Properties of Sobolev spaces $W^{k,\infty}(\Omega)$

Question

I'm looking for different properties of spaces $W^{k,\infty}(\Omega)$ for bounded domain $\Omega \subset \mathbb R^n$ and $k \geq 1$ that I couldn't find in literature. References are wery welcome.

1. Extension. Theorem 12.15 of Giovanni Leoni "A first course in Sobolev spaces" tells us that if $\Omega$ has Lipshitz boundary then there exists a continuous linear extension operator from $W^{1,\infty}(\Omega)$ to $W^{1,\infty}(\mathbb R^n)$. What about spaces $W^{k,\infty}$ for $k \geq 2$? Does there exist a continuous extension operator in the case of $C^k$ boundary $\partial \Omega$?

2. Trace. Theorem 5, p. 131 of Evans "Measure theory and fine properties of functions" tells us that $f \in W^{1,\infty}_\text{loc}(\Omega)$ if and only if $f$ is locally Lipshitz in $\Omega$. If I'm not mistaken, combining this with the above extension result we obtain that for any $f \in W^{1,\infty}(\Omega)$ we can define its trace on $\partial D$ as an element of $W^{1,\infty}(\partial \Omega)$ if $\partial \Omega \in C^1$. To do this we extend $f$ to $\widetilde f \in W^{1,\infty}(\mathbb R^n)$ so that $\widetilde f|_{\partial D}$ is the restriction of the Lipshitz function which is Lipshitz and hence $W^{1,\infty}(\partial D)$. Is it true in general that the trace operator continuously maps $W^{k,\infty}(\Omega)$ to $W^{k,\infty}(\partial \Omega)$ if $k \geq 1$, $\partial \Omega \in C^k$? (Intuitively it must be true since taking trace on $W^{k,p}$ "costs" 1/p smoothness).

3. Extension from boundary. Let $f \in W^{1,\infty}(\partial D)$. Then (as above) $f$ is Lipshitz. Then by the Kirszbraun theorem $f$ can be extended to a Lipshitz $\widetilde f$ (the extension not necessarily linear) on $\mathbb R^n$ and hence to $\widetilde f|_\Omega \in W^{1,\infty}(\Omega)$. Does the same result hold for $f \in W^{k,\infty}(\partial D)$, $k \geq 2$?

Any comments are very welcome.

Answer 1

Q1: Yes there is one. Please check this book, section $5.17$

Q2: I don't have a reference for this but as you said, this is proved by extension...

Q3: Yes, I think we can try induction.

Answer 2

Look in Chapter VI of Singular Integrals and Differentiability Properties of Functions by Elias Stein.