**Note**: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a separate question because, if I just edit my previous question, it will not get any attention.

**Question**: Consider
$$
v(x)\equiv e^x\int_x^\infty \frac{f(x')}{e^{x'}}dx'.
$$
where $f\in H^1(\mathbb{R}^+)\cap C^1(\mathbb{R}^+)$. I want to prove that
$v\in L^2(\mathbb{R}^+)$.

**Context**: I am computing the resolvent set of a differential operator and I computed the solution of the resolvent equation. Using variation of parameters, I end up with expressions similar to what is above. Among other things, I need to determine if those expressions give me functions that are in $L^2(\mathbb{R}^+)$.

**Reasoning**: Considering the behavior at $x=\infty$, it "seems" fine to me also since the integral will go to zero exponentially fast and fast enough to cancel the behavior of the exponential outside and make $v$ be in $L^2(\mathbb{R})$. Now, obviously, this last sentence is a heuristic argument, which needs to be made formal, if true at all. I need help with that.

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