Some one-sided estimates for oscillatory singular integrals

Abstract: The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H<sup>{1}_{+}(w)$ is proved. It is here also show that the $H<sup>{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H<sup>{q}_{+}(w)$ when $0<q<1$ and $w\in A_{p}<sup>{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L<sup>{p}$-boundednesss of the oscillatory singular integral is given.