Local behavior of traces of Besov functions: Prevalent results

TL;DR

This paper investigates the local and global regularity of traces of Besov functions on affine subspaces, revealing that for most functions, these traces exhibit multifractal behavior and are more regular than classical theorems suggest.

Contribution

It provides a prevalence-based analysis of the singularity spectrum of traces of Besov functions, showing enhanced regularity and multifractality for almost all such functions.

Findings

Most traces are more regular than standard trace theorems predict.

Traces exhibit multifractal behavior for almost all functions in the Besov space.

The singularity spectrum of traces is explicitly computed for prevalent functions.

Abstract

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