Fractional and non-fractional Hardy inequality on a lattice $\Z^d$

TL;DR

This paper provides simple proofs for discrete fractional and non-fractional Hardy inequalities on lattice $ ext{Z}^d$, with explicit constants and a comprehensive characterization of when these inequalities hold for power weights.

Contribution

It introduces straightforward proofs for Hardy inequalities on lattices, with explicit constants and a complete characterization for power weights across all dimensions.

Findings

Explicit constants for Hardy inequalities are provided.

Complete characterization of when inequalities hold for power weights.

Applicable to any lattice dimension $d$ and exponent $p$.

Abstract

We present simple proofs of a discrete fractional and non-fractional Hardy inequality, Our constants are explicit, but not optimal. In the class of power weights, we get a complete picture of when the non-fractional Hardy inequality holds, for any dimension $d$ of the lattice $\Z^d$ and exponent $0<p<\infty$.