Fix a positive integer $n$ and and an non-negative integer $k$. The Arzela-Ascoli theorem guarantees that for a given positive integer $k$ and a given $L>0$ the set $$ Ball_{C^{k,1}([0,1]^n)}(0,L) := \{f\in C(\mathbb{R}^n):\,\|f\|_{C^{k,1}}\leq L\} , $$ is compact in $C([0,1]^n)$ for the topology of uniform convergence on compacts. Furthermore, by virtue of the inclusion $Ball_{C^{k,1}([0,1]^n)}(0,L)\subseteq Ball_{C^{0,1}(\mathbb{R}^n)}(0,L)$ and the metric entropy estimates of $Ball_{C^{0,1}(\mathbb{R}^n)}(0,L)$ we know that $Ball_{C^{k,1}(\mathbb{R}^n)}(0,L)$ has $\epsilon$-covering number at-most $$ \epsilon \propto \epsilon^{-n}. $$
However, if $k>0$ then we should be able to do much better. Does any one know where to find tight covering number estimates for $Ball_{C^{k,1}([0,1]^n)}(0,L)$ for there case of general $k$?
I should note that, the case where $k=0$ is addressed in these MO posts 1 and 2; but what about the general case?
I'm also curious about the analogous question in $L^1$, with the $C^{k,1}$ norm replaced by the Sobolev norm $W^{k}$.
