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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 351597, 14 pages doi:10.1155/2010/351597 Research Article Hardy-Littlewood and Caccioppoli-Type Inequalities for A-Harmonic Tensors Peilin Shi 1 and Shusen Ding 2 1 Department of Epidemiology, Harvard School of Public Health, Harvard University, Boston, MA 02115, USA 2 Department of Mathematics, Seattle University, Seattle, WA 98122, USA Correspondence should be addressed to Peilin Shi, pshi@hsph.harvard.edu Received 21 December 2009; Revised 17 March 2010; Accepted 19 March 2010 Academic Editor: Yuming Xing Copyright q 2010 P. Shi and S. 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 prove the new versions of the weighted Hardy-Littlewood inequality and Caccioppoli-type inequality for A-harmonic tensors. We also explore applications of our results to K-quasiregular mappings and p-harmonic functions in R n . 1. Introduction The purpose of this paper is to prove the new versions of the weighted Hardy-Littlewood and Caccioppoli-type inequalities for the A-harmonic tensors. Our results may have applications in different fields, particularly, in the study of the integrability of solutions to the A-harmonic equation in some domains. Roughly speaking, the A-harmonic tensors are solutions of the A-harmonic equation, which is intimately connected to the fields, including potential theory, quasiconformal mappings, and the theory of elasticity. The investigation of the A-harmonic equation has developed rapidly in the recent years see 1–11. In this paper, we still keep using the standard notations and symbols. All notations and definitions involved in this paper can be found in 1 cited in the paper. We always assume that M is a bounded and convex domain in R n , n ≥ 2. We write R  R 1 . Let e 1 ,e 2 , ,e n be the standard unit basis of R n and ∧ l  ∧ l R n  the linear space of l-vectors, generated by the exterior products e I  e i 1 ∧ e i 2 ∧···e i l , corresponding to all ordered l-tuples I i 1 ,i 2 , ,i l , 1 ≤ i 1 <i 2 < ···<i l ≤ n, l  0, 1, ,n. The Grassman algebra ∧  ⊕∧ l is a graded algebra with respect to the exterior products. For α   α I e I ∈∧and β   β I e I ∈∧, the inner product in ∧ is given by α, β   α I β I , with summation over all l-tuples I i 1 ,i 2 , ,i l  and all integers l  0, 1, ,n. We define the Hodge star operator : ∧→∧by the rule 1  e 1 ∧ e 2 ∧···∧e n 2 Journal of Inequalities and Applications and α ∧ β  β ∧ α  α, β1 for all α, β ∈∧. The norm of α ∈∧is given by the formula |α| 2  α, α  α ∧ α ∈∧ 0  R. The Hodge star is an isometric isomorphism on ∧ with  : ∧ l →∧ n−l and −1 ln−l : ∧ l →∧ l . It is well known that a differential l-form ω on M is a de Rham current see 12, Chapter III on M with values in ∧ l R n . Let Λ l M be the lth exterior power of the cotangent bundle. We use D  M, Λ l  to denote the space of all differential l-forms and L p Λ l M to denote the l-forms ω  x    I ω I  x  dx I   ω i 1 i 2 ···i l  x  dx i 1 ∧ dx i 2 ∧···∧dx i l 1.1 on M satisfying  M |ω I | p < ∞ for all ordered l-tuples I, where I i 1 ,i 2 , ,i l ,1 ≤ i 1 < i 2 < ··· <i l ≤ n,andω i 1 i 2 ···i l x are differentiable functions. Thus, L p Λ l M is a Banach space with norm ||ω|| p,M   M |ωx| p dx 1/p   M   I |ω I x| 2  p/2 dx 1/p . Here, |ux|    I |ω I x| 2  1/2   I |ω i 1 i 2 ···i l x| 2  1/2 . We denote the exterior derivative by d : D  M, Λ l  → D  M, Λ l1  for l  0, 1, ,n. The Hodge codifferential operator d  : D  M, Λ l1  → D  M, Λ l  is given by d  −1 nl1 don D  M, ∧ l1 , l  0, 1, ,n.WeuseB to denote aballandσB, σ>0, is the ball with the same center as B and with diamσBσ diamB. We do not distinguish the balls from cubes in this paper. For any measurable set E ⊂ R n ,we write |E| for the n-dimensional Lebesgue measure of E. We call w a weight if w ∈ L 1 loc R n  and w>0 a.e For 0 <p<∞, we write f ∈ L p Λ l E, w α  if the weighted L p -norm of f over E satisfies ||f|| p,E,w α   E |fx| p wx α dx 1/p < ∞, where α is a real number. See 1 or 13 for more properties of differential forms. For any differential k-form ux  I ω I xdx I   ω i 1 i 2 ···i k xdx i 1 ∧ dx i 2 ∧···∧dx i k , k  1, 2, ,n, the vector-valued differential form ∇u is defined by ∇u   ∂u ∂x 1 , , ∂u ∂x n     I ∂u I ∂x 1 dx I ,  I ∂u I ∂x 2 dx I , ,  I ∂u I ∂x n dx I  , | ∇u |  ⎛ ⎝ n  j1      ∂u ∂x j      2 ⎞ ⎠ 1/2  ⎛ ⎝ n  j1  I      ∂u I ∂x j      2 ⎞ ⎠ 1/2 . 1.2 Also, we all know that du  x   n  k1  1≤i 1 <i 2 <···<i k ∂ω i 1 i 2 ···i k  x  ∂x k dx k ∧ dx i 1 ∧ dx i 2 ∧···∧dx i k ,k 0, 1, ,n− 1, | du  x  |   n  k1  1≤i 1 <i 2 <···<i k     ∂ω i 1 i 2 ···i k  x  ∂x k     2  1/2 . 1.3 There has been remarkable work in the study of the A-harmonic equation d  A  x, dω   0 1.4 Journal of Inequalities and Applications 3 for differential forms, where A : M ×∧ l R n  →∧ l R n  satisfies the following conditions: | A  x, ξ  | ≤ a | ξ | p−1 ,  A  x, ξ  ,ξ  ≥ | ξ | p 1.5 for almost every x ∈ M and all ξ ∈∧ l R n .Herea>0 is a constant and 1 <p<∞ is a fixed exponent associated with 1.4. A solution to 1.4 is an element of the Sobolev space W 1 p,loc Ω, ∧ l−1  such that  Ω Ax, dω,dϕ  0 for all ϕ ∈ W 1 p M, ∧ l−1  with compact support. Definition 1.1. We call u an A-harmonic tensor on M if u satisfies the A-harmonic equation 1.4 on M. Adifferential l-form u ∈ D  M, ∧ l  is called a closed form if du  0onM. Similarly, a differential l  1-form v ∈ D  M, ∧ l1  is called a coclosed form if d  v  0. The equation A  x, du   d  v 1.6 is called the conjugate A-harmonic equation. Suppose that u is a solution to 1.4 in Ω. Then, at least locally in a ball B, there exists a form v ∈ W 1 q B, ∧ l1 ,1/p  1/q  1, such that 1.6 holds. Definition 1.2. When u and v satisfy 1.6 on M,andA −1 exists on M, we call u and v conjugate A-harmonic tensors on M. Let Q ⊂ R n be a cube or a ball. To each y ∈ Q there corresponds a linear operator K y : C ∞ Q, ∧ l  → C ∞ Q, ∧ l−1  defined by K y ωx; ξ 1 , ,ξ l   1 0 t l−1 ωtx  y − ty; x − y, ξ 1 , ,ξ l−1 dt and the decomposition ω  dK y ωK y dω. The linear operator T Q : C ∞ Q, ∧ l  → C ∞ Q, ∧ l−1  is defined by averaging K y over all points y in QT Q ω   Q ϕyK y ωdy, where ϕ ∈ C ∞ 0 Q is normalized by  Q ϕydy  1. See 1 for more property for the operator T Q . We define the l-form ω Q ∈ D  Q, ∧ l  by ω Q  |Q| −1  Q ωydy, l  0, and ω Q  dT Q ω,l 1, 2, ,n,for all ω ∈ L p Q, ∧ l ,1≤ p<∞. 2. The Local Hardy-Littlewood Inequality We first introduce the following two-weight class which is an extension of A r -weight and A r λ-weights. Definition 2.1. We say the weight w 1 x,w 2 x satisfies the A r λ, M condition for r>1and 0 <λ<∞, write w 1 ,w 2  ∈ A r λ, M,ifw 1 x > 0, w 2 x > 0 a.e., and sup B  1 | B |  B w λ 1 dx   1 | B |  B  1 w 2  1/r−1 dx  r−1 < ∞ 2.1 for any ball B ⊂ M. If we choose w 1  w 2 in Definition 2.1, we obtain the usual A r λ-weights introduced in 7. Also, if λ  1andw 1  w 2 , the above weight reduces to the well-known A r -weight. 4 Journal of Inequalities and Applications See 1, 14, 15 for more properties of weights. We will also need the following generalized H ¨ older inequality. Lemma 2.2. Let 0 <α<∞, 0 <β<∞, and s −1  α −1  β −1 .Iff and g are measurable functions on R n ,then   fg   s,M ≤   f   α,M ·   g   β,M 2.2 for any M ⊂ R n . The following two versions of the Hardy-Littlewood integral inequality Theorem A and Theorem B appear in 16 and 9, respectively. Theorem A. For each p>0, there is a constant C such that  D | u − u  0  | p dx dy ≤ C  D | v − v  0  | p dx dy 2.3 for all analytic functions f  u  iv in the unit disk D. Theorem B. Let u and v be conjugate A-harmonic tensors in M ⊂ R n , σ>1, and 0 <s,t<∞. Then there exists a constant C, independent of u and v, such that  u − u B  s,B ≤ C | B | β  v − c  q/p t,σB 2.4 for all balls B with σB ⊂ M.Herec is any form in W 1 p,loc M, Λ with d  c  0 and β  1/s  1/n − 1/t  1/nq/p. Now we prove the following local two-weight Hardy-Littlewood integral inequality. Theorem 2.3. Let u and v be conjugate A-harmonic tensors on M ⊂ R n and w 1 ,w 2  ∈ A r λ, M for some r>1 and λ>0.Let0 <s,t<∞. Then there exists a constant C, independent of u and v, such that   B | u − u B | s w λ/α 1 dx  1/s ≤ C | B | γ   σB | v − c | t w pt/αqs 2 dx  q/pt 2.5 for all balls B with σB ⊂ M ⊂ R n , σ>1 and α>1.Herec is any form in W 1 q,loc M, Λ with d ∗ c  0 and γ  1/s  1/n − 1/t  1/nq/p. Note that 2.5 can be written as the following symmetric form:  1 | B |  B | u − u B | s w λ/α 1 dx  1/qs ≤ C | B | 1/q−1/p/n  1 | B |  σB | v − c | t w pt/αqs 2 dx  1/pt . 2.6   Journal of Inequalities and Applications 5 Proof. Let k  αs/α − 1. Since α>1, then k>0andk>s. Applying the H ¨ older inequality, we have   B | u − u B | s w λ/α 1 dx  1/s    B  | u − u B | w λ/αs 1  s dx  1/s ≤  u − u B  k,B   B w kλ/αk−s 1 dx  k−s/ks   u − u B  k,B   B w λ 1 dx  1/αs . 2.6 Choose m  αqst/αqs  ptr − 1, then m<t. By Theorem B we have  u − u B  k,B ≤ C 1 | B | β  v − c  q/p m,σB , 2.7 where β  1/k 1/n−1/m1/nq/p. Since 1/m  1/tt−m/mt,bytheH ¨ older inequality again, we obtain  v − c  m,σB    σB  | v − c | w p/αqs 2 w −p/αqs 2  m dx  1/m ≤   σB |v − c| t w pt/αqs 2 dx  1/t   σB  1 w 2  pmt/αqst−m dx  t−m/mt    σB |v − c| t w pt/αqs 2 dx  1/t   σB  1 w 2  1/r−1 dx  pr−1/αqs . 2.8 Hence  v − c  q/p m,σB ≤   σB  1 w 2  1/r−1 dx  r−1/αs   σB | v − c | t w pt/αqs 2 dx  q/pt . 2.9 Combining 2.6, 2.7,and2.9 yields   B | u − u B | s w λ/α 1 dx  1/s ≤ C 1 | B | β   B w λ 1 dx  1/αs   σB  1 w 2  1/r−1 dx  r−1/αs   σB | v − c | t w pt/αqs 2 dx  q/pt . 2.10 6 Journal of Inequalities and Applications Using the condition that w 1 ,w 2  ∈ A r λ, M,weobtain   B w λ 1 dx  1/αs   σB  1 w 2  1/r−1 dx  r−1/αs ≤ | σB | r/αs   1 | σB |  B w λ 1 dx   1 | σB |  σB  1 w 2  1/r−1 dx  1/αs ≤ C 2 | σB | r/αs  C 3 | B | r/αs . 2.11 Putting 2.11 into 2.10 and noting that β  r/αs  1/k  1/n − 1/m  1/nq/p  r/αs  1/s  1/n − 1/t  1/nq/p, we have   B | u − u B | s w λ/α 1 dx  1/s ≤ C | B | γ   σB | v − c | t w pt/αqs 2 dx  q/pt , 2.12 where γ  1/s  1/n − 1/t  1/nq/p. We have completed the proof of Theorem 2.3. Note that in Theorem 2.3, α>1 is arbitrary. Hence, if we choose α to be some special values, we will have some different versions of the Hardy-Littlewood inequality. For example, if we let α  λ, λ>1. By Theorem 2.3, we have   B | u − u B | s w 1 dx  1/s ≤ C | B | γ   σB | v − c | t w pt/λqs 2 dx  q/pt 2.13 for all balls B with σB ⊂ M ⊂ R n , σ>1, and γ  1/s  1/n − 1/t  1/nq/p. If we choose α  p in Theorem 2.3, we obtain the following result:   B | u − u B | s w λ/p 1 dx  1/s ≤ C | B | γ   σB | v − c | t w t/qs 2 dx  q/pt 2.14 for all balls B with σB ⊂ M ⊂ R n , σ>1, and γ  1/s  1/n − 1/t  1/nq/p. As an application of Theorem 2.3, we have the following example. Example 2.4. Let fxf 1 ,f 2 , ,f n  be K-quasiregular in R n , then u  f l df 1 ∧ df 2 ∧···∧df l−1 ,v ∗f l1 df l2 ∧···∧df n , 2.15 Journal of Inequalities and Applications 7 l  1, 2, ,n− 1, are conjugate A-harmonic tensors with p  n/l and q  n/n − l, where A is some operator satisfying 1.5. Then by Theorem 2.3,weobtain   B    f l df 1 ∧ df 2 ∧···∧df l−1 −  f l df 1 ∧ df 2 ∧···∧df l−1  B    s w λ/α 1 dx  1/s ≤ C | B | γ   σB |∗f l1 df l2 ∧···∧df n − c| t w pt/αqs 2 dx  q/pt , 2.16 where C is independent of f, γ  1/s  1/n − 1/t  1/nq/p and d ∗ c  0. For more examples of conjugate harmonic tensors, see 3. We will have different versions of the global two-weight Hardy-Littlewood inequality if we choose α and λ to be some special values as we did in the local case. Recently, Xing and Ding introduced the following Aα, β, γ; E-weights in 17. Definition 2.5. We say that a measurable function gx defined on a subset E ⊂ R n satisfies the Aα, β, γ; E-condition for some positive constants α, β, γ, write gx ∈ Aα, β, γ; E if gx > 0 a.e., and sup B  1 | B |  B g α dx  1 | B |  B g −β dx  γ/β < ∞, 2.17 where the supremum is over all balls B ⊂ E.Wesaygx satisfies the Aα, β; E-condition if 2.17 holds for γ  1 and write gx ∈ Aα, β; EAα, β, 1; E. We should notice that there are three parameters in the definition of the Aα, β, γ; E- weights. If we choose some special values for these parameters, we may obtain some existing weighted classes. For example, it is easy to see that the Aα, β, γ; E-class reduces to the usual A r E-class if α  γ  1andβ  1/r − 1. Moreover, it has been proved in 17 that the A r E-weight is a proper subset of the Aα, β, γ; E-weight. Using the similar method to the proof of Theorem 1.5.5in1, we can prove the following version of the Hardy-Littlewood inequality. Considering the length of the paper, we do not include the proof here. Theorem 2.6. Let u and v be conjugate A-harmonic tensors on M ⊂ R n and gx ∈ Aα, β, α; M with α>1 and β>0.Let0 <s,t<∞. Then, there exists a constant C, independent of u and v,such that   B | u − u B | s gdx  1/s ≤ C | B | γ   σB | v − c | t g pt/qs dx  q/pt 2.18 for all balls B with σB ⊂ M ⊂ R n and σ>1.Herec is any form in W 1 q,loc M, Λ with d ∗ c  0 and γ  1/s  1/n − 1/t  1/nq/p. 8 Journal of Inequalities and Applications Example 2.7. Let u  x   3  x 2 1  x 2 2  x 2 3 2.19 be a harmonic function in R 3 and v a 2-form in R 3 defined by v  v 3 dx 1 ∧ dx 2  v 2 dx 1 ∧ dx 3  v 1 dx 2 ∧ dx 3 , 2.20 where v 1 ,v 2 ,andv 3 are defined as follows: v 1  x 2 x 3   x 2 i x 4 2 − x 4 3  i<j  x 2 i  x 2 j   x 2 x 3  x 2 1  x 2 2  x 2 3 x 2 2 − x 2 3  x 2 1  x 2 2  x 2 1  x 2 3  , v 2  x 1 x 3   x 2 i x 4 1 − x 4 3  i<j  x 2 i  x 2 j   x 1 x 3  x 2 1  x 2 2  x 2 3 x 2 1 − x 2 3  x 2 1  x 2 2  x 2 2  x 2 3  , v 3  x 1 x 2   x 2 i x 4 1 − x 4 2  i<j  x 2 i  x 2 j   x 1 x 2  x 2 1  x 2 2  x 2 3 x 2 1 − x 2 2  x 2 1  x 2 3  x 2 2  x 2 3  . 2.21 Then u and v are a pair of conjugate harmonic tensors; see 3. Hence, the Hardy-Littlewood inequality is applicable. Using inequality 2.5 with w 1  w 2  1andc  0 over any ball B, we can obtain the norm comparison inequality for u and v defined by 2.19 and 2.20, respectively. 3. The Local Caccioppoli-Type Inequality The purpose of this section is to obtain some estimates which give upper bounds for the L p - norm of ∇u or du in terms of the corresponding norm u or u − c, where u is a differential form satisfying the A-harmonic equation 1.4 and c is any closed form. These kinds of estimates are called the Caccioppoli-type estimates or the Caccioppoli inequalities. From 9, we can obtain the following Caccioppoli-type inequality. Theorem C. Let u be an A-harmonic tensor on M and let σ>1. Then there exists a constant C, independent of u, such that  du  s,B ≤ C diam  B  −1  u − c  s,σB 3.1 for all balls or cubes B with σB ⊂ M and all closed forms c.Here1 <s<∞. The following weak reverse H ¨ older inequality appears in 9. Journal of Inequalities and Applications 9 Theorem D. Let u be an A-harmonic tensor in Ω, σ>1 and 0 <s,t<∞. Then there exists a constant C, independent of u, such that  u  s,B ≤ C | B | t−s/st  u  t,σB 3.2 for all balls or cubes B with σB ⊂ Ω. Now, we prove the following local two-weight Caccioppoli-type inequality for A- harmonic tensors. Theorem 3.1. Let u ∈ D  M, ∧ l , l  0, 1, ,n,beanA-harmonic tensor on M ⊂ R n , ρ>1 and 0 <α<1. Assume that 1 <s<∞ is a fixed exponent associated with the A-harmonic equation and w 1 ,w 2  ∈ A r λ, M for some r>1 and λ>0. Then there exists a constant C, independent of u, such that   B | du | s w αλ 1 dx  1/s ≤ C diam  B    ρB | u − c | s w α 2 dx  1/s 3.3 for all balls B with ρB ⊂ M and all closed forms c. Proof. Choose t  s/1 − α, then 1 <s<t. Since 1/s  1/t t − s/st,byH ¨ older inequality and Theorem C, we have   B | du | s w αλ 1 dx  1/s    B  | du | w αλ/s 1  s dx  1/s ≤   B | du | t dx  1/t   B  w αλ/s 1  st/t−s dx  t−s/st ≤  du  t,B ·   B w λ 1 dx  α/s  C 1 diam  B  −1  u − c  t,σB   B w λ 1 dx  α/s 3.4 for all balls B with σB ⊂ Ω and all closed forms c. Since c is a closed form and u is an A- harmonic tensor, then u − c is still an A-harmonic tensor. Taking m  s/1  αr − 1,wefind that m<s<t. Applying Theorem D yields  u − c  t,σB ≤ C 2 | B | m−t/mt  u − c  m,σ 2 B  C 2 | B | m−t/mt  u − c  m,ρB , 3.5 where ρ  σ 2 . Substituting 3.5 in 3.4, we have   B | du | s w αλ 1 dx  1/s ≤ C 3 diam  B  −1 | B | m−t/mt  u − c  m,ρB   B w λ 1 dx  α/s . 3.6 10 Journal of Inequalities and Applications Now 1/m  1/s s − m/sm,bytheH ¨ older inequality again, we obtain  u − c  m,ρB    ρB | u − c | m dx  1/m    ρB  | u − c | w α/s 2 w −α/s 2  m dx  1/m ≤   ρB | u − c | s w α 2 dx  1/s   ρB  1 w 2  1/r−1 dx  αr−1/s 3.7 for all balls B with ρB ⊂ Ω and all closed forms c. Combining 3.6 and 3.7,weobtain   B | du | s w αλ 1 dx  1/s ≤ C 3 diam  B  −1 | B | m−t/mt  w 1  αλ/s λ,B     1 w 2     α/s 1/r−1,ρB   ρB | u − c | s w α 2 dx  1/s . 3.8 Since w 1 ,w 2  ∈ A r λ, M, then we have  w 1  αλ/s λ,B ·     1 w 2     α/s 1/r−1,ρB ≤ ⎛ ⎝   ρB w λ 1 dx   ρB  1 w 2  1/r−1 dx  r−1 ⎞ ⎠ α/s  ⎛ ⎝   ρB   r  1   ρB    ρB w λ 1 dx  1   ρB    ρB  1 w 2  1/r−1 dx  r−1 ⎞ ⎠ α/s ≤ C 4 | B | αr/s . 3.9 Substituting 3.9 in 3.8,wefindthat   B | du | s w αλ 1 dx  1/s ≤ C diam  B    ρB | u − c | s w α 2 dx  1/s 3.10 for all balls B with ρB ⊂ M and all closed forms c. This ends the proof of Theorem 3.1. [...]... choose λ and α to be some special values in Theorem 4.1 Considering the length of the paper, we do not list these similar results here 14 Journal of Inequalities and Applications References 1 R P Agarwal, S Ding, and C Nolder, Inequalities for Differential Forms, Springer, New York, NY, USA, 2009 2 S Ding, “Lipschitz and BMO norm inequalities for operators,” Nonlinear Analysis: Theory, Methods and Applications,... 1274–1293, 2007 6 S Ding and B Liu, “A singular integral of the composite operator,” Applied Mathematics Letters, vol 22, no 8, pp 1271–1275, 2009 7 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 8 B Liu, “Aλ Ω -weighted imbedding inequalities for A-harmonic tensors,”... Journal of Mathematical r Analysis and Applications, vol 273, no 2, pp 667–676, 2002 9 C A Nolder, Hardy-Littlewood theorems for A-harmonic tensors,” Illinois Journal of Mathematics, vol 43, no 4, pp 613–632, 1999 10 Y Wang and C Wu, “Global Poincar´ inequalities for Green’s operator applied to the solutions of e the nonhomogeneous A-harmonic equation,” Computers and Mathematics with Applications,... c| ρB 1/s w2 dx 3.12 for all balls B with ρB ⊂ M and all closed forms c If we choose α 1/s in Theorem 3.2, then 0 < α < 1 since 1 < s < ∞ Thus, Theorem 3.2 reduces to the following version Theorem 3.4 Let u ∈ D M, ∧l , l 0, 1, , n, be an A-harmonic tensor in a domain M ⊂ Rn and ρ > 1 Assume that 1 < s < ∞ is a fixed exponent associated with the A-harmonic equation and w ∈ Ar M for some r > 1 Then... differential forms,” Journal of Mathematical Analysis and Applications, vol 227, no 1, pp 251–270, 1998 4 S Ding, “New weighted integral inequalities for differential forms in some domains,” Pacific Journal of Mathematics, vol 194, no 1, pp 43–56, 2000 5 S Ding, “Local and global norm comparison theorems for solutions to the nonhomogeneous Aharmonic equation,” Journal of Mathematical Analysis and Applications,... c| w 1/s dx 3.13 ρB for all balls B with ρB ⊂ M and all closed forms c Example 3.5 Let A : M × ∧l Rn → ∧l Rn be an operator defined by A x, ξ ξ|ξ|p−2 Then A satisfies the condition 1.5 Equation 1.4 reduces to the p-harmonic equation d du|u|p−2 0 3.14 12 Journal of Inequalities and Applications and 1.6 reduces to the conjugate p-harmonic equation du|u|p−2 3.15 dv for differential forms, respectively... is the Euclidean distance between ξ and ∂Ω Using the properties of John domain and the well-known Covering Lemma, we can prove the following global two-weight Hardy-Littlewood inequality Theorem 4.1 Let u ∈ D Ω, Λ0 and v ∈ D Ω, Λ2 be conjugate A-harmonic tensors in a John domain Ω Assume that q ≤ p, v − c ∈ Lt Ω, Λ2 , w1 , w2 ∈ Ar λ, Ω , and w1 ∈ Ar Ω for some r > 1 and λ > 0 If s is defined by s npt/... fixed exponent associated with the A-harmonic equation and w ∈ Ar M for some r > 1 Then there exists a constant C, independent of u, such that s |du| w dx α 1/s B C ≤ diam B 1/s s |u − c| w dx α 3.11 ρB for all balls B with ρB ⊂ M and all closed forms c We also need to note that in Theorem 3.1α is a parameter with 0 < α < 1 Thus, we will obtain different versions of the Caccioppoli-type inequality if we... Caccioppoli-type inequality if we let α be some particular values For example, putting α 1/s, we have the following result Theorem 3.3 Let u ∈ D M, ∧l , l 0, 1, , n, be an A-harmonic tensor in a domain M ⊂ Rn and ρ > 1 Assume that 1 < s < ∞ is a fixed exponent associated with the A-harmonic equation and w1 , w2 ∈ Ar λ, M for some r > 1 and λ > 0 Then there exists a constant C, independent of u, such...Journal of Inequalities and Applications 11 Ar 1, M becomes the usual Ar M weight See Note that if λ 1, then Ar λ, M 14 for the properties of Ar M weights Thus, choosing λ 1 and w1 w2 in Theorem 3.1, we have the following Ar M -weighted Caccioppoli-type inequality Theorem 3.2 Let u ∈ D M, ∧l , l 0, 1, , n, be an A-harmonic tensor in a domain M ⊂ Rn , ρ > 1 and 0 < α < 1 Assume that . Corporation Journal of Inequalities and Applications Volume 2010, Article ID 351597, 14 pages doi:10.1155/2010/351597 Research Article Hardy-Littlewood and Caccioppoli-Type Inequalities for A-Harmonic Tensors Peilin. Journal of Inequalities and Applications References 1 R. P. Agarwal, S. Ding, and C. Nolder, Inequalities for Differential Forms, Springer, New York, NY, USA, 2009. 2 S. Ding, “Lipschitz and BMO. purpose of this paper is to prove the new versions of the weighted Hardy-Littlewood and Caccioppoli-type inequalities for the A-harmonic tensors. Our results may have applications in different