| Singular integral operators along subvarieties, with rough kernels have attracted a considerable amount of attention over the last few years. In this thesis, we establish Lp estimates of a variety of these operators.; In the first part of this thesis, we deal with singular integrals along subvarieties with kernels in the space L(log+ L)(). A general method concerning these operators is presented. As applications of this method, we improve many earlier results that have been established when the kernel is in Lq(), q > 1. We prove Lp boundedness of those singular integral operators with kernels in the space L(log+ L) () that have singularities spread over: (a) Submanifolds of finite type; (b) Surfaces of revolutions determined by flat curves; and (c) Subvarieties determined by a special class of polynomial mappings. Our approach is quite general and it is conceivable that it can be used to handle many other problems.; In the second part of this thesis, we study singular integral operators along subvarieties with kernels satisfying Grafakos-Stefanov's condition. We establish an Lp boundedness result for singular integral operators along subvarieties determined by polynomial mappings. Moreover, we establish Lp boundedness results for many other operators. |
