Discrete weighted Hardy inequality in difference form

Abstract

In this paper we find necessary and sufficient conditions for the fulfillment of the discrete Hardy inequality with weights written in the difference form. The problem is investigated on the set of finitely supported sequences. The key result of this article is finding of estimates for the exact constant of this inequality. These estimates will be later used to establish qualitative characteristics, such as oscillation and non - oscillation conditions, of certain difference equations. Moreover, as a consequence of the main results, we find criteria for the embedding of some spaces and for the compactness of this embedding.