Multilinear Operators In Variable Space

With the development of some actual problems on elastic mechanics, current rheol-ogy and to image restoration, the variable exponent nonlinear problems with nonstandard growth conditions become an emerging research topic. Therefore, characterizations of variable exponent function spaces and the boundedness of related operators and commu-tators naturally become the subjects of more intensive study. In this dissertation, multi-linear integral operators and their commutators in variable exponent function spaces are mainly studied. It is arranged as follows:The background and significance of variable exponent spaces are given in Chapter 1. At the same time, some basic knowledge and notations in this paper are also introduced.In Chapter 2, the definition of multilinear singular integral operators is firstly given. Then using the replacement of variable index to regular index, combining with the defini-tion of equivalent norms, the studied spaces are changed to the familiar variable Lebesgue spaces from Herz-Morre spaces. Finally, the boundedness of commutators generated by multilinear singular operators and BMO functions in Herz-Morrey spaces with variable exponent are obtained by Holder inequality.Fractional integral operators with variable exponents are considered in the third chapter. In this chapter, the classical Riesz operators are extended to fractional integral operators with variable exponents. By using the characterizations of variable fractional maximal operators, the boundedness of fractional integral operators are discussed and it is proved that it satisfied the weak-type inequality in variable Lebesgue spaces.