Put, $C\_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim\_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\infty$) and $A:= L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C\_{0}(\mathbb R).$ 



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> My Question: Given $|f|\in A$. Can we always expect $|f|\in H\_{1}(\mathbb R)=\{ f\in \mathcal{S'}(\mathbb R): (1+|\xi|^{2})^{1/2}\hat{f}\in L^{2}(\mathbb R) \}$ ([Sobolev spaces](http://en.wikipedia.org/wiki/Sobolev_space)), Or, we we can produce a counter example ?
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Thanks,



$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?
No. A simple family of counter examples come from the Sobolev embeddings. Since
$$
W^{1,2}(\mathbb{R})\hookrightarrow C^{0,\frac{1}{2}}(\mathbb{R}),
$$
your identity would imply that continuous functions are always Hölder continuous.


