Norm Inequalities For Some Operators On Differential Forms

The differential form is the natural extension of the function,which develops the classical calculus theory in Euclidean space.As a powerful mathematical tool for the calculation of differential and integral calculus on manifolds,differential forms play an important role in partial differential equations,differential geometry and some physics fields,such as electromagnetism and mechanics.On the other hand,operator theory performs an indispensable role in many fields,such as mathematics,physics,engineering and computer science.In the past twenty years,the theory of differential forms,including the equation theory,the operator theory,L~ptheory and the characterization of domains for differential forms,is developed rapidly and becomes one of the hot field of modern scientific research.The operator theory for differential forms has a very significant position in modern scientific research and is applied widely in many fields.This dissertation mainly studies some classical operators in the harmonic analysis and partial differential equations,such as maximal operator,singular integral operator,Green's operator,Dirac operator and their composite operators on differential form.Especially,it studies the boundedness of the operators defined on the nonhomogeneous A-harmonic tensor and conjugate A-harmonic tensors in-depth.The contents of this dissertation are as follows:Firstly,by using the decomposition property and some basic inequalities,along with the boundedness of the maximal operator and potential operator,the boundedness of the composite operator Ms? P for the maximal operator and potential operator is discussed,the comparison among L~pnorm,BMO norm and Lipschitz norm of the composite operator Ms? P is established.Secondly,the definition of multilinear Calderón–Zygmund operator on differential forms is set forth.By combining the Calderón–Zygmund decomposition with some skillful techniques,end-point weak type boundedness of multilinear Calderón–Zygmund operator on differential forms is established,which provides an important support for proving the strong boudedness of operators.Based on the nonhomogeneous A-harmonic tensor,by applying H?lder inequality and characteristic function,the estimates for multilinear Calderón–Zygmund operator on differential forms are studied in terms of L~pnorm.Moreover,some composite operators related with Hodge-Dirac operator is investigated.Specifically,the upper boundedness for the composite operator M?s? D ? G are shown in terms of L~pnorm,BMO norm and Lipschitz norm,as well as the weighted BMO norm and weighted Lipschitz norm.Subdeteminant of Jacobian determinant and K-quasiregular mapping are displayed as applications for the results about the composite operator M?s? D ? G.The definitions of BMO norm and Lipschitz norm are generalized,and the norm comparison is made between generalized BMO norm and Lipschitz norm for the iterated operators Dk? Gkand Dk+1? Gk.Finally,due to the significance of Orlicz function theory on modern analysis,by combining Orlicz function with the bounded mean oscillation spacce,the definitions about Orlicz Lipschitz norm and Orlicz BMO norm are introduced.By using a class of Young function,G(p,q,C)-class,the L~?-Lipschtiz norm and L~?-BMO norm inequalities on differential forms for homotopy operator T are discussed.At this end of this dissertation,the estimates of L~?-BMO norm are proven with respect to the conjugate A-harmonic tensors,which generalizes the norm comparison inequalities for the conjugate A-harmonic tensors.