Embedding Inequalities Of Composition Operators With Orlicz Norm

Since the combination with the thoughts of manifold, differential form has been a quite effective tool for the research of contemporary mathematical problems. And now, it has been applied to research the partial differential equations, algebraic topology and differential geometry. Specifically, various operators play a vital role in the partial differential equations, so it is very necessary to establish the operator theory of differential forms.The main work of this thesis can be summarized as follows:The first part is to establish the integral estimates of the Green operator G, the potential operator P, and their composite Operators for differential forms, such as Poincaré inequalities, Caccioppoli inequalities. Moreover, similarly with the Poincaré inequalities for functions, we get Poincaré inequalities for differential forms with Orlicz norm, which are based on the results in the theory of Orlicz space, the properties of doubling weight of Young functions. Then, by using the bounded estimates of operators, such as Green operator, homotopy operator, potential operators, etc, and the existed properties, the thesis establishes the Poincaré estimates of Orlicz norm for the composite operators, which act on the A- harmonic tensor. Finally, according to the theoretical results of Sobolev spaces and combined with the established Orlicz norm inequalities, on the one side, this thesis establishes the imbedding inequalities of operators for differential form. On the other side, considering the related properties about the average domain, it generalizes the results showed above to the L(m)?-average domain, and then obtains the global Poincaré estimates.For the second part, in accordance with the Caccioppoli inequalities of operator du in pL-space and the properties of Young function, it initially shows the Caccioppoli inequalities for differential forms with Orlicz norm. After that, by using the related properties about weight function A(α, β,γ; M), it correspondingly gives the single and two weighted norm inequalities. And finally, we get single weighted Caccioppoli inequalities in Orlicz space.For the third part, based on the Poincaré inequalities of composite operators G oP in pL-space appearing in the second part, and the related properties of the weight function A(α, β,γ; M), it derives the single and two weighted norm inequalities. Finally, we utilize the propositions of Young function to get single weighted Caccioppoli inequalities in Orlicz space.