JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 971595 10.1155/2014/971595 971595 Review Article Advances in Study of Poincaré Inequalities and Related Operators http://orcid.org/0000-0002-0525-9703 Xing Yuming 1 Ding Shusen 2 Shi Peilin 1 Department of Mathematics Harbin Institute of Technology Harbin 150001 China hit.edu.cn 2 Department of Mathematics Seattle University Seattle WA 98122 USA seattleu.edu 2014 652014 2014 11 01 2014 06 03 2014 6 5 2014 2014 Copyright © 2014 Yuming Xing and Shusen Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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  proved the following Poincaré inequality for Sobolev functions in Ls-averaging domains. If D is an Ls-averaging domain, sn, then there exists a constant C, such that (1)(1m(D)D|u-uD|sdm)1/sC(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  for more versions of the Poincaré inequality.

Throughout this paper, we assume that Ω is a domain in n, n2, 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 En. Differential forms are extensions of functions in n. 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 n 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 {dxi1dxi2dxik}, k=1,2,,n, that is, u(x)=IuI(x)dxI=ui1i2ik(x)dxi1dxi2dxik, where I=(i1,i2,,ik), 1i1<i2<<ikn. Let l=l(n) be the set of all l-forms in n, 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 : kn-k as follows. If u=ui1i2ik(x1,x2,,xn)dxi1dxi2dxik=uIdxI, i1<i2<<ik, is a differential k-form, then u=(ui1i2ikdxi1dxi2dxik)=(-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+1d on D(Ω,l+1), l=0,1,,n-1. We write us,E=(E|u|s)1/s and us,E,μ=(E|u|sdμ(x))1/s, where EΩ, 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 uW1,p(Ω,k) over EΩ is denoted by (2)uW1,p(E,μ)=(diam(E))-1up,E,μ+up,E,μ. We consider here the solutions to the nonlinear partial differential equation (3)dA(x,du)=B(x,du),   which is called nonhomogeneous A-harmonic equation, where A: Ω×l(n)l(n) and B: Ω×l(n)l-1(n) 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 xΩ and all ξl(n). 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 Wloc1,p(Ω,l-1) such that ΩA(x,du)·dφ+B(x,du)·φ=0 for all φWloc1,p(Ω,l-1) with compact support. If u is a function (0-form) in n, (3) reduces to (4)divA(x,u)=B(x,u). If the operator B=0, (3) becomes dA(x,du)=0, which is called the (homogeneous) A-harmonic equation. Let A:Ω×l(n)l(n) be defined by A(x,ξ)=ξ|ξ|p-2 with p>1. Then, A satisfies the required conditions and dA(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, 1218] for recent results on the solutions to the different versions of the A-harmonic equation.

Let Dn be a bounded and convex domain. The linear operator Ky: C(D,l)C(D,l-1) was first introduced in , and then it was generalized to the following version in . For any yD, 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 φC0(D) is normalized by Dφ(y)dy=1. The l-form ωDD(D,l) is defined by ωD=|D|-1Dω(y)dy, l=0, and   ωD=d(Tω),  l=1,2,,n, for all ωLp(D,l), 1p<. From , we know that, for any bounded and convex domain D, we have (Tu)s,DC|D|us,D and Tus,DC|D|diam(D)us,D. From , any open subset Q in n 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=1Qk, (ii) Qj0Qk0= if jk, (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 n. For any xQ, xQk 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=1TQkχQk. Hence, for any bounded domain Q and any differential form uLlocs(Q,l), we have (5)(Tu)s,QC|Q|us,Q,Tus,QC|Q|diam(Q)us,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 , which will be used repeatedly later.

Definition 1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

A weight w satisfies Ar(Ω)-condition in a subset Ωn, where r>1, and write wAr(Ω) when (6)supB(1|B|Bwdx)(1|B|Bw1/(1-r)  dx)r-1<, where supremum is over all BΩ.

Definition 2 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

A pair of weights (w1,w2) satisfy the Arλ(E)-condition in a set En, and write (w1,w2)Arλ(E) for some r>1 and λ>0, if (7)supB(1|B|Bw1dx)(1|B|B(1w2)1/(1-r)dx)r-1< for any ball BE.

Lemma 3.

Let u 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)us,BC|B|(t-s)/stut,σB for all balls or cubes B with σBΩ 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 the A-harmonic equation.

Definition 4 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

We call a proper subdomain Ωn an Ls(μ,0)-averaging domain, s1, if μ(Ω)< and there exists a constant C such that (9)(1μ(B0)Ω|u-uB0|sdμ)1/sCsup2BΩ(1μ(B)B|u-uB|sdμ)1/s for some ball B0Ω and all uLlocs(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 2BΩ.

In 1993, the following Poincaré-Sobolev inequality was proved in , which can be used to generalize the theory of Sobolev functions to that of differential forms.

Theorem 5.

Let uD(Q,l) and duLp(Q,l+1). Then, u-uQ is in Lnp/(n-p)  (Q,l) and (10)(Q|u-uQ|np/(n-p)dx)(n-p)/npCp(n)(Q|du|pdx)1/p for Q a cube or a ball in n, l=0,1,,n-1, and 1<p<n.

From Corollary 4.1 in , we have the following version of Poincaré inequality for differential forms.

Theorem 6.

Let uD(Q,l) and duLp(Q,l+1). Then, u-uQ is in W1,p(Q,l) with 1<p< and (11)u-uQp,QC(n,p)|Q|1/ndup,Q for Q a cube or a ball in n, l=0,1,,n-1.

The above Poincaré-Sobolev inequalities are about differential forms. We know that the A-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 , Ding and Nolder proved the following symmetric Poincaré-Sobolev inequalities for solutions of the nonhomogeneous A-harmonic equation (3).

Theorem 7.

Let uD(Ω,l) be a solution of the nonhomogeneous A-harmonic equation (3) in a domain Ωn and duLs(Ω,l+1), l=0,1,,n-1. Assume that σ>1, 0<s<, and wAr for some r>1. Then (12)u-uBs,B,wC|B|1/ndus,σB,w for all balls B with σBΩ. Here C is a constant independent of u and du.

Note that (12) is equivalent to (12)(1μ(B)B|u-uB|sdμ)1/sC|B|1/n(1μ(B)σB|du|sdμ)1/s.

Theorem 8.

Let uD(Ω,l) be a solution of the nonhomogeneous A-harmonic equation (3) in a domain Ωn and duLs(Ω,l+1), l=0,1,,n-1. Assume that σ>1, 0<α1, 1+α(r-1)<s<, and wAr for some r>1. Then (13)u-uBs,B,wαC|B|1/ndus,σB,wα for all balls B with σBΩ. 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/sC|B|1/n(1|B|σB|du|swαdx)1/s.

Next, we will prove the following global weighted Poincaré-Sobolev inequality in Ls(μ)-averaging domains.

Theorem 9.

Let wAr with wτ>0, r>1, where τ is a constant. Assume that uD(Ω,0) is an A-harmonic tensor and duLs(Ω,1); then (14)(1μ(Ω)Ω|u-uB0|sdμ)1/sC(Ω|du|sdμ)1/s for any Ls(μ)-averaging domain Ω and some ball B0 with 2B0Ω. 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-uB0s,Ω,wCμ(Ω)1/sdus,Ω,w.

In , 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.

Let uD(Ω,l) be an A-harmonic tensor in a domain Ωn and duLlocs(Ω,l+1), l=0,1,,n-1. Assume that σ>1, 1<s<, and wAr for some r>1. Then, there exists a constant C, independent of u, such that (16)(1μ(B)B|u-uB|sdμ)1/sC|B|1/n(1|B|σB|du|sdx)1/s for all balls B with σBΩ. Here, the measure μ is defined by dμ=w(x)dx.

Theorem 11.

Let uD(Ω,l) be an A-harmonic tensor in a domain Ωn and duLlocs(Ω,l+1), l=0,1,,n-1. Assume that σ>1, 1<s<, and wAr for some r>1. Then (17)(1|B|B|u-uB|sdx)1/sC|B|1/n(1μ(σB)σB|du|sdμ)1/s for all balls B with σBΩ. Here, the measure μ is defined by dμ=w(x)dx and C is a constant independent of u and du.

Theorem 12.

Let wAr for some r>1, uD(Ω,0), and duLs(Ω,1). If sn, then (18)(1μ(Ω)Ω|u-uB0|sdμ)1/sC|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 2B0Ω. Here, the measure μ is defined by dμ=w(x)dx and C is a constant independent of u and du.

Theorem 13.

Let wAr with w(x)α>0, r>1, uD(Ω,0), and duLs(Ω,1). If sn, then (19)(1|Ω|Ω|u-uB0|sdx)1/sC|Ω|1/n(1|Ω|Ω|du|sdμ)1/s for any Ls-averaging domain Ω and some ball B0 with 2B0Ω. 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, 2325] for example.

Theorem 14.

Let uD(B,l) and duLt(B,l+1), l=0,1,,n-1. Then, there exists a constant β>1 such that if w1Ar1 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/sC(B|du|tw2dx)1/t for all balls Bn. Here C is a constant independent of u and du.

Theorem 15.

Let uD(B,l) and duLn(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/sC(B|du|nw2dx)1/n for any ball or cube Bn.

We remark that the exponents t 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.

Let uD(B,l) and duLt(B,l+1), l=0,1,,n-1. Then, there exists a constant β>1 such that if w1Ar1 and (w1,w2)Arαs/t, where 1<s<n, t=s+αs(r-1) and r>1, we have (22)(1|B|βB|u-uB|sw1dx)1/sC|B|α(r-1)/t(B|du|tw2αdx)1/t for all balls Bn and any constant α>0. Here C is a constant independent of u and du.

Clearly, in this result if α0, then ts.

Theorem 17.

Let uD(Ω,0) and duLt(Ω,1). Then there exists a constant β>1 such that if w1Ar1 and (w1,w2)At/ss/t, where 1<s<n, t=sβ, r>1, and w1>w2η>0, we have (23)(1μ1(Ω)Ω|u-uB0|sdμ1)1/sC(Ω|du|tdμ2)1/t for any Ls(μ1)-averaging domain Ω and some ball B0 with 2B0Ω. 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) Theorems 15 and 16 can be extended to the global versions. (2) From , 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.

Let uD(Ω,l) be a differential form satisfying the A-harmonic equation (3) in a domain Ωn and duLs(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/sC|B|1/n(σB|du|sw2αλdx)1/s for all balls B with σBΩ. Here uB is a closed form.

If we choose λ=1/α in Theorem 19, we get the following version of the Arλ-weighted Poincaré inequality.

Corollary 20.

Let uD(Ω,l) be a differential form satisfying the A-harmonic equation (3) in a domain Ωn and duLs(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/sC|B|1/n(σB|du|sw2dx)1/s for all balls B with σBΩ. Here uB is a closed form.

Selecting α=1/s in Theorem 19, we have the following two-weighted Poincaré inequality.

Corollary 21.

Let uD(Ω,l) be a differential form satisfying the A-harmonic equation (3) in a domain Ωn and duLs(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/sC|B|1/n(σB|du|sw2λ/sdx)1/s for all balls B with σBΩ. Here uB is a closed form.

When λ=1 in Corollary 21, we obtain the following symmetric two-weighted inequality.

Corollary 22.

Let uD(Ω,l) be a differential form satisfying the A-harmonic equation (3) in a domain Ωn and duLs(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/sC|B|1/n(σB|du|sw21/sdx)1/s for all balls B with σBΩ. 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  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.

Let 0<α<, 0<β<, and s-1=α-1+β-1. If f and g are measurable functions on n, then (28)fgs,Efα,E·gβ,E for any En.

Now, we prove the following local Poincaré inequality with the Radon measure which will be used to establish the global inequality.

Theorem 24.

Let uLlocs(M,l) be a solution of the nonhomogeneous A-harmonic equation (3) in a bounded domain M, duLlocs(M,l+1), l=0,1,,n-1, and 1<s<. Then, for any ball B with σBM, there exists a constant C, independent of u, such that (29)(B|u-uB|sdμ)1/sC|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,hLloc1(M), 0<g(x)K1/|x-xB|α, and K2/|x-xB|λh(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 that ε(0,1) is small enough so that εn<α-λ and BM is any ball with center xB and radius rB. Also, let δ>0 be small enough, Bδ={xB:|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-uBs,B=T(du)s,BC1|B|diam(B)dus,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/su-uBt,Dδ(Dδ(C2|x-xB|)tα/(t-s)dx)(t-s)/st=u-uBt,Dδ(DδC3|x-xB|-αβdx)1/βsu-uBt,B·|x-xB|-αβ,Dδ1/sC4|B|diam(B)dut,B·|x-xB|-αβ,Dδ1/s. We may suppose that xB=0. Otherwise, we can move the center to the origin by a simple transformation. Thus, for any xB, |x-xB||x|-|xB|=|x|. Using the polar coordinate substitution, we find that (33)|x-xB|-αβ,Dδ1/s=(Dδ|x-xB|-αβdx)1/sβ(C5δrBρ-αβρn-1dρ)1/sβ=|C5n-αβ(rBn-αβ-δn-αβ)|1/sβC6|rBn-αβ-δn-αβ|1/sβ. Set m=nst/(ns+αt-λt)  ; then 0<m<s. From Lemma 3, we have (34)dut,BC7|B|(m-t)/mtdum,σB, where σ>1 is a constant. Using the Hölder inequality again, we obtain (35)dum,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)/msC9(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/sC10|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/sβC10|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/sβC10|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/sβC12|B|1+1/n-(α-λ)/ns(σB|du|sdν)1/sC12|B|γ(σB|du|sdν)1/s, that is, (37)(Dδ|u-uB|sdμ)1/sC12|B|γ(σB|du|sdν)1/s. Notice that limδ0(Dδ|u-uB|sdμ)1/s=(B|u-uB|sdμ)1/s. Letting δ0 in (37), we obtain (29). The proof of Theorem 24 has been completed.

Let g(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.

Let uLlocs(M,l) be a solution of the nonhomogeneous A-harmonic equation (3) in a bounded domain M, duLlocs(M,l+1), l=0,1,,n-1, and 1<s<. Then, there exists a constant C, independent of u, such that (38)u-uBs,B,μC|B|γdus,σB,ν for all balls B with σBM, 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.

Let uLlocs(M,0) be a solution of (4). From (2), we have (39)u-uBW1,s(B,μ)  =(diam(B))-1u-uBs,B,μ+(u-uB)  s,B,μ for any ball BM. Note that us,B,μ=dus,B,μ and d(uB)s,B,μ=0. Hence, (40)(u-uB)s,B,μ=d(u-uB)s,B,μdus,B,μ+duBs,B,μ=dus,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-uBW1,s(B,μ)=C2|B|-1/nu-uBs,B,μ+dus,B,μC3|B|-1/n|B|1+1/n-(α-λ)/nsdus,σ1B,ν+dus,B,μC3|B|1-(α-λ)/nsdus,σ1B,ν+dus,B,μ. Using the same method as we developed in the proof of Theorem 24, we have (42)dus,B,μC4|B|(λ-α)/nsdus,σ2B,ν. Combining (41) and (42) and noticing that M is bounded and BM, we find that (43)u-uBW1,s(B,μ)(C3|B|+C4)|B|(λ-α)/nsdus,σB,νC5|B|(λ-α)/nsdus,σB,ν, where σ=max{σ1,σ2}. Hence, we obtain the following Sobolev-Poincaré imbedding inequality with the Radon measure.

Corollary 26.

Let uLlocs(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-uBW1,s(B,μ)C|B|(λ-α)/nsdus,σB,ν for all balls B with σBM, 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 subdomain Ωn is called a δ-John domain, δ>0, if there exists a point x0Ω, which can be joined with any other point xΩ by a continuous curve γΩ, so that (45)d(ξ,Ω)δ|x-ξ| for each ξγ. Here d(ξ,Ω) is the Euclidean distance between ξ and Ω.

Lemma 28 (see [<xref ref-type="bibr" rid="B16">16</xref>] (Covering Lemma)).

Each Ω has a modified Whitney cover of cubes 𝒱={Qi} such that iQi=Ω,  Qi𝒱χ(5/4)QiNχΩ and some N>1, and if  QiQj, then there exists a cube R (this cube need not be a member of 𝒱) in QiQj such that QiQjNR. Moreover, if Ω is δ-John, then there is a distinguished cube Q0𝒱 which can be connected with every cube Qm𝒱 by a chain of cubes Q0,Q1,,Qk=Qm from 𝒱 and such that QmρQi, i=0,1,2,,k, for some ρ=ρ(n,δ).

Theorem 29.

Let Ω be any bounded and convex δ-John domain Ωn and let uLs(Ω,0) be a solution of the nonhomogeneous A-harmonic equation (4), duLs(Ω,1), and 1<s<. Then, there exists a constant C, independent of u, such that (46)(Ω|u-uQ0|sdμ)1/sC|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,hL1(Ω), 0<g(x)K1/dα(x,Ω), and iχQiK2/|x-xQi|λh(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 Q0Ω, the constant N>1, and the cubes Qi appeared in Lemma 28.

Proof.

We may assume g(x)1 a.e. Otherwise, let Ω1=Ω{xΩ:0<g(x)<1} and Ω2=Ω{xΩ:g(x)1}; then Ω=Ω1Ω2. We define the new function G(x) by (47)G(x)={1,xΩ1g(x),xΩ2. Also, we choose the constant K1diam  α(Ω); then K1/dα(x,Ω)1 for any xΩ. 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)dxQdx=|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 Qi𝒱χ(5/4)QiNχΩ for some N>1. Since Ω=Qi, for any xΩ, it follows that xQi 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/sC1|Qi|γ(σQi|du|sdν)1/s, where σ>1 is a constant. Using the elementary inequality (a+b)s2s(|a|s+|b|s), s0, we have (51)(Ω|u-uQ0|sdμ)1/s=(Qi|u-uQ0|sdμ)1/s(Qi𝒱(2sQi|u-uQi|sdμ+2sQi|uQi-uQ0|sdμ))1/sC2(s)((Qi𝒱Qi|u-uQi|sdμ)1/s+(Qi𝒱Qi|uQi-uQ0|sdμ)1/s) for a fixed Q0Ω. The first sum in (51) can be estimated by using Theorem 24 and the Covering Lemma (52)(Qi𝒱Qi|u-uQi|sdμ)1/s(Qi𝒱C3|Qi|sγσQi|du|sdν)  1/s(C3|Ω|sγQi𝒱σQi|du|sdν)1/sC4|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 Qm𝒱 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 QiQi+1, there exists a cube Di such that DiQiQi+1 and QiQi+1NDi, N>1. Consider (54)max{|Qi|,|Qi+1|}|QiQi+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|}|QiQi+1|max{|Q0|,|Q1|,,|Qk|}|D|C6. By (49), (53), (55), and (50), we have (56)|uQi-uQi+1|s=1μ(QiQi+1)QiQi+1|uQi-uQi+1|sdμ1|QiQi+1|QiQi+1|uQi-uQi+1|sdμC7max{|Qi|,|Qi+1|}QiQi+1|uQi-uQi+1|sdμC8j=ii+11|Qj|Qj|u-uQj|sdμC9j=ii+1|Qj|γs|Qj|σQj|du|sdν=C9j=ii+1|Qj|γs-1σQj|du|sdν. Since QmρQj for j=i, i+1, 0ik-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)C10j=ii+1χρQj(x)|Qj|γs-1σQj|du|sdνC11j=ii+1χρQj(x)|Ω|γs-1σQj|du|sdνC11|Ω|γs-1j=ii+1χρQj(x)σQj|du|sdν. Taking the sth root both sides in (57) and using (a+b)1/s21/s(|a|1/s+|b|1/s), (53), and (55), we obtain (58)|uQm-uQ0|χQm(x)C12|Ω|γ-1/sDi𝒱(σDi|du|sdν)1/s·χρDi(x) for every xn. Raising both sides of inequality (58) to s powers and then integrating over n both sides, we have (59)Qm𝒱Qm|uQm-uQ0|sdμC13|Ω|γs-1n|Di𝒱(σDi|du|sdν)1/sχρDi(x)|sdμ. Notice that (60)Di𝒱χρDi(x)Di𝒱χρNDi(x)NχΩ(x). Using elementary inequality |i=1Mti|sMs-1i=1M|ti|s for s>1, we finally have (61)(Qm𝒱Qm|uQm-uQ0|sdμ)1/sC14|Ω|γ-1/s×(n(Di𝒱(σDi|du|sdν)χρDi(x))dμ)1/s=C15|Ω|γ-1/s(Di𝒱(σDi|du|sdν))1/sC16|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, choose g(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.

Let uLs(Ω,0) be a solution of the nonhomogeneous A-harmonic equation (4), duLs(Ω,1), 1<s<. Then, there exists a constant C, independent of u, such that (62)u-uQ0s,Ω,μC|Ω|γdus,Ω,ν for any bounded and convex δ-John domain Ωn 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 Q0Ω, 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-uQ0W1,s(Ω,μ)=(diam(Ω))-1u-uQ0s,Ω,μ+(u-uQ0)s,Ω,μ(diam(Ω))-1C1|Ω|γdus,Ω,ν+d(u-uQ0)s,Ω,μC1|Ω|1-(α-λ)/nsdus,Ω,ν+dus,Ω,μC2dus,Ω,ν+dus,Ω,μ. Thus, we have the following global Sobolev-Poincaré imbedding inequality with the Radon measure.

Corollary 31.

Assume that all conditions in Theorem 29 are satisfied. Then, there exists a constant C, independent of u, such that (64)u-uQ0W1,s(Ω,μ)Cdus,Ω,ν+dus,Ω,μ for any bounded and convex δ-John domain Ωn.

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 function φ: [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)fφ(Ω)=  inf{λ>0:Ωφ(|f|λ)dx1}. 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 [<xref ref-type="bibr" rid="B28">28</xref>]).

We say a Young function φ lies in the class G(p,q,C), 1p<q<, C1, if (i) 1/Cφ(t1/p)/g(t)C and (ii) 1/Cφ(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 , 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)C1tqh-1(φ(t))C2tq,C1tpg-1(φ(t))C2tp, where C1 and C2 are constants. Also, for all 1p1<p<p2 and α, 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 te, and log+(t)=log(t) for t>e. Particularly, if α=0, we see that φ(t)=tp lies in G(p1,p2,C), 1p1<p<p2.

The following results were proved in .

Theorem 33.

Let φ be a Young function in the class (p,q,C), 1p<q<, C1 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 σBΩ, there exists a constant C, independent of u, such that (67)Bφ(|u-uB|)dmCσBφ(|du|)dm.

Proof.

From (3.5) in , we have (68)u-uBs,BC1|B|1+1/ndus,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/qC2|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)/qσB|du|pdm)1/p)C4g(C3p|B|p(1+1/n)+(p-q)/qσB|du|pdm)=C4g(σBC3p|B|p(1+1/n)+(p-q)/q|du|pdm)C4σBg(C3p|B|p(1+1/n)+(p-q)/q|du|p)dm. Since p1, then 1+1/n+(p-q)/pq>0. Hence, we have |B|1+1/n+(p-q)/pq|Ω|1+1/n+(p-q)/pqC5. 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)dmC6σBφ(C3|B|1+1/n+(p-q)/pq|du|)dmC6σBφ(C7|du|)dm. Combining (70) and (71) yields (72)φ((B|u-uB|qdm)1/q)C8σBφ(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(C9B|u-uB|qdm)  C10φ((C9B|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|)dmC12σBφ(C7|du|)dmC13σBφ(|du|)dm. We have completed the proof of Theorem 33.

Since each of φ,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)dmCσBφ(|du|λ)dm for all balls B with σBΩ and any constant λ>0. From (65) and (75), the following Poincaré inequality with the Luxemburg norm (76)u-uBφ(B)Cduφ(σB) holds under the conditions described in Theorem 33.

Theorem 34.

Let φ be a Young function in the class (p,q,C), 1p<q<, C1, Ω a bounded domain, and q(n-p)<np. Assume that uD(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 BΩ, there exists a constant C, independent of u, such that (77)Bφ(|u-uB|)dmCBφ(|du|)dm.

Proof.

From (73), it follows that (78)Bφ(|u-uB|)dmC1φ((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)/npC2(B|du|pdm)1/p, we find that (80)(B|u-uB|qdm)1/qC3(B|du|pdm)1/p. Note that the Lp-norm of |u-uB| increases with p and np/(n-p) as pn; it follows that (80) still holds when pn. Since φ is increasing, from (78) and (80), we obtain (81)Bφ(|u-uB|)dmC1φ(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|)dmC1φ(C3(B|du|pdm)1/p)C1g(C4(B|du|pdm))C5Bg(|du|p)dm. Using (i) in Definition 32 again yields (83)Bg(|du|p)dmC6Bφ(|du|)dm. Combining (82) and (83), we obtain (84)Bφ(|u-uB|)dmC7Bφ(|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-uBφ(B)Cduφ(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 following Lφ(m)-averaging domains.

Definition 35 (see [<xref ref-type="bibr" rid="B31">31</xref>]).

Let φ be an increasing convex function on [0,) with φ(0)=0. We call a proper subdomain Ωn an Lφ(m)-averaging domain, if m(Ω)< and there exists a constant C such that (86)Ωφ(τ|u-uB0|)dmCsupBΩBφ(σ|u-uB|)dm for some ball B0Ω and all u such that φ(|u|)Lloc1(Ω,m), where τ,σ are constants with 0<τ<, 0<σ< and the supremum is over all balls BΩ.

From above definition, we see that Ls-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.

Let φ be a Young function in the class G(p,q,C), 1p<q<, C1, 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|)dmCΩφ(|du|)dm, where B0Ω is some fixed ball.

Similar to the local case, the following global Poincaré inequality with the Luxemburg norm (88)u-uBφ(Ω)Cduφ(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.

Let φ be a Young function in the class G(p,q,C), 1p<q<, C1, Ω a bounded Lφ(m)-averaging domain, and q(n-p)<np. Assume that uD(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|)dmCΩφ(|du|)dm, where B0Ω is some fixed ball.

Note that (89) can be written as (90)u-uB0φ(Ω)Cduφ(Ω).

It has been proved that any John domain is a special Lφ(m)-averaging domain. Hence, we have the following results.

Corollary 38.

Let φ be a Young function in the class G(p,q,C), 1p<q<, C1, 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|)dmCΩφ(|du|)dm, where B0Ω is some fixed ball.

Choosing φ(t)=tplog+αt in Theorems 36 and 37, respectively, we obtain the following Poincaré inequalities with the Lp(log+αL)-norms.

Corollary 39.

Let φ(t)=tplog+αt, p1, and α. 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|)dmCΩ|du|plog+α(|du|)dm for any bounded Lφ(m)-averaging domain Ω and B0Ω is some fixed ball.

Note that (92) can be written as the following version with the Luxemburg norm (93)u-uB0Lp(log+αL)(Ω)CduLp(log+αL)(Ω) provided the conditions in Corollary 39 are satisfied.

Corollary 40.

Let φ(t)=tplog+αt, 1p1<p<p2, and α and let Ω be a bounded Lφ(m)-averaging domain and p2(n-p1)<np1. Assume that uD(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|)dmCΩ|du|plog+α(|du|)dm, where B0Ω is some fixed ball.

5. Inequalities for Green’s Operator

In this section, we say that uLloc1(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 . We write (95)𝒲(lΩ)={uLloc1(lΩ):u  has  generalized  gradient}. As usual, the harmonic l-fields are defined by (96)(lΩ)={u𝒲(lΩ):du=du=0,uLp  for  some  1<p<}. The orthogonal complement of in L1 is defined by (97)={uL1:  <u,h>=0  h}. We define Green’s operator (98)G:C(lΩ)C(lΩ) by setting G(u) equal to the unique element of 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  that for 1<p< and ωLp(lM), ΔG(ω)=ω-H(ω).

We will need the following lemma about Ls-estimates for Green’s operator which appeared in .

Lemma 41.

Let uC(lΩ), l=0,1,,n. For 1<s<, there exists constant C, independent of u, such that (100)ddG(u)s,Ω+ddG(u)s,Ω+dG(u)s,Ω+dG(u)s,Ω+G(u)s,ΩCus,Ω.

The following results about the composition of the Laplace-Beltrami operator and Green’s operator were proved in .

Theorem 42.

Let uC(lΩ), l=0,1,,n, and 1<s<. Then, there exists a constant C, independent of u, such that (101)Δ(G(u))s,ΩCus,Ω.

Theorem 43.

Let uC(lΩ), l=0,1,,n. Assume that 1<s<. Then, there exists a constant C, independent of u, such that (102)G(Δu)s,ΩCus,Ω.

Using Minkowski’s inequality and combining Theorems 42 and 43, we obtain the following corollary immediately.

Corollary 44.

Let uC(lΩ), l=0,1,,n. For 1<s<, there exists a constant C, independent of u, such that (103)(GΔ+ΔG)us,ΩCus,Ω.

Theorem 45.

Let uC(lΩ), l=0,1,,n. If 1<s<, then there exists a constant C, independent of u, such that (104)(G(u))Ds,DCus,D for any convex and bounded D with DΩ.

Corollary 46.

Let uC(lΩ), l=0,1,,n. Assume that 1<s<. Then, for any convex and bounded D with DΩ, there exists a constant C, independent of u, such that (105)G(u)-(G(u))Ds,DCG(u)-cs,D for any closed form c, and (106)G(u)-(G(u))Ds,DCus,D.

For ωD(Ω,l), the vector-valued differential form (107)ω=(ωx1,,ωxn) consists of differential forms ω/xiD(Ω,l), where the partial differentiation is applied to the coefficients of ω. The notations Wloc1,p(Ω,) and Wloc1,p(Ω,l) are self-explanatory. For 0<p< and a weight w(x), the weighted norm of ωW1,p(Ω,l) over Ω is denoted by (108)ωW1,p(Ω),wα=diam(Ω)-1ωp,Ω,wα+ωp,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.

Let uC(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))Bs,BCdiam(B)dus,B for all balls B with BΩ.

As an application of Theorem 47, now we will prove the following Sobolev-Poincaré imbedding theorem for Green’s operator G applied to a differential form u.

Theorem 48.

Let uC(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))BW1,s(B)Cdus,B for all balls B with BΩ.

Proof.

Since uD 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 us,B=dus,B. Using (108) and (111), we obtain (112)G(u)-(G(u))BW1,s(B)=diam(B)-1G(u)-(G(u))Bs,B+(G(u)-(G(u))B)s,B=  diam(B)-1G(u)-(G(u))Bs,B+d(G(u)-(G(u))B)s,B=  diam(B)-1G(u)-(G(u))Bs,B+d(G(u))s,B. From (109) and (112), we have (113)G(u)-(G(u))BW1,s(B)diam(B)-1G(u)-(G(u))Bs,B+d(G(u))s,Bdiam(B)-1·C1diam(B)dus,B+d(G(u))s,BC1dus,B+d(G(u))s,BC1dus,B+C2dus,BC3dus,B.

Remark 49.

Since Green’s operators can commute with d and d*, Theorem 48 can be proved by applying Corollary  4.1 in  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 form u 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.

Let G(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 wAr(Ω) for some r>1. Then, there exists a constant C, independent of u, such that (114)G(u)-(G(u))BW1,s(B),wCdus,ρB,w for all balls B with ρBΩ.

Similarly, we can extend inequalities (104) and (106) to the following Ar(Ω)-weighted version.

Theorem 51.

Let uC(lΩ), l=0,1,2,,n, be an A-harmonic tensor on a manifold Ω. Assume that ρ>1, 1<s<, and wAr(Ω) for some r>1. Then, there exists a constant C, independent of u, such that (115)(G(u))Bs,B,wαC1us,ρB,wα,G(u)-(G(u))Bs,B,wαC2us,ρB,wα for all balls B with ρBΩ and any real number α with 0<α1.

In , 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.

Let uD(Ω,0) be a solution of the nonhomogeneous A-harmonic equation (3) and wAr(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/sC(Ω|du|swdx)1/s for any bounded δ-John domain Ωn. Here Q0Ω is a fixed cube appearing in Lemma 28.

Proof.

From Theorem 51, 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 wAr, then (118)μ(NQ)MNnrμ(Q) for each cube Q with NQn (see ) and (119)max(μ(Qi),μ(Qi+1))MNnrμ(QiQi+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|s2s(|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)2sQ𝒱  Q|G(u)-(G(u))Q|sdμ(x)+2sQ𝒱  Q|(G(u))Q0-(G(u))Q|sdμ(x).

The first sum can be estimated by (117) and the Covering Lemma (121)Q𝒱  Q|G(u)-(G(u))Q|sdμ(x)C1Q𝒱  (σQ|du|swdx)C1N(Ω|du|swdx). Now we will estimate the second sum in (120). Fix a cube Q𝒱 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(QiQi+1)QiQi+1|(G(u))Qi-(G(u))Qi+1|sdμ(x)MNnrmax(μ(Qi),μ(Qi+1))×QiQi+1|(G(u))Qi-(G(u))Qi+1|sdμ(x)C2j=ii+11μ(Qj)Qj|G(u)-(G(u))Qj|sdμ(x)C3j=ii+1diam(Qj)μ(Qj)σQj|du|swdx. Since QNQj for j=i,i+1, 0ik-1, we have (124)|(G(u))Qi-(G(u))Qi+1|sχQ(x)C3j=ii+1χNQj(x)diam(Ω)μ(Qj)(σQj|du|swdx). According to (122) and diam(Ω)<, we have (note |a+b|1/s21/s(|a|1/s+|b|1/s)) (125)|(G(u))Q0-(G(u))Q|χQ(x)C4R𝒱(1μ(R)(σR|du|swdx))1/s·χNR(x) for every xn. Hence, (126)Q𝒱Q|(G(u))Q0-(G(u))Q|sdμ(x)C5n|R𝒱(1μ(R)(σR|du|swdx))1/sχNR(x)|sdμ(x). Now from (126), it follows that (127)3Q𝒱Q|(G(u))Q0-(G(u))Q|sdμ(x)C6n|R𝒱(1μ(R)(σR|du|swdx))1/sχR(x)|sdμ(x). Notice that (128)R𝒱χR(x)R𝒱χσR(x)NχΩ(x). Finally, using the elementary inequality |i=1Nti|s    Ns-1i=1N|ti|s, we conclude that (129)Q𝒱Q|(G(u))Q0-(G(u))Q|sdμ(x)C7n(R𝒱1μ(R)(σR|du|swdx)χR(x))dμ(x)=C7R𝒱(σR|du|swdx)C8(Ω|du|swdx) by the Covering Lemma. Combining (120), (121), and (129), we obtain the required inequality.

Using Theorem 52 and the proof of Theorem  3.2.10 in , 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.

Let uD(Ω,0) be a solution of the nonhomogeneous A-harmonic equation (3) and wAr(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))Q0W1,s(Ω),wCdus,Ω,w for any δ-John domain Ωn. Here Q0Ω is a fixed cube, which appears in Lemma 28.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Staples S. G. L p -averaging domains and the Poincaré inequality Annales Academiae Scientiarum Fennicae A 1989 14 1 103 127 MR997974 ZBL0706.26010 Agarwal R. P. Ding S. Nolder C. Inequalities for Differential Forms 2009 New York, NY, USA Springer MR2552910 Beckner W. A generalized Poincaré inequality for Gaussian measures Proceedings of the American Mathematical Society 1989 105 2 397 400 10.2307/2046956 MR954373 ZBL0677.42020 Belloni M. Kawohl B. A symmetry problem related to Wirtinger's and Poincaré's inequality Journal of Differential Equations 1999 156 1 211 218 10.1006/jdeq.1998.3603 MR1701790 ZBL0954.26005 Benguria R. D. Depassier M. C. A reversed Poincaré inequality for monotone functions Journal of Inequalities and Applications 2000 5 1 91 96 10.1155/S1025583400000060 MR1740214 ZBL0949.26006 Chavel I. Feldman E. A. An optimal Poincaré inequality for convex domains of non-negative curvature Archive for Rational Mechanics and Analysis 1977 65 3 263 273 MR0448457 10.1007/BF00280444 ZBL0362.35059 Acosta G. Durán R. G. An optimal Poincaré inequality in L1 for convex domains Proceedings of the American Mathematical Society 2004 132 1 195 202 10.1090/S0002-9939-03-07004-7 MR2021262 ZBL1057.26010 Ding S. Nolder C. A. Weighted Poincaré inequalities for solutions to A-harmonic equations Illinois Journal of Mathematics 2002 46 1 199 205 MR1936085 Hebey E. Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds Transactions of the American Mathematical Society 2002 354 3 1193 1213 10.1090/S0002-9947-01-02913-0 MR1867378 ZBL1044.58031 Ding S. Shi P. Weighted Poincaré-type inequalities for differential forms in Ls(μ)-averaging domains Journal of Mathematical Analysis and Applications 1998 227 1 200 215 10.1006/jmaa.1998.6096 MR1652939 Payne L. E. Weinberger H. F. An optimal Poincaré inequality for convex domains Archive for Rational Mechanics and Analysis 1960 5 286 292 MR0117419 ZBL0099.08402 Wang Y. Wu C. Global Poincaré inequalities for Green's operator applied to the solutions of the nonhomogeneous A-harmonic equation Computers & Mathematics with Applications 2004 47 10-11 1545 1554 10.1016/j.camwa.2004.06.006 Ding S. Liu B. Generalized Poincaré inequalities for solutions to the A-harmonic equation in certain domains Journal of Mathematical Analysis and Applications 2000 252 2 538 548 10.1006/jmaa.2000.6951 MR1800194 Ding S. Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds Proceedings of the American Mathematical Society 2004 132 8 2367 2375 10.1090/S0002-9939-04-07347-2 MR2052415 Liu B. A r λ ( Ω ) -weighted imbedding inequalities for A-harmonic tensors Journal of Mathematical Analysis and Applications 2002 273 2 667 676 10.1016/S0022-247X(02)00331-1 MR1932514 Nolder C. A. Hardy-Littlewood theorems for A-harmonic tensors Illinois Journal of Mathematics 1999 43 4 613 632 MR1712513 Stroffolini B. On weakly A-harmonic tensors Studia Mathematica 1995 114 3 289 301 MR1338833 Xing Y. Weighted integral inequalities for solutions of the A-harmonic equation Journal of Mathematical Analysis and Applications 2003 279 1 350 363 10.1016/S0022-247X(03)00036-2 MR1970511 Cartan H. Differential Forms 1970 Boston, Mass, USA Houghton Mifflin Co. MR0267477 Iwaniec T. Lutoborski A. Integral estimates for null Lagrangians Archive for Rational Mechanics and Analysis 1993 125 1 25 79 10.1007/BF00411477 MR1241286 ZBL0793.58002 Stein E. M. Singular Integrals and Differentiability Properties of Functions 1970 30 Princeton, NJ, USA Princeton University Press xiv+290 Princeton Mathematical Series MR0290095 Lieb E. H. Seiringer R. Yngvason J. Poincaré inequalities in punctured domains Annals of Mathematics 2003 158 3 1067 1080 10.4007/annals.2003.158.1067 MR2031861 ZBL1048.26012 Hurri-Syrjanen R. A weighted Poincaré inequality with a doubling weight Proceedings of the American Mathematical Society 1998 126 2 545 552 10.1090/S0002-9939-98-04059-3 MR1415588 ZBL0893.46022 Wang Y. Two-weight Poincaré-type inequalities for differential forms in LS(μ)-averaging domains Applied Mathematics Letters 2007 20 11 1161 1166 10.1016/j.aml.2007.02.003 MR2358794 Sawyer E. T. A characterization of two weight norm inequalities for fractional and Poisson integrals Transactions of the American Mathematical Society 1988 308 2 533 545 10.2307/2001090 MR930072 ZBL0665.42023 Ding S. Nolder C. A. L S ( μ ) -averaging domains Journal of Mathematical Analysis and Applications 2003 283 1 85 99 10.1016/S0022-247X(03)00216-6 MR1994174 ZBL1027.30053 Xing Y. Ding S. Poincaré inequalities with the Radon measure for differential forms Computers & Mathematics with Applications 2010 59 6 1944 1952 10.1016/j.camwa.2009.11.012 MR2595969 ZBL1189.35011 Buckley S. M. Koskela P. Orlicz-Hardy inequalities Illinois Journal of Mathematics 2004 48 3 787 802 MR2114252 ZBL1070.46018 Xing Y. Poincaré inequalities with Luxemburg norms in Lφ(m)-averaging domains Journal of Inequalities and Applications 2010 2010 11 241759 10.1155/2010/241759 MR2592866 Xing Y. A new weight class and Poincaré inequalities with the Radon measure Journal of Inequalities and Applications 2012 2012, article 32 10.1186/1029-242X-2012-32 MR2897464 ZBL1279.26038 Ding S. L φ ( μ ) -averaging domains and the quasi-hyperbolic metric Computers & Mathematics with Applications 2004 47 10-11 1611 1618 10.1016/j.camwa.2004.06.016 MR2079867 ZBL1063.30022 Staples S. G. Averaging domains: from Euclidean spaces to homogeneous spaces Proceedings of the Conference on Differential and Difference Equations and Applications 2006 Hindawi Publishing 1041 1048 Stein E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals 1993 43 Princeton University Press, Princeton, NJ Princeton Mathematical Series MR1232192 Scott C. L p theory of differential forms on manifolds Transactions of the American Mathematical Society 1995 347 6 2075 2096 10.2307/2154923 MR1297538 ZBL0849.58002 Ding S. Integral estimates for the Laplace-Beltrami and Green's operators applied to differential forms on manifolds Journal for Analysis and its Applications 2003 22 4 939 957 10.4171/ZAA/1181 MR2036938 ZBL1044.58002 Riesz M. Sur les fonctions conjuguées Mathematische Zeitschrift 1927 27 1 218 227 10.1007/BF01171098