On restrictions of Besov functions

TL;DR

This paper investigates the restrictions of Besov functions, revealing that for certain parameters, restrictions do not preserve Besov space membership, but belong to a generalized smoothness space instead, with optimal conditions identified.

Contribution

It demonstrates the failure of classical restriction properties for Besov spaces when p<q and introduces the concept of restrictions belonging to Besov spaces of generalized smoothness.

Findings

Restrictions do not always preserve Besov space membership when p<q.

Restrictions belong to Besov spaces of generalized smoothness for almost every y.

Optimal conditions on the function Ψ are identified for the case q=∞.

Abstract

In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any $f\in B\_{p,q}^s(\mathbb{R}^N)$ with $q\leq p$ we have $f(\cdot,y)\in B\_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$. We prove that this is no longer true when $p\<q$. Namely, we 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. $y\in \mathbb{R}^{N-d}$. We show that, in fact, $f(\cdot,y)$ belong to $B\_{p,q}^{(s,\Psi)}(\mathbb{R}^d)$ for a.e. $y\in\mathbb{R}^{N-d}$, a Besov space of generalized smoothness, and, when $q=\infty$, we find the optimal condition on the function $\Psi$ for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated.