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 $H_{+}^1(w)$ 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.

24 pages, Nonlinear Anal. 2014