The Research On Norm Inequalities Of Some Operators Of Differential Forms

Differential form is an antisymmetric covariant tensor field defined on a differential manifold,has been widely used in many fields,such as general relativity theory,theory of elasticity,electromagnetism,and differential geometry.Hence,differential forms are valuable tools in many fields.In recent years,the research on the operator theory of differential forms,such as Dirac operator,Green operator,homotopy operator and Hardy-Littlewood maximal operator,has a lot of achievements.The composite operators of these several operators in this paper have been further studied.And several norm inequalities in different differential form spaces for these composite operators have been obtained.Firstly,we introduce the definitions of the homotopy operator,the Dirac operator and Green's operator on differential forms.Then,through applying the Ls norm inequality for the composition of the homotopy operator,the Dirac operator and Green's operator on differential forms,we prove the Lipschitz and BMO norm inequalities for the composite operator T(?)D(?)G.Finally,by the strictly increasing convex condition and the reverse Holder inequality,we establish the comparison inequality acting on the solutions of the A-harmonic equations in terms of the Lipschitz and BMO norms.Secondly,we use the decomposition of differential form for the homotopy operator and the norm inequality for the Green's operator to prove the strong type(p,q)norm comparison inequalities for the composition of the homotopy operator and Green's operator.Then,by using the properties of continuous functions and the strong type(p,q)norm comparison inequalities for the composite operator T(?)G,we establish the norm estimates with the power-type weights acting on the solutions of the nonhomogeneous A-harmonic equations.Finally,by using a weak-type weighted inequality for the Hardy-Littlewood maximal operator,we develop a reverse weighted inequality for theHardy-Littlewood maximal operator to differential forms of Orlicz spaces and obtain a sufficient and necessary condition.Furthermore,It is further proved that when l? s<p<? in the weighted Lp space,the Lp norm of u(x)can be controlled by Msu(x).In the sense that we establish a reverse weighted inequality for the Hardy-Littlewood maximal operator on differential forms.The last,we prove when the Hardy-Littlewood maximal operator acts on the weight function,an embedding inequality for the Hardy-Littlewood maximal operator on differential forms holds.