  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 $H^{1}_{+}(w)$ is proved. It is here also
show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above
cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\in
A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class.
Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the
oscillatory singular integral is given.
