Multilinear Fourier Multipliers On Variable Lebesgue Spaces

Variable exponent Lebesgue spaces were lirst studied by Orlicz. about 60 years ago. They can be regarded as the generalization of classical Lebesgue spaces. People interest on this type of spaces has been growing since 1991 with the publication of Kovacik and Rakosnik’s foundational paper. From then on. mathematicians started to study these spaces systematically and specially. And since 90s, the theory of variable exponent spaces has been playing important roles in more and more fields, such as the research of electrorheological fluids, image processing and so on.The main work of this paper is to study some properties of the bilinear multi-plier space and the boundedness of multilinear Fourier multipliers on variable expo-nent Lebesgue spaces. This paper is composed of three chapters. The lirst one is the introduction. Its first section talks about a brief history background of variable expo-nent Lebesgue spaces; the second section gives some basic definitions and preliminary results in this field.In the second chapter, we lirst study certain properties of bilinear multipliers on variable exponent Lebesgue spaces such as the localization property, etc. Then a neces-sary condition for a continuous bounded function to be a bilinear multiplier on variable exponent Lebesgue spaces is obtained.In the third chapter, we propose a Mihlin-Hormander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give the corresponding proof. Moreover, applications of this theorem are also discussed.