Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

In this paper we introduce the one-sided weighted spaces ${\mathcal L}^{-}_w (\beta)$, $-1 < \beta < 1$. The purpose of this definition is to obtain an extension of the Weyl fractional integral operator $I_{\alpha}^+$ from $L^p_w$ into a suitable weighted space.

Under certain condition on the weight $w$, we have that ${\mathcal L}^{-}_w (0)$ coincides with the dual of the Hardy space $H_{-}^1(w)$. We prove for $0 <\beta < 1$, that ${\mathcal L}^{-}_w (\beta)$ consists of all functions satisfying a weighted Lipschitz condition. In order to give another characterization of ${\mathcal L}^{-}_w (\beta)$, $0 \leq \beta < 1$, we also prove a one-sided version of John-Nirenberg Inequality.

Finally, we obtain necessary and sufficient conditions on the weight $w$ for the boundedness of an extension of $I_{\alpha}^+$ from $L^p_w$ into ${\mathcal L}^{-}_w (\beta)$, $-1 < \beta < 1$, and its extension to a bounded operator from ${\mathcal L}^{-}_w (0)$ into ${\mathcal L}^{-}_w (\alpha)$.