How smooth are restrictions of Besov functions?
Julien Brasseur
TL;DR
This work resolves a sharpness question for the restriction of Besov functions: while Besov spaces B_{p,q}^s generally fail to restrict to a fixed dimension when p<q, their partial maps can be controlled by Besov spaces of generalised smoothness B_{p,p}^{(s,Ψ)} with a slowly varying modulation Ψ, provided a simple summability condition holds. The authors prove that this condition is also necessary, by constructing f ∈ B_{p,q}^s(R^N) whose partial maps f(·,y) fail to lie in B_{p,∞}^{(s,Ψ)}(R^d) for almost every y whenever Ψ violates the summability criterion. The main technical tool is a carefully crafted sequence (λ_{j,k}) whose global Besov norm remains finite, but whose dyadic-locally weighted sums diverge along every dyadic line when weighted by Ψ, enabled by a rearrangement argument. The result completes the picture by providing a complete, sharp dichotomy for the restriction behavior of Besov functions in terms of generalized smoothness and slowly varying modulations, clarifying the role of the parameter q relative to p and the impact of Ψ on restriction properties.
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 $B_{p,q}^{(s,Ψ)}(\mathbb{R}^d)$ provided the function $Ψ$ satisfies a simple summability condition involving $p$ and $q$. This short note completes the picture by showing that this characterisation is sharp.