Weighted Estimates Of The A –harmonic Tensors And Related Operators

The results about the A-harmonic equations have been applied widely in many areas of natural sciences and engineering technology. The weighted integral inequalities of it is one of the heat topics in the study of contemporary harmonic analysis. And its result has important values in nonlinear analysis, partial differential equations.the weighted integral inequalities of A-harmonic tensors about differential forms is established. And it is on the basis of the research work about C.Nolder, Shusen Ding,Gejun Bao,Hongya Gao and so on.Firstly, using of Holder inequality, Sobolev space theory and all kinds of weight functions (such as:Ar(λ,Ω), Arλ(Ω), Ar,λ(Ω)-weight),giving two-weight norm estimates for conjugate A-harmonic tensors and Sobolev inequality.Secondly,achieving Arλ3 (λ1,λ2,Ω)-weight Poincare -inequalities for composition operator H(?)G and Arλ3 (λ1,λ2,Ω)-weight about BMO norms and Lipschitz norms inequalities for H(?)G with the help of the relevant conclusion about G, BMO norms, H, Lipschitz norms and new weight function Arλ3(λ1,λ2,Ω)-weight.At last, obtaining Arλ3(λ1,λ2,Ω)-weight Poincare -inequalities for T(?)G as soon as promoting it to δ-John domain in combination with the relevant conclusion about δ-John domain, T, G and Whitney cover Lemma, partition of unity.Main results as follows:1) Ar(λ,Ω), Arλ(Ω), Ar,λ(Ω) weight norms estimates for conjugate A-harmonic tensors;2) Two-weights Sobolev imbedding inequality for conjugate A-harmonic tensors;3) Arλ3(λ1,λ2,Ω)-weight Poincare-inequalities for composition operator H(?)G;4)Arλ3(λ1,λ2,Ω)-weight about BMO and Lipschitz norms inequalities for H(?)G;5) Arλ3(λ1,λ2,Ω)-weight Poincare-inequalities for composition operator T(?)G;6) Norms estimates for composition operator T(?)G in δ-John domain.The above results substantiate the related problems of nonlinear elliptical partial differential equations.