Hardy inequalities on variable exponent spaces and some applications

In this thesis, information about Lebesgue space, Hardy inequalities and variable exponents spaces will be given, and Hardy's inequality which obtain in the space $L^{p(x)}$ is investigated with it's proof. This thesis consist of six chapters The first chapter is devoted to introduction. In the second chapter, we give information about some space knowledge, definition of Lebesgue space and it's basic properties, which is very important for the other sections. The information about $L_p$ space, it's norm and it's basic properties is given and also the definitions of Orlicz space and Modular space is given. The information about the structures' of these spaces is given. In the third chapter, we talk about the beginning of Hardy inequality and also developing from beginning to today. And we talk about Hardy's inequalities' the other forms. In the fourth chapter, we give information about $L^{p(x)}$ space, it's norm and it's basic properties like $L_{p}$ In the fifth chapter, Hardy's inequality which obtain in the space $L^{p(x)}$ will be given with it's proof. Finally we will present our opinions and suggestions. In the six chapter, some applications is introduced.