Endpoint estimates of discrete fractional operators on discrete weighted Lebesgue spaces

Let $0<\alpha <1$ and $\frac{1}{q}=1-\alpha$. We first obtain that the function $\omega :\mathbb{Z}\to (0,\mathrm{\infty })$ 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 $l_{{\omega }^q}^{q,weak}(\mathbb{Z})$. Then for $p=\frac{1}{\alpha }$, we further obtain that the function $\omega$ belongs to weight class of $\mathcal{A}(p,\mathrm{\infty })(\mathbb{Z})$ if and only if discrete Riesz potential $I_{\alpha }$ has a property resembling discrete bounded mean oscillation. Moreover, we give another simple proof of $I_{\alpha }:l_{{\omega }^p}^p(\mathbb{Z})\to l_{{\omega }^q}^q(\mathbb{Z})$ for $\omega \in \mathcal{A}(p,q)(\mathbb{Z})$, $1<p<\frac{1}{\alpha }$ and $\frac{1}{q}=\frac{1}{p}-\alpha$. As applications, more weighted norm inequalities for $M_{\alpha }$ and $I_{\alpha }$ are established when $\omega \in \mathcal{A}(1,q)(\mathbb{Z})$ or $\omega \in \mathcal{A}(p,\mathrm{\infty })(\mathbb{Z})$, and some of them are new even in continuous setting.}