Discrete weighted Hardy inequality in 1-D

Abstract

In this paper we consider weighted versions of one dimensional discrete Hardy's inequality on the half-line with power weights of the form nα; namely, we consider:(0.1)∑n=1∞|u(n)−u(n−1)|2nα≥c(α)∑n=1∞|u(n)|2n2nα.We prove the above inequality when α∈[0,1)∪[5,∞) with the sharp constant c(α). Furthermore, when α∈[1/3,1)∪{0}, we prove an improved version of (0.1) by adding infinitely many positive lower order terms in the RHS. More precisely, we prove(0.2)∑n=1∞|u(n)−u(n−1)|2nα≥c(α)∑n=1∞|u(n)|2n2nα+∑k=3∞bk(α)∑n=2∞|u(n)|2nknα, for non-negative constants bk(α).