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 $B^{\frac{\theta}{p}}_{p,1}(\mathbb{R}^n,\gamma)$ to $E$. In particular, our results cover the case of power-type weights $\gamma(x)=|x|^\alpha$ with $-n<\alpha<n(p-1)$, $\alpha\neq -(n-1)$, when $E=\mathbb{R}^{n-1}$. This extends earlier results obtained by Haroske and Schmeisser.