On restrictions of Besov functions

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 $q\le p$ 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.