Maximal oscillatory singular integrals with kernels in L log L $(S^{n-1})$

Abstract

In this paper, we study the $L^p$ mapping properties of a certain class of maximal oscillatory singular integral operators. We establish the $L^p$ boundedness of our operators provided that their kernels belong to the natural space L log +L $(S^{n-1})$. Our result substantially improves a previously known result. Moreover, the approach developed in this paper can be applied to handle more general maximal oscillatory singular integral operators.