Discrete weighted Hardy Inequality in 1-D

TL;DR

This paper establishes sharp weighted discrete Hardy inequalities on the half-line for specific power weights, providing improved bounds and constants in certain parameter ranges.

Contribution

It proves the validity of weighted discrete Hardy inequalities with sharp constants for certain weight exponents and introduces improved inequalities with additional correction terms.

Findings

Proved inequalities for lpha  [0,1)  [5,) with sharp constants.

Established improved inequalities with correction terms for lpha  [1/3,1)  .

Derived explicit constants c(lpha) and coefficients b_k(lpha).

Abstract

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form $n^\alpha$. Namely we consider: \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq c(\alpha) \sum_{n=1}^\infty \frac{|u(n)|^2}{n^2}n^\alpha \end{equation} We prove the above inequality when $\alpha \in [0,1) \cup [5,\infty)$ with the sharp constant $c(\alpha)$. Furthermore when $\alpha \in [1/3,1) \cup \{0\}$ we prove an improved version of the above inequality. More precisely we prove \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq c(\alpha) \sum_{n=1}^\infty \frac{|u(n)|^2}{n^2} n^\alpha + \sum_{k=3}^\infty b_k(\alpha) \sum_{n=2}^\infty \frac{|u(n)|^2}{n^k}n^\alpha. \end{equation} for non-negative constants $b_k(\alpha)$.