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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 $\mathbb{T}^n$.

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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.

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  • $\begingroup$ The example is nice. Any idea about the case of $p=\infty$? $\endgroup$
    – Housen
    Oct 16, 2013 at 7:50
  • $\begingroup$ @Housen Li: see the extended answer above $\endgroup$
    – username
    Oct 16, 2013 at 12:57

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