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

S. S. Rabie,S. H. Saker,R. P. Agarwal

doi:10.1186/s13660-021-02702-y

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.

Highlights

The study of regularity and boundedness of the discrete operators on p and higher summability of sequences was considered in the literature; see for example [2, 19,20,21, 28, 29] and the references cited therein
Our aim in this paper is to first prove some new refinements of discrete weighted inequalities with negative powers on finite intervals, we use these inequalities to prove that the self-improving property of the weighted discrete Muckenhoupt classes holds
We deduce the self-improving property of the discrete Gehring class, i.e., we prove that if v ∈ Gp(K), there exist constants, K1 > 0 such that v ∈ Gp+ (K1)

Summary

Introduction

The study of regularity and boundedness of the discrete operators on p and higher summability of sequences was considered in the literature; see for example [2, 19,20,21, 28, 29] and the references cited therein. For a given exponent q > 1 and a constant K > 1, a discrete nonnegative weight v belongs to the discrete Gehring class Gq(K) (or satisfies a reverse Hölder inequality) on the interval I ⊂ Z+ if for every subinterval J ⊆ I we have 1 |J |. A nonnegative discrete weight v defined on a a fixed interval I is called an Apλ(C)-Muckenhoupt weight for p > 1 if there exists a constant C < ∞ such that. Our aim in this paper is to first prove some new refinements of discrete weighted inequalities with negative powers on finite intervals, we use these inequalities to prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds.

Main inequalities

Main results

11 C p–1 p–1

Conclusion