Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$. Here $\Omega$ can be a subset of $\mathbb{R}^n$ with desired boundary, $\mathbb{R}^n$ or torus $\mathbb{T}^n$.
1 Answer
If $\Omega=B(0,1)$, take a radial function $f(r)=\frac{1}{r^{n/p}\left(\log r-1\right)^{2/p}}$ Then $$ \int f^p dx = \omega_n \int_0^1 \frac{1}{r\left(\log r-1\right)^2} dr =\omega_n, $$ so $f\in L^p$, but not in $L^q$ for $q>p$, and therefore not in any W^{s,p}, $s>0$.
Now integrate $f$ k times in $r$, say $F_k(r)=\int_0^r \int_0^{t_1} ..\int_0^{t_{k-1}} f(u)du$. Then $F_k\in W^{k,p}(\Omega)$, but not in $W^{s,p}(\Omega)$ for $s>k$.
In the $p=\infty$ case, choose $$ f(r)=1 \mbox{ for } r\leq \frac{1}{2} \mbox{ and } 0 \mbox{ otherwise.} $$ Then $f$ is bounded but not continuous and therefore not in any $W^{s,p}(\Omega)$ space for $s>0$, $q>1$ and $sq>n$. Then proceed as above.