Boundedness Of A Class Of Operators On Lebesgue Space With Variable Exponent In Differential Form

The research background of this paper is to explore the general meaning of variable exponent Lebesgue space and Musialek-Orlicz space.Their properties in the differential form are not yet clear.Whether we try to get the important properties of the space or not,the space operator is especially significant.The main research content of this paper is whether the operator of variable exponent Lebesgue space is well bounded in the differential form.The research methods of this paper include introducing the definition of variable exponent Lebesgue space,analyzing the basic properties of maximal operators,singular integral operators,and commutators;introducing basic theorems about the study of function pairs,through the study of weighted norm inequalities,reveals the magnitude relationship of the norm in the case of variable exponents and the vector value inequality in the case of function sequences.In addition,the denseness of functions is also one of the important methods for studying problems.Through the boundedness theorem,transforming the inequality of the function into the inequalit y of the sum of functions is an important research method in this article.This paper is using the weighted inequality of the operator in the function space and the extrapolation theorem to transform it into the vector value inequality of the sum,which reveals the new and good properties of the correspond ing operator in the differential form.By studying this aspect of the problem and giving a suitable answer,it can provide help for studying the properties of the operators in differential form,and can also better understand the role of operators in space.At the same time,the study of the boundedness of a class of operators can provide convenience for the study of continuity problems related to operators,and extend the good properties of the function space to the differential form,so as to better develop the differential form Research with Musialek-Orlicz space.It enriches the research theory of operators in function spaces and differential forms,expands the understanding of operators in differential forms,and lays a foundation for further research o n the properties of operators in differential forms,as well as the applications of o perators in differential forms provided a great help.