Endpoint estimates of discrete fractional operators on discrete weighted Lebesgue spaces
Let $0<α<1$ and $\frac{1}{q}=1-α$. We first obtain that the function $ω:\mathbb{Z} \rightarrow (0,\infty)$ belongs to weight class of $\mathcal{A} (1,q)(\mathbb{Z})$ if and only if discrete fractional maximal operator $M_α$ or discrete Riesz potential $I_α$ is bounded from $l_ω^{1}(\mathbb{Z})$ to $l_{ω^q}^{q,weak}(\mathbb{Z})$. Then for $p=\frac{1}α$, we further obtain that the function $ω$ belongs to weight class of $\mathcal{A} (p,\infty)(\mathbb{Z})$ if and only if discrete Riesz potential $I_α$ has a property resembling discrete bounded mean oscillation. Moreover, we give another simple proof of $I_α:l_{ω^p}^{p}(\mathbb{Z}) \rightarrow l_{ω^q}^{q}(\mathbb{Z})$ for $ω\in \mathcal{A}(p,q)(\mathbb{Z})$, $1<p<\frac{1}α$ and $\frac{1}{q}=\frac{1}{p}-α$. As applications, more weighted norm inequalities for $M_α$ and $I_α$ are established when $ω\in \mathcal{A}(1,q)(\mathbb{Z})$ or $ω\in \mathcal{A}(p,\infty)(\mathbb{Z})$, and some of them are new even in continuous setting.}