I stumbled with this very unique issue, which seems trivial, but I was not able to prove of disprove the claim. Here it is.

Let $f\in L^2(\mathbb{R}^+)$, such that $f\in H^1(a,\infty)$ for all $a>0$. Hence, $f$ is an $L^2(\mathbb{R}^+)$ function that is nice in the sense that it is absolutely continuous for $x>0$.

Claim: there exists an $\epsilon>0$ such that
$$\frac{f(x)}{x^\epsilon}\in L^2(\mathbb{R}^+).$$

In other words, there is always "room" to make $f$ just a little more "bad" at the origin, while staying in $L^2(\mathbb{R}^+)$. That claim would help me in some eigenvalue computation I am currently performing. I believe it is true but I have not found a way to prove or disprove it.