Discrete weighted Hardy Inequality in 1-D

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^α\geq c(α) \sum_{n=1}^\infty \frac{|u(n)|^2}{n^2}n^α\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^α\geq c(α) \sum_{n=1}^\infty \frac{|u(n)|^2}{n^2} n^α+ \sum_{k=3}^\infty b_k(α) \sum_{n=2}^\infty \frac{|u(n)|^2}{n^k}n^α. \end{equation} for non-negative constants $b_k(\alpha )$.

minor changes in v2: corrected some typos and added some references. The paper has been accepted in the Journal of mathematical analysis and applications

References in corpus (1)

• On weighted remainder form of Hardy-type inequalities

Cited by in corpus (4)

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• Improvement of the discrete Hardy inequality
• One-dimensional discrete Hardy and Rellich inequalities on integers
• On the improvements of Hardy and Copson inequalities