Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 397340, 15 pages doi:10.1155/2008/397340 Research Article The Reverse H ¨ older Inequality for the Solution to p-Harmonic Type System Zhenhua Cao, 1 Gejun Bao, 1 Ronglu Li, 1 and Haijing Zhu 2 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 College of Mathematics and Physics, Shan Dong Institute of Light Industry, Jinan 250353, China Correspondence should be addressed to Gejun Bao, baogj@hit.edu.cn Received 6 July 2008; Revised 9 September 2008; Accepted 5 November 2008 Recommended by Shusen Ding Some inequalities to A-harmonic equation Ax, dud ∗ v have been proved. The A-harmonic equation is a particular form of p-harmonic type system Ax, a  dub  d ∗ v only when a  0 and b  0. In this paper, we will prove the Poincar ´ e inequality and the reverse H ¨ older inequality for the solution to the p-harmonic type system. Copyright q 2008 Zhenhua Cao et al. This is an open access article d istributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Recently, amount of work about the A-harmonic equation for the differential forms has been done. In fact, the A-harmonic equation is an important generalization of the p-harmonic equation in R n , p>1, and the p-harmonic equation is a natural extension of the usual Laplace equation see 1 for the details. The reverse H ¨ older inequalities have been widely studied and frequently used in analysis and related fields, including partial differential equations and the theory of elasticity see 2. In 1999, Nolder gave the reverse H ¨ older inequality for the solution to the A-harmonic equation in 3,anddifferent versions of Caccioppoli estimates have been established in 4–6. In 2004, D’Onofrio and Iwaniec introduced the p-harmonic type system in 7, which is an important extension of the conjugate A-harmonic equation. In 2007, Ding proved the following inequality in 8. Theorem A. Let u, v be a pair of solutions to Ax, g  duh  d ∗ v in a domain Ω ⊂ R n .If g ∈ L p B, Λ L  and h ∈ L q B, Λ L ,thendu ∈ L p B, Λ L  if and only if d ∗ v ∈ L q B, Λ L . Moreover, there exist constants C 1 ,C 2 independent of u and v, such that   d ∗ v   q q,B ≤ C 1  h q q,B  g p p,B  du p p,B  , du p p,B ≤ C 2  h q q,B  g p p,B  d ∗ v q q,B  ∀B ⊂ σB ⊂ Ω. 1.1 2 Journal of Inequalities and Applications In this paper, we will prove the Poincar ´ e inequality see Theorem 2.5 and the reverse H ¨ older inequality for the solution to the p-harmonic type system see Theorem 3.5.Nowlet us see some notions and definitions about the p-harmonic type system. Let e 1 ,e 2 , ,e n denote the standard orthogonal basis of R n . For l  0, 1, ,n, we denote by Λ l Λ l R n  the linear space of all l-vectors, spanned by the exterior product 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. The Grassmann algebra Λ⊕Λ l is a graded algebra with respect to the exterior products. For α   α I e I ∈ Λ and β   β I e I ∈ Λ, then its inner product is obtained by α, β   α I β I , 1.2 with the summation over all I i 1 ,i 2 , ,i l  and all integers l  0, 1, ,n. The Hodge star operator ∗: Λ → Λ is defined by the rule ∗1  e i 1 ∧ e i 2 ∧···∧e i n , α ∧∗β  β ∧∗α  α, β∗1 ∀α, β ∈ Λ. 1.3 Hence, the norm of α ∈ Λ can be given by |α| 2  α, α  ∗α ∧∗α ∈ Λ 0  R. 1.4 Throughout this paper, Ω ⊂ R n is an open subset, for any constant σ>1, Q denotes a cube such that Q ⊂ σQ ⊂ Ω, where σQ denotes the cube whose center is as same as Q and diamσQσ diam Q.Wesayα   α I e I ∈ Λ is a differential l-form on Ω, if every coefficient α I of α is Schwartz distribution on Ω. The space spanned by differential l-form on Ω is denoted by D  Ω, Λ l . We write L p Ω, Λ l  for the l-form α   α I dx I on Ω with α I ∈ L p Ω for all ordered l-tuple I.ThusL p Ω, Λ l  is a Banach space with the norm α p,Ω    Ω |α| p dx  1/p    Ω   I |α I | 2  p/2 dx  1/p . 1.5 Similarly W k,p Ω, Λ l  denotes those l-forms on Ω with all coefficients in W k,p Ω. We denote the exterior derivative by d : D   Ω, Λ l  −→ D   Ω, Λ l1  , for l  0, 1, 2, ,n, 1.6 and its formal adjoint operator the Hodge codi fferential operator d ∗ : D   Ω, Λ l  −→ D   Ω, Λ l−1  . 1.7 The operators d and d ∗ are given by the formulas dα   I dα I ∧ dx I ,d ∗ −1 nl1 ∗d∗. 1.8 Zhenhua Cao et al. 3 The following two definitions appear in 7. Definition 1.1. The Hodge system holds: Ax, a  dub  d ∗ v, 1.9 where a ∈ L p Ω, Λ l  and b ∈ L q Ω, Λ l ,isap-harmonic type system if A is a mapping from Ω × Λ l to Λ l satisfying 1 x → Ax, ξ is measurable in x ∈ Ω for every ξ ∈ Λ l ; 2 ξ → Ax, ξ is continuous in ξ ∈ Λ l for almost every x ∈ Ω; 3 Ax, tξt p−1 Ax, ξ for every t ≥ 0; 4 KAx, ξ − Ax, ζ,ξ− ζ≥|ξ − ζ| 2 |ξ|  |ζ| p−2 ; 5 |Ax, ξ − Ax, ζ|≤K|ξ − ζ||ξ|  |ζ| p−2 for almost every x ∈ Ω and all ξ, ζ ∈ Λ l , where K ≥ 1 is a constant. It should be noted that Ax, ∗ : Ω × Λ l → Λ l is invertible and its inverse denoted by A −1 satisfies similar conditions as A but with H ¨ older conjugate exponent q in place of p. Definition 1.2. If 1.9 is a p-harmonic type system, then we say the equation d ∗ Ax, a  dud ∗ b 1.10 is a p-harmonic type equation. The following definition appears in 9. Definition 1.3. Adifferential form u is a weak solution for 1.10 in Ω if u satisfies  Ω  Ax, a  du,dϕ    d ∗ b, ϕ  ≡ 0 1.11 for every ϕ ∈ W k,p Ω, Λ l−1  with compact support. We can find that if we let a  0andb  0, then the p-harmonic type system Ax, a  dub  d ∗ v 1.12 becomes Ax, dud ∗ v. 1.13 It is the conjugate A-harmonic equation, where the mapping A : Ω × Λ l → Λ l satisfies the following conditions:   Ax, ξ   ≤ a|ξ| p−1 ,  Ax, ξ,ξ  ≥|ξ| p . 1.14 4 Journal of Inequalities and Applications If we let Ax, ξ|ξ| p−2 ξ, then the conjugate A-harmonic equation becomes the form |du| p−2 du  d ∗ v. 1.15 It is the conjugate p-harmonic equation. So we can see that the conjugate p-harmonic equation and the conjugate A-harmonic equation are the specific p-harmonic type system. Remark 1.4. It should be noted that the mapping Ax, ∗ in p-harmonic system Ax, a  du b  d ∗ v, is invertible. If we denote its inverse by A −1 x, ∗, then the mapping A −1 x, ∗ : Λ l → Λ l satisfies similar conditions as A but with H ¨ older conjugate exponent q in place of p. 2. The Poincar ´ e inequality In this section, we will introduce the Poincar ´ e inequality for the differential forms. Now first let us see a lemma, which can be found in 9,Section4 for the details. Lemma 2.1. Let D be a bounded, convex domain in R n .Toeachy ∈ D there corresponds a linear operator K y : C ∞ D, Λ l  → C ∞ D, Λ l−1  defined by  K y ω  x; ξ 1 , ,ξ l−1    1 0 t l−1 ω  tx  y − ty; x − y, ξ 1 , ,ξ l−1  dt, 2.1 and the decomposition ω  d  K y ω   K y dω2.2 holds at any point y ∈ D. We construct a homotopy operator T : C ∞ D, Λ l  → C ∞ D, Λ l−1  by averaging K y over all points y ∈ D: Tω   D ϕyK y ωdy, 2.3 where ϕ form C ∞ D is normalized so that  ϕydy  1. It is obvious that ω  dK y ωK y dω remains valid for the operator T : ω  dTωTdω. 2.4 We define the l-forms ω D ∈ D  D, Λ l  by ω D  |D| −1  D ωydy for l  0 and ω D  dTω for l  1, 2, ,n,and all ω ∈ W 1,p D, Λ l , 1 <p<∞. The following definition can be found in [9, page 34]. Definition 2.2. For ω ∈ D  D, Λ l , the vector valued differential form ∇ω   ∂ω ∂x 1 , , ∂ω ∂x n  2.5 Zhenhua Cao et al. 5 consists of differential forms ∂ω/∂x i ∈ D  D, Λ l , where the partial differentiation is applied to coefficients of ω. The proof of 9, Proposition 4.1 implies the following inequality. Lemma 2.3. For any ω ∈ L p D, Λ l , it holds that ∇Tω p,D ≤ Cn, pω p,D 2.6 for any ball or cube D ∈ R n . The following Poincar ´ e inequality can be found in [2]. Lemma 2.4. If u ∈ W 1,p 0 Ω, then there is a constant C  Cn, p > 0 such that  1 |B|  B |u| pχ dx  1/pχ ≤ Cr  1 |B|  B |∇u| p dx  1/p , 2.7 whenever B  Bx 0 ,r is a ball in R n ,wheren ≥ 2 and χ  2 for p ≥ n, χ  np/n − p for p<n. Theorem 2.5. Let u ∈ D  D, Λ l , and du ∈ L p D, Λ l1 . Then, u − u D is in L χp D, Λ l  and  1 |D|  D |u − u D | pχ dx  1/pχ ≤ Cn, p, ldiamD  1 |D|  D |du| p dx  1/p 2.8 for any ball or cube D ∈ R n ,whereχ  2 for p ≥ n and χ  np/n − p for 1 <p<n. Proof. We know Tduu − u D . Now we suppose u − u Q  Tdu  I u I dx I , where I  i 1 , ,i l1  take over all l  1-tuples. So we have ∇Tdu  ∂u ∂x 1 , , ∂u ∂x n     I ∂u I ∂x 1 dx I , ,  I ∂u I ∂x n dx I  . 2.9 So we have  1 |D|  D   u − u D   pχ dx  1/pχ   1 |D|  D       I u I dx I      pχ dx  1/pχ   1 |D|  D   I   u I   2  pχ/2 dx  1/pχ . 2.10 6 Journal of Inequalities and Applications By the inequality  n  i1  a i  2  1/2 ≤ n  i1 a i ≤ n 1/2  n  i1  a i  2  1/2 2.11 for any a i ≥ 0, and the Minkowski inequality, we have  1 |D|  D   I   u I   2  pχ/2 dx  1/pχ ≤  I  1 |D|  D   u I   pχ dx  1/pχ . 2.12 According to the Poincar ´ e inequality, we have  I  1 |D|  D |u I | pχ dx  1/pχ ≤ C 1 n, pdiamD  I  1 |D|  D |∇u I | p dx  1/p . 2.13 Combining 2.10, 2.12,and2.13, we can obtain  1 |D|  D   u − u D   pχ dx  1/pχ ≤ C 1 n, pdiamD  I  1 |D|  D |∇u I | p dx  1/p . 2.14 By 2.9 we have ∇Tdu p,D       ∂u ∂x 1 , , ∂u ∂x n      p,D    D      ∂u ∂x 1 , , ∂u ∂x n      p dx  1/p    D  n  i1     ∂u ∂x i     2  p/2 dx  1/p    D  n  i1  I     ∂u I ∂x i     2  p/2 dx  1/p    D   I n  i1     ∂u I ∂x i     2  p/2 dx  1/p . 2.15 Zhenhua Cao et al. 7 Combining 2.11 and 2.15, then we have ∇Tdu p,D    D   I n  i1     ∂u I ∂x i     2  p/2 dx  1/p ≥  C l1 n  −1/2   D   I  n  i1     ∂u I ∂x i     2  1/2  p dx  v 1/p ≥  C l1 n  −1/2   D  I  n  i1     ∂u I ∂x i     2  p/2 dx  1/p ≥  C l1 n  −1/2  C l1 n  −p−1/p  I   D  n  i1     ∂u I ∂x i     2  p/2 dx  1/p ≥  C 2 n, p, l  −1  I   D   ∇u I   p dx  1/p , 2.16 where C 2 n, p, lC l1 n  1/2p−1/p . Now combining 2.14, 2.16,and2.6, we can get  1 |D    D   u − u D | pχ dx  1/pχ ≤ C 1 n, pdiamD  I  1 |D|  D   ∇u I   p dx  1/p ≤ C 1 n, p, lC 2 n, p, l  1 |D|  1/p ∇Tdu p,D ≤ C 3 n, p, ldiamD  1 |D|  D |du| p dx  1/p . 2.17 3. The reverse H ¨ older inequality In this section, we will prove the reverse H ¨ older inequality for the solution of the p-harmonic type system. Before we prove the reverse H ¨ older inequality, let us first see some lemmas. Lemma 3.1. If f, g ≥ 0 and for any nonnegative η ∈ C ∞ 0 Ω, it holds  Ω ηf dx ≤  Ω gdx, 3.1 then for any h ≥ 0:  Ω ηfh dx ≤  Ω ghdx. 3.2 8 Journal of Inequalities and Applications Proof. Let μ be a measure in X, f be a nonnegative μ-measurable function in a measure space X, using the standard representation theorem, we have  X f q dμ  q  ∞ 0 t q−1 μx : fx >tdt 3.3 for any 0 <t<q.Now, we let μE  E ηf dx and νE  E gdxthen, we can obtain  Ω ηfh dx   ∞ 0  h>t ηf dx dt ≤  ∞ 0  h>t gdxdt  Ω ghdx. 3.4 So Lemma 3.1 is proved. Lemma 3.2. If u, v is a pair of solution to the p-harmonic type system 1.9, then it holds  Ω |ηda| p dx ≤ C  Ω |a  dudη| p dx 3.5 for any nonnegative η ∈ C ∞ 0 Ω and where C C l1 n  p . Proof. Since u, v is a pair of solutions to Ax, a  dub  d ∗ v, it is also the solution to A −1 x, b  d ∗ va  du, where A −1 x, ∗ is the inverse Ax, ∗. Now, we suppose that da   I ω I dx I and let ϕ 1  −  I η signω I dx I .Byusingϕ  ϕ 1 and dϕ 1   I signω I dη ∧ dx I in 1.11, we can obtain  Ω  A −1  x, b  d ∗ v  ,dϕ 1    da, ϕ 1  dx ≡ 0. 3.6 That is,  Ω  da,  I η sign  ω I  dx I  dx   Ω  A −1  x, b  d ∗ v  , −  I sign  ω I  dη ∧ dx I  dx. 3.7 In other words,  Ω  I η   ω I   dx   Ω  A −1  x, b  d ∗ v  , −  I sign  ω I  dη ∧ dx I  dx. 3.8 By the elementary inequality  n  i1 a i 2  1/2 ≤ n  i1   a i   , 3.9 Zhenhua Cao et al. 9 we have  Ω η|da|dx   Ω η   I ω I 2  1/2 dx≤  Ω  I η|ω I |dx   Ω  A −1  x, b  d ∗ v  , −  I sign  ω I  dη ∧ dx I  dx. 3.10 Using the inequality |a, b| ≤ |a||b|, 3.11 3.10 becomes  Ω η|da|dx ≤  Ω   A −1  x, b  d ∗ v          I sign  ω I  dη ∧ dx I      ≤  Ω   A −1  x, b  d ∗ v     I   sign  ω I    |dη|dx  C l1 n  Ω   A −1  x, b  d ∗ v    |dη|dx  C l1 n  Ω |a  du||dη|dx, 3.12 where I takes over all l  1-tuples for dη ∈ Λ l1 ,thusithasC l1 n numbers at most. Now we let f  |da| and g  C l1 n |a  du||dη|. In the subset {x : fη  g}, we have  {x:fηg} |ηda| p dx ≤  {x:fηg} |a  dudη| p dx. 3.13 In the subset {x : fη /  g},leth |fη| p −|g| p /fη − g, then we easily obtain h>0. So by Lemma 3.1, we have  {x:fη /  g} hfη dx ≤  {x:fη /  g} hg dx. 3.14 That is to say  {x:fη /  g} hfη − gdx ≤ 0, 3.15 10 Journal of Inequalities and Applications that is,  {x:fη /  g} |fη| p dx ≤  fη /  g |g| p dx. 3.16 Combining 3.13 and 3.16, we have  Ω |fη| p dx ≤  Ω |g| p dx, 3.17 that is,  Ω |ηda| p dx ≤  Ω   C l1 n a  dudη   p dx. 3.18 So Lemma 3.2 is proved. The following lemma appears in 2. Lemma 3.3. Suppose that 0 <q<p<s≤∞, ξ ∈ R, and that B  Bx 0 ,r is a ball. If a nonnegative function v ∈ L p B, dμ satisfies  1 μ  λB    λB  v s dμ  1/s ≤ C1 − λ ξ  1 μ  B    B  v p dμ  1/p 3.19 for each ball B   Bx 0 ,r   with r  ≤ r and for all 0 <λ<1,then  1 μλB  λB v s dμ  1/s ≤ C1 − λ ξ/θ  1 μB  B v q dμ  1/q ∀0 <λ<1. 3.20 Here C>0 is a constant depending on p, q, s and θ ∈ 0, 1 is such that 1/p  θ/q 1 − θ/s. The following lemma appears in 10. Lemma 3.4. Let u, v be a pair of solutions of the p-harmonic type system on domain Ω,thenwe have a constant C only depending on K, n, p, and l, such that ηdu p,Ω ≤ C  u − cdη p,Ω  ηa p,Ω  , 3.21 [...]... and Applications, vol 335, no 2, pp 1274–1293, 2007 9 T Iwaniec and A Lutoborski, “Integral estimates for null Lagrangians,” Archive for Rational Mechanics and Analysis, vol 125, no 1, pp 25–79, 1993 10 Z Cao, G Bao, and R Li, The Caccioppoli estimate for the solution to the p-harmonic type system,” to appear in the Proceedings of the 6th International Conference on Differential Equations and Dynamical... integral inequalities for A-harmonic tensors,” Journal of Mathematical Analysis and Applications, vol 247, no 2, pp 466–477, 2000 5 S Ding, “Weighted Caccioppoli -type estimates and weak reverse Holder inequalities for A-harmonic ¨ tensors,” Proceedings of the American Mathematical Society, vol 127, no 9, pp 2657–2664, 1999 6 X Yuming, “Weighted integral inequalities for solutions of the A-harmonic equation,”...Zhenhua Cao et al 11 ∞ 0) and for any η ∈ C0 Ω Also we have a constant C only where c is any closed form (i.e., dc depending on K, n, q, such that η d∗ v q,Ω v − c dη ≤C where c is any coclosed form (i.e., d∗ c ηb q,Ω q,Ω , 3.22 0) and q is the conjugate exponent of p Theorem 3.5 If u, v is a pair of solutions to the p-harmonic type system, then there exists a constant C > 0 dependent on... equation,” Journal of Mathematical Analysis and Applications, vol 279, no 1, pp 350–363, 2003 7 L D’Onofrio and T Iwaniec, The p-harmonic transform beyond its natural domain of definition,” Indiana University Mathematics Journal, vol 53, no 3, pp 683–718, 2004 8 S Ding, “Local and global norm comparison theorems for solutions to the nonhomogeneous Aharmonic equation,” Journal of Mathematical Analysis and... p-harmonic differential forms,” Journal of Mathematical Analysis and Applications, vol 227, no 1, pp 251–270, 1998 2 J Heinonen, T Kilpel¨ inen, and O Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, a Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1993 3 C A Nolder, “Hardy-Littlewood theorems for A-harmonic tensors,” Illinois Journal of Mathematics, vol 43,... we have d u − uQ ≤ ∇T du 3.28 ∞ For any η ∈ C0 Ω , according to 2.6 , we have η∇T dω p,D ≤ C n, p max η dω p,D x∈D 3.29 By the similar method as Lemma 3.1, we can prove the following inequality: d u − uQ p u − uQ p ∇T du a ∞,Q t 1/p p ηm dx rm Q ≤ ηm p u − uQ a ∞,Q t 1/p 3.30 dx rm Q p p ηm |du|p u − uQ ≤ C n, p max ηm x∈D rm Q a ∞,Q t 1/p dx Zhenhua Cao et al 13 ∞ for any η ∈ C0 Ω By Lemma 3.1 and... set κ 1 rm 1 Q p t, then by computation, we obtain u − uQ a κχ ∞,Q 1/κχ ≤ C3 dx p/κ p/κ 1−λ κ −p/κ pm/κ 2 r 1 u − uQ a ∞,Q p/κ rm 1 Q 1 rm Q × κ 1/κ dx rm Q 3.34 Since this inequality holds for all κ > p, it can be applied with κ κm pχm And we 1/p is increasing with p and its limit is ess supQ |f| So by can easily prove 1/|Q| Q |f|p dx iterating we arrive at the desired inequality for q p: ess sup u... and let ωm then we have dum t p 1 u − uQ a ∞,Q t/p ηm d u − uQ u − uQ a ∞,Q 1 t/p dηm 3.25 By the Minkowski inequality, we can obtain p dum dx 1/p u − uQ ≤ rm Q ∞,Q p t p 1/p dηm dx rm Q p t p We assume that u − uQ we have |d|u − uQ || 0 a d u − uQ p u − uQ a ∞,Q t p ηm dx 3.26 1/p rm Q 2 1/2 then we have |u − uQ | If u − uQ is zero, then I aI |∇T du | If u − uQ is not equal zero, and the proof of... σ −1 diam Q 1 χ/ χ−1 Q 1 |σQ| × u − uσQ a ∞,σQ t 3.23 1/t dx σQ for any 0 < s, t < ∞, σ > 1 and all cubes with Q ⊂ σQ ⊂ Ω, where χ > 1 is the Poincar´ constant e Proof Suppose that the center of Q is x0 and diam Q rm 1 − λ 2−m , λ m σ −1 < 1 Let r, 0 < λ 0, 1, 2, 3.24 Then rm is decreasing and λ < rm < 1 So we have uQ |rm Q urm Q , for any m ∈ 0, 1, 2, ∞ 1 in rm 1 Q, 0 ≤ ηm ≤ 1 in Let ηm ∈ C0... et al 15 for any 0 < s < ∞ and 1 < p < t < ∞ Combining 3.36 and 3.37 , we have 1 |Q| u − uQ a ∞,Q s 1/s dx ≤ C6 1 − λ −tχ/p χ−1 r 1 χ/ χ−1 Q × 1 |σQ| |u − uσQ | a ∞,σQ t 1/t 3.38 dx σQ for any 0 < s, t < ∞ and σ > 1 such that σQ ⊂ Ω Theorem 3.5 is proved Acknowledgment This work is supported by the NSF of China Grants nos 10671046 and 10771044 References 1 S Ding, “Some examples of conjugate p-harmonic . ldiamD  1 |D|  D |du| p dx  1/p . 2.17 3. The reverse H ¨ older inequality In this section, we will prove the reverse H ¨ older inequality for the solution of the p-harmonic type system. Before we prove the reverse H ¨ older inequality, . Applications In this paper, we will prove the Poincar ´ e inequality see Theorem 2.5 and the reverse H ¨ older inequality for the solution to the p-harmonic type system see Theorem 3.5.Nowlet us see some. and Applications Volume 2008, Article ID 397340, 15 pages doi:10.1155/2008/397340 Research Article The Reverse H ¨ older Inequality for the Solution to p-Harmonic Type System Zhenhua Cao, 1 Gejun