The Study Of Differential Forms On Several Classes Of Space

In the past few years, differential forms, as the extension of functions, have been be-come powerful tools to be studied and used in many mathematical fields, such as partial differential equations, differential geometry, algebraic topology and mathematical physic-s. The theory of the A-harmonic equations provide a effective theoretical tool for the study of both quantitative and qualitative properties of solutions to the related differential system, which is used in many branches of science and engineering, such as quasicon-formal analysis, quantum field theory, theory of elasticity and nonlinear potential theory, etc. Especially, in the study of quasiconformal analysis, it has been proved that in higher space, the coordinate functions be the solutions to the A-harmonic equation for differ-ential forms. Hence, the theory of A-harmonic equation for differential forms has been widely investigated by many mathematical researchers. And in recent years, the subject of A-harmonic equation for differential forms has undergone a vast development and received increasing attention in the fields of mathematics and engineering.In this paper, we will study the properties of differential forms and the solutions to several classes of nonhomogeneous A-harmonic equations in several special spaces. Re-spectively, in Banach spaces Lφ(Ω, Λl,μ) with different measure using differen weights, bounded sharp BMO spaces, local Lipschitz spaces, Orlicz spaces Lφ(Ω, Λl) which sat-isfies φp condition, we will establish many weighted integral inequalities for differential forms or A-harmonic tensors. We also prove the boundedness and embedding for some operators, such as homotopy operator T, Green’s operatorG, Laplace-Beltrami operator△, harmonic projection operator H and composite operators, etc. We will obtain all re-sults both locally and globally. The main work of the dissertation is as follows:1. Several classes of two-weight functions are studied. We introduce the definitions of two-weights, such as Ar,λ(Ω)-weight, Arλ(Ω)-weight, Ar(λ, Ω)-weight, and Arλ3(λ1, λ2, Ω)-weight in Lp(Ω) spaces, base on the work of Iwaniec, Nolder and Ding, we generalize the main results of the nonhomogeneous A-harmonic equation A(x, g+du)=h+d*v for differential forms to the different two-weighted versions. Meantime, we establish the weighted Poincare inequalities for the composite operator Tο△οG, and we obtain the global results in bounded convex domain and Ls-averaging domain, respectively. 2. A(φ1(x), φ2(x),τ, Ω)-weight is denoted in Orlicz spaces, taking different Young functions φ1,φ2and the different parameter τ, some existing versions of two-weight in Lp(Ω) spaces are the special cases of A(φ1(x), φ2(x),τ,Ω)-weight, then the form of the two-weight will be made more general and utility. As the application of the weight, we establish two-weighted Poincare inequality, Caccioppoli inequality and weak reverse Holder inequality for the solutions to the nonhomogeneous A-harmonic equation d*A(x, dω)=B(x, dω). We also prove the Orlicz two-weighted Sobolev-Poincare em-bedding theorem for the composite operator T o H, and generalize the local results to compact Riemann manifolds globally.3. The subject of the bounded mean oscillation space-BMO and the Lipschitz space of functions has been widely investigated and some results have been proved. In this paper, the BMO space and local Lipschitz spaces of differential forms and A-harmonic tensors will be discussed. First, we study the properties of the solutions to the equation d*A(x, dω)=B(x,dω) and give some results for the solutions with BMO norm and Lipschitz norm, such as establishing weighted Poincare inequalities for A-harmonic ten-sors of the equation d*A(x, dω)=B(x, dω) with BMO norm and Lipschitz norm; giving some equivalences for the solutions to the equation d*A(x, dω)=B(x, dω) with BMO norm and Lipschitz norm. Secondly, we generalize Hardy-Littlewood inequalities for the conjugate A-harmonic tensors, we obtain some versions of weighted Hardy-Littlewood inequalities for the conjugate A-harmonic tensors of the equation A(x, du)=d*v with Lp-norm, BMO norm and Lipschitz norm, respectively.4. The theory of the Orlicz spaces is the important tool in studying PDE, so in this paper, the integral estimates of differential forms with Lp norm will be replaced by the large norm estimates in the Orlicz spaces. According to a Young function which satis-fies the φp condition, a particular Orlicz space Lφ(Ω,Λl) for differential forms is defined, which satisfies the φp condition, then we study the integrability of differential forms and A-harmonic tensors in Lφ(Ω, Λl). First, using the boundedness of the Orlicz maximal operator, we discuss the boundedness of the homotopy operator T in Lφ(Ω,Λl), and the local Poincare inequalities, Caccioppoli estimates and weak reverse Holder-type inequal-ities for the solutions to the equation d*A(x, dω)=B(x, dω) are also established, then we get the global estimates with Orlicz norm in Lφ(x)-averaging domains. Meantime, we study a two-weight class which satisfies φp condition with Orlicz norm, from the def-inition of C-Z singular integral operator P for differential forms, we obtain two-weighted strong (p,p) type inequalities. Finally, we give two examples of Young functions which satisfies the φp condition.