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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 589040, 14 pages doi:10.1155/2010/589040 Research Article Aλ3 λ1 , λ2 , Ω -Weighted Inequalities with Lipschitz r and BMO Norms Yuxia Tong,1 Juan Li,2 and Jiantao Gu1 College of Science, Hebei Polytechnic University, Tangshan 063009, China Department of Mathematics, Ningbo University, Ningbo 315211, China Correspondence should be addressed to Yuxia Tong, tongyuxia@126.com Received 29 December 2009; Revised 25 March 2010; Accepted 31 March 2010 Academic Editor: Shusen Ding Copyright q 2010 Yuxia Tong et al 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 first define a new kind of Aλ3 λ1 , λ2 , Ω two-weight, then obtain some two-weight integral r inequalities with Lipschitz norm and BMO norm for Green’s operator applied to differential forms Introduction Green’s operator G is often applied to study the solutions of various differential equations and to define Poisson’s equation for differential forms Green’s operator has been playing an important role in the study of PDEs In many situations, the process to study solutions of PDEs involves estimating the various norms of the operators Hence, we are motivated to establish some Lipschitz norm inequalities and BMO norm inequalities for Green’s operator in this paper In the meanwhile, there have been generally studied about Ar Ω -weighted 1, and Aλ Ω -weighted 3, different inequalities and their properties Results for more r applications of the weight are given in 5, The purpose of this paper is to derive the new weighted inequalities with the Lipschitz norm and BMO norm for Green’s operator applied to differential forms We will introduce Aλ3 λ1 , λ2 , Ω -weight, which can be considered as a r further extension of the Aλ Ω -weight r We keep using the traditional notation Let Ω be a connected open subset of Rn , let e1 , e2 , , en be the standard unit basis of l n Rn be the linear space of l-covectors, spanned by the exterior products R , and let l i , i , , il , ≤ i < i < · · · < eI ei1 ∧ ei2 ∧ · · · ∧ eil , corresponding to all ordered l-tuples I ⊕ l is a graded algebra with il ≤ n, l 0, 1, , n We let R R1 The Grassman algebra I βI eI ∈ , the inner product respect to the exterior products For α α eI ∈ and β Journal of Inequalities and Applications i1 , i2 , , il and all in is given by α, β αI βI with summation over all l-tuples I integers l 0, 1, , n We define the Hodge star operator : → by the rule e1 ∧ e2 ∧ · · · ∧ en and α∧ β β∧ α α, β for all α, β ∈ The norm of α ∈ is given by the formula α, α α ∧ α ∈ R The Hodge star is an isometric isomorphism on with |α|2 −1 l n−l : l → l : l → n−l and Balls are denoted by B and ρB is the ball with the same center as B and with diam ρB ρ diam B We not distinguish balls from cubes throughout this paper The n-dimensional Lebesgue measure of a set E ⊆ Rn is denoted by |E| We call w x a weight if w ∈ L1 Rn and w > a.e For < p < ∞ and a weight w x , we denote the loc weighted Lp -norm of a measurable function f over E by f p f x p,E,wα wα dx 1/p , 1.1 E where α is a real number Differential forms are important generalizations of real functions and distributions Specially, a differential l-form ω on Ω is a de Rham current 7, Chapter III on Ω with values in l Rn ; note that a 0-form is the usual function in Rn A differential l-form ω on Ω is a Schwartz distribution on ω with values in l Rn We use D Ω, l to denote the space of all ωi1 i2 ···il x dxi1 ∧dxi2 ∧· · ·∧dxil We write Lp Ω, l differential l-forms ω x I ωI x dxI p for the l-forms with ωI ∈ L Ω, R for all ordered l-tuples I Thus Lp Ω, l is a Banach space with norm ω For ω ∈ D Ω, l p,Ω Ω |ω x |p dx 1/p |ωI x |2 Ω p/2 1/p dx 1.2 the vector-valued differential form ∂ω ∂ω , , ∂x1 ∂xn ∇ω 1.3 consists of differential forms ∂ω ∈D ∂xi Ω, l , 1.4 where the partial differentiations are applied to the coefficients of ω As usual, W 1,p Ω, l is used to denote the Sobolev space of l-forms, which equals p Lp Ω, l ∩ L1 Ω, l with norm ω W 1,p Ω, l ω W 1,p Ω, l diam Ω −1 ω p,Ω ∇ω p,Ω 1.5 Journal of Inequalities and Applications 1,p 1,p The notations Wloc Ω, R and Wloc Ω, l are self-explanatory For < p < ∞ and a weight w x , the weighted norm of ω ∈ W 1,p Ω, l over Ω is denoted by ω W 1,p Ω, l ,wα ω W 1,p Ω, l ,wα diam Ω −1 ω p,Ω,wα ∇ω p,Ω,wα , 1.6 where α is a real number We denote the exterior derivative by d : D Ω, l → D Ω, l for l 0, 1, , n Its formal adjoint operator d : D Ω, l → D Ω, l is given by d −1 nl d on l l ,l 0, 1, , n Let Ω be the lth exterior power of the cotangent bundle and D Ω, l {u ∈ L1 Ω : let C∞ l Ω be the space of smooth l-forms on Ω We set W l Ω loc l Ω {u ∈ W l Ω : u has generalized gradient} The harmonic l-fields are defined by H p 0, u ∈ L for some < p < ∞} The orthogonal complement of H in L1 is du du ⊥ {u ∈ L1 : u, h for all h ∈ H} Then, Green’s operator G is defined as defined by H l ∞ Ω → H ⊥ ∩ C∞ l Ω by assigning G u to be the unique element of H ⊥ ∩ C∞ l Ω G:C satisfying Poisson’s equation ΔG u u − H u , where H is the harmonic projection operator that maps C∞ l Ω onto H, so that H u is the harmonic part of u See for more properties of Green’s operator is called the The nonlinear elliptic partial differential equation d A x, du homogeneous A-harmonic equation or the A-harmonic equation, and the differential equation d A x, du B x, du 1.7 is called the nonhomogeneous A-harmonic equation for differential forms, where A : Ω × l Rn → l Rn and B : Ω × l Rn → l−1 Rn satisfy the following conditions: |A x, ξ | ≤ a|ξ|p−1 , A x, ξ , ξ ≥ |ξ|p , |B x, ξ | ≤ b|ξ|p−1 1.8 for almost every x ∈ Ω and all ξ ∈ l Rn Here a, b > are constants and < p < ∞ is a fixed exponent associated with 1.7 A solution to 1.7 is an element of the Sobolev space 1,p Wloc Ω, l−1 such that Ω A x, du · dϕ B x, du · ϕ 1.9 1,p for all ϕ ∈ Wloc Ω, l−1 with compact support ξ|ξ|p−2 with p > Then, A Let A : Ω × l Rn → l Rn be defined by A x, ξ becomes the p-harmonic equation satisfies the required conditions and d A x, du d du|du|p−2 1.10 for differential forms If u is a function a 0-form , 1.10 reduces to the usual p-harmonic equation div ∇u|∇u|p−2 for functions We should notice that if the operator B equals in Journal of Inequalities and Applications 1.7 , then 1.7 reduces to the homogeneous A-harmonic equation Some results have been obtained in recent years about different versions of the A-harmonic equation; see 9–11 Let u ∈ L1 Ω, l , l 0, 1, , n We write u ∈ loc Lipk Ω, l , ≤ k ≤ 1, if loc u sup |Q|− n loc Lipk ,Ω k /n u − uQ σQ⊂Ω 1,Q 0, w2 x > a.e r and sup B for any ball B ⊂ Ω |B| B λ w1 dx |B| B w2 λ3 r−1 λ2 / r−1 dx and C, independent of w, such that w β,B ≤ C|B| 1−β /β w 2.2 1,B for all balls B ⊂ Rn We need the following generalized Holder inequality ă Lemma 2.3 Let < α < ∞, < β < ∞ and s−1 Rn , then fg s,E α−1 ≤ f β−1 If f and g are measurable functions on α,E · g 2.3 β,E for any E ⊂ Rn The following version of weak reverse Holder inequality appeared in 13 ă Lemma 2.4 Suppose that u is a solution to the nonhomogeneous A-harmonic equation 1.7 in Ω, σ > and q > There exists a constant C, depending only on σ, n, p, a, b and q, such that du p,Q ≤ C|Q| q−p /pq du 2.4 q,σQ for all Q with σQ ⊂ Ω Lemma 2.5 see 14 Let du ∈ Ls Ω, l be a smooth form and let G be Green’s operator, l 1, , n, and < s < ∞ Then, there exists a constant C, independent of u, such that G u − G u B s,B ≤ C|B| diam B du s,B 2.5 for all balls B ⊂ Ω We need the following Lemma 2.6 Caccioppoli inequality that was proved in Lemma 2.6 see Let u ∈ D Ω, l be a solution to the nonhomogeneous A-harmonic equation 1.7 in Ω and let σ > be a constant Then, there exists a constant C, independent of u, such that du p,B ≤ C diam B −1 u−c p,σB for all balls or cubes B with σB ⊂ Ω and all closed forms c Here < p < ∞ 2.6 Journal of Inequalities and Applications Lemma 2.7 see 14 Let du ∈ Ls Ω, l , l 1, 2, , n, < s < ∞, be a smooth form in a domain Ω Then, there exists a constant C, independent of u, such that G u loc Lipk ,Ω ≤ C du s,Ω , 2.7 where k is a constant with ≤ k ≤ Lemma 2.8 see 14 Let du ∈ Ls Ω, l , l 1, 2, , n, < s < ∞, be a smooth form in a bounded domain Ω and let G be Green’s operator Then, there exists a constant C, independent of u, such that G u ,Ω ≤ C du s,Ω 2.8 Main Results and Proofs Theorem 3.1 Let du ∈ Ls Ω, l , υ , l 1, 2, , n, < s < ∞, be a solution of the nonhomogeneous A-harmonic equation 1.7 in a bounded domain Ω and let G be Green’s operator, where the Radon αλ αλ measures μ and υ are defined by dμ w1 x , dυ w2 λ3 x Assume that w1 x , w2 x ∈ Aλ3 λ1 , λ2 , Ω for some r > 1, < λ1 , λ2 , λ3 < ∞ Then, there exists a constant C, independent of u, r such that G u − G u B 1,B,wαλ1 ≤ C|B| diam B du αλ2 λ3 s,σB,w2 , 3.1 where k is a constant with ≤ k ≤ 1, and α is a constant with < α < Proof Choose t s/ − α where < α < 1; then < s < t and αt/ t − s 1/s 1/t t − s /st, by Lemmas 2.3 and 2.5, we have Gu − G u B |G u − G u t B | dx B ≤ B G u − G u B t,B ≤ C1 |B| diam B du for all ball B ⊂ Ω Choosing m 1/s s αλ1 B | w1 dx |G u − G u αλ s,B,w1 s/ αλ3 r − du t,B λ w1 t,B Since 1/t B αλ w1 /s st/ t−s t−s /st dx 3.2 α/s 1,B λ w1 α/s 1,B , then m < s From Lemma 2.4, we have ≤ C2 |B| m−t /mt du m,σB , 3.3 Journal of Inequalities and Applications where σ > and σB ⊂ Using Holder inequality with 1/m ă du m,B B ≤ σB du m αλ −αλ |du|w2 λ3 /s w2 λ3 /s αλ3 r−1 /s λ2 / r−1 w2 σB dx 3.4 λ2 αλ3 /s w2 αλ λ s,σB,w2 1/m dx 1/s αλ |du|s w2 λ3 dx αλ3 r − /s, we have 1/s 1/ r−1 ,σB Since w1 , w2 ∈ Aλ3 λ1 , λ2 , Ω , then r α/s λ w1 1,B λ2 αλ3 /s w2 · 1/ r−1 ,σB ⎡ ≤⎣ λ w1 dx σB σB λ3 r−1 λ2 / r−1 w2 ⎤α/s ⎦ dx ⎡ ⎣|σB|λ3 r−1 ≤ C3 |σB|αλ3 ≤ C4 |B|αλ3 Since m − t /mt G u − G u |σB| σB |σB| λ w1 dx w2 σB λ3 r−1 λ2 / r−1 dx ⎦ r−1 /s α/s r−1 /s α/s αλ3 r − B s,B,wαλ1 α /s 0, combining with 3.2 , 3.3 , 3.4 , and 3.5 , we have ≤ C1 |B| diam B C2 |B| m−t /mt du C5 |B| diam B du αλ λ s,σB,w2 αλ2 λ3 s,σB,w2 C4 |B|αλ3 |G u − G u B 1,B,wαλ1 B ≤ B s−1 /s 3.6 1/s s − /s, αλ1 B |w1 dx |G u − G u μB r−1 /s α/s Notice that |Ω| < ∞, − 1/s > 0; from 3.6 and the Holder inequality with ă we nd that G u − G u 3.5 ⎤α/s s αλ1 B | w1 dx ≤ C6 |B| diam B du We have completed the proof of Theorem 3.1 B s,B,wαλ1 αλ2 λ3 s,σB,w2 αλ2 λ3 s,σB,w2 B αλ 1s/ s−1 w1 dx s−1 /s 3.7 G u − Gu ≤ |Ω|1−1/s C5 |B| diam B du 1/s Journal of Inequalities and Applications Remark Specially, choosing λ2 λ3 λ1 and w1 G u − G u B 1,B,wαλ1 w2 in Theorem 3.1, we have ≤ C6 |B| diam B du αλ1 s,σB,w1 3.8 Next, we will establish the following weighted norm comparison theorem between the Lipschitz and the BMO norms Theorem 3.2 Let u ∈ Ls Ω, l , υ , l 1, 2, , n, < s < ∞, be a solution of the nonhomogeneous A-harmonic equation 1.7 in a bounded domain Ω and let G be Green’s operator, where the Radon αλ αλ measures μ and υ are defined by dμ w1 x , dυ w2 λ3 /s x Assume that w1 x , w2 x ∈ Aλ3 λ1 , λ2 , Ω for some r > 1, < λ1 , λ2 , λ3 < ∞ with w1 x ≥ ε > for any x ∈ Ω Then, there r exists a constant C, independent of u, such that G u ≤C u αλ1 loc Lipk ,Ω,w1 αλ2 λ3 /s ,Ω,w2 , 3.9 where k is a constant with ≤ k ≤ 1, and α is a constant with < α < Proof Choose t s/ − α where < α < 1; then < s < t and αt/ t − s 1/s 1/t t − s /st, by Lemma 2.3, we have Since 1/s du αλ s,σ1 B,w1 σ1 B αλ |du|s w1 dx 1/t αλ w1 /s |du|t dx ≤ σ1 B du σ1 B t,σ1 B λ w1 st/ t−s t−s /st dx α/s 1,σ1 B for any ball B and some constant σ1 > with σ1 B ⊂ Ω Choosing c find that du t,σ1 B 3.10 −1 ≤ C1 diam B u − uB uB in Lemma 2.6, we t,σ2 B , 3.11 where σ2 > σ1 is a constant and σ2 B ⊂ Ω Combining 3.8 , 3.10 , and 3.11 , it follows that G u − Gu B 1,B,wαλ1 ≤ C2 |B| diam B du ≤ C2 |B| diam B C3 |B| u − uB αλ1 s,σ1 B,w1 λ w1 t,σ2 B α/s 1,σ1 B λ w1 C1 diam B α/s 1,σ1 B −1 u − uB t,σ2 B 3.12 Journal of Inequalities and Applications Choosing m s/ αλ3 r −1 s , then m < s < t Applying the weak reverse Holder inequality ă for the solutions of the nonhomogeneous A-harmonic equation, we obtain u − uB t,σ2 B ≤ C4 |B| m−t /mt u − uB m,σ3 B , 3.13 where σ3 > σ2 is a constant and σ3 B ⊂ Ω Substituting 3.13 into 3.12 , we have G u − G u B 1,B,wαλ1 ≤ C3 |B|C4 |B| m−t /mt u − uB C5 |B| Using Holder inequality with 1/m ă u − uB m,σ3 B σ3 B m−t /mt u− αλ |u − uB |w2 λ3 /s dx u − uB αλ λ /s 1,σ3 B,w2 α/s 1,σ1 B 3.14 α/s λ uB m,σ3 B w1 1,σ1 B αλ3 r − /s, we have αλ −αλ |u − uB |w2 λ3 /s w2 λ3 /s σ3 B ≤ 1/1 λ w1 m,σ3 B m σ3 B w2 1/m dx αλ3 r−1 /s λ2 / r−1 w2 3.15 dx αλ3 /s λ2 1/ r−1 ,σ3 B Since w1 , w2 ∈ Aλ3 λ1 , λ2 , Ω , then r α/s λ w1 1,σ1 B w2 · λ2 αλ3 /s 1/ r−1 ,σ3 B ⎡ ≤⎣ σ3 B λ w1 dx σ3 B w2 λ3 r−1 λ2 / r−1 ⎤α/s ⎦ dx ⎡ ⎣|σ3 B|λ3 r−1 ≤ C6 |σ3 B|αλ3 ≤ C7 |B|αλ3 1 |σ3 B| r−1 /s α/s r−1 /s α/s σ3 B λ w1 dx |σ3 B| σ3 B w2 λ3 r−1 λ2 / r−1 dx ⎤α/s ⎦ 3.16 10 Journal of Inequalities and Applications αλ3 r − Since m − t /mt have G u − G u B 1,B,wαλ1 α /s ≤ C5 |B|1 1/s, combining with 3.14 , 3.15 , and 3.16 , we m−t /mt C7 |B|αλ3 C8 |B|1/s u − uB Since μ B B αλ w1 dx ≥ B εαλ1 dx r−1 /s α/s αλ λ /s 1,σ3 B,w2 u − uB αλ2 λ3 /s 1,σ3 B,w2 3.17 C9 |B|, we have C10 ≤ μ B |B| 3.18 for all ball B Notice that − k/n > and |Ω| < ∞; from 3.17 , we have G u αλ1 loc Lipk ,Ω,w1 − n k /n sup μ B G u − Gu σ4 B⊂Ω ≤ C8 sup μ B −1/s−k/n B 1,B,wαλ1 |B|1/s u − uB αλ2 λ3 /s 1,σ3 B,w2 σ4 B⊂Ω ≤ C11 sup |B|−1/s−k/n |B|1 1/s |B|−1 u − uB σ4 B⊂Ω 1−k/n ≤ C11 sup |Ω| |B| −1 u − uB σ4 B⊂Ω ≤ C12 sup |B|−1 u − uB σ4 B⊂Ω C12 u αλ2 λ3 /s ,Ω,w2 αλ2 λ3 /s 1,σ3 B,w2 3.19 αλ2 λ3 /s 1,σ3 B,w2 αλ2 λ3 /s 1,σ3 B,w2 , where σ4 > σ3 is a constant and σ4 B ⊂ Ω We have completed the proof of Theorem 3.2 Now, we will prove the following weighted inequality between the BMO norm and the Lipschitz norm for Green’s operator Theorem 3.3 Let u ∈ Ls Ω, l , υ , l 1, 2, , n, < s < ∞, be a solution of the nonhomogeneous A-harmonic equation 1.7 in a bounded domain Ω and let G be Green’s operator, where the Radon αλ αλ λ measures μ and υ are defined by dμ w1 x , dυ w2 λ3 /s x Assume that w1 x ∈ Ar Ω λ3 and w1 x , w2 x ∈ Ar λ1 , λ2 , Ω for some r > 1, < λ1 , λ2 , λ3 < ∞ with w1 x ≥ ε > for any x ∈ Ω Then, there exists a constant C, independent of u, such that Gu where α is a constant with < α < αλ1 ,Ω,w1 ≤C u αλ2 λ3 /s loc Lipk ,Ω,w2 , 3.20 Journal of Inequalities and Applications 11 λ Proof Since w1 ∈ Ar Ω , using Lemma 2.2, there exist constants β > and C1 > 0, such that λ w1 for any ball B ⊂ Rn Since 1/s G u −G u λ ≤ C1 |B| 1−β /β w1 3.21 1,B s − /s, by Lemma 2.3, we have B 1,B,wαλ1 B αλ |G u − G u B |w1 dx αλ |G u − G u B |s w1 dx ≤ B s−1 /s μ B Choose t 1/s 1/t β,B G u −G u 1/s B αλ w1 dx s−1 /s B s,B,wαλ1 s/ − α/β where < α < 1, β > 1; then < s < t and αt/ t − s t − s /st, by Lemma 2.3 and 3.21 , we have G u −G u B s,B,wαλ1 B ≤ B s αλ |G u − G u B |w1 /s |G u − G u B |t dx 1/s 1/t α/ βs λ β B w1 dx 3.23 α/s B t,B λ · w1 ≤ G u −G u Gu − G u β Since dx G u −G u B t,B λ · C2 |B| 1−β α/ βs w1 From Lemmas 2.5 and 2.6 with c 3.22 β,B α/s 1,B uB , we have B t,B ≤ C3 |B| diam B du t,B ≤ C3 |B| diam B C4 diam B C5 |B| u − uB −1 u − uB t,σ1 B 3.24 t,σ1 B , where σ1 > is a constant and σ1 B ⊂ Ω Applying the weak reverse Holder inequality for the ă solutions of the nonhomogeneous A-harmonic equation, we obtain u − uB t,σ1 B ≤ C6 |B| m−t /mt u − uB m,σ2 B , 3.25 12 Journal of Inequalities and Applications where σ2 > σ1 is a constant and σ2 B ⊂ Ω Choosing m s/ αλ3 r − Using Holder inequality with 1/m 1/1 αλ3 r − /s, we have ¨ u − uB |u − m,σ2 B σ2 B ≤ |u − σ2 B u − uB 1/m m αλ −αλ uB |w2 λ3 /s w2 λ3 /s dx αλ uB |w2 λ3 /s dx αλ2 λ3 /s 1,σ2 B,w2 1/ r−1 ,σ2 B Since w1 , w2 ∈ Aλ3 λ1 , λ2 , Ω and m−t /mt αλ3 r −1 /s α/s r combining with 3.22 , 3.23 , 3.24 , and 3.25 , we have G u −G u ≤μ B 3.26 dx αλ3 /s λ2 w2 αλ3 r−1 /s λ2 / r−1 w2 σ2 B s , then m < < s s−1 /s 1−β α/ βs 0, B 1,B,wαλ1 s−1 /s C5 |B|C6 |B| m−t /mt C2 |B| 1−β α/ βs C7 |B|αλ3 ≤ C8 |B||B| 1−β α/ βs |B| m−t /mt C8 |B| u − uB αλ3 r−1 /s α/s s−1 /s r−1 /s α/s u − uB u − uB αλ2 λ3 /s 1,σ2 B,w2 αλ2 λ3 /s 1,σ2 B,w2 αλ2 λ3 /s 1,σ2 B,w2 3.27 From the definitions of the Lipschitz and BMO norms, we obtain G u αλ1 ,Ω,w1 sup |B|−1 G u − G u σ3 B⊂Ω sup |B|k/n |B|− n k /n B 1,B,wαλ1 G u −G u B 1,B,wαλ1 σ3 B⊂Ω ≤ sup |M|k/n |B|− n k /n G u −G u σ3 B⊂Ω ≤ C9 sup |B|− n k /n G u −G u σ3 B⊂Ω 3.28 B 1,B,wαλ1 B 1,B,wαλ1 for all balls B with σ3 > σ2 and σ3 B ⊂ Ω Substituting 3.27 into 3.28 , we have G u αλ1 ,Ω,w1 ≤ C9 sup |B|− n k /n G u −G u k /n C8 |B| u − uB σ3 B⊂Ω ≤ C9 sup |B|− n σ3 B⊂Ω ≤ C10 sup |B| αλ2 λ3 /s 1,σ2 B,w2 3.29 − n k /n u − uB σ3 B⊂Ω C10 u B 1,B,wαλ1 αλ2 λ3 /s loc Lipk ,Ω,w2 We have completed the proof of Theorem 3.3 αλ2 λ3 /s 1,σ2 B,w2 Journal of Inequalities and Applications · 13 Using the same methods, and by Lemmas 2.7 and 2.8, we can estimate Lipschitz norm and BMO norm · ,Ω,wα of Green’s operator in terms of Ls norm loc Lipk ,Ω,wα Theorem 3.4 Let du ∈ Ls Ω, l , μ , l 1, 2, , n, < s < ∞, be a solution of the nonhomogeneous A-harmonic equation 1.7 in a bounded domain Ω and let G be Green’s operator, where the Radon αλ αλ λ measures μ and υ are defined by dμ w1 x , dυ w2 λ3 /s x Assume that w1 x ∈ Ar Ω λ3 and w1 x , w2 x ∈ Ar λ1 , λ2 , Ω for some r > 1, < λ1 , λ2 , λ3 < ∞ with w1 x ≥ ε > for any x ∈ Ω Then, there exists a constant C, independent of u, such that G u αλ1 loc Lipk ,Ω,w1 ≤ C du αλ2 λ3 /s s,Ω,w2 , 3.30 where k is a constant with ≤ k ≤ 1, and α is a constant with < α < Theorem 3.5 Let du ∈ Ls Ω, l , μ , l 1, 2, , n, < s < ∞, be a solution of the nonhomogeneous A-harmonic equation 1.7 in a bounded domain Ω and let G be Green’s operator, where the Radon αλ αλ λ measures μ and υ are defined by dμ w1 x , dυ w2 λ3 /s x Assume that w1 x ∈ Ar Ω λ3 and w1 x , w2 x ∈ Ar λ1 , λ2 , Ω for some r > 1, < λ1 , λ2 , λ3 < ∞ with w1 x ≥ ε > for any x ∈ Ω Then, there exists a constant C, independent of u, such that G u αλ1 ,Ω,w1 ≤ C du αλ2 λ3 /s s,Ω,w2 , 3.31 where α is a constant with < α < Remark Note that the differentiable functions are special differential forms 0-forms Hence, the usual p-harmonic equation div ∇u|∇u|p−2 for functions is the special case of the A-harmonic equation for differential forms Therefore, all results that we have proved for solutions of the A-harmonic equation in this paper are still true for p-harmonic functions Acknowledgments The first author is supported by NSFC No:10701013 , NSF of Hebei Province A2010000910 and Tangshan Science and Technology projects 09130206c The second author is supported by NSFC 10771110 and 60872095 and NSF of Nongbo 2008A610018 References J B Garnett, Bounded Analytic Functions, vol 96 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1981 J Heinonen, T Kilpelă inen, and O Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, a Oxford Mathematical Monographs, Oxford University Press, Oxford, UK, 1993 S Ding and Y Ling, “Weighted norm inequalities for conjugate A-harmonic tensors,” Journal of Mathematical Analysis and Applications, vol 203, no 1, pp 278–288, 1996 S Ding, “Estimates of weighted integrals for differential forms,” Journal of Mathematical Analysis and Applications, vol 256, no 1, pp 312–323, 2001 D Cruz-Uribe and C P´ rez, “Two-weight, weak-type norm inequalities for fractional integrals, e Calderon-Zygmund operators and commutators,” Indiana University Mathematics Journal, vol 49, no ´ 2, pp 697–721, 2000 14 Journal of Inequalities and Applications J Garc´a-Cuerva and J M Martell, “Two-weight norm inequalities for maximal operators and ı fractional integrals on non-homogeneous spaces,” Indiana University Mathematics Journal, vol 50, no 3, pp 1241–1280, 2001 T Iwaniec and A Lutoborski, “Integral estimates for null Lagrangians,” Archive for Rational Mechanics and Analysis, vol 125, no 1, pp 25–79, 1993 S Ding, “Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds,” Proceedings of the American Mathematical Society, vol 132, no 8, pp 2367–2375, 2004 Y Wang and C Wu, “Sobolev imbedding theorems and Poincar´ inequalities for Green’s operator on e solutions of the nonhomogeneous A-harmonic equation,” Computers & Mathematics with Applications, vol 47, pp 1545–1554, 2004 10 Y Xing, “Weighted integral inequalities for solutions of the A-harmonic equation,” Journal of Mathematical Analysis and Applications, vol 279, no 1, pp 350–363, 2003 11 Y Xing, “Two-weight imbedding inequalities for solutions to the A-harmonic equation,” Journal of Mathematical Analysis and Applications, vol 307, no 2, pp 555–564, 2005 12 S Ding and P Shi, “Weighted Poincar´ -type inequalities for differential forms in Ls μ -averaging e domains,” Journal of Mathematical Analysis and Applications, vol 227, no 1, pp 200–215, 1998 13 S Ding and C A Nolder, “Weighted Poincar´ inequalities for solutions to A-harmonic equations,” e Illinois Journal of Mathematics, vol 46, no 1, pp 199–205, 2002 14 Y Xing and S Ding, “Inequalities for Green’s operator with Lipschitz and BMO norms,” Computers & Mathematics with Applications, vol 58, no 2, pp 273–280, 2009  ... of ω ∈ W 1,p ? ?, l over Ω is denoted by ω W 1,p ? ?, l ,wα ω W 1,p ? ?, l ,wα diam Ω −1 ω p ,Ω, wα ? ?ω p ,Ω, wα , 1.6 where α is a real number We denote the exterior derivative by d : D ? ?, l → D ? ?, l for... the Aλ3 λ1 , λ2 , Ω condition for r some r > and < λ1 , λ2 , λ3 < ∞; let w1 , w2 ∈ Aλ3 λ1 , λ2 , Ω , if w1 x > 0, w2 x > a.e r and sup B for any ball B ⊂ Ω |B| B λ w1 dx |B| B w2 λ3 r? ??1 λ2 / r? ??1... Lipk ? ?, l for those forms whose coefficients are in the usual Lipschitz space with exponent k and write u Lipk ,? ? for this norm Similarly, for u ∈ L1 ? ?, loc l ,l l 0, 1, , n, we write u ∈ BMO Ω,