Discrete weighted Hardy Inequality in 1-D

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form $n^α$. 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 $α\in [0,1) \cup [5,\infty)$ with the sharp constant $c(α)$. Furthermore when $α\in [1/3,1) \cup \{0\}$ 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 $b_k(α)$.

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