Uniform estimates for oscillatory integral operators

Abstract

In this paper we prove the uniform L p boundedness of oscillatory singular integral operators with smooth phase functions which satisfy a certain finite-type condition. These phase functions include the class of real-analytic functions. The uniform boundedness does not hold if the finite-type condition is removed. We also obtained estimates for related operators, including an optimal L 2 estimate for operators (with singular kernels) whose phase functions have non-singular Hessians.