Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces

Abstract

Let $\mathcal{D}$ be the normalized ridge dictionary generated by ${\mathrm{ReLU}}^k$ on a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^d$. We establish sharp embeddings between isotropic Besov spaces and the associated variation space ${\mathcal{L}}_1(\mathcal{D})$ in the quasi-Banach range $0<p\le 1$. Specifically, \[B^s_{p,q}(Ω)\hookrightarrow \mathcal L_1(\mathcal D)\] when $s\ge k+d/p$ for $0<q\le 1$, and when $s>k+d/p$ for $1<q\le \infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathcal D)\hookrightarrow B^{k+1}_{p,2}(Ω), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts known Besov regularity, in particular for solutions of partial differential equations, into controlled approximation error bounds and convergence of greedy algorithms based on shallow ${\text{ReLU}}^k$ neural networks. The proofs combine Littlewood--Paley localization, Fourier--Radon representations, measure-valued derivatives, and vector-valued singular-integral estimates.