Local behavior of traces of Besov functions: Prevalent results

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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 ᵈ, of functions belonging to the Besov space Bˢₚ,q(ᵈ). Given a d-dimensional subspace H ⊂ ᵈ, for almost all functions in Bˢₚ,q(ᵈ) (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 ∈ ᵈ⁻ᵈ. In particular, we prove that for Lebesgue-almost every a ∈ ᵈ⁻ᵈ, these traces fₐ are more regular than what could be expected from standard trace theorems, and that fₐ enjoys a multifractal behavior.