I have given a density $f: \mathbb{R} \rightarrow \mathbb{R}_{+}$ with the properties
- $f\in L_1$
- $f\in L_2$
- $\int f\ dx=1$
- $\exists L,\beta>0:\int|\mathcal F(f)(u)|^2|u|^{2 \beta} d u \leq L$
- $\exists A,\beta^\prime>0:A\le|\mathcal F(f)(u)||u|^{\beta^\prime}$ for large $|u|$.
Can I follow from these properties that $f$ is bounded? I.e. $$\exists C>0\ \forall x\in\mathbb R:|f(x)|\le C.$$ Or can someone give a counterexample?