Traces of weighted Besov spaces to Ahlfors-David regular sets: the limiting case

Abstract

Given $n\in \mathbb{N}$, $p\in [1,\infty )$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_p$ condition, we introduce a weakened version of the Ahlfors--David codimension-$\theta$ regularity condition for Ahlfors--David $d$-regular sets $E\subset {\mathbb{R}}^n$, where $d\in (0,n)$ and $\theta \in (0,p)$. Under this assumption, we provide a complete intrinsic description of the trace-space of the weighted Besov space $B_{p,1}^{\frac{\theta }{p}}({\mathbb{R}}^n,\gamma )$ to $E$. In particular, our results cover the case of power-type weights $\gamma (x)=|x{|}^{\alpha }$ with $-n<\alpha <n(p-1)$, $\alpha \ne -(n-1)$, when $E={\mathbb{R}}^{n-1}$. This extends earlier results obtained by Haroske and Schmeisser.