Norm Estimates Of Composite Operators

The A- harmonic equations belong to the nonlinear elliptic partial differential equations that have received much investication. The theory of the A-harmonic equations provide a powerful and effective tool for the study of both qualitative and quantitative properties of solutions of the related differential system, which appear in many branches of science and engineering, such as physics, theory of elasticity and quasiconformal analysis. The different versions of A- harmonic equation can be considered as a bridge connecting these branches with mathematics. Hence the results about them are of interest in this field. The integral inequalities are widely studied and used in different fields, such as partial differential equations, potential theory, nonlinear analysis, etc. They have been serving as effective tools in studying the integrability of differential forms and estimating the integral of differential forms.In this paper we research the composite of the homotopy operator T , differential operator d and the Green operator G under the weighted A-harmonic tensors inequality. Then established the norm inequalities of the composite operatorT ? G, as an extension of the present result, we obtain the composition operators are restricted to the A_r (Ω)-weight and A_r (λ,Ω)? double weight inequality. Known by some of the inherent nature of the Whitney cover Lemma, we extend the local integral inequalities to bounded domain,to get the global promotion of local integral inequality. In the third chapter we first introduces the introduction of mean oscillation functions and functions of bounded mean oscillation (BMO) space and in bounded convex domains of the composite operator norm integral estimates,include the definition of the BMO norm and the Lipschitz norm ,after the composition operator T ? G on smooth differential forms is given the role of under the Ls norm,BMO norm and the Lipschitz norm of inequality on bounded convex domain , including the norm under certain conditions, then extend to the A_r,λ(Ω)-double weight inequality. And finally, use the inherent nature of L^s (μ)-average domain, established the norm estimate in the average domain to do the extention.