On discrete weighted Hardy type inequalities and properties of weighted discrete Muckenhoupt classes

Abstract

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property (forward propagation property) of the discrete Gehring class.

Self-Improving Properties of Continuous and Discrete Muckenhoupt Weights: A Unified Approach

“…In [28], Bennett and Gross-Erdmann improved the result of Heing and Kufner by excluding the conditions on v. In [29], the authors proved that the discrete Hardy operator is bounded in p (v) d for p > 1 if and only if v ∈ A p . The discrete weight v is said to be belong to the discrete Muckenhoupt A 1 −class if there exists a constant A > 0 such that the inequality Hu(n) ≤ A inf n∈J u(n), or equivalently Mu(n) ≤ Au(n), holds for all n ∈ J.…”

“…The discrete weight v is said to be belong to the discrete Muckenhoupt A 1 −class if there exists a constant A > 0 such that the inequality Hu(n) ≤ A inf n∈J u(n), or equivalently Mu(n) ≤ Au(n), holds for all n ∈ J. In [29], the authors proved the self-improving property of the weighted discrete Muckenhoupt classes. They established also the exact values of the limit exponents as well as new constants of the new classes.…”