الملخص الإنجليزي
This work is focused on studying the LP boundedness of various classes of integral operators with rough homogeneous kernels. Integral operators with homogeneous kernels are important part of harmonic analysis. There applications spread over many branches of mathematics and related fields such as partial differential equations, complex analysis and potential theory. During the last 15 years, a significant development of the theory of integral operators with homogeneous kernels has occurred. The presence of rough kernels and the type of singularties are among the main features of this theory.
In this study, we discussed three types of singular integral operators that have been the core of the investigation of many researchers. In fact, we established LP boundedness results for: Singular integrals associated to polynomial mappings; singular integrals along surfaces of revolution, and singular integrals along surfaces determined by convex functions. We proved the LP boundedness of certain classes of singular integral operators whose kernels satisfying Grafakos-Stefanov conditions. We also discussed the LP boundedness of maximal functions that are related to singular integral operators which carry oscillatory factors in their kernels. The class of maximal operators here carries the common properties of the classes of operators discussed in the previous chapters.
The Marcinkiewicz integral operators also appear in this study, we proved the LP boundedness of Marcinkiewicz integral operators and discussed three types of Marcinkiewicz integral operators that correspond to the same classes of singular integrals discussed before. However, we extended a recent result given by AL-Salman in 2012 to the case when the operators are associated to the surfaces determined by functions satisfying certain growth conditions and kernels satisfy Grafakos and Stefanov conditions. In addition, we proved the analogous parametric Marcinkiewicz integral operator on the product space. The obtained results is an extension and improvement of some earlier results established under Grafakos-Stefanov conditions. Throughout this thesis, many of the ideas come from the work of J. Duoadikoetxea and J. L. Rubio de Francia.
