My question is in fact, if $f \in C^1(\mathbb{T})$ where $\mathbb{T}$ is the torus, is there exist a constant $C>0$ such that $|\widehat{f}(k)|\geq \frac{C}{|k|}$, for all $k \in \mathbb{Z}\setminus\{0\}$?
I think that is not true because $f' \in C(\mathbb{T})$, then $f'\in L^1(\mathbb{T})$, so $|\widehat{f'}(k)|=|2\pi i k \widehat{f}(k)|$, and by the Riemann-Lebesgue Lemma, $|\widehat{f'}(k)|\to 0$ as $|k| \to \infty$. It is to say, $|k||\widehat{f}(k)| \to 0$ as $|k| \to \infty$. So that $C>0$ cannot be exist.
What I did, is actually correct? Thank you!