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Best approximation of L1 function by Lipschitz function

Fix constant $L,C>0$ and $k\geq 1$ and let $f\in W^{1,k}(\mathbb{R}^d,\mathbb{R}^n)$ with $\|f\|_{W^{1,k}}\leq C$. Is there a known estimate on the distance $$ \|f - \operatorname{Lip}_L(\mathbb{R}^...