On restrictions of Besov functions

In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any f is an element of B-p,q(s) (R-N) with q <= p we have f(center dot, y) is an element of B-p,q(s) (R-d) for a e. y is an element of RN-d. We prove that this is no longer true when p < q. Namely, we construct a function f is an element of B-p,q(s) (R-N) such that f(center dot, y) (sic) B-p,q(s) (R-d) for a.e. y is an element of RN-d. We show that, in fact, f(center dot, y) belong to B-p,q((s, psi)) (R-d) for a.e. y is an element of RN-d, a Besov space of generalized smoothness, and, when q = infinity, we find the optimal condition on the function (psi) over bar for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated. (c) 2018 Elsevier Ltd. All rights reserved.