On restrictions of Besov functions

Julien Brasseur (1, 2)

1 BioSP - Biostatistique et Processus Spatiaux
2 I2M - Institut de Mathématiques de Marseille

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 $f(\cdot,y)\notin B_{p,q}^s(\R^d)$ 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.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.

Domaines: Analyse fonctionnelle [math.FA]

Julien Brasseur. On restrictions of Besov functions. 2017. ⟨hal-01538362v1⟩