A note on the spaces of variable integrability and summability of Almeida and Hästö

Abstract

We address an open problem posed recently by Almeida and Hästö in \cite{AlHa10}. They defined the spaces $\ellqp$ of variable integrability and summability and showed that $\|\cdot|\ellqp\|$ is a norm if $q$ is constant almost everywhere or if $\esssup_{x\in\R^n}1/p(x)+1/q(x)\le 1$. Nevertheless, the natural conjecture (expressed also in \cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\ge 1$ almost everywhere. We show, that $\|\cdot|\ellqp\|$ is a norm, if $1\le q(x)\le p(x)$ for almost every $x\in {\mathbb{R}}^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\min (p(x),q(x))\ge 1$ for every $x\in {\mathbb{R}}^n$ such that the triangle inequality does not hold for $\|\cdot|\ellqp\|$.