Let $a, b \in \mathbb R$ such that $ab> 1$ ; put

$$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$

and

$$FL^{1}_{b}(\mathbb R)= \{ f:\mathbb R \to \mathbb C \ \text {measurable} : ||(1+|w|)^{b} \hat {f}||_{L^{1}(\mathbb R)}< \infty \}.$$

We consider a Foureier-Lebsgue space, $$FL:=L^{1}_{a}(\mathbb R) \cap FL^{1}_{b}(\mathbb R).$$

**My question is**: Let $f\in FL$.
Can we expect $|f|\in FL$ ; Or we get a counter example ?

Thanks to math fraternity;-)