Norm Inequalities For Operators Applied To Differential Forms And Applications

Di-erential forms, as powerful tools in modern mathematics, have been widely stud-ied and used in many fields, such as partial di-erential equations, algebraic topology anddi-erential geometry. Meantime, differential forms also bring a revolutionary change ofresearch methods for mathematical physics, including quantum field theory, elementaryparticle physics, etc. Especially in the past few years, from the perspective of PDEs, theinvestigation of A-harmonic equations for di-erential forms has been rapidly developed.Usually, the operator A of the A-harmonic equation for di-erential forms is a mappingthat satisfies certain conditions. Thus, some famous PDEs, such as p-harmonic equa-tions, can be all regarded as the special cases of A-harmonic equations. Hence, recentyears A-harmonic equations have received much attention. However, the study of thecorresponding operator theory just began.In this dissertation, we mainly study the integrability of di-erential forms applied bythe homotopy operator T, Green's operator G, the potential operator P and the compositeoperator T - H of the homotopy operator T and the projection operator H in the weightedLp spaces and Orlicz spaces. At the same time, we also prove some weighted integralinequalities for these operators acting on the solutions of the A-harmonic equations. Themain work of the dissertation can be summarized as follows:First, based on the work of R. P. Agarwal and S. Ding, etc., the local Poincare′-typeinequalities for Green's operator in terms of Lp(log L)α-norms are proved. Then, for theweighted inequalities for Green's operator acting on the solutions of the nonhomogeneousA-harmonic equations, the corresponding local parametric versions are established insome more general measure spaces. At last, by these local results the weighted Lp(log L)α-norm estimates for Green's operator are further developed to the Lφ(μ)-domains.Secondly, in a class of more general measure spaces, the weighted norm comparisoninequalities are established for the composite operator T - H defined in a bounded convexdomain. At the same time, for the Young functionφin the class G(p, q,C), the localweighted Orlicz norm inequalities and the global versions in Lφ(μ)-averaging domainsfor the composite operator are proved. At last as applications, several specific Youngfunctions that belong to G(p, q,C) are given, by which some estimates for the solutionsof the A-harmonic equations are obtained. Then, noting that the homotopy operator T plays a critical role in Lp theory for dif-ferential forms, we prove the local Lp-norm inequalities with Ar weights for the operatorT. Furthermore, under the condition of n < p <∞, these results are extended to theglobal cases that implies the boundedness of the homotopy operator in the weighted Lpspaces. Meantime, we also establish some strong type (p, q) inequalities and the versionswith the power-type weights for the homotopy operator. As applications, we further studythe integrability for some specific di-erential forms applied by the homotopy operator.Finally, after successfully extending the definition of the potential operator P to dif-ferential forms, we establish the corresponding two-weight weak type (p, p) inequalities.Then, through applying the strong (p, p) type estimates for the potential operator actedon functions, we prove the local and global Ar,λtwo-weight strong type (p, p) inequali-ties and their parametric versions, and hence obtain some global norm estimates for thepotential operator with a special kernel.