Norm Estimates Of P·T Operator

Differential forms as the forms of the traditional function of extension, it has beenrapid development in many areas of modern mathematics. Operator norm theory basedon differential forms,not only in traditional partial differential equations、harmonicanalysis、nonlinear potential theory、quasi-conformal mapping has in depth researchbut also in modern theoretical physics of general relativity、elasticity theory has a widerange of applications. Nonhomogeneous A-harmonic equation as a special kinds ofnonlinear elliptic partial differential equation, it has a deep background in physics andmechanics. Therefore, based on the nonhomogeneous A-harmonic equations in different-ial forms, operator norm theory has wide range of research significance and value.This major works is summarized as follows:The first part has respectively introduces the convolution type potential operatorand homotopy operator for the definition and the basic properties; furthermore, linkingwith above contents, it gives the basis of strong type （p, p） estimates for P。 Toperatoron differential forms.The second part has established the Caccioppoli type estimate based in the strongtype （p, p） estimates for P Toperator on differential forms. In order to make theresults more widely, using the definition of the weighted function and natures to give th-e corresponding weighted norm estimator.Such as single-weight A_r （Ω）forms Caccio-ppoli type estimate、double-weight A_r^λ （E）forms Caccioppoli type estimate.The third part has been established the Poincaré type estimates for P Toperatoron differential forms. Point to non-homogeneous harmonic tensors, it given the Poincaré-type estimators single-type A_r （λ,E）forms and two-type A_r,λ（E）forms.Combined with the P Toperator Poincaré type estimates, the fourth part makeuse of the Lipschitz norm and BMO norm definition and nature in differential forms toestablish the Lipschitz norm estimate and BMO norm inequalities for P Toperator inA harmonic tensor from.