proved in [29].Theorem 33.
Letφ be a Young function in the class (p,q,C), 1≤p<q<∞, C≥1 and let Ω be a bounded domain. Assume that φ(|u|)∈Lloc1(Ω,m) and u is a solution of the nonhomogeneous A-harmonic equation (3) in Ω, φ(|du|)∈Lloc1(Ω,m). Then, for any ball B with σB⊂Ω, there exists a constant C, independent of u, such that
(67)∫B‍φ(|u-uB|)dm≤C∫σB‍φ(|du|)dm.Proof.
From (3.5) in [30], we have
(68)∥u-uB∥s,B≤C1|B|1+1/n∥du∥s,B
for any s>0. Note that if u is a solution of the nonhomogeneous A-harmonic equation (3), then u-uB is also a solution of (3). Since uB is a closed form, from Lemma 3, it follows that
(69)(∫B‍|u-uB|qdm)1/q≤C2|B|(p-q)/pq(∫σB‍|u-uB|pdm)1/p
for any positive numbers p and q. From (69), (i) in Definition 32, and using the fact that φ is an increasing function, Jensen’s inequality, and noticing that φ and g are doubling, we have
(70)φ((∫B‍|u-uB|qdm)1/q)≤φ(C2|B|(p-q)/pq(∫σB‍|u-uB|pdm)1/p)≤φ(C3|B|1+1/n+(p-q)/pq(∫σB‍|du|pdm)1/p)≤φ((C3p|B|p(1+1/n)+(p-q)/q∫σB‍|du|pdm)1/p)≤C4g(C3p|B|p(1+1/n)+(p-q)/q∫σB‍|du|pdm)=C4g(∫σB‍C3p|B|p(1+1/n)+(p-q)/q|du|pdm)≤C4∫σB‍g(C3p|B|p(1+1/n)+(p-q)/q|du|p)dm.
Since p≥1, then 1+1/n+(p-q)/pq>0. Hence, we have |B|1+1/n+(p-q)/pq≤|Ω|1+1/n+(p-q)/pq≤C5. From (i) in Definition 32, we find that g(t)≤C6φ(t1/p). Thus,
(71)∫σB‍g(C3p|B|p(1+1/n)+(p-q)/pq|du|p)dm≤C6∫σB‍φ(C3|B|1+1/n+(p-q)/pq|du|)dm≤C6∫σB‍φ(C7|du|)dm.
Combining (70) and (71) yields
(72)φ((∫B‍|u-uB|qdm)1/q)≤C8∫σB‍φ(C7|du|)dm.
Using Jensen’s inequality for h-1, (66), and noticing that φ and h are doubling, we obtain
(73)∫B‍φ(|u-uB|)dm=h(h-1(∫B‍φ(|u-uB|)dm))≤h(∫B‍h-1(φ(|u-uB|))dm)≤h(C9∫B‍|u-uB|qdm)≤C10φ((C9∫B‍|u-uB|qdm)1/q)≤C11φ((∫B‍|u-uB|qdm)1/q).
Substituting (72) into (73) and noticing that φ is doubling, we have
(74)∫B‍φ(|u-uB|)dm≤C12∫σB‍φ(C7|du|)dm≤C13∫σB‍φ(|du|)dm.
We have completed the proof of Theorem 33.Since each ofφ,g, and h in Definition 32 is doubling, from the proof of Theorem 33 or directly from (67), we have
(75)∫B‍φ(|u-uB|λ)dm≤C∫σB‍φ(|du|λ)dm
for all balls B with σB⊂Ω and any constant λ>0. From (65) and (75), the following Poincaré inequality with the Luxemburg norm
(76)∥u-uB∥φ(B)≤C∥du∥φ(σB)
holds under the conditions described in Theorem 33.Theorem 34.
Letφ be a Young function in the class (p,q,C), 1≤p<q<∞, C≥1, Ω a bounded domain, and q(n-p)<np. Assume that u∈D′(Ω,∧l) is any differential l-form, l=0,1,…,n-1, φ(|u|)∈Lloc1(Ω,m), and φ(|du|)∈Lloc1(Ω,m). Then, for any ball B⊂Ω, there exists a constant C, independent of u, such that
(77)∫B‍φ(|u-uB|)dm≤C∫B‍φ(|du|)dm.Proof.
From (73), it follows that
(78)∫B‍φ(|u-uB|)dm≤C1φ((∫B‍|u-uB|qdm)1/q).
If 1<p<n, by assumption, we have q<np/(n-p). Using the Poincaré-type inequality for differential forms
(79)(∫B‍|u-uB|np/(n-p)dm)(n-p)/np≤C2(∫B‍|du|pdm)1/p,
we find that
(80)(∫B‍|u-uB|qdm)1/q≤C3(∫B‍|du|pdm)1/p.
Note that the Lp-norm of |u-uB| increases with p and np/(n-p)→∞ as p→n; it follows that (80) still holds when p≥n. Since φ is increasing, from (78) and (80), we obtain
(81)∫B‍φ(|u-uB|)dm≤C1φ(C3(∫B‍|du|pdm)1/p).
Applying (81), (i) in Definition 32, Jensen’s inequality, and noticing that φ and g are doubling, we have
(82)∫B‍φ(|u-uB|)dm≤C1φ(C3(∫B‍|du|pdm)1/p)≤C1g(C4(∫B‍|du|pdm))≤C5∫B‍g(|du|p)dm.
Using (i) in Definition 32 again yields
(83)∫B‍g(|du|p)dm≤C6∫B‍φ(|du|)dm.
Combining (82) and (83), we obtain
(84)∫B‍φ(|u-uB|)dm≤C7∫B‍φ(|du|)dm.
The proof of Theorem 34 has been completed.Similar to (76), from (65) and (77), the following Luxemburg norm Poincaré inequality
(85)∥u-uB∥φ(B)≤C∥du∥φ(B)
holds if all conditions of Theorem 34 are satisfied.Based on the above discussion, we extend the local Poincaré inequalities into the global cases in the followingLφ(m)-averaging domains.Definition 35 (see [31]).
Letφ be an increasing convex function on [0,∞) with φ(0)=0. We call a proper subdomain Ω⊂ℝn an Lφ(m)-averaging domain, if m(Ω)<∞ and there exists a constant C such that
(86)∫Ω‍φ(τ|u-uB0|)dm≤Csup⁡B⊂Ω∫B‍φ(σ|u-uB|)dm
for some ball B0⊂Ω and all u such that φ(|u|)∈Lloc⁡1(Ω,m), where τ,σ are constants with 0<τ<∞, 0<σ<∞ and the supremum is over all balls B⊂Ω.From above definition, we see thatLs-averaging domains and Ls(m)-averaging domains are special Lφ(m)-averaging domains when φ(t)=ts in Definition 35. Also, uniform domains and John domains are very special Lφ(m)-averaging domains; see [2, 26, 32, 33] for more results about domains.Theorem 36.
Letφ be a Young function in the class G(p,q,C), 1≤p<q<∞, C≥1, and Ω any bounded Lφ(m)-averaging domain. Assume that φ(|u|)∈L1(Ω,m) and u are a solution of the nonhomogeneous A-harmonic equation (4) in Ω, φ(|du|)∈L1(Ω,m). Then, there exists a constant C, independent of u, such that
(87)∫Ω‍φ(|u-uB0|)dm≤C∫Ω‍φ(|du|)dm,
where B0⊂Ω is some fixed ball.Similar to the local case, the following global Poincaré inequality with the Luxemburg norm(88)∥u-uB∥φ(Ω)≤C∥du∥φ(Ω)
holds if all conditions in Theorem 36 are satisfied. Also, by the same way, we can extend Theorem 34 into the following global result in Lφ(m)-averaging domains.Theorem 37.
Letφ be a Young function in the class G(p,q,C), 1≤p<q<∞, C≥1, Ω a bounded Lφ(m)-averaging domain, and q(n-p)<np. Assume that u∈D′(Ω,∧0) and φ(|u|)∈L1(Ω,m) and φ(|du|)∈L1(Ω,m). Then, there exists a constant C, independent of u, such that
(89)∫Ω‍φ(|u-uB0|)dm≤C∫Ω‍φ(|du|)dm,
where B0⊂Ω is some fixed ball.Note that (89) can be written as
(90)∥u-uB0∥φ(Ω)≤C∥du∥φ(Ω).It has been proved that any John domain is a specialLφ(m)-averaging domain. Hence, we have the following results.Corollary 38.
Letφ be a Young function in the class G(p,q,C), 1≤p<q<∞, C≥1, and Ω a bounded John domain. Assume that φ(|u|)∈L1(Ω,m) and u is a solution of the nonhomogeneous A-harmonic equation (4) in Ω, φ(|du|)∈L1(Ω,m). Then, there exists a constant C, independent of u, such that
(91)∫Ω‍φ(|u-uB0|)dm≤C∫Ω‍φ(|du|)dm,
where B0⊂Ω is some fixed ball.Choosingφ(t)=tplog⁡+αt in Theorems 36 and 37, respectively, we obtain the following Poincaré inequalities with the Lp(log⁡+αL)-norms.Corollary 39.
Letφ(t)=tplog⁡+αt, p≥1, and α∈ℝ. Assume that φ(|u|)∈L1(Ω,m) and u is a solution of the nonhomogeneous A-harmonic equation (4), φ(|du|)∈L1(Ω,m). Then, there exists a constant C, independent of u, such that
(92)∫Ω‍|u-uB0|plog⁡+α(|u-uB0|)dm≤C∫Ω‍|du|plog⁡+α(|du|)dm
for any bounded Lφ(m)-averaging domain Ω and B0⊂Ω is some fixed ball.Note that (92) can be written as the following version with the Luxemburg norm
(93)∥u-uB0∥Lp(log⁡+αL)(Ω)≤C∥du∥Lp(log⁡+αL)(Ω)
provided the conditions in Corollary 39 are satisfied.Corollary 40.
Letφ(t)=tplog⁡+αt, 1≤p1<p<p2, and α∈ℝ and let Ω be a bounded Lφ(m)-averaging domain and p2(n-p1)<np1. Assume that u∈D′(Ω,∧0), φ(|u|)∈L1(Ω,m), and φ(|du|)∈L1(Ω,m). Then, there exists a constant C, independent of u, such that
(94)∫Ω‍|u-uB0|plog⁡+α(|u-uB0|)dm≤C∫Ω‍|du|plog⁡+α(|du|)dm,
where B0⊂Ω is some fixed ball.


## 5. Inequalities for Green’s Operator

In this section, we say thatu∈Lloc⁡1(∧lΩ) has a generalized gradient, if, for each coordinate system, the pullbacks of the coordinate function of u have generalized gradient in the familiar sense; see [21]. We write
(95)𝒲(∧lΩ)={u∈Lloc⁡1(∧lΩ):uhasgeneralizedgradient}.
As usual, the harmonic l-fields are defined by
(96)ℋ(∧lΩ)={u∈𝒲(∧lΩ):du=d⋆u=0,u∈Lpforsome1<p<∞}.
The orthogonal complement of ℋ in L1 is defined by
(97)ℋ⊥={u∈L1:<u,h>=0∀h∈ℋ}.
We define Green’s operator
(98)G:C∞(∧lΩ)⟶ℋ⊥∩C∞(∧lΩ)
by setting G(u) equal to the unique element of ℋ⊥∩C∞(∧lΩ) satisfying Poisson’s equation
(99)ΔG(ω)=ω-H(ω),
where H is either the harmonic projection or the harmonic part of ω. It has been proved in [34] that for 1<p<∞ and ω∈Lp(∧lM), ΔG(ω)=ω-H(ω).We will need the following lemma aboutLs-estimates for Green’s operator which appeared in [34].Lemma 41.
Letu∈C∞(∧lΩ), l=0,1,…,n. For 1<s<∞, there exists constant C, independent of u, such that
(100)∥dd⋆G(u)∥s,Ω+∥d⋆dG(u)∥s,Ω+∥dG(u)∥s,Ω+∥d⋆G(u)∥s,Ω+∥G(u)∥s,Ω≤C∥u∥s,Ω.The following results about the composition of the Laplace-Beltrami operator and Green’s operator were proved in [35].Theorem 42.
Letu∈C∞(∧lΩ), l=0,1,…,n, and 1<s<∞. Then, there exists a constant C, independent of u, such that
(101)∥Δ(G(u))∥s,Ω≤C∥u∥s,Ω.Theorem 43.
Letu∈C∞(∧lΩ), l=0,1,…,n. Assume that 1<s<∞. Then, there exists a constant C, independent of u, such that
(102)∥G(Δu)∥s,Ω≤C∥u∥s,Ω.Using Minkowski’s inequality and combining Theorems42 and 43, we obtain the following corollary immediately.Corollary 44.
Letu∈C∞(∧lΩ), l=0,1,…,n. For 1<s<∞, there exists a constant C, independent of u, such that
(103)∥(GΔ+ΔG)u∥s,Ω≤C∥u∥s,Ω.Theorem 45.
Letu∈C∞(∧lΩ), l=0,1,…,n. If 1<s<∞, then there exists a constant C, independent of u, such that
(104)∥(G(u))D∥s,D≤C∥u∥s,D
for any convex and bounded D with D⊂Ω.Corollary 46.
Letu∈C∞(∧lΩ), l=0,1,…,n. Assume that 1<s<∞. Then, for any convex and bounded D with D⊂Ω, there exists a constant C, independent of u, such that
(105)∥G(u)-(G(u))D∥s,D≤C∥G(u)-c∥s,D
for any closed form c, and
(106)∥G(u)-(G(u))D∥s,D≤C∥u∥s,D.Forω∈D′(Ω,∧l), the vector-valued differential form
(107)∇ω=(∂ω∂x1,…,∂ω∂xn)
consists of differential forms ∂ω/∂xi∈D′(Ω,∧l), where the partial differentiation is applied to the coefficients of ω. The notations Wloc⁡1,p(Ω,ℝ) and Wloc⁡1,p(Ω,∧l) are self-explanatory. For 0<p<∞ and a weight w(x), the weighted norm of ω∈W1,p(Ω,∧l) over Ω is denoted by
(108)∥ω∥W1,p(Ω),wα=diam⁡(Ω)-1∥ω∥p,Ω,wα+∥∇ω∥p,Ω,wα,
where α is a real number.We have made necessary preparation in the previous section to prove the following Poincaré-type inequality for Green’s operator.Theorem 47.
Letu∈C∞(∧lΩ), l=0,1,…,n. Assume that 1<s<∞. Then, there exists a constant C, independent of u, such that
(109)∥G(u)-(G(u))B∥s,B≤Cdiam⁡(B)∥du∥s,B
for all balls B with B⊂Ω.As an application of Theorem47, now we will prove the following Sobolev-Poincaré imbedding theorem for Green’s operator G applied to a differential form u.Theorem 48.
Letu∈C∞(∧lΩ), l=0,1,…,n. Assume that 1<s<∞. Then, there exists a constant C, independent of u, such that
(110)∥G(u)-(G(u))B∥W1,s(B)≤C∥du∥s,B
for all balls B with B⊂Ω.Proof.
SinceuD is a closed form for any form u, it follows that (G(u))B is a closed form and
(111)∥d(G(u)-(G(u))B)∥s,B=∥d(G(u))∥s,B.
Note that ∥∇u∥s,B=∥du∥s,B. Using (108) and (111), we obtain
(112)∥G(u)-(G(u))B∥W1,s(B)=diam⁡(B)-1∥G(u)-(G(u))B∥s,B+∥∇(G(u)-(G(u))B)∥s,B=diam⁡(B)-1∥G(u)-(G(u))B∥s,B+∥d(G(u)-(G(u))B)∥s,B=diam⁡(B)-1∥G(u)-(G(u))B∥s,B+∥d(G(u))∥s,B.
From (109) and (112), we have
(113)∥G(u)-(G(u))B∥W1,s(B)≤diam⁡(B)-1∥G(u)-(G(u))B∥s,B+∥d(G(u))∥s,B≤diam⁡(B)-1·C1diam⁡(B)∥du∥s,B+∥d(G(u))∥s,B≤C1∥du∥s,B+∥d(G(u))∥s,B≤C1∥du∥s,B+C2∥du∥s,B≤C3∥du∥s,B.Remark 49.
Since Green’s operators can commute withd and d*, Theorem 48 can be proved by applying Corollary  4.1 in [20] to G(u) and using (100).
We notice that all the results developed so far in this section are about differential forms. We do not require that differential formu must satisfy any differential equation. However, if u satisfies some version of harmonic equation, we can extend above inequalities into the weighted cases. In fact, now we will prove the following Ar(M)-weighted Sobolev-Poincaré imbedding theorem for Green’s operator G.Theorem 50.
LetG(u)∈C∞(∧lΩ), l=0,1,2,…,n-1, be an A-harmonic tensor on a manifold Ω. Assume that ρ>1, 1<s<∞, and w∈Ar(Ω) for some r>1. Then, there exists a constant C, independent of u, such that
(114)∥G(u)-(G(u))B∥W1,s(B),w≤C∥du∥s,ρB,w
for all balls B with ρB⊂Ω.Similarly, we can extend inequalities (104) and (106) to the following Ar(Ω)-weighted version.Theorem 51.
Letu∈C∞(∧lΩ), l=0,1,2,…,n, be an A-harmonic tensor on a manifold Ω. Assume that ρ>1, 1<s<∞, and w∈Ar(Ω) for some r>1. Then, there exists a constant C, independent of u, such that
(115)∥(G(u))B∥s,B,wα≤C1∥u∥s,ρB,wα,∥G(u)-(G(u))B∥s,B,wα≤C2∥u∥s,ρB,wα
for all balls B with ρB⊂Ω and any real number α with 0<α≤1.In [12], Wang and Wu proved the following global weighted Poincaré-type inequality for Green’s operator applied to the solutions of the nonhomogeneous A-harmonic equation (3).Theorem 52.
Letu∈D′(Ω,∧0) be a solution of the nonhomogeneous A-harmonic equation (3) and w∈Ar(Ω) for some 1<r<∞. Assume that s is a fixed exponent associated with the A-harmonic equation (3), r<s<∞. Then, there exists a constant C, independent of u, such that
(116)(∫Ω‍|G(u)-(G(u))Q0|swdx)1/s≤C(∫Ω‍|du|swdx)1/s
for any bounded δ-John domain Ω⊂ℝn. Here Q0⊂Ω is a fixed cube appearing in Lemma 28.Proof.
From Theorem51, we have
(117)∫Q‍|G(u)-(G(u))Q|sdμ(x)≤C1diam⁡(B)∫σQ‍|du|sdμ(x),
where the measure μ(x) is defined by dμ(x)=w(x)dx. We use the notation and the covering 𝒱 described in Lemma 28 (Covering Lemma) and the properties of the measure μ(x). If w∈Ar, then
(118)μ(NQ)≤MNnrμ(Q)
for each cube Q with NQ⊂ℝn (see [36]) and
(119)max⁡(μ(Qi),μ(Qi+1))≤MNnrμ(Qi∩Qi+1)
for the sequence of cubes Qi,Qi+1,i=0,1,…,k-1 described in the Covering Lemma. From the elementary inequality |a+b|s≤2s(|a|s+|b|s), s>0, we find that
(120)∫Ω‍|G(u)-(G(u))Q0|swdx=∫Ω‍|G(u)-(G(u))Q0|sdμ(x)≤2s∑Q∈𝒱‍∫Q‍|G(u)-(G(u))Q|sdμ(x)+2s∑Q∈𝒱‍∫Q‍|(G(u))Q0-(G(u))Q|sdμ(x).
The first sum can be estimated by (117) and the Covering Lemma
(121)∑Q∈𝒱‍∫Q‍|G(u)-(G(u))Q|sdμ(x)≤C1∑Q∈𝒱‍(∫σQ‍|du|swdx)≤C1N(∫Ω‍|du|swdx).
Now we will estimate the second sum in (120). Fix a cube Q∈𝒱 and let Q0,Q1,…,Qk=Q be the chain in the Covering Lemma. We have
(122)|(G(u))Q0-(G(u))Q|≤∑i=0k-1‍|(G(u))Qi-(G(u))Qi+1|.
Using (117) and (118), we get
(123)|(G(u))Qi-(G(u))Qi+1|s=1μ(Qi∩Qi+1)∫Qi∩Qi+1‍|(G(u))Qi-(G(u))Qi+1|sdμ(x)≤MNnrmax⁡(μ(Qi),μ(Qi+1))×∫Qi∩Qi+1‍|(G(u))Qi-(G(u))Qi+1|sdμ(x)≤C2∑j=ii+1‍1μ(Qj)∫Qj‍|G(u)-(G(u))Qj|sdμ(x)≤C3∑j=ii+1‍diam⁡(Qj)μ(Qj)∫σQj‍|du|swdx.
Since Q⊂NQj for j=i,i+1, 0≤i≤k-1, we have
(124)|(G(u))Qi-(G(u))Qi+1|sχQ(x)≤C3∑j=ii+1‍χNQj(x)diam⁡(Ω)μ(Qj)(∫σQj‍|du|swdx).
According to (122) and diam⁡(Ω)<∞, we have (note |a+b|1/s≤21/s(|a|1/s+|b|1/s))
(125)|(G(u))Q0-(G(u))Q|χQ(x)≤C4∑R∈𝒱‍(1μ(R)(∫σR‍|du|swdx))1/s·χNR(x)
for every x∈ℝn. Hence,
(126)∑Q∈𝒱‍∫Q‍|(G(u))Q0-(G(u))Q|sdμ(x)≤C5∫ℝn‍|∑R∈𝒱‍(1μ(R)(∫σR‍|du|swdx))1/sχNR(x)|sdμ(x).
Now from (126), it follows that
(127)3∑Q∈𝒱‍∫Q‍|(G(u))Q0-(G(u))Q|sdμ(x)≤C6∫ℝn‍|∑R∈𝒱‍(1μ(R)(∫σR‍|du|swdx))1/sχR(x)|sdμ(x).
Notice that
(128)∑R∈𝒱‍χR(x)≤∑R∈𝒱‍χσR(x)≤NχΩ(x).
Finally, using the elementary inequality |∑i=1N‍ti|s≤Ns-1∑i=1N‍|ti|s, we conclude that
(129)∑Q∈𝒱‍∫Q‍|(G(u))Q0-(G(u))Q|sdμ(x)≤C7∫ℝn‍(∑R∈𝒱‍1μ(R)(∫σR‍|du|swdx)χR(x))dμ(x)=C7∑R∈𝒱‍(∫σR‍|du|swdx)≤C8(∫Ω‍|du|swdx)
by the Covering Lemma. Combining (120), (121), and (129), we obtain the required inequality.Using Theorem52 and the proof of Theorem  3.2.10 in [2], we obtain the following global Sobolev imbedding inequality for Green’s operator applied to the solutions of the nonhomogeneous A-harmonic equation in the John domain.Theorem 53.
Letu∈D′(Ω,∧0) be a solution of the nonhomogeneous A-harmonic equation (3) and w∈Ar(Ω) for some 1<r<∞. Assume that s is a fixed exponent associated with the A-harmonic equation (3), r<s<∞. Then, there exists a constant C, independent of u, such that
(130)∥G(u)-(G(u))Q0∥W1,s(Ω),w≤C∥du∥s,Ω,w
for any δ-John domain Ω⊂ℝn. Here Q0⊂Ω is a fixed cube, which appears in Lemma 28.



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*Source: 971595-2014-05-06.xml*