Local behavior of traces of Besov functions: Prevalent results

Abstract

Let 1⩽d<D and (p,q,s) satisfying 0<p<∞, 0<q⩽∞, 0<s−d/p<∞. In this article we study the global and local regularity properties of traces, on affine subsets of RD, of functions belonging to the Besov space Bp,qs(RD). Given a d-dimensional subspace H⊂RD, for almost all functions in Bp,qs(RD) (in the sense of prevalence), we are able to compute the singularity spectrum of the traces fa of f on affine subspaces of the form a+H, for Lebesgue-almost every a∈RD−d. In particular, we prove that for Lebesgue-almost every a∈RD−d, these traces fa are more regular than what could be expected from standard trace theorems, and that fa enjoys a multifractal behavior.