How smooth are restrictions of Besov functions?

Abstract

In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p<q$, then it is always possible to construct a function $f\in B_{p,q}^s(\mathbb{R}^N)$ such that $f(\cdot,y)\notin B_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$, while this "pathology" does not happen if $q\leq p$. We showed that the partial maps belong, in fact, to the Besov space of generalised smoothness $B_{p,q}^{(s,\Psi)}(\mathbb{R}^d)$ provided the function $\Psi$ satisfies a simple summability condition involving $p$ and $q$. This short note completes the picture by showing that this characterisation is sharp.