Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$.
The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$.
I stumbled on an example like that and my intuition was that, the more regularity is demanded, the higher the lower bound on $p$. It is just "funny" that as the regularity increases, some values of $p$ are "left-behind". So I am not looking for a precise answer. Rather, I am looking for an insight as to why it is like that. Just to be sure, I do understand how to show something is in those spaces. I am just wondering if someone has an intuitive idea as to why it works like that. Of course, there is the possibility my statements are wrong!