A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \mathcal{X}\}_{n=1}^\infty$ such that $$\sup_{f \in K}\|f - U_n f\|_\mathcal{X} \to 0 \quad \text{as} \quad n \to \infty.$$ Let $\Omega \subset \mathbb{R}^d$ be a bounded, open set. I am wondering how much is known about standard function spaces possessing AP. I am particularly interest in the Lebesgue spaces $L^p(\Omega)$ for $1 \leq p \leq \infty$, the Sobolev spaces $W^{k,p}(\Omega)$ with $k \in \mathbb{N}$ and $1 \leq p \leq \infty$, the uniformly continous functions $C^k(\bar{\Omega})$ for $k \in \mathbb{N}_0$, and maybe even the Besov and Holder spaces, but the previous three are the main ones. The best I could find is this which says that $W^{k,p}(\Omega)$ with $1 < p < \infty$ and $\partial \Omega$ is class $C^\infty$ has AP. Or this which explicitly constructs a Schauder basis for $C^k ([0,1]^d)$ and $W^{k,p}([0,1]^d)$ with $k \in \mathbb{N}_0$ and $1 \leq p < \infty$. What results are known for more general domains?

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