A counterexample concerning weighted convolution on the integers

Abstract

Let $1<p<\infty$ and let $w:\mathbb{Z}\to(0,\infty)$ be a weight. When is the weighted space $\ell^p(\mathbb{Z},w)$ an algebra with respect to convolution? A well-known sufficient condition is that $(w^{-q}*w^{-q})/w^{-q}$ be bounded on $\mathbb{Z}$, where $q:=p/(p-1)$ is the index conjugate to $p$. We present an example to show that this sufficient condition is not necessary.