I know that if the decreasing of fourier transform of a function implies regularity and that the converse is more technical.
For random variable, I know that if the characteristic function $\varphi$ of a random variable $X$ is $C^2(\mathbb{R})$, then $\operatorname{E}[X^2] < \infty$. But, there I remember that there is counter-example if $\varphi$ is only $C^1(\mathbb{R})$ : it doesn't imply that $\operatorname{E}[|X|] < \infty$.
My problem is the following. Let $g \in L^1(\mathbb{R})$ with $g \geq 0$ and $g(x) = g(-x)$. Let $f(k) = \int_{\mathbb{R}} g(x) e^{ikx} dx$. Is the following true :
$$f \in C^1(\mathbb{R}) \quad \implies \quad \int_{\mathbb{R}} g(x) |x| dx < \infty$$
I can prove that : $f \in C^2(\mathbb{R})\implies \int_{\mathbb{R}} g(x) |x|^2 dx < \infty$, using Fatou's lemma with $\frac{f(h) + f(-h) - 2 f(0)}{2h}$. But, I struggle to generalize it for $f \in C^2(\mathbb{R})$.
Do you have any counter-example or hint ?
Thanks,