The Norm Comparison Inequalities For The Composite Operators

As extensions of functions, differential forms have been widely studied and used in many fields, such as general relativity, theory of elasticity, electromagnetism. They can be used to describe various systems of partial differential equations and to express different geometrical structures on manifolds. As a special nonlinear elliptic partial differential equation, the A-harmonic equation for differential forms have been extensively investigated and developed. Especially with the Development of differential forms on manifold, we can apply the notation and calculus theory in Rn to differential forms, thereby the integral estimate for differential forms become a hot topic. In many situations, the process to solve a PDE involves and related operators integral estimate, and the norms for differential forms are mostly related with integration. Therefore it is very critical to carry on the estimates for different norms of the involved operators and their compositions.Present theoretical studies in differential form has confirmed the findings on the boundedness of the maximal operator and the homotopy operator. This paper establishs the norm comparison inequalities for composite operators Ms#oT of sharp maximal operator and homotopy operator T which act on the nonlinear A-harmonic tensors, we get the related norm inequalities for compsite operators Ms oT of Hardy-Littlewood maximal operator and homotopy operator T. Firstly, we establish the Lp-norm estimate for composite operator Ms oT of sharp maximal operator and homotopy operator T, then we obtain the norm comparison inequalities among Lp-norm, Lipschitz norm and BMO norm. Using the similar method we also obtain the related results for the composite operator Ms oT of Hardy-Littlewood maximal operator and homotopy operator T. In order to apply these results more flexibly, we establish the A(α,β,γ,Ω)-weighted comparison inequalities for the related operators by using the main properties of weight function. Furthermore we obtain the Aλ3(λ1,λ2,Ω)-two weight version for the composite operators. Finally, using the properties of δ-John domain and Lφ(μ)-averaging domain, we generalize the related results to δ-John domain and Lφ(μ)-averaging domain.