(In what follows, $s, \varepsilon\in\mathbb R$). I know that the Dirac delta is in the fractional Sobolev space $H^s(\mathbb R^n)$ for $s<-n/2$. Besides, I think I have proved that negative-order Sobolev spaces $H^{-\varepsilon}(\mathbb R^n)$, $\varepsilon>0$ do not contain exclusively locally integrable functions, even for $\varepsilon$ arbitrarily close to $0$ (but my proof relies on Banach-Steinhaus and I don’t have an explicit counterexample yet; I can maybe post it as an answer if someone could find it interesting). Roughly speaking, I would like to know how bad a distribution can behave locally if I know that it lies in $H^{-\varepsilon}(\mathbb R^n)$.
I would like to know if there are simple, explicit, or remarkable examples of distributions that lie in $H^{-\varepsilon}(\mathbb R^n)$ but not in $L^1_{loc}(\mathbb R^n)$ for small $\varepsilon>0$. Or even just some $0<\varepsilon<n/2$.
Also, everything that can be interesting and references about the local behaviour of distributions in negative order Sobolev spaces is welcome.
