Boundedness Of Operators In Several Morrey-type Spaces On Differential Forms

The theory of classical Morrey spaces is a powerful mathematical tool to solve partial differential equations.It also plays an important role in operator theory and harmonic analysis.The differential form as the generalization of function is an important tool to study the calculus in differential manifold and it is also applied to many fields,such as differential equations,Riemann manifold,differential geometry.With the development of theoretical research,the theory of operators and equations becomes one of the hot field of modern scientific research,and it has attracted great attention from home and abroad.Combining with classical Morrey space and differential form,the problem for equations which appeared in differential forms can be solved better.Therefore,Morrey space on differential forms has been the focus point nowadays.This dissertation mainly studies the Morrey space on differential forms and discusses the boundedness and integrability for some classical operators,such as homotopy operator,Hardy operator,singular integral operator.By using the theory of A-harmonic equations,we establish the inequalities about the solution of A-harmonic equations with Morrey norms.Furthermore,we extend the differential form to the filed of p-adic numbers and L（φ_s,μ）-averaging domains and research some properties for operators.The contents of this dissertation are as follows:Firstly,the definition of generalized Morrey spaces on differential forms is given,and the boundedness for commutator of fractional maximal operator and commutator for frac-tional integral operator are discussed.Moreover,we show BMO^p,φ（Ω,∧^l）on differential forms and establish the strong（p,p）-inequality for Calderón-Zygmund singular integral commutator.By the double phase functional of Musielak-Orlicz-Morrey spaces and the decomposition theorem on differential forms,the H（?）lder continuity for commutator of fractional integral operator is derived.We establish the Caccioppoli-type inequality and Poincaré-type inequality for A-harmonic equations with Morrey norms.Secondly,the definition of H（?）lder-Morrey spaces on differential forms is given,and the boundedness for homotopy operator T is derived.On this basis,the Poincaré-type inequality is derived with H（?）lder-Morrey norms.We introduce the concept of envelope function from physics and show the H（?）lder continuity for Riesz potential operator with envelope function in Morrey norms.Furthermore,the inequalities for homotopy operator T and conjugate A-harmonic equations are studied with Morrey norms.Next,the differential forms was defined in the filed of p-adic numbers.We obtain the decomposition theorem for differential forms in the filed of p-adic numbers,and the imbedding inequalities for homotopy operator are derived.By using PoincaréLemma,some properties for closed form are obtianed.Then,we show the definitions of p-aidc weak central Morrey spaces and p-adicλ-central BMO spaces CMO^q,λ（∧^l,Q_pⁿ）on differ-ential forms and give the weak boundedness of fractional p-adic Hardy operator H_αand its adjoint operator H_α^*on differential forms.Finally,we focus on s-convex function and L^p-averaging domain and show the def-inition of L（φ_s,μ）-averaging domain,which is a extension of L^p-averaging domain.We give the equivalence theorem of L（φ_s,μ）-averaging domain.By the Whitney cover the-ory,we discuss the relationship between local and global L（φ_s,μ）-averaging domain.The invariant of L（φ_s,μ）-averaging domains under the quasi-conformal mappings and quasi-isometric mappings are given.We define the generalized Orlicz-Morrey spaces on differential forms and establish the Poincaré-type inequality for the solution which relate to A-harmonic equations with Orlicz-Morrey norms.