L p estimates of rough maximal functions along surfaces with applications

In this paper, we study the L p mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the L p boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(S n.1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.