Ar(λ)-WEIGHTED CACCIOPPOLI-TYPE AND POINCARÉ-TYPE INEQUALITIES FOR A-HARMONIC TENSORS

We prove a local version of weighted Caccioppoli-type inequality, then we prove a version of weighted Poincaré-type inequality for A-harmonic tensors both locally and globally.


Introduction.
There have been many studies for the integrability of differential forms, and the estimations of the integrals of differential forms.As extensions of those studies, the integrability and estimations of integrals for A-harmonic tensors are also studied and applied in many fields such as in tensor analysis, potential theory, partial differential equations, and quasiregular mappings, see [1,2,6,7,8,9,10,11,12].There are many studies about Caccioppoli-type and Poincaré-type inequalities for Aharmonic tensors, see [3,4,5,12].We state the following specific results from [12].Theorem 1.1.Let u be an A-harmonic tensor in Ω and let σ > 1.Then there exists a constant C, independent of u and du, such that du s,B ≤ C|B| −1 u − c s,σ B (1.1) for all balls or cubes B with σ B ⊂ Ω.
Our work is to give new versions of Theorems 1.1 and 1.2 with A r (λ) weight.When λ = 1, the properties of A r (1)-weight can be found in [6].
We first introduce some related definitions and notations which are adopted from [12].
We assume that Ω is a connected open subset of R n .The Lebesgue measure of a set E ⊂ R n is denoted by |E|.Balls in R n are denoted by B and σ B is the ball with the same center as B and with diam(σ B) = σ diam(B).We call w a weight if w ∈ L 1 loc (R n ) and w > 0 a.e.
Let e 1 ,e 2 ,...,e n be the standard unit basis of R n .Assume that ∧ l = ∧ l (R n ) is the linear space of l-vectors, spanned by the exterior products e 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 l ) and all integers l = 0, 1,...,n.The Hodge star operator * : ∧ → ∧ is defined by where ω i on Ω is a locally integrable function or more generally, a Schwartz distribution on Ω with values in ∧ l (R n ).We denote D (Ω, ∧ l ) as a space of all differential l-forms and L p (Ω, ∧ l ) as a space of differential l-forms with coefficients in the L p (Ω, R n ).The space L p (Ω, ∧ l ) is a Banach space with the norm We also denote W 1 p (Ω, ∧ l ) as a space of differential l-forms on Ω whose coefficients are in Sobolev space W 1 p (Ω, R).An A-harmonic equation for differential forms is where d : D (Ω, ∧ l+1 ) → D (Ω, ∧ l ), as the formal adjoint operator of d, is given by

,n, and
for almost every x ∈ Ω and all ξ ∈ ∧ l (R n ).Here a > 0 is a constant and 1 < p < ∞ is a fixed exponent associated with (1.5).Let , where the intersection is for all Ω compactly contained in Ω.A solution to (1.5) is an element of the Sobolev space W 1 p,loc (Ω, ∧ l−1 ) such that for all ϕ ∈ W 1 p (Ω, ∧ l−1 ) with compact support, see [8,9,12].
The following definition belongs to Ding and Shi [5].
The following lemma is from Nolder [12].
Lemma 1.5.Each Ω has a modified Whitney cover of cubes W = {Q i } which satisfy for all x ∈ R n and some N > 1 and if We also need the following generalized Hölder's inequality. (1.11) for any Ω ⊂ R n .

Main results
Theorem 2.1.Let u ∈ D (Ω, ∧ l ), l = 0, 1,...,n, be an A-harmonic tensor in a domain Ω ⊂ R n and ρ > 1. Assume that 1 < s < ∞ is a fixed exponent associated with the A-harmonic equation and weight w ∈ A r (λ) for some r > 1 and λ > 0. Then there exists a constant C, independent of u and du, such that for all balls B with ρB ⊂ Ω and all closed forms c.
Proof.We only need to prove the following: For any 1 < s < ∞, choose t such that t = s 2 /(s − 1), then 1 < s < t.By Hölder's inequality we have

.16)
Proof.By Lemma 1.5, there exists a Whitney cover F = {Q i } of Ω.In particular, we can choose 1 < σ ≤ 5/4 in Theorem 2.2, so that Since parameter λ > 0 can be chosen arbitrarily, the inequalities in our theorems can be used to estimate a relatively broad class of integrals.
If we chooseλ = s + 1 in Theorem 2.1, (2.1) is in the form of B |du| s w s dx 1/s dxNχ Ω (x) Remark 2.4.Similar to Theorem 2.1, we have different versions of global results for Poincaré-type inequality by choosing different values of λ.For instance, as λ = 1, (2.16) reduces to