Studies Of Hardy Operators And Related Properties In Variable Exponet Spaces

Function spaces with variable exponents play an effective role in sciences,fluid me-chanics especially electrorheological fluids,image restoration,the problems with non-standard growth conditons in engineering technology and so on.Then they have been inten-sively studied by numerous authors in recent two-decade and developed rapidly with a lot of importent results.Hardy operator,as the classicial operator in harmonic analysis,has been studied all the time.In this dissertation we studied a small part of Hardy operator,mainly about the boundness of Hardy operator and its weighted form in variable exponent spaces.The organization framework of this paper is arranged as follows:Chapter 1 is the introduction includes the background and significance about the spaces with variable exponents.we also provide some basic knowledge,notations and related lem-mas and show the main work of this paper.The boundness of Cesaro operator what is the dual Hardy operator is proofed in Chapter 2.The proof idea is similar to Hardy operator's.We obtain our results by an estimate of x-?+?(x)+1/p(x)which is almost increasing and Holder inequality.Chapter 3 firstly introduces Hardy operator.Then gives a necessary and sufficient con-dition for Hardy's operator(retain x?(x)in the variable Herz space.Using characterization of Herz space,the sudied space can be changed to familiar variable Lebesgue space.And an estimate for x-1/p'(x)+?(x)+? is alomst decreasing,derived from the relation of Herz space contains Lebesgue space,can obtain the result.In Chapter 4,the Hardy operator is extended to weighted Hardy operator and obtain the bondness condition in variable Lebesgue space.Insert two test functions and we can get the estimate about ?(x)and ?(x).Then using Holder inequality and Minkowskin inequality,the boundness of weighted power Hardy operator is proved.