Laplace transform and majorant

Question

I am looking for a majorant such that for every $t>0$ we have that for all $x>0: |x^ne^{-xt}|\le F(x)$ such that $\int_0^\infty F(x) dx < \infty$? I guess this one does not exist, but the excercise is the following:

$f \in L^1$ I am supposed to show that $G(t):=\int_0^\infty e^{-st}f(s) ds$ is a $C^{\infty}$ function. The problem is, when I differentiate this, I can no longer use Lebesgue's convergence theorem, which I need to differentiate it once more?

So what I want to use here is 2nd theorem but I don't see how to find a suitable majorant.

Answer 1

Differentiability is a local property. To see that $G$ is differentiable in $t_0 > 0$, choose a $t_1 \in (0,t_0)$. Then for all $t \geqslant t_1$, you have

$$0 \leqslant s^ne^{-st} \leqslant s^n e^{-st_1}$$

for $s \in [0,\infty)$, and that produces the uniform majorisation of the differentiated integrand [uniform in $s\in [0,\infty)$ and $t \in [t_1,\infty)$, it depends on $n$].