On the singular integral operators with variable kernel

This thesis consists of four chapters. The first chapter devoted to the fundamental concepts. In the second chapter, the definitions of of weights and the weighted Hardy spaces togerher with their properties is given. In Chapter 3, the class of kernels with smoothness in both variables are discussed and Kurtz-Wheeden's method to get the boundedness of singular integral operators with kernel in this category are investigated. In additional, the - boundedness of the operator are studied. In the final chapter, the weighted boundedness of the singular integral operator with variable kernels are investigated.Keywords: Weights, - Boundedness, Singular Integral, Singular Integral Operator With Variable Kernel.