Estimate for L1 funcction

Question

Suppose for every $r\in \mathbb{R}$, $\int_{\mathbb{R}}\lvert f(x)\rvert e^{rx}dx<\infty$. Then is it true that there exists $C>0$ and $\alpha>1$, such that $\lvert f(x)\rvert\leq Ce^{\lvert x\rvert^{-\alpha}}$ almost everywhere.

Answer 1

Counter-example

Let us consider a function $f: \mathbb R \to \mathbb R$, such that for every $n \in \mathbb Z$, $f(x)=e^n$ on $[n, n+c_n]$, with $c_n>0$.

Let $r \in \mathbb R_{-}$, then $$\int_{\mathbb R} |f(x)| e^{rx} \mathrm{d}x \leq \sum_{n \in \mathbb Z} c_n e^{(r+1)n}$$ The last sum converges if $c_n$ = $e^{-n^2}$ for example.

Let $r \in \mathbb R_{+}$, then $$\int_{\mathbb R} |f(x)| e^{rx} \mathrm{d}x \leq \sum_{n \in \mathbb Z} c_n e^{r(n+1) + rc_n}$$ And the sum still converges.