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∈Bp,qs(RN) with q≤p we have f(⋅,y)∈Bp,qs(Rd) for a.e. y∈RN−d. We prove that this is no longer true when p<q. Namely, we construct a function f∈Bp,qs(RN) such that f(⋅,y)∉Bp,qs(Rd) for a.e. y∈RN−d. We show that, in fact, f(⋅,y) belong to Bp,q(s,Ψ)(Rd) for a.e. y∈RN−d, a Besov space of generalized smoothness, and, when q=∞, we find the optimal condition on the function Ψ for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated.