The Boundedness Of Singular Integral Operators On Function Spaces With Variable Exponents

With the development of elasticity and fluid dynamics, since 1991, Kovacik and J.Rakosnik first proposed the theory of variable Lebesgue space, more and more scholars are interested in the application of function spaces with variable exponents. The study on boundedness of operators has been one of key research subjects in the harmonic analysis. But until 2004 the Hardy-Littlewood maximal operators is bounded on Lp(·) by Diening[6] when Ω∈Rn is bounded. At present, it attracted much attention on the boundedness of other operators and many results have been obtainedIn this paper, we mainly research some function spaces with variable exponents and the boundedness of a class of some operators. For the function spaces, we introduce definitions and some basic propertise of variable Lebesgue spaces, variable Herz spaces and variable Herz-type Hardy spaces, and discuss the boundedness of a class of singular integral operators in these variable exponents function spaces.Chapter 1 is preface, which presents that the background, current situation and some basic notations, what’s more, the basic knowledge and main contents of related function spaces are also given.Chapter 2 introduces the definition of fractional integral operators with variable kernel and the known conclusions. In this chapter, we will discuss the boundedness of TΩ,μ with variable kernel on the variable Lebesgue space.In Chapter 3, firstly we transform the norm of fractional integral operator with variable kernel on Kq2()α,p space to Lq2(·) spaces by using the definition of variable Herz space. Then, combining with the atomic characterizations of the variable Herz-type Hardy spaces, we prove this operator is bounded from HKq1(·)α,p, to Kq2(·)α,pIn Chapter 4, we mainly give the definition, some properties and conclusions of the multilinear fractional integral operators with variable kernel. By transforming multilinear fractional integral operators to T|Ω|,μf(x), we show that the operators TΩ,μ,A1,A2 and corre-sponding fractional maximal operator MΩ,μ,A1,A2 with two remainder terms is bounded on the variable Lebesgue spaces.