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The purpose of this paper is to establish some one-sided estimates foroscillatory singular integrals. The boundedness of certain oscillatory singularintegral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here alsoshow that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals abovecannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\inA_{p}^{+}$, a wider weight class than the classical Muckenhoupt class.Furthermore, a criterion on the weighted $L^{p}$-boundednesss of theoscillatory singular integral is given.

Some one-sided estimates for oscillatory singular integrals