Discrete Weighted Hardy Inequalities with Different Kinds of Boundary Conditions

TL;DR

This paper investigates weighted Hardy inequalities on discrete intervals under four boundary conditions, providing a unified estimate for the optimal constants and introducing techniques like dual and splitting methods.

Contribution

It introduces a unified approach to estimate optimal constants for discrete weighted Hardy inequalities with various boundary conditions, including new techniques for complex cases.

Findings

Unified expression for optimal constants across boundary conditions

Dual method translation between boundary cases

Splitting technique for complex boundary conditions

Abstract

This paper studies the weighted Hardy inequalities on the discrete intervals with four different kinds of boundary conditions. The main result is the uniform expression of the basic estimate of the optimal constant with the corresponding boundary condition. Firstly, one-side boundary condition is considered, which means that the sequences vanish at the right endpoint (ND-case). Based on the dual method, it can be translated into the case vanishing at left endpoint (DN-case). Secondly, the condition is the case that the sequences vanish at two endpoints (DD-case). The third type of condition is the generality of the mean zero condition (NN-case), which is motivated from probability theory. To deal with the second and the third kinds of inequalities, the splitting technique is presented. Finally, as typical applications, some examples are included.