Orlicz Norm Inequalities Of A-harmonic Tensor

In recent years, from the perspective of PDEs, the investigation of A-harmonic equa-tions for differential forms has been rapidly developed. As an extension of function, Differential forms have been received valuable result. In many situations, the process to study solutions to PDEs involves estimating various norms of operators. The estimates for norms of different operators and their compositions are critical to investigate the prop-erties of the solutions of the partial differential equations, or a system of the partial differ-ential equations. Differential forms and the relevant operator have been widely used not only in the analysis and partial differential equations, but also in physics. Therefore, it is crucial to establish some operator norm inequality.In this dissertation, we mainly study the integrability of differential forms applied by the homotopy operator T, projection operator H, Green’s operator G, and their com-posite operator. At the same time corresponding operators of Poincare and Caccioppoli inequalities are given. The main work of the dissertation can be summarized as follows:1. Firstly, the local Poincare-type inequalities for T o H o G operator acting on the solution of nonhomogneous A-harmonic equations in terms of Lp(log L)α norms are proved. By these local results the Lp(log L)α norms estimates are further developed to the domains. Secondly, the weighted Poincare-type inequalities are established for Luxem-burg norms and for these results are extended to the global cases of Luxemburg norms estimates in Lφ(M)-domains.2.We prove the weighted conjugate Orlicz space of nonhomogeneous A-harmonic tensor Caccioppoli estimate. And to the Young functions that belong to G(p, q, C), we also establish the weighted Orlicz norm of local and global inequality.3.We proved the nonhomogeneous A-harmonic tensor on bounded convex domain of Lipschitz norm inequalities of composite operator, and then proves that the BMO and Lipschitz norm comparison inequalities of composite operator ToHoG, then respectively proves that composite operator T o H o G weighted norm in the form of inequality.