标题： 加权离散$p$- Hardy 不等式及其稳定性 显示英文标题

标题： Sharp Weighted Discrete $p$-Hardy Inequality and Stability

摘要： 本文中，我们证明了在离散半直线上带权的$p$-Hardy 不等式，其中权函数为$n^{\alpha}$，且适用于所有实数$p > 1$。 基于 Miclo 对$p = 2$的工作和 Muckenhoupt 在连续情形下的研究，我们发展了一种定量方法来探讨涉及两个测度$p$和$\mu$的离散半直线上$\nu$-Hardy 不等式的存在性。 此外，我们还研究了离散与连续情形下尖常数之间的比较，并探索了离散情形下不等式的稳定性。 显示英文摘要

摘要： In this paper, we prove a $p$-Hardy inequality on the discrete half-line with weights $n^{\alpha}$ 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 $\mu$ and $\nu$ 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.