Almost-every-direction trace conjecture for prevalent Besov functions

Let $D=d+d'$ and let $B^{s}_{p,q}(\mathbb{R}^D)$ be the Besov space under the hypotheses on $p,q,s$ of Theorem~. For a $d$-dimensional subspace $\mathcal{H}$ of $\mathbb{R}^D$, write $\mathcal{H}+a$ for its affine translate by $a\in\mathcal{H}^{\perp}$, and let $\operatorname{Gr}_d(D)$ be the Grassmannian of such subspaces, equipped with Haar measure $\mu_{d,D}$. Almost-every-direction trace conjecture. For almost all $f$ in $B^{s}_{p,q}(\mathbb{R}^D)$, for $\mu_{d,D}$-almost all $\mathcal{H}\in\operatorname{Gr}_d(D)$, and for Lebesgue-almost all $a\in\mathcal{H}^{\perp}$, the trace of $f$ on $\mathcal{H}+a$ has the properties stated in Theorem~. The statement is presented as a natural generalization of the preceding result for a countable collection of directions; the authors note that prevalence lacks a Fubini theorem, so they leave this generalization for subsequent studies.