Does a continuous compactly supported function $f : \mathbb R \to \mathbb R$ satisfy the inequality $$\lvert f(x)\rvert+\lvert \widehat{f}(x)\rvert \leq C(1+\lvert x\rvert)^{-1-\epsilon},\quad x \in \mathbb{R}$$ for some $C,\epsilon >0$ ?
Here, $\hat f$ is the Fourier transform of $f$. Presumably everything is defined since $f \in L^1(\mathbb{R})$.
Edit 1:
For any $\epsilon >0\,$ I can bound $\lvert f(x)\rvert \leq \max \left(1,\frac{\Vert f\Vert_\infty}{\min_K \left(1+\lvert x \rvert\right)^{-1-\epsilon}}\right)$, then I want to use the following (but my function is just $\mathcal{C}^0$...)
Theorem (Paley-Wiener) If $f \in \mathcal{C}_o^\infty$ and $f(z)=0$ for $ \lvert z \rvert > R$, then $$\lvert \widehat{f}(z) \rvert \leq C_n(1+ \lvert z\rvert)^{-n}e^{2\pi\lvert \text{Im}(z)\rvert R} \quad \forall n \in \mathbb{Z}^+.$$
I am referring to what they say in page 4 of the paper https://www.sciencedirect.com/science/article/pii/S0022247X19309813