Boundedness Of Singular Integral Operators

The singular integrals with rough kernels are important operators. A lot of results on boundedness of the singular integrals with rough kernels are obtained by mathematicians. Some extensions of them are widespread researched in last twenty years. Specially, many mathematicians concentrated on the problem of the endpoint estimate of the singular integrals with rough kernels. At the end of the twentieth Century, a series of important results of these problems are obtained. But until now, there are some open problems on the endpoint estimates. In this thesis, we get some results in two dimensions.This PH.D thesis will concentrate on the endpoint estimate of singular integral TΩ with rough kernel p.v(Ω(x/|x|)/(|x|n) when Ω ∈H1(S1). When Ω is odd functions on S1, we get some results. Also, we show that TΩ is bounded form H1(R2) to L1(R2) when Ω has the property of "monotonous"On the other hand, this PH. D thesis focuses on rearrangement and its appli-cation on singular integral. Distribution is an important term which can be found in many branches of mathematics such as Real analysis, Probability and Statistics. A rearrangement of the function f has the same distribution with f. This property can used to characterize Lebesgue space and Lorentz space. Also using this proper-ty, we show that singular integral operators are bounded when restricted to a class of functions from Hp(Rn) (0< p< 1).In Chapter 1 of this paper, we will introduce the background and the main results obtained in this thesis. In Chapter 2, we will review the definitions of distribution, rearrangement, Lorentz spaces and so on. In Chapter 3, we focus on the boundedness of singular integral operators with rough kernels. The proof of relative theorems are given. In Chapter 4, we focus on the properties of rearrangement and its application and we give the proofs of main theorems.