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

Abstract

Let $\mathcal D$ be the normalized ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\subset\mathbb R^d$. 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.