Local behavior of traces of Besov functions: Prevalent results

Abstract

Let $1\le d<D$ and $(p,q,s)$ satisfying $0<p<\infty$, $0<q\le \infty$, $0<s-d/p<\infty$. In this article we study the global and local regularity properties of traces, on affine subsets of ${\mathbb{R}}^D$, of functions belonging to the Besov space $B_{p,q}^s({\mathbb{R}}^D)$. Given a $d$-dimensional subspace $H\subset {\mathbb{R}}^D$, for almost all functions in $B_{p,q}^s({\mathbb{R}}^D)$ (in the sense of prevalence), we are able to compute the singularity spectrum of the traces $f_a$ of $f$ on affine subspaces of the form $a+H$, for Lebesgue-almost every $a\in {\mathbb{R}}^{D-d}$. In particular, we prove that for Lebesgue-almost every $a\in {\mathbb{R}}^{D-d}$, these traces $f_a$ are more regular than what could be expected from standard trace theorems, and that $f_a$ enjoys a multifractal behavior.

arXiv comment25 Pages, 3 figures