Poincar\'e type and spectral gap inequalities with fractional Laplacians on Hamming cube

TL;DR

This paper establishes dimension-free Poincaré inequalities for functions with fractional Laplacians on the Hamming cube, emphasizing $L^1$ estimates and illustrating the necessity of spectral assumptions.

Contribution

It introduces new Poincaré-type inequalities for fractional Laplacians on the Hamming cube, highlighting spectral conditions needed for these inequalities.

Findings

Dimension-free Poincaré inequalities for fractional Laplacians

Counterexamples showing spectral assumptions are essential

Focus on $L^1$ norm estimates on the Hamming cube

Abstract

We prove here some dimension free Poincar\'e-type inequalities on Hamming cube for functions with different spectral properties and for fractional Laplacians. In this note the main attention is paid to estimates in $L^1$ norm on Hamming cube. We build the examples showing that our assumptions on spectral properties of functions cannot be dropped in general.