Global Caccioppoli-Type and Poincaré Inequalities with Orlicz Norms

The L-theory of solutions of the homogeneous A-harmonic equation d A x, dω 0 for differential forms has been very well developed in recent years. Many L-norm estimates and inequalities, including the Hardy-Littlewood inequalities, Poincaré inequalities, Caccioppoli-type estimates, and Sobolev imbedding inequalities, for solutions of the homogeneous A-harmonic equation have been established; see 1–11 . Among these results, the Caccioppoli-type inequalities and the Poincaré inequalities for differential forms have become more and more important tools in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equation d A x, dω B x, dω just began 4, 6 . Roughly, the Caccioppoli-type inequalities or estimates provide upper bounds for the norms of ∇u or du in terms of the corresponding norm of u or u−c, where u is a differential form or a function satisfying certain conditions. For example, umay be a solution of an A-harmonic equation or a minimizer of a functional, and c is some constant if u is a function or a closed form if u is a differential form. Different versions of the Caccioppoli-type inequalities and the Poincaré inequalities have been established during the past several decades. For instance, Sbordone proved in 12 the following version of the Caccioppoli-type inequality:


Introduction
The L p -theory of solutions of the homogeneous A-harmonic equation d A x, dω 0 for differential forms has been very well developed in recent years. Many L p -norm estimates and inequalities, including the Hardy-Littlewood inequalities, Poincaré inequalities, Caccioppoli-type estimates, and Sobolev imbedding inequalities, for solutions of the homogeneous A-harmonic equation have been established; see 1-11 . Among these results, the Caccioppoli-type inequalities and the Poincaré inequalities for differential forms have become more and more important tools in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equation d A x, dω B x, dω just began 4, 6 . Roughly, the Caccioppoli-type inequalities or estimates provide upper bounds for the norms of ∇u or du in terms of the corresponding norm of u or u−c, where u is a differential form or a function satisfying certain conditions. For example, u may be a solution of an A-harmonic equation or a minimizer of a functional, and c is some constant if u is a function or a closed form if u is a differential form. Different versions of the Caccioppoli-type inequalities and the Poincaré inequalities have been established during the past several decades. For instance, Sbordone proved in 12 the following version of the Caccioppoli-type inequality: Journal of Inequalities and Applications for a quasiminimizer u of the functional F Ω; v Ω A |dv| dx, where A is a continuous, convex, and strictly increasing function satisfying the so-called Δ 2 -condition, B R is a ball with radius R > 0, and u R − B R u dx; see 12 . Using the above Caccioppoli-type inequality, Fusco and Sbordone obtained in 13 the higher integrability result for the gradient of minimizers of the functional I Ω; v , where r > 1 is some constant. In 14 , Greco et al. studied the variational integrals whose integrand grows almost linearly with respect to the gradient and the related equation div A x, f ∇u 0, where A is slowly increasing to ∞. For instance, A t log α 1 t , α > 0, or A t log log e t . They proved that the minimizer u subject to the Dirichlet data v satisfies the estimate at least for some small ε > 0. In 15 , Cianchi and Fusco investigated the higher integrability properties of the gradient of local minimizers of an integral functional of the form J u, Ω Ω f x, u, du dx, where Ω is an open subset of R n , n ≥ 2, and f is a Carathodory function defined in Ω × R N × R nN satisfying some growth conditions. Using a new form of the Caccioppoli inequality and some other tools, such as the Sobolev inequality and a generalized version of the Gehring lemma, they proved that if u is a local minimizer of J u, Ω , for Ω 0 ⊂⊂ Ω there exists δ > 0 such that where A satisfies the so-called Δ 2 -condition. However, all versions of the Caccioppoli-type inequality developed or used in [12][13][14][15] are about the minimizer u of some functional. In this paper, we will prove the Caccioppoli-type inequalities and the Poincaré inequalities with the L s log L α -norm for differential forms satisfying the nonhomogeneous A-harmonic equation. The method developed in this paper could be used to establish other L s log L α -norm inequalities for solutions of the homogeneous A-harmonic equation or the nonhomogeneous A-harmonic equation. Throughout this paper, we always assume that Ω is an open subset of R n , n ≥ 2. The ndimensional Lebesgue measure of a set E ⊆ R n is denoted by |E|. We say that w is a weight if w ∈ L 1 loc R n and w > 0 a.e. For 0 < p < ∞, we denote the weighted L p -norm of a measurable function f over E by f p,E,w α E |f x | p w α x dx 1/p , where α is a real number. We write Journal of Inequalities and Applications   3 for some k k f > 0. L ϕ Ω is equipped with the nonlinear Luxemburg functional A convex Orlicz function ϕ is often called a Young function. If ϕ is a Young function, then · ϕ defines a norm in L ϕ Ω , which is called the Luxemburg norm or Orlicz norm. The Orlicz space L ψ Ω with ψ t t p log α e t/c will be denoted by L p log L α Ω and the corresponding norm will be denoted by f L p log L α Ω , where 1 ≤ p < ∞, α ≥ 0, and c > 0 are constants. The spaces L p log L 0 Ω and L 1 log L 1 Ω are usually referred as L p Ω and L log L Ω , respectively. From 16 , we have the equivalence Similarly, we have where μ is a measure defined by dμ w x dx and w x is a weight. In this paper, we simply write and f L p log L α E f L p log L α E,1 , where w is a weight. We keep using the traditional notations related to differential forms in this paper. Let Λ Λ R n be the linear space of the -covectors on R n , 1, 2, . . . , n. It is a normed space of dimension n . A differential -form ω on Ω is a Schwartz distribution on Ω with values in Λ R n . We write D Ω, Λ for the space of all differential -forms and L p Ω, Λ for all -forms ω x

1.10
We use L p log L α Ω, Λ to denote the space of all differential -forms u on Ω with

Journal of Inequalities and Applications
We use d : D Ω, Λ → D Ω, Λ 1 to denote the differential operator and d : D Ω, Λ 1 → D Ω, Λ to denote the Hodge codifferential operator given by d −1 nl 1 d on D Ω, ∧ l 1 , 0, 1, . . . , n. Here is the well-known Hodge star operator. We use B to denote a ball and σB, σ > 0, is the ball with the same center as B and with diameter σ diam B . A differential form u is called closed if du 0 and a differential form v is called coclosed if d v 0. Definition 1.1. Let A : Ω×Λ R n → Λ R n and B : Ω×Λ R n → Λ −1 R n be two operators satisfying the conditions: for almost every x ∈ Ω and all ξ ∈ Λ R n . Then the nonlinear elliptic equation is called the nonhomogeneous A-harmonic equation for differential forms. Here a, b > 0 are constants and 1 < p < ∞ is a fixed exponent associated with 1.13 .
We should notice that if the operator B equals 0 in 1.

Preliminaries
The purpose of this section is to establish some preliminary results that will be used in the proof of our main theorems. In 6 , the weighted Poincaré inequality for solutions of the nonhomogeneous A-harmonic equation was established. From 7 , we have the following local Poincaré inequality.
for all balls B with σB ⊂ Ω and diam B ≥ d 0 . Here d 0 > 0 is a constant and c is any closed form.
for some σ > 1. Similar to 3.4 in the proof of Theorem 3.1, we may assume that From 2.6 and 2.7 , it follows that for α > 0, we obtain This ends the proof of inequality 2.4 . If c is a closed differential form, from 1. 16 for any closed form c. The proof of Proposition 2.4 has been completed.
Next, extend the weak reverse Hölder inequality above to the case of Orlicz norms.

Lemma 2.5.
Let u be a solution of 1.13 in Ω, σ > 1, and 0 < s, t < ∞. Then there exists a constant C, independent of u, such that 14 for any constants α > 0 and β > 0, and all balls B with σB The proof of Lemma 2.5 is similar to that of Proposition 2.4. For completeness, we prove Lemma 2.5 as follows.
Proof. For any ball B ⊂ Ω with diam B ≥ d 0 > 0, we may choose ε > 0 small enough and a constant C 1 such that

2.19
This ends the proof of Lemma 2.5.
Using a similar method developed in the proof of Lemma 2.5 and from Lemma 2.9 in 6 , we can prove the following version of the weak reverse Hölder inequality with Orlicz norms. Note that the following version of the weak reverse Hölder inequality cannot be obtained by replacing u by du in Lemma 2.5 since du may not be a solution of 1.13 . Lemma 2.6. Let u be a solution of 1.13 in Ω, σ > 1, and 0 < s, t < ∞. Then there exists a constant C, independent of u, such that for all balls B with σB ⊂ Ω and diam B ≥ d 0 > 0. Here d 0 is a fixed constant, and α > 0 and β > 0 are any constants.
It is easy to see that for any constant k, there exist constants m > 0 and M > 0, such that From the weak reverse Hölder inequality Lemma 2.3 , we know that the norms u s,B and u t,B are comparable when 0 Thus, we have for any s > 0 and t > 0, where C i is a constant, i 1, 2, 3, 4. Using 2.22 , we obtain for any ball B and any s > 0, t > 0, and α > 0. Consequently, we see that We recall the Muckenhoupt weights as follows. More properties and applications of Muckenhoupt weights can be found in 1 .
for any ball B ⊂ E.

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We will need the following reverse Hölder inequality for A r E -weights.
Lemma 2.8. If w ∈ A r E , r > 1, then there exist constants k > 1 and C, independent of w, such that for all balls or cubes Q ⊂ E.

Caccioppoli-Type Estimates
In recent years different versions of Caccioppoli-type estimates have been established; see 1, 2, 4, 12-15, 17-19 . The Caccioppoli-type estimates have become powerful tools in analysis and related fields. The purpose of this section is to prove the following Caccioppoli-type estimates with L p log L α -norms for solutions to the nonhomogeneous A-harmonic equation.
for some constant σ > 1 and all balls B with σB ⊂ Ω and diam B ≥ d 0 > 0. Here d 0 , 1 < p < ∞ and α > 0 are constants, and c ∈ L p log L α Ω, Λ is any closed form.
Let ε > 0 be small enough and a constant C 1 large enough such that Applying Lemma 2.9 in 6 , we have where σ 2 > 1 is a constant. From 3.4 , 3.6 , and 3.6 , we have where σ 3 max{σ 1 , σ 2 }. By Lemma 2.2, we obtain for some σ 4 > σ 3 and all closed forms c. Note that Combining last three inequalities, we obtain The proof of Theorem 3.1 has been completed.
If we revise 3.5 and 3.5 in the proof of Theorem 3.1, we obtain the following version of Caccioppoli-type estimate. Proof. Let B be a ball with σB ⊂ Ω and diam for any solution u of 1.13 and any constants s, t > 0, where 0 < ρ 1 < 1, ρ 2 > 1, 0 < m 1 < 1, and m 2 > 1 are some constants. By Lemma 2.8, there exist constants k > 1 and C 0 > 0, such that 3.14 Choose s pk/ k − 1 , then 1 < p < s and k s/ s − p . We know that u ∈ L p log L α Ω, Λ implies u ∈ L p Ω, Λ . Then, for any closed form c ∈ L p log L α Ω, Λ , it follows that u − c ∈ L p log L α Ω, Λ . Thus, u − c ∈ L p Ω, Λ . By Caccioppoli inequality with L p -norms, we know that du ∈ L p Ω, Λ which gives du p,Ω N < ∞.

3.19
Since w ∈ A r Ω , then

3.20
Substituting the last inequality into 3.19 it follows obviously that This ends the proof of Theorem 3.3.
Let α 1 in Theorem 3.3; we obtain the following corollary. We know that if w ∈ A r E and 0 < λ ≤ 1, then w λ ∈ A r E . Thus, under the same conditions of Theorem 3.3, we also have the following estimate: where c is any closed form, and 0 < λ ≤ 1 and α > 0 are any constants. Choose λ 1/p, 1 < p < ∞, in 3.23 . Then, for closed form c and any constant α > 0, we have du L p log L α B,w 1/p ≤ C|M| −1/n u − c L p log L α σB,w 1/p .

3.24
We have proved Caccioppoli-type inequalities with L p log L α -norms for solutions to the nonhomogeneous A-harmonic equation. Using the same method developed in 12 , we can obtain the more general version of the Caccioppoli-type inequality for differential forms satisfying certain conditions. A special useful Young function ψ : 0, ∞ → 0, ∞ , termed an N-function, is a continuous Young function such that ψ x 0 if and only if x 0 and lim x → 0 ψ x /x 0, lim x → ∞ ψ x /x ∞. We say that a differential form u ∈ W 1,1 loc Ω, Λ is a k-quasiminimizer for the functional if and only if, for every φ ∈ W 1,1 loc Ω, Λ with compact support, where k > 1 is a constant. We say that ψ satisfies the so-called Δ 2 -condition if there exists a constant p > 1 such that ψ 2t ≤ pψ t for all t > 0, from which it follows that ψ λt ≤ λ p ψ t 3.27 for any t > 0 and λ ≥ 1; see 12 .
We will need the following lemma which can be found in 19 or 12 .

Lemma 3.5. Let f t be a nonnegative function defined on the interval a, b with
where c is any closed form.
The proof of Theorem 3.6 is the same as that of Theorem 6.1 developed in 12 . For the complete purpose, we include the proof of Theorem 3.6 as follows.
Proof. Let B R B x 0 , R ⊂ be a ball with radius R and center x 0 , R/2 < t < s < R.

3.32
Let v x u x η x p c − u x . We find that Since ψ is an increasing convex function satisfying the Δ 2 -condition, we obtain Using the definition of the k-quasiminimizer and 3.27 , it follows that

Journal of Inequalities and Applications
Adding k B t ψ |du| dx to both sides of inequality 3.36 yields Next, write f t B t ψ |du| dx, f s B s ψ |du| dx, A 2pR p B s ψ |u − c|/R dx, and B 0. From 3.37 , we find that the conditions of Lemma 3.5 are satisfied. Hence, using Lemma 3.5 with ρ R/2 and α p, we obtain 3.30 immediately. The proof of Theorem 3.6 has been completed.
It should be noticed that c ∈ L p log L α Ω, Λ is any closed form on the right side of each version of the Caccioppoli-type inequality. Hence, we may choose c 0 in each of the above Caccioppoli-type inequalities. For example, if we choose c 0 in Theorem 3.1 and Theorem 3.6, we obtain the following Corollaries 3.7 and 3.8, respectively, which can be considered as the special version of the Caccioppoli-type inequality. 3.39

Poincaré Inequalities
In this section, we focus our attention on the local and global Poincaré inequalities with L p log L α -norms. The main result for this section is Theorem 4.2, the global Poincaré inequality for solutions of the nonhomogeneous A-harmonic equation. The following definition of L ϕ μ -domains can be found in 1 .
Definition 4.1. Let ϕ be a Young function on 0, ∞ with ϕ 0 0. We call a proper subdomain Ω ⊂ R n an L ϕ μ -domain, if there exists a constant C such that for all u such that ϕ |u| ∈ L 1 loc Ω; μ , where the measure μ is defined by dμ w x dx, w x is a weight and τ, σ are constants with 0 < τ ≤ 1, 0 < σ ≤ 1, and the supremum is over all balls B ⊂ Ω.

Theorem 4.2.
Assume that Ω ⊂ R n is a bounded L ϕ μ -domain with ϕ t t p log α e t/k , where k u − u B 0 p,Ω , 1 < p < ∞, and B 0 ⊂ Ω is a fixed ball. Let u ∈ D Ω, Λ 0 be a solution of the nonhomogeneous A-harmonic equation in Ω and du ∈ L p Ω, Λ 1 as well as w ∈ A r Ω for some r > 1. Then, there is a constant C, independent of u, such that for any constant α > 0.
To prove Theorem 4.2, we need the following local Poincaré inequalities, Theorems 4.3 and 4.4, with Orlicz norms. Proof. Let B ⊂ Ω be a ball with diam B ≥ d 0 > 0. Choose ε > 0 small enough and a constant C 1 such that for some ρ 1 > 1. Similar to the proof of Theorem 3.1, we may assume that |u−u B |/ u−u B p,B ≥ 1. Hence, for above ε > 0, there exists C 3 > 0 such that 4.6 From 4.5 and 4.6 and Lemma 2.1, we have ≤ C 6 |B| −ε/p 2 |B| 1/n du L p log L α ρ 2 B ≤ C 7 |B| 1/n du L p log L α ρ 2 B .

4.9
The proof of Theorem 4.3 has been completed. Proof. Choose s kp/ k − 1 , where k > 1 is a constant involved in 3.14 . Using the Hölder inequality with 1/p 1/s s − p /sp, 3.14 , and 2.23 , we obtain for some σ 2 > σ 1 . Using the Hölder inequality again with 1/t 1/p p − t /pt, we obtain

4.15
Note that w ∈ A r Ω , then

4.16
Substituting 4.16 into 4.15 it follows obviously that u − u B L p log L α B,w ≤ C 6 |B| 1/n du L p log L α σ 2 B,w .

4.17
This ends the proof of Theorem 4.4.
Proof of Theorem 4.2. For any constants k i > 0, i 1, 2, 3, there are constants C 1 > 0 and C 2 > 0 such that for any t > 0. Therefore, we have

4.19
Journal of Inequalities and Applications 23 By properly selecting constants k i , we will have different inequalities that we need. Using the definition of L ϕ μ -domains with τ 1, σ 1 and ϕ t t p log α e t/k , where k u−u B 0 p,Ω , Theorem 4.4 and 4.19 , we obtain

4.20
which is equivalent to We have completed the proof of Theorem 4.2. for each ξ ∈ γ. Here d ξ, ∂Ω is the Euclidean distance between ξ and ∂Ω.
We know that any δ-John domain is an L ϕ μ -domain 1 . Hence, Theorem 4.2 holds if Ω is a δ-John domain. Specifically, we have the following theorem. Theorem 4.6. Let u ∈ D Ω, Λ 0 be a solution of the nonhomogeneous A-harmonic equation in a δ-John domain Ω ⊂ R n and du ∈ L p Ω, Λ 1 . Assume that 1 < p < ∞ and w ∈ A r Ω for some r > 1. Then, there is a constant C, independent of u, such that u − u Ω L p log L α Ω,w ≤ C|Ω| 1/n du L p log L α Ω,w 4.23 for any constant α > 0.

24
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Applications
In this section, we explore some applications of the results obtained in previous sections. which is equivalent to the following partial differential equation: Let Each version of the Caccioppoli-type inequality proved in Section 3 can be used to study the properties of the solutions of the different A-harmonic equations, particularly, the equations 5.1 -5.3 . For example, using Corollary 3.7, we have the following integrability result.
Corollary 5.2. Let u be a solution to one of the equations 1.13 -1.14 , or 5. 1 -5.3 in Ω ⊂ R n . If u is locally L p log L α -integrable in Ω, then du is also locally L p log L α -integrable in Ω.
From Theorem 3 in 20, page 16 , we know that any open subset of R n is the union of a sequence of mutually disjoint Whitney cubes. Also, cubes are convex. Thus, the definition of the homotopy operator T can be extended into any domain Ω in R n . Using the same method developed in the proof of Theorem 4.4, we can extend inequality 2.5 into the weighted case. Then, similar to the proof of Theorem 4.2, we can generalize the local weighted result into the following global estimate. Proposition 5.3. Let u ∈ D Ω, Λ 0 be a solution of the nonhomogeneous A-harmonic equation 1.13 in an L ϕ μ -domain. Assume that α > 0, 1 < p < ∞ and w ∈ A r Ω for some r > 1. Then there exists a constant C, independent of u, such that u − u Ω L p log L α Ω,w ≤ C u − c L p log L α Ω,w 5.4 for any closed form c.

25
Using 5.4 with c 0 and the triangle inequality, we have u Ω L p log L α Ω,w ≤ u − u Ω L p log L α Ω,w u L p log L α Ω,w ≤ C 1 u L p log L α Ω,w u L p log L α Ω,w C 1 1 u L p log L α Ω,w C 2 u L p log L α Ω,w .

5.5
Thus, u Ω L p log L α Ω,w ≤ C u L p log L α Ω,w .

5.6
Theorem 5.4. Let u ∈ D Ω, Λ 0 be a solution of the nonhomogeneous A-harmonic equation 1.13 in a bounded L ϕ μ -domain and let T be the homotopy operator. Assume that α > 0, 1 < p < ∞, and w ∈ A r Ω for some r > 1. Then there exists a constant C, independent of u, such that T du L p log L α Ω,w ≤ C|Ω| 1/n du L p log L α Ω,w , T du L p log L α Ω,w ≤ C u − c L p log L α Ω,w

5.7
for any closed form c.
Proof. For any differential form u, from 1.16 and 1.17 , we obtain u d Tu T du u Ω T du .

5.8
Hence, by 5.8 and Theorem 4.2, it follows that T du L p log L α Ω,w u − u Ω L p log L α Ω,w ≤ C|Ω| 1/n du L p log L α Ω,w .

5.9
Next, combining 5.8 and 5.4 yields T du L p log L α Ω,w u − u Ω L p log L α Ω,w ≤ C u − c L p log L α Ω,w .

5.10
This ends the proof of Theorem 5.4.
The general theory of solutions to above equations is known as potential theory. In the study of heat conduction, the Laplace equation is the steady-state heat equation. Considering the length of the paper, we only discuss applications to the homotopy operator T ; see 1 for more results about this operator. We leave it to readers to find similar applications to other operators, such as the Laplace-Beltrami operator Δ dd d d and Green's operator G applied to differential forms. Note that there is a parameter α in our main results. For different 26 Journal of Inequalities and Applications choices of this parameter, we will have different versions of global inequalities. For example, selecting α 1 in Theorem 4.2, we have u − u Ω L p log L Ω,w ≤ C|Ω| 1/n du L p log L Ω,w . 5.11