Sharp Weighted Discrete $p$-Hardy Inequality and Stability
Ali Barki
TL;DR
This work extends sharp discrete $p$-Hardy inequalities to the half-line with power weights $n^{\alpha}$ for all $p>1$, and links the discrete constants to their continuous counterparts via a two-measure, Muckenhoupt-type framework. It develops a discrete version of Muckenhoupt's criterion, adapts the two-measure approach of Miclo and Muckenhoupt to general $p$, and establishes when the discrete constants are finite and sharp, including a discrete Leray-type inequality at the critical exponent $\alpha=p-1$. The authors further formulate Hardy weights for $p$-Schrödinger operators on graphs, prove optimality and criticality results, and specialize to the discrete half-line to obtain precise asymptotics for the optimal weights $w_{\nu}$, with $w_{\nu}(n) \sim n^{\alpha-p}$ for $\alpha\neq p-1$ and $w_{\nu}(n) \sim 1/(n \log^{p} n)$ for $\alpha=p-1$, recovering known continuous and discrete cases (e.g., $\alpha=0$ corresponds to KPP18). Finally, a discrete sharp inequality is proved for all $\alpha\neq p-1$, with sharp constants matching the continuous ones for $\alpha\ge0$, together with subcriticality, stability (via a $\Psi$-stability with $\Psi(t)=t^{p}$), and explicit bounds for negative $\alpha$ and the critical regime.
Abstract
In this paper, we prove a $p$-Hardy inequality on the discrete half-line with weights $n^α$ for all real $p > 1$. Building on the work of Miclo for $p = 2$ and Muckenhoupt in the continuous settings, we develop a quantitative approach for the existence of a $p$-Hardy inequality involving two measures $μ$ and $ν$ on the discrete half-line. We also investigate the comparison between sharp constants in the discrete and continuous settings and explore the stability of the inequality in the discrete case.