L(p) estimates for singular integrals with kernels belonging to certain block spaces.

摘要

In this thesis we study the mapping properties of homogenous singular integral operators with rough kernels. We deal with two kinds of singular integrals: singular integrals in the one-parameter setting and singular integrals in the multi-parameter setting. Our work in the one-parameter theory sharpens and generalizes significant results obtained by many authors. On the other hand, our work in the multi-parameter theory generalizes many earlier results as well as resolves a long-standing open problem on the boundedness of the maximal truncated singular integrals. As a matter of fact, our results in the product theory of singular integrals go a substantial distance towards understanding this theory under very weak conditions on the kernel.; To prove our results, we develop a general theory which allows us to derive norm inequalities for very important classes of maximal functions and rough singular integral operators. This general theory is based on the method of J. Duoandikoetxea and J. L. Rubio de Francia as well as the techniques developed by Fan and Pan in their study of the mapping properties of singular integral operators.