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
$$
\|f - \operatorname{Lip}\_L(\mathbb{R}^d,\mathbb{R}^n)\|\_{L^1(\mathbb{R}^d,\mathbb{R}^n)},
$$
*depending on the constants $L,C,$ and $k$*, where $\operatorname{Lip}\_L(\mathbb{R}^d,\mathbb{R}^n)$ is the set of Lipschitz functions from $\mathbb{R}^d$ to $\mathbb{R}^n$ with Lipschitz constant *at-most* $L$?



Best approximation of L1 function by Lipschitz function
Let $(\rho\_\epsilon)\_{\epsilon>0}$ be a standard family of mollifiers, with $\rho\_\epsilon$ supported in the ball $B\_\epsilon(0)$. Since $\|\rho\_\epsilon\|\_{L^1}=1$ and $\|\rho\_\epsilon\|\_{L^\infty}=c\_d\epsilon^{-d}$, by interpolation we get $\|\rho\_\epsilon\|\_{L^{k'}}=c\_d^{1/k}\epsilon^{-d/k}$ (for the dual exponent $k'=\frac{k}{k-1}$). Hence,
$$\|\nabla(\rho\_\epsilon\*f)\|\_{L^\infty}=\|\rho\_\epsilon\*\nabla f\|\_{L^\infty}\le c\_d^{1/k}\epsilon^{-d/k}C$$
for your function $f$ (by Holder). In order to get an $L$-Lipschitz function you can take
$$\epsilon:=c\_d^{1/d}(C/L)^{k/d}=c\_d'(C/L)^{k/d}$$
for another constant $c\_d'$ depending only on $d$.
Then you can bound the distance of $\rho\_\epsilon\*f$ from $f$ as follows:
$$\|\rho\_\epsilon\*f-f\|\_{L^{k}}\le\epsilon\|\nabla f\|\_{L^{k}}\le c\_d'(C/L)^{k/d}.$$


Note: the question of measuring the distance in $L^1$ seems ill-posed, since $W^{1,k}$ does not embed into $L^1$ (unless $k=1$).


