One-sided Estimates For Integral Operators With Kernels Satisfying A Variant Of LIPSCHITZ Condition

During the recent half century, the theories of modern harmonic analysis have made great process and its ideas, methods and techniques are widely used in mathe-matical fields. Due to the foundings of the operator theory, the Calderon-Zygmund theory of singular integral operators plays a basic role in harmonic analysis and lies at the heart of much of the work being done in this area nowadays. In this paper, we mainly study one-sided estimates for integral operators with kernels satisfying a variant of Lipschitz condition, Lipschitz estimates for one-sided Cohen’s commu-tators on weighted one-sided Triebel-Lizorkin spaces and weighted boundedness of some operators satisfying size condition on one-sided weighted Morrey space. The main content of this paper is arranged as follows:In the first chapter, it has six parts:we firstly introduce research background and the present research situation of singular integral operators about the kernels. Then, we introduce the classical C-Z singular integral theory and the definition of Ap weights. We propose a variant of Hormander condition. Secondly, we introduce the one-sided Ap classes and the one-sided Calderon-Zygmund(C-Z) singular integrals. The definitions and properties of commutator are given. Thirdly, we study the definitions of some one-sided spaces and necessary lemmas which will be used in this paper. Finally, we state the main work of this paper.In the second chapter, we introduce one-sided C-Z singular integral operators with kernels satisfying the variant Lipschitz condition. Then, we will use the sharp maximal operator as a bridge to pass from weighted estimates of the operator to the weighted estimates of the functions. We can also prove that T+is of weak type-(1,1) with ω E A+1 using the modification of the Calderon-Zygmund decomposition.The third chapter is divided into two parts, we introduce the one-sided Cohen’s commutators of singular integral operators and fractional integral operators, respec-tively. Using the extrapolation of one-sided weights, we establish the boundedness of these operators from weighted Lebesgue spaces to weighted one-sided Triebel-Lizorkin spaces.In the fourth chapter, we introduce a version of one-sided sublinear operators satisfying size condition. The weighted weak type-(1,1) estimates for these operators are considered on one-sided weighted Morrey space.