Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $A:= L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R).$

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), Or, we we can produce a counter example ?


share|cite|improve this question
$H^1$ functions are almost everywhere differentiable, so any nowhere differentiable continuous function with compact support is a counterexample. – Alexander Shamov Feb 17 '14 at 13:07

1 Answer 1

up vote 3 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.