Higher Estimates Of The Solutions To Harmonic Equations And The Related Integral Operators On Differential Forms

The differential form as the generalization of function has the advantage of being coordinate system independent.The emergence of differential form is closely related to the calculus theory and many problems of differential manifold,now it has become an important tool in the study of the modern differential geometry.With the development of geometry,differential form has been playing an indispensable role in many areas,such as physics,thermodynamics,electromagnetism and the theory of relativity,which makes the study of the theory of differential form more significant.In recent years,the research on the theory of operators and equations on differential forms has developed rapidly and also been the focus point of studies home and abroad.This dissertation mainly discusses some operators on differential forms including homotopy operator,projection operator,singular integral operators and their commutators.We proceed to the study of the boundedness and integrability of the related operators and establish some inequalities with different norms.On this basis,we further study the higher order estimates of the related operators.Especially,we consider of the higher integrability of the weak solutions and very weak solutions to non-homogeneous A-harmonic equations and Dirac-harmonic equations,respectively.The primary contents of the dissertation are as follows:Firstly,it considers the composition of homotopy operator T and projection operator H which are two key operators on differential forms,and investigates the embedding prop-erty and higher order property of the composite operator T?H.On one hand,by making use of the decomposition property and elementary inequalities on differential forms,and choosing a special Young functionφ∈NG(p,q)-class,some inequalities with L~φnorm for the composite operator T?H are established.Furthermore,when u satisfies the A-harmonic equations,the main L~φembedding theorems and inequalities with L~φ-Lipschitz and L~φ-BMO norms for the composite operator T?H are derived by using the elementary inequalities of the solution to A-harmonic equation.On the other hand,the higher order estimates of the composite operator T?H are studied and the L~phigher order Poincaré-type inequalities for the composite operator T?H are proved by applying the properties of homotopy operator T and projection operator H.Secondly,it introduces the singular integral operators on differential forms including the Calderón-Zymund singular integral operators and the fractional integral operators,and then it defines their commutators when b∈BMO(R~n)and discusses the L~pboundedness for the two types of commutators.The strong type inequalities for the two types of com-mutators are established and the weighted Caccioppoli-type inequalities with L~φnorm for commutator[b,T_?]are derived.Moreover,with aid of the results on boundedness,the higher order integrability of the commutator[b,T_?]with L~pnorm is studied systematical-ly.By using the Poincaré-Sobolev inequality on differential forms,the local and global higher integrability and higher order Poincaré-type inequalities for commutator[b,T_?]are established with two cases,that is,1<p<n and p≥n and the related applications are demonstrated.It also makes a primary study on the higher order commutator on differen-tial forms,the definition of the higher order commutator is given and the L~pboundedness for the higher order commutator is proved.Finally,the higher order estimates for the solutions to harmonic equations on dif-ferential forms are discussed.By using the basic inequalities for the solutions to non-homogenous A-harmonic equation,along with the properties of a class of Young functionφ∈N G(p,q)-class,the L~φhigher order Poincaréinequality and Caccioppoli inequality for the solution to non-homogenous A-harmonic equation are derived.As applications,the L~φhigher order Caccioppoli-type inequality and a class of weak type inequality for homotopy operator T are obtained.Moreover,it introduces the definition of the very weak solution to homogenous Dirac-harmonic equation under some basic assumptions and stud-ies the higher integrability of the very weak solution.By applying Hodge-decomposition theorem and skillful method,the higher integrability of the very weak solution to homoge-nous Dirac-harmonic equation is proved rigorously.