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. They defined the spaces ${\ell _{q(\cdot )}(L_{p(\cdot )})}$ of variable integrability and summability and showed that $\|\cdot |{\ell _{q(\cdot )}(L_{p(\cdot )})}\|$ is a norm if $q\ge 1$ is constant almost everywhere or if $1/p(x)+1/q(x)\le 1$ for almost every $x\in \mathbb {R}^n$. Nevertheless, the natural conjecture (expressed also by Almeida and Hästö) is that the expression is a norm if $p(x),q(x)\ge 1$ almost everywhere. We show that $\|\cdot |{\ell _{q(\cdot )}(L_{p(\cdot )})}\|$ 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 $\|\cdot |{\ell _{q(\cdot )}(L_{p(\cdot )})}\|$.