Xiong Hu, Xuebing Hao, Baode Li
Let $0<\alpha<1$, $1<p<\frac{1}{\alpha}$, $\frac{1}{q}=\frac{1}{p}- \alpha$. We first obtain that the function $\omega :\mathbb{Z} \rightarrow (0,\infty)$ belongs to weight class of $\mathcal{A} (1,q)(\mathbb{Z})$ if and only if discrete fractional maximal operator $M_{\alpha}$ or discrete Riesz potential $I_\alpha$ is bounded from $l_{\omega}^{1}(\mathbb{Z})$ to $l_{\omega^q}^{q, {\rm weak}}(\mathbb{Z})$. Then for $q=\infty$, we further obtain that the function $\omega$ belongs to weight class of $\mathcal{A} (p,\infty)(\mathbb{Z})$ if and only if discrete Riesz potential $I_\alpha$ has an upper estimate for a resembling discrete BMO norm. Moreover, we give another simple proof of $I_{\alpha}:l_{\omega ^p}^{p}(\mathbb{Z}) \rightarrow l_{\omega ^q}^{q}(\mathbb{Z})$ for $\omega \in \mathcal{A}(p,q)(\mathbb{Z})$. As applications, more weighted norm inequalities for $M_{\alpha}$ and $I_\alpha$ are established when $\omega \in \mathcal{A}(1,q)(\mathbb{Z})$ or $\omega \in \mathcal{A}(p,\infty)(\mathbb{Z})$, and some of them are new even in continuous setting.
