Endpoint estimates of discrete fractional operators on discrete weighted Lebesgue spaces

TL;DR

This paper characterizes endpoint weighted inequalities for discrete fractional operators on weighted Lebesgue spaces, establishing new boundedness criteria and simplifying proofs, with applications to weighted norm inequalities.

Contribution

It provides new characterizations of weight classes for boundedness of discrete fractional operators and simplifies existing proofs, extending results to weighted Lebesgue spaces.

Findings

Characterization of weights in (1,q) for boundedness of M_lpha and I_lpha

Equivalence between (p,inite) weights and bounded mean oscillation for I_lpha

New weighted norm inequalities for discrete fractional operators

Abstract

Let $0<\alpha<1$ and $\frac{1}{q}=1-\alpha$. 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 $l_{\omega^q}^{q,weak}(\mathbb{Z})$. Then for $p=\frac{1}{\alpha}$, 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 a property resembling discrete bounded mean oscillation. Moreover, we give another simple proof of $I_{\alpha}:l_{\omega ^p}^{p}(\mathbb{Z}) \rightarrow l_{\omega ^q}^{q}(\mathbb{Z})$ for $\omega \in \mathcal{A}(p,q)(\mathbb{Z})$, $1<p<\frac{1}{\alpha}$ 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 $\omega \in \mathcal{A}(p,\infty)(\mathbb{Z})$, and some of them are new even in continuous setting.}