**Note**: This is a simplified version of the following [question](https://math.stackexchange.com/questions/4504786/determine-if-an-integral-expression-is-in-l2-mathbbr). 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.



Determine if an integral expression is in $L^2(\mathbb{R})$
From Theorem 0.3.1 of [1], we have that if a kernel operator
$$
T(f)\equiv \int\_0^\infty K(x,x') f(x')dx',\;\;K(x,y)=e^{x-x'}\chi\_{y>x>0}
$$
is such that
$$
\sup\_{x'}\left(\int\_0^\infty K(x,x') dx\right),\;\;
\sup\_{x}\left(\int\_0^\infty K(x,x') dx'\right)\leq C
$$
then
$$
\|T(f)\|\_{L^2(\mathbb{R}^+)}\leq C \|f\|\_{L^2(\mathbb{R}^+)}.
$$
Thus $T$ defines a continuous operator on $L^2(\mathbb{R}^+)$. In my specific case, $K(x,y)=e^{x-x'}\chi\_{y>x>0}$, $C=1$ and thus that answers my question!


A big thank you to Christian Remling to point me toward that theorem.


[1] *Sogge, Christopher D.*, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics. 105. Cambridge: Cambridge University Press. x, 237 p. (1993). [ZBL0783.35001](https://zbmath.org/?q=an:0783.35001).


